Bioenergetics and Kinetics of Microtubule and Actin Filament Assembly–Disassembly

INTERNATIONAL REVIEW OF CYTOLOGY, VOL. 78

Bioenergetics and Kinetics of Microtubule and Actin Filament Assembly-Disassembly TERRELL L. HILLA N D MARCW. KIRSCHNER Lohorcrtory of Molecrilar Biology, Nritionrrl Institiite of Arthritis, Diabetes, rind Digestive (rnd Kidney Diseases, Nationtrl Itistitrites of Herrlth, Bethesdri, Maryland, and Department of' Biochemistry nrid Biophysics. Scliool of Medicine, University of Criltfornin, Son Francisco, C(il(foriiia I. Introduction . . . . . . . . . . . . . . . . . . . . . 11. Polymer with Free Ends . . . . . . . . . . . . . . . A. Equilibrium Polymer . . . . . . . . . . . . . . . B. Steady-State Polymer . . . . . . . . . . . . . . . 111. Polymer with End or Ends Capped or Anchored . . . . A. Equilibrium Polymer . . . . . . . . . . . . . . . B. Steady-State Polymer . . . . . . . . . . . . . . . IV. Polymer under a Moveable Force . . . . . . . . . . . A. Equilibrium Polymer . . . . . . . . . . . . . . . B. Steady-State Polymer . . . . . . . . . . . . . . . V. Polymer between Two Barriers . . . . . . . . . . . . A. Equilibrium Polymer . . . . . . . . . . . . . . . B. Steady-State Polymer . . . . . . . . . . . . . . . VI. Fluctuations and Stochastics . . . . . . . . . . . . . A. Equilibrium Polymer . . . . . . . . . . . . . . . B. Steady-State Polymer . . . . . . . . . . . . . . . VII. Afterword , , , . . , , , . . , . , . . , . . , , . . VIII. Appendix 1. Comparison of Rate Constant Notations . . . IX. Appendix 2. Fluctuations in Polymer Length Distribution . . . . . . . . . . . . . . . . . . . . . X. Appendix 3. Persistence of NTP at Polymer Ends . . . . References . . . . . . . . . . . . . . . . . . . . .

1

6 7 12 29 29 34 43 44 60 71 72 81 92 93

100

108 11 1 111 113

123

I. Introduction

The cytoskeleton of all eukaryotic cells must have several special properties. In vitro the polymers that comprise the cytoskeleton are of indefinite length, while in vivo length and orientation are controlled in some manner. In the cell the cytoskeletal filaments interact with each other, the cell membrane, and other cytoplasmic organelles. These filaments are often under compressive or extensive forces owing to their involvement in the motility of the whole cell or of organelles within the cell. The ar1 Copyright B 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-364478-X

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TERRELL L. HILL AND MARC W. KIRSCHNER

rangement of cytoskeletal filaments is often very dynamic. They must change their organization with cell growth, and undergo particularly major changes during cell division. The filament arrays are also drastically remodeled during cell differentiation, and there is a major reorganization of local parts of the cytoskeleton during cell movement and phagocytosis. It is the goal of modern cell biology to explain the properties of the whole cell in terms of the biochemical properties of the individual components. In the case of the cytoskeleton this will be a difficult undertaking because many of the properties depend on a large number of specific interactions spanning large distances in the cell. Many of these interactions are mediated by specific proteins, of which more than 50 have been identified for the actin system alone. However, the very dynamic nature of the cytoskeleton encourages one to believe that the detailed history of each cell may not be necessary for describing the properties of the filamentous arrays within cells, and that much can be explained in terms of the energetics and kinetics of elementary processes of spontaneous assembly. This is demonstrated clearly by the ability of the microtubule system to regrow a normal array after drug-induced depolymerization (Brinkley et al., 1976; Osborn and Weber, 1976), or the ability of actin arrays to reform cable patterns after trypsinization or viral induced disorganization (Lazarides, 1976; Pollack et al., 1975). Although many of the detailed properties of these systems will require knowledge of the specific properties of many individual associated proteins, many important results can be obtained by looking at the pure polymers themselves. This is partially because the effects of associated proteins can be understood in terms of their modifying existing properties of the polymers in rather simple ways, such as by binding to one end or the other, binding to the monomer but not the polymer, binding to the polymer but not the monomer, or by cross-linking the polymer. Thus the rules for assembly of the polymer itself can be extended easily to include many of the properties of associated proteins. We have to expect, however, that effector molecules may be found that will alter considerably the chemistry of the polymers and which could require major changes in existing theoretical treatments. Two of the major filamentous systems in the cell, actin filaments (microfilaments) and microtubules, share several interesting biochemical, physical chemical, and cellular properties. The third major filamentous system, intermediate filaments, is less well studied, but seems much less dynamic and may assemble by mechanisms different than actin and tubulin (Steinert et al., 1978; Renner et al., 1981). Both actin and tubulin assemble from globular subunits into helically ordered surface lattices in semiinfinite linear polymers (see reviews by Kirschner, 1978; Timasheff

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

3

and Grisham, 1980; Pollard and Weihing, 1974; Clarke and Spudich, 1977). Both show structural and kinetic polarity so that the two ends are different. Both bind and hydrolyze nucleoside triphosphates (ATP for actin, GTP for tubulin). The kinetics of assembly from purified subunits for both systems can be described roughly in terms of the condensation polymerization model of Oosawa and Kasai (1971a), with a lag phase for nucleation, an exponential phase for growth, and a slow phase for length redistribution. In the cell both actin filaments and microtubules are found in many different locations. The ordered array of actin filaments in muscle is a special case, but the different spatial organization of actin in different regions of other cells is well documented (Lazarides, 1976; Heuser and Kirschner, 1980; Small, 1981). For microtubules, the highly ordered arrays in protozoa, neurons, and platelets are special examples and the dynamic and stereotyped arrays during mitosis of all cells are well known (see Weber and Osborn, 1979). In the case of microtubules, several wellknown organizing centers have been identified such as the centriole, basal body, phragmoplast in plants, and the kinetochore of metaphase chromosomes. In actin the only known nucleating structure is a special structure in echinoderm sperm called the actomere (Tilney, 1976), but several proteins have been described which could serve to nucleate actin polymerization (for recent papers see 1981 Cold Spring Harbor Symposium). Both actin and microtubules are involved in motility and must undergo either extensive or compressive forces that could affect the properties of the filaments themselves. This is again clear for actin in muscle but also in the contractile ring of dividing cells. For microtubules the best examples are in mitosis where the poles are moved apart relative to each other and the chromosomes are moved relative to the poles. In some cases movement in these filamentous systems may be explained solely by the forces of polymerization and depolymerization (Inoue and Ritter, 1975). However, in other cases where other proteins act on the filaments (e.g., myosin on actin filaments or dynein on microtubules), it follows that these external forces should cause changes in the polymerization of the filaments. (We are not referring here to muscle or cilia.) These forces, for example, could deform or compress the filament and alter the association of the filament with free subunits. Until recently no general theoretical treatment of the polymerization of microtubules and actin filaments had been given that takes into account two very important properties: forces acting on the polymers and simple interactions of the ends of the filaments with other components. However, recently such problems have become even more interesting with the further experimental and theoretical investigations of the role of nucleoside

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TERRELL L. HILL AND MARC W. KIRSCHNER

triphosphate hydrolysis in assembly. This added chemistry of tubulin and actin complicates considerably the energetics and kinetics of assembly, but more importantly allows for several interesting and unique properties of the system that may be very important in specifying the position of these filaments in the cell and their capacities for doing work. Although it was recognized early that tubulin and actin bind and hydrolyze nucleoside triphosphates (Weisenberg et a/., 1968; Straub and Fever, 1950; Oosawa and Kasai, 1971b) it was unclear until recently what function this hydrolysis might have. It was clearly demonstrated that nucleotide hydrolysis was not required for assembly (Cooke and Murdoch, 1973; Penningroth and Kirschner, 1977; Arai and Kaziro, 1977), since rapid and efficient polymerization would occur with nonhydrolizable ATP and GTP analogs. However it was also demonstrated that when the natural triphosphates were used, the stoichiometry of hydrolysis was approximately 1 mole per each mole of subunit assembled (Oosawa and Kasai, 1971b; David-Pfeuty et a/., 1977), suggesting that hydrolysis was coupled to assembly. A striking theoretical and experimental paper by Wegner (1976) argued that nucleoside triphosphate hydrolysis could be used to drive head-to-tail polymerization of actin at steady state. This property, now also called “treadmilling,” involves the net assembly of subunits of the filaments at one end and the net disassembly at the other end, at steady state (i.e., when the polymer mass remains unchanged). This is a consequence of the ATP free energy being utilized to make the effective affinity of the two ends of the polymer for the monomer different. Margolis and Wilson (1978) then demonstrated that treadmilling also exists in microtubules by using a direct method for measuring the flux. In their initial studies Margolis and Wilson (1978) assumed that there was an exclusive addition of subunits to one end and exclusive loss at the other. However, the measured flux of 0.28 dimers s-l, or 0.31 dimers s-l (Terry and Purich, 1980), was small compared to measured dissociation rates under pre-steady state conditions of 154 dimers s-l (Karr et a / ., 1980) and thus was inconsistent with exclusive assembly at one end and loss at the other, as pointed out by Zeeberg et a/. (1980). In microtubule protein preparations containing associated proteins, the flux owing to treadmilling measured from pre-steady-state rates was 1.5 dimers s-l (Bergen and Borisy, 1980). Zeeberger a/. (1980) also measured a flux of 2 dimers s-l at steady!state in a microtubule system where the dissociation rate at steady state was found to be 119 dimers s-l, which is somewhat high compared to other measured values (Johnson and Borisy, 1977). Thus, although treadmilling was demonstrable, it was inefficient, and questions were even raised by Zeeberget d.(1980) as to experimental and theoretical problems

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

5

in showing it. However, recently in a very complete study, Cote and Borisy (1981) measured a treadmilling flux of 28 dimers s-l for microtubules, depleted of most of the associated proteins, which have dissociation rates measured under the same steady-state conditions of about 100 dimers s-l. Thus, under these conditions, treadmilling occurs to an appreciable extent and amounts to more than one translocation event for every four association or dissociation events at steady state. As expected from the theoretical treatments of Wegner (1976) and Hill (1980a), treadmilling does not occur with nonhydrolyzable analogs that support microtubule polymerization (Terry and Purich, 1980; Margolis, 1981; Cote and Borisy, 1981). For a recent review of experimental studies of treadmilling, see Margolis and Wilson (1981); see also Pollard and Mooseker (1981). The clear demonstration of the phenomenon of treadmilling in v i m prompted an evaluation of the cellular consequences of having the two ends of the filament different and the possible role of treadmilling to do work. Margolis et ul. (1978) described a model for mitosis where differential polymerization and depolymerization at the two ends played a key role but treadmilling itself played a minor one. Various experimental observations from Inoue’s laboratory have long suggested that force generation could be achieved by polymerization and depolymerization (Inoue and Ritter, 1975), but treadmilling at this time could not be considered. The apparent stable steady-state distribution of organized filaments within cells led to the proposal that an important consequence of treadmilling could be that the cell could use this property to selectively stabilize filaments attached at one end in nucleating structures. Treadmilling could then be used as a mechanism of suppressing spontaneous filament assembly (Kirschner, 1980). This focused attention on the theoretical effects of proteins or structures which might cap one end of a filament at the same time that such proteins were being described in the actin system. Finally, it was possible to show that treadmilling could actually be made to do work under conditions which might be expected to exist in cells (Hill and Kirschner, 1982). The need to examine the kinetics and energetics of linear polymerizing systems while at the same time taking into account nucleotide hydrolysis, external forces acting on the polymer, and fluctuations has led to a reformulation of polymerization theory in terms of general models utilized previously to explain other metabolic and mechanochemical cycles that use ATP hydrolysis (Hill, 1977a). In this article we will consider the kinetics and bioenergetics of polymers like actin filaments and microtubules that utilize ATP and GTP hydrolysis and, for comparison, we will also consider the kinetics and bioenergetics of those that do not. We will also

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TERRELL L. HILL AND MARC W. KIRSCHNER

consider important boundary conditions such as forces acting in various ways on the filaments and the effect of materials that interact at the ends of filaments. We will be mostly concerned with the biological implications of these properties in terms of capacity to do work, regulate length, and regulate spatial distribution. Although this article reviews the material given in five previous articles (Hill, 1980a, 1981a,b; Kirschner, 1980; Hill and Kirschner, 1982) we go into these problems here in much greater depth, with more examples, and with more biological commentary. In addition, much of the material, including that dealing with the effects of capping and specific properties of the ends of filaments, has not been previously published. It is hoped that this unified and comprehensive format will clarify some of the characteristics of actin and microtubule assembly and stimulate further studies of the way other cellular materials interact with and modify these systems. 11. Polymer with Free Ends

In this and in each of the remaining sections we consider first an “equilibrium polymer,” or aggregate, by which we mean a rod-shaped, linear polymer comprised of physically aggregated monomers (subunits) with no enzymatic activity. We then turn, in each section, to the more complicated problem of a “steady-state polymer,” which refers to polymers whose subunits contain bound nucleotide diphosphate (NDP) and whose terminal subunits are enzymatically active (see Section II,B for details). Sickle-cell hemoglobin (HbS) is an example of an equilibrium polymer while microtubules (tubulin) and microfilaments (actin) are steady-state polymers. In addition to possible application to HbS, etc., the prior equilibrium treatment in each section provides necessary background for the steady-state problem. In Sections I1 through V, only macroscopic thermodynamics and the corresponding kinetics are used. This treatment is applicable to very long polymers. Topics that relate to statistical mechanical partition functions, fluctuations, stochastics, and finite systems (polymers) are all reserved for Section VI. This will make it convenient for readers so inclined to omit the subjects included in Section VI. We discuss in the present section polymers (aggregates) in solution with free ends. That is, the ends are not in contact with cellular barriers nor are they capped with foreign substances or structures: the terminal subunits of the polymer have direct and uninhibited access to the solution. The above-mentioned cases that are excluded here are treated in Sections 111, IV, and V.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

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A. EQUILIBRIUM POLYMER We consider a long (essentially macroscopic) rod-shaped aggregate in solution, in equilibrium with dilute free monomers at concentration c , (Oosawa and Asakura, 1975). For simplicity, we do not include the solvent explicitly in the thermodynamics nor do we include pressure-volume effects (but see Hill, 1964, for a treatment). The polymer, then, can be characterized thermodynamically by the temperature T and by the chemical potential po(T)of the monomers in the polymer. Because the polymer is open with respect to addition of monomers and its ends are unrestrained, its length L and number of monomers N are thermodynamically indeterminate at c = ce (Hill, 1964). That is, any L (if large enough) is consistent with p o and c,. But if c is just less than c,, the polymer has a definite mean finite length, L(c) (Hill, 1980a). Monomers in solution at an arbitrary concentration c have a chemical potential p s = p!(T)

+ kTln c ,

(1)

where p!(T) is a standard free energy. ( p sis the chemical potential per molecule, and k is the Boltzmann constant.) Because c is of order 1 p M for the tubulin and actin cases of interest, we omit an activity coefficient in Eq. (1). However, this would not be a good approximation for HbS (Ross and Minton, 1977). Because of the assumed equilibrium at c = c,, The polymer can be considered to be a one-dimensional crystal with solubility c e . The concentration c e is also referred to as the critical concentration of monomer: starting with monomer at c G c,, if c is increased, linear aggregates begin to form as c nears c, and essentially infinite polymers are produced at the “critical” concentration c = c, (Hill, 1964; Oosawa and Asakura, 1975). If we denote the monomer by A , then c e is also the equilibrium constant for the process A(po1ymer) + A(so1ution). The equilibrium constant ce is related to the standard free energy change for this process in the conventional way by That is, on a per mole basis, the right-hand side is -RTln K. The more stable the monomers are in the one-dimensional crystal (e.g., from strong intermolecular attractive forces), the lower po(T) and, consequently, from Eq. (2), the lower c,. Turning now to related kinetic aspects, we assume that monomer exchange between solution and polymer occurs only via the two polymer

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TERRELL L. HILL AND MARC W. KIRSCHNER

ends and not through the bulk of the polymer. It should be noted that, in the macroscopic thermodynamic discussion above, the number and nature of the polymer ends are of no consequence and were not mentioned. However, the ends control the aggregation kinetics. The polymers of interest have a polarity (are not isotropic). This is demonstrated structurally by the directional binding of a fragment of myosin to actin filaments (Huxley, 1963) and by the directional binding of dynein or tubulin ribbons to microtubules (Heidemann and McIntosh, 1980). Because the polymer is polar, the two ends are different. In general, then, the rate constants for the addition or loss of subunits at the two ends will be different. The on and off rate constants at one end are denoted a and a ' , respectively, and at the other end, p and p' (a and p are secondorder constants, a' and p' are first-order). This is shown schematically in Fig. 1A. In any system at a true equilibrium, there can be no net flux or flow in any process, even at the most elementary level. This is the principle of detailed balance at equilibrium (i.e., balance, or equality, of inverse rates). If the polymer is in equilibrium with free subunits at concentration c , , the on rate must equal the off rate at both ends of the polymer ace = a ' ,

pc, =

p',

or

c e = a'la

=

p'lp

(4)

In general a # p and a' # p ' , but the ratios must be equal [Eq. (4)]. As we shall see later, in steady-state polymers, where detailed balance is not required, it is possible to have zero total flux of subunits onto the polymer with nonzero flux at both ends (one flux negative, the other positive). The significance of detailed balance for this system can also be examined thermodynamically. The addition, at equilibrium, of a monomer to the polymer at a particular end does not in any way alter the equilibrium state of the end itself but rather simply has the effect of increasing the number of bulk (nonterminal) monomers in the polymer by one: the polymer free energy increases by po and the solution free energy decreases by p:(=po). After the addition of the monomer, the polymer would be in exactly the same state regardless of which end the addition was made to, even though the ends are different. Thus the equilibrium constant for monomer addition (llc,) must be the same at the two ends. This alternative argument confirms Eq. (4). Although there are obvious similarities between ligand binding and polymer aggregation, there is also a fundamental difference. When a ligand is bound on a site on another molecule or on a surface, the state of the site itself is changed by the binding. An empty site becomes an occupied site, and is no longer available for binding. But when a monomer is added to the end of a polymer (made of the same monomers), the state of

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

9

A (solution)

A

A A A A A Polymer A A A A A

A

B FIG.1 . (A) Equilibrium polymer in solution with ends that are different and with on-off transitions at the ends. (B) Net rate of adding monomers at the two ends as a function of free monomer concentration c .

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TERRELL L. HILL AND MARC W. KIRSCHNER

the polymer end (at equilibrium) does not change. The attachment site remains an attachment site. Thus, whether a monomer adds to the a or p end of a linear polymer, there is no change in the capacity of the polymer to add or lose subunits. Polymers that are multistranded, such as actin, tubulin, and HbS, have many possible detailed surface configurations at their ends. Some of these have been seen during the course of assembly of microtubules in vivo and in vitro (Kirschner et al., 1974; Erickson, 1974; Dentler and Rosenbaum, 1977). The surface configurations may be in dynamic equilibrium with each other via on and off transitions, or diffusion transitions, where subunits move from one location to another while still attached to the polymer. Included in each surface configuration are several nonequivalent subunits that can escape from the polymer end to the solution and several nonequivalent sites to which new subunits can be added from the solution. Thus the observable rate constants a and a' (also, of course, p and p ' ) are really composites of more microscopic rate constants. This can be expressed formally, as would be important for any theoretical analysis of a and a' in a particular case. Let a{]be the on rate constant for the addition of a subunit to that site in surface configuration i that converts configuration i into configurationj. Let a;{ be the inverse off rate constant. There must be detailed balance in this elementary process at equilibrium: ~5ceat1 =~ f a h

(5)

9

where p f is the equilibrium probability of surface configuration i, etc. If we sum both sides of Eq. (9,first, over all configurationsj that can be reached from i (i.e., over all addition sites in i ) and then over all i , we obtain ace = a',as in Eq. (4), where a

=

&lPfa,l,

=

&*jPw;i

(6)

This exhibits the more detailed nature of a and a ' . We also have, at equilibrium, pf/ple =

-(C,-

Cj)/ kT

Y

(7)

where Gi- GI is the difference in surface free energies (primarily owing to different intermolecular interactions) between configurations i andj. Consequently, Eq. (5) can be rewritten in the form .;{/afj =

c,e-'ci-cj)/kT

(8)

where ce may be replaced here by a ' h , if desired. This is the microscopic equilibrium constant for the release of a subunit from the polymer to the solution, from configuration j to give configuration i . A simple explicit example of the above analysis is included in Section II1,A.

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At an arbitrary free monomer concentration c ,

J p = pc - p‘ (9) are the net rates of addition of monomers to the two ends, per polymer molecule. The total rate of addition is Jon= J , + J p . At equilibrium, J , = J p = 0 and c = c,. An example of J,(c) and Jp(c)is shown in Fig. 1B in which the a end is more active than the p end (i.e., a > p , a’> p’). The two lines necessarily cross at c = c, [Eq. (4)l. This type of diagram (Bergen and Borisy, 1980) is very useful in more complicated cases (below). If a large number of these (Fig. 1) polymers are present in a relatively small volume of solution with, initially, c > c e , aggregation will occur at both ends of the polymers (Ja> 0, J p > 0) but, as a result of loss of free monomers from the solution to the polymers, c will steadily decrease. This will continue until c reaches c,, at which point growth of polymers will cease. Similarly, if c < c, at the outset, polymers will lose monomers from both ends to the solution ( J , < 0, J p < 0) causing c to increase until, again, the stable value c , is finally reached. An explicit, but more complicated, example of this kind of behavior will be presented in Section II1,B. In writing Eqs. (9), one usually assumes the rate constants are independent of c . This in turn implies that the equilibrium averaging in Eq. (6) is valid at any c. That is, we are assuming that an internal equilibrium among the many surface configurations is maintained even under conditions of steady subunit gain or loss (c # c,). This would require subunit surface diffusion transitions that are relatively fast compared to on-off transitions. The simple example treated in Section II1,A shows that without surface diffusion we would expect the rate constants themselves to be functions of c. Because the distribution among surface configurations at the ends would depend on the relative rates of subdnit addition, which depends on c , and subunit diffusion, which does not, this problem may also arise in the case of proteins that interact with the subunits in the polymer. Such proteins are well known for both actin filaments and microtubules. Since all the known proteins bind substoichiometrically, the exact ratio of these proteins to the monomers at the ends can vary. If transitions among configurations of bound proteins are slow compared to the rates of assembly, the observed average rate constants can again be dependent on the rate of subunit addition and hence on the monomer concentration. Incidentally, the above is a special case of a general problem in biochemical kinetics (Hill, 1980b): whenever rate constants are assigned to transitions between pairs of discrete states in a biochemical cycle, the implicit assumption is made that the individual states of the cycle are all in J , = ac - a’

and

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TERRELL L. HILL AND MARC W. KIRSCHNER

internal equilibrium among their own substates even when the states are not in equilibrium with each other (e.g., in steady-state cycling).

B. STEADY-STATE POLYMER The two known cases of this type, so far, are microtubules with the

a * p tubulin dimer considered as the aggregation monomer or subunit, and microfilaments with actin molecules as aggregation monomers. Mi-

crotubules generally have 13 strands; microfilaments have 2 strands (but microfilaments often collect into bundles). The monomers, whether free in solution or as part of a polymer, are enzymes; they are GTPases (tubulin) or ATPases (actin). We use NTPase, below, to refer to either case. In general, owing to different nearest-neighbor numbers, one would expect (Hill, 1977b,c,d, 1978; Hill and Levitski, 1980) different NTPase activity by monomers that are (a) free, (b) in the interior of a polymer, or (c) on either end of a polymer. As is well-known, this is indeed observed in these systems, as follows. If hydrolysis occurs rapidly as subunits add (see Appendix 3), interior monomers would be frozen in the cycle state A D(A refers to a monomer, D to NDP). Unlike free monomers, they are unable to exchange their NDP with NTP. In Fig. 2A, the principal pathway involves the attachment of monomers in state A T and the hydrolysis of NTP while on the polymer end. In Fig. 2B, the NTP is hydrolyzed to NDP and Pi on the free monomer (Brenner and Korn, 1980), but release of the phosphate occurs from the polymer. End monomers and free monomers can both pass readily through parts of the NTPase cycle, but only very slowly through the complete cycle; however, the two parts complement In Solution

.....................

A

On Polymer End o or /l

B

FIG.2. Two possible NTPase cycles involving a combination of monomer states from free monomers in solution and from monomers on the end of a polymer. T = NTP; D = NDP; P = PI. Circular arrows show dominant directions. The squared species are dominant in the overall NTPase cycle.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

13

each other, thus allowing complete enzymatic activity in combination with on off transitions at each end of the polymer. The effect of this is to link the hydrolysis of NTP closely with the on-off transitions of monomers. Of the two specific possibilities shown in Fig. 2, we shall refer below, for concreteness, to the scheme in Fig. 2A. Corresponding remarks about Fig. 2B will be obvious to the reader. A four-state cycle for the hydrolysis of NTP by a monomer in solution is shown above the horizontal line in Fig. 2A. This is the minimal cycle required to include all necessary molecular events. The cycle may be more complicated than this because of conformational changes in A. The dotted lines indicate that the transitions A T + AD.p + A D in the lower half of this cycle are so slow that they can be ignored. The same four-state cycle, at the bottom of the figure, refers to NTPase activity on a terminal bound monomer at either end of a polymer. A different set of transitions, A D A AT (dotted lines), is assumed to be negligibly slow in this cycle. The vertical transitions in Fig. 2A represent reversible attachment and release of monomers, in either the AD or A T form, from either end of the polymer. The overall six-state cycle that remains (if we ignore the dotted lines), operating in the clockwise direction, is a complete NTPase cycle: NTP is bound, NTP is hydrolyzed, and products are released. This complete cycle is put together from two partial cycles that are separately ineffective in hydrolyzing NTP. Incidentally, this feature of combining complementary partial enzymatic cycles, along with on G off transitions of the enzyme, is not novel. Essentially the same concept is used in most muscle contraction models (Eisenberg et al., 1980): part of the complete myosin-ATPase cycle is traversed when myosin-ATP is free and part when it is bound to an actin monomer of a thin filament. If either pair of dotted transitions in Fig. 2A is, in fact, used to a significant extent, a new NTPase cycle is introduced that relaxes the assumed tight coupling between NTP hydrolysis and on off transitions. We now simplify the above kinetic model (Fig. 2A) considerably by assuming that AD and A in solution and A T and A0.p on either polymer end are unimportant transient intermediate states. Hence the only significant states remaining are A T in solution and A Don the polymer (as indicated by the boxes in Fig. 2A): the six-state NTPase cycle becomes a two-state cycle. Under some conditions A D in solution is also a significant species and should be included (Niedl and Engel, 1979). This produces a reduced three-state, rather than two-state, cycle. We shall confine ourselves in this article to the simpler and presumably more important two-state case. However, much of the corresponding and not very different three-state theory has been published (Hill, 1981b). There is also recent evidence (Carlier and Pantaloni, 1981) that the state

--

*

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TERRELL L. HILL AND MARC W. KIRSCHNER

A T on the polymer can exist transiently. This is more difficult to include in the theory. Appendix 3 is concerned with this subject. Figure 3 shows, schematically, the polymer with its two-state enzymatic activity at each end. The rate constant notation for the two different ends is shown in Fig. 3A while the corresponding NTPase events (from Fig. 2) are given in Fig. 3B. Thus a1and p1 are second-order rate constants for the overall process (Fig. 2A) that leads from A T in solution to A D on the polymer, AT(^)

+

AT(P)

+

AD.P(P)+ AD(P)+ Pi,

(10)

where s = solution and p = polymer end, while a2 and p2 are first-order constants that refer to the transition from An on the polymer to AT in solution, A&)

+ T + &(s) + T + A(s) + D + T

A,(s)

+D

(1 1)

The inverse rate constants (negative subscripts) are probably negligible under ordinary conditions, where the concentration of AD(s)is small compared to that of AT(s) and the hydrolysis of NTP on A&) is fast compared AT (solution)

Polymer

AT (solution)

AD AD

AD

AD AD

CAD 1 AD

132

13-2

13-1

81

I / AT (solution)

AT 6olution)

A B FIG.3. (A) Rate constant notation for two-state NTPase cycles at the two ends of a steady-state polymer. (B) Gain and loss of ligands in the course of the two-state NTPase cycles.

MICROTUBULE AND ACTlN FILAMENT ASSEMBLY-DISASSEMBLY

15

to the release of A T from the polymer. However, these inverse rate constants must be included in order to understand the connections between the kinetics and the thermodynamics of the problem. The two inverse pairs of rate constants a , a’ and p, p ‘ , each operative at one end of an equilibrium polymer (Section II,A), are replaced here by at the a end two NTPase cycles, with two inverse pairs a l , and a2, and two inverse pairs pl, and p2,pP2at the p end. There are now two biochemically distinct modes of attachment and detachment of monomers for at each end (e.g., at the a end, at and a s for attachment and a2and detachment). Also, there is a thermodynamic force (the negative of the NTPase free energy of hydrolysis) driving each of the NTPase cycles. Thus, there are altogether new conceptual features present in the steadystate polymer compared to the equilibrium polymer (Wegner, 1976; Hill, 1980a). Wegner was the first to recognize that “treadmilling” (see below) is a consequence of these features. As an addendum, we summarize and comment here on the above notation system for the rate constants. We have introduced this system, which has the virtue of economy and clarity, for use in our later analysis. All other treatments of the kinetics have used different schemes (see Appendix 1). However, none has been as complete. We refer to the two polymer ends as a and p. In general, the a end is assumed to have a lower critical concentration than the p end and hence corresponds to the A end of Margolis and Wilson (1978) and the plus end of Borisy (1978). The monomer attachment transition involving subunits in the NTP form, AT, uses a1for the forward (on) second-order rate constant and a-l for the inverse (of€) first-order rate constant, because the on rate predominates. The monomer transition involving the release of A D uses a2 for the firstorder off rate constant and a* for the on second-order rate constant because here the off rate is dominant. Thus in many cases we need consider only a1and a2 and not and a*. Similarly for the p end, p1 is the second-order rate constant for addition Of A T , p-l is the inverse first-order rate constant, p2 is the first-order rate constant for the removal of A D , and p-2 is the inverse second-order rate constant. Again p1and p2 predominate over p-l and p-2. 1. Thermodynamic Force and Rate Constants The eight individual rate constants for the processes shown in Fig. 3 are related to the free energy of NTP hydrolysis. In this section we examine this relationship explicitly (Hill, 1980a). Because we make the simplifying assumption that the concentrations of A D and A in solution are negligible, the free monomer concentration c now

16

TERRELL L. HILL AND MARC W. KIRSCHNER

refers to A T in solution. As in Eq. (l), we denote the chemical potential of AT, at c , by P.\T

= p!lT

+ kTln c

(12)

Similarly, the chemical potentials of NTP, NDP, and Pi in solution are written

The only thermodynamic force driving the cycles in Figs. 2 and 3 is the free energy of hydrolysis of NTP. At concentrations cT, cDrand c p (the concentration c of A T is not involved), this force is

XT

/AT

- p~ -

p p

=

&

- p! - p$

+ kTln (cTIcDcP)

(16)

This quantity is usually of order 12-14 kcal mole-'. The standard free energy of NTP hydrolysis is ,ub + p0p - p'+, which is of order -7 kcal mole-'. XT can obviously be varied by changing cT, c D ,and c p . The monomers in the polymer are in state A,. The chemical potential of which is a property of the bulk polymer, these monomers is denoted pUlD, not of its ends. Any subunits at or near the ends that are in state A T transiently would contribute only to the negligible end effects (but see is the analogue of po in Section Appendix 3). The chemical potential p..\,, I1,A. We are now in a position to derive the fundamental relation between the NTP force XT and the rate constants in Fig. 3A. We first consider the hypothetical but possible situation in which all transitions in Fig. 3A are processes, that is, blocked by inhibitors except the (inverse) a1 and AT(s) AD(p) + Pi. If the concentration of A T in solution is now adjusted ( c = c y ) so that there is equilibrium between A T and the polymer (as in Section II,A), at some fixed concentrationcpof Pi, then we have, as in Eq. (2) , p . + ~,up= ,uh = polT+ kTln c',) (17) The only new feature here, compared to Eq. (2), is the extra ,up term on the left, which arises because, on attachment of a A T molecule to the polymer, A T becomes A D (thus increasing the number of bulk A D by one) and one Pi is released to the solution [Eq. (lo)]. Also, because there is a true monomer-polymer attachment equilibrium and detailed balance, we have alc(el)= a_'. If we use this relation to eliminate c'," from Eq. (17), we obtain kTln (al/a-l)

= PoIT

- @.\D

+ PUP)

(18)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

17

This is a relation between intrinsic molecular properties that obviously does not depend on the free monomer concentration c . Although the condition c = c'," was used as a convenient device to derive Eq. (18), the equation provides a property of the rate constants a1and that is valid under any conditions, including steady states and transients (but see the end of Section 11,A). If we apply the same argument to the PI,p-l pair at the other end of the polymer, the free energy difference on the right-hand side of Eq. (18) and hence c'," [Eq. (17)] are necessarily the same as at the a end because initial ( A T ) and final (A,), Pi) states in the attachment process [Eq. (lo)] are the same. (It is not necessary that the intermediate biochemical details be the same at the two ends; see, for example, Fig. 2B.) Therefore a l / a - l = /31/p-l.This is of course equivalent to & / a ' = p / p ' in Section II,A, even though there is some biochemistry involved in the attachment process in al, and pl, p-l [Eq. (lo)]. The key point here is that each end participates in the same equilibrium reaction and hence by detailed balance the equilibrium constant must be the same for both ends. As in Section II,A, once a subunit ( A o ) has been added to the polymer molecule, free in solution, at either end (a or p ) , by the same overall biochemical process, the polymer "cannot tell" which end was used-it is in the same state in either case. (This is not true, of course, if a radioactive label is employed.) The a2,a-2 (and p2,p-2) transitions, for the process AD(p)+ T A T ( s ) + D, can be treated in essentially the same way. At given concentrations of cT and cD, we start with

*

P.\D

+ PT

=

P.e\T+

instead of Eq. (17), and sac'," IkTln

(a2/a-2)

PD =

=

PO\T + k T h ci2' +

PD9

(19)

a 2 ,and obtain

(P?T - PT + PD) + PI') - (Po\T - XT),

= P.\D = (P.\D

(20)

where XT is the NTP thermodynamic force defined in Eq. (16). By the same argument as above, we also have a 2 / a P 2= p2/p-2. If we now add Eqs. (18) and (20), which apply to the two biochemically different [Eqs. (10) and (1 l)] parts of the NTP cycle, there results kTln ( ~ ~ a ~ / a -=~kTln a - (/31p2/p-1p-2) ~)

=

XT

(21)

These are the desired relationships between NTP cycle rate constants and XT. They hold under any conditions (transients, steady state, etc.); they are self-consistency requirements of the two-state aggregation model we are using. Figure 4 expresses Eqs. (18), (20), and (21) graphically, in terms of free energy levels for a single cycle. The total free energy drop XTin a cycle is

18

TERRELL L. HILL AND MARC W. KIRSCHNER

broken down into its two component parts, for the two steps in the cycle [Eqs. (10) and (1 l)]. The levels i, ii, and iii in Fig. 4 may be termed “basic free energy levels,” for convenience, because of their close analogy to the basic free energy levels (Hill, 1977a) introduced for the biochemical cycles of independent enzyme molecules. However, special treatment has been required here because of the aggregation of the enzyme molecule. Because XT is usually of order 12-14 kcal mole-’, e X T r k T is usually of order lo9 or 10 lo. The separate dimensionless factors in = a-1

a-2c

P&.

p-1

-,X,/kT

P2 p-2c

are then perhaps of order 104 to 106, though they obviously depend on c . Consequently, the reverse rate constants (negative subscripts) are presumably negligible for kinetic (not thermodynamic) purposes. Equation (16) shows that XT depends on the concentrations of NTP, NDP, and Pi through the term kTln ( C T I C D c p ) . Thus, from Eq. (21), we have aIff2Ia-1~-2

= PlP2lP-1P-2

-

(22)

CTICDCP

Correspondingly, for the reactions (partial cycles) represented by Eqs. (18) and (20), or Fig. 4,

%lcLl= P I I P - 1 aZIa-2 =

P2/p-2

-.

11CP

(23)

cT/cD

If a particular detailed biochemical scheme is adopted, such as the two shown in Fig. 2, then it is possible to relate the individual rate constants a l ,

kTln a l l @ -1 = kTln 8118- 1

ii

i

PAD+Pp

AD(P)+ P

XT Basic Free Energy

t

FIG.4. Schematic basic free energy levels for the two-state NTPase cycle at either polymer end.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

19

etc. of the two-state cycle (which considers certain states negligible) to the more elementary rate constants (not shown in Fig. 2) of the complete biochemical scheme. In this way the separate explicit dependences of al,a - l , etc., on cT, cD, and c p can be found. Examples have been published elsewhere (Hill, 1980a, 1981b) but we shall not pursue this subject here. Associated Protein Attached to Monomer or Polymer. Proteins that bind preferentially to either the polymer or monomer form of tubulin or actin have been suggested to play important roles in regulating polymerization. In the case of actin, a protein called profilin has been shown to be widespread and to bind exclusively to the monomer of actin in a 1 : 1 complex (Carlsson et al., 1977). In the case of microtubules two major proteins, called tau (Weingarten et al., 1975) and HMW (MAP2) protein (Murphy and Borisy, 1975), have been described that induce polymerization, presumable by binding preferentially to the polymer. There is no essential complication in the above thermokinetic discussion if an associated protein molecule (e.g., profilin) is invariably attached to a free monomer but not to an aggregated monomer. In this case the basic free energy levels in Fig. 4 become i:

ii:

p!iTM

p,iD

+ pP + p

iii:

M

~ % T M-

XT,

where M refers to the associated protein and p Mis the chemical potential of M free in solution at its actual concentration. Formally, M and P are analogous. In the opposite case (e.g., tau) where M is invariably attached to a monomer in the polymer but not to a free monomer, the basic free energy levels are i:

pp\T

+

pM

ii:

+

p . 1 ~ ~ pup

... AT+

111:

FM

- XT.

Of course if M is invariably attached to a monomer in both the free and aggregated state it can be considered part of the monomer and need not be acknowledged explicitly.

2. Treadmilling Rate and NTP Flux The rate of treadmilling (defined below) and the rate of NTP hydrolysis are easily measured experimental quantities, and can be related to the individual rate constants. It is simplest to consider these rates as a composite of contributions from the four individual reversible pairs of transitions. These are introduced by means of the illustration in Fig. 5 , where several fluxes are plotted against the monomer concentration. The two pairs of broken lines in Fig. 5 correspond to reversible transitions between A T ( s ) and AD(p)and are analogous to the single pair of lines in Fig. 1B.

20

TERRELL L. HILL AND MARC W. KIRSCHNER

Scale

J

- 82 - a2 - (a2 + 821

FIG. 5. Various equilibrium and steady-state fluxes, as described in the text, for a steady-state polymer with free ends. The critical concentration c:') is very small, but not zero. Note the scale change needed because e:'] is very large.

and p-l are very small, the lines a l c - a-l and plc Because ando, are very small, intersect at c'," = 0. Correspondingly, because the lines C Y - ~ C- a2 and PSc - PZare almost horizontal and intersect at a very large value of c, c',").As in Fig. lB, fluxes for the same reaction at the two ends of the polymer must intersect at the same critical concentration. The rate of adding monomers at the a end of the polymer is simply the sum of the fluxes for the a l , and az, reactions: J,

= (a1

= a1c

+ a&

- (a2

+ a-1)

- a2,

(24) (25)

where Eq. (25) is presumably a very accurate approximation, as explained above. J , is shown as a solid line in Fig. 5 , with slope ( a l )the same as for the line a l c and intercept ( - a & the same as for the line ( Y - ~ C- as. Similarly, at t h e p end, JP=

(Pi + P a ) c - ( P z + P-1)

=PlC

- P2

(26) (27)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

21

J pis also included as a solid line in Fig. 5 , with properties analogous to J,.

J , = 0 at c = c,, where c, = a 2 / a 1 This . can be called the critical concentration for the a end. Also, J p = 0 at c = c p , where c p = p2/p1,which is the critical concentration for the P end. Unlike c‘,“ and c?), c, and c p are not equilibrium properties because al and a2 do not pertain to inverse transitions, nor do P1 and P2 (see Fig. 3A). There are no thermodynamic relations that determine the ratios a 2 / a 1and p2/p1 [compare Eqs. (18) and (20)l. Whereas c t ) and cf) have extreme values, c, and c p are in the measurable range, usually of order 1 p M (al and p1are of order 10 p M - ’ s-l; a2 and p2 are of order 10 s-l). Because c , < c p , the a end in the illustration in Fig. 5 is the so-called + end of the polymer and the P end is the - end: in the free monomer concentration range of primary interest, c , Ic 5 c p , J, is positive (monomers add to the + or a end) and J p is negative (monomers depart from the - or /3 end). Thus, in this concentration range there is a net flux of monomers through the polymer. The net total rate of addition of monomers to the polymer is Jon

J,

+ Jp

= (a1 + P l k

-

(a2

+ P2)

(28) (29)

This is included as a solid line in Fig. 5 . Jonintersects J p at c, where the rate of addition to the a end is zero. Jonintersects J, at c p , where the rate of addition to the P end is zero. Jon= 0 at c = c, , where J , = - J p (vertical line). That is, this is the definition of c,. Necessarily, c, < c , < c p . Because Jon= 0 at c = c,, the mean number of subunits in a polymer remains constant with time, at this concentration (except for large fluctuations-see Section VI), though monomers are being added at the +(a)end and are being lost at the - ( p ) end. This phenomenon is usually referred to as treadmilling (or head-to-tail polymerization) (Wegner, 1976). The explicit expression for c,, which is a joint steady-state property of both ends of the polymer, from Eqs. (24), (26), and (28), is cm =

(a2

= (a2

+ P 2 + a-1 + P - 1 > / ( ~ 1 + P1 + a-2 + P - 2 ) + P2)/(a1 + P1)

(30) (31)

As in the discussion following Eqs. (9), if in aclosed system with many polymer molecules we start with c > c, ,there will be net growth (Jon> 0) of the polymers and the free monomer concentration c will decrease to c, . Similarly, if c < c , at the outset, there will be net loss of monomers from the polymers and c will increase to cm. Thus c, is the stable value of c in a closed system of this type. We shall usually use the term “treadmilling” to refer to the particular case Jon= 0 (polymer of constant length). But, from a more general point

22

TERRELL L. HILL AND MARC W. KIRSCHNER

of view, treadmilling is a meaningful concept in the entire monomer concentration range c, < c < c p , where J , is positive a n d J Bis negative. In this range, excluding fluctuations, a monomer added to the a end will make its way through the polymer and leave at the P end. However, the monomer itself is not moving; rather, the a end is growing and the P end receding. The action is similar to that of the caterpillar tread on a tractor. The rate at which a monomer, newly added at the a end, approaches the P end is called (here) the treadmilling rate, J,. This is, of course, also the rate at which the p end approaches the added monomer, that is, the rate at which the p end recedes. Thus J , (in units of monomers s-* per polymer molecule) is defined as -Jp, in the interval c, < c < c p . Hence J m = P2 - plc . This is shown as a heavy line in Fig. 6 , which represents the same hypothetical system as in Fig. 5 . The value of J , at c = cm is of special interest; this is denoted (also called the monomer flux). Here the “tractor” maintains a tread of constant length. At c = c,,

Ji

=

-Jp(cm)

(32)

J,(Cm)

= ( f f l P 2- f f 2 P M f f l

+ PA,

(34)

where Eq. (33) follows on substituting Eq. (30) into Eq. (24) or (26), and using ffllff-1

=

PlIP-1,

ff2lff-2

= P2lP-2

(35)

and Eq. (21). The steady-state treadmilling rate J ; [Eq. (33)] is zero if the two ends of the polymer are alike (a1 = P1, a2 = P2) or if (hypothetically) cT, cD and c p have values such that XT = 0 (Lea, at NTP hydrolysis equilibrium). Thus, a nonzero NTP driving force XT is a necessary condition for steady-state treadmilling in solution. Incidentally, this is not a necessary condition in other circumstances (Sections IV,A and V,A). The approximate Eq. (34) follows because usually p-2, and = O(lO-’O) are all negligible. J,” is necessarily positive because we chose c, < cp: cp

= P2IP1 > c, = f f Z l f f 1 ,

fflP2

> ff2Pl

If a polymer is either growing or shrinking, the total net flux (from both ends) for subunits participating in part 1 of the NTP cycle, J1, involving the addition of AT(s)[Eq. (lo)], is not equal to the total net flux, J2, in part 2 of the cycle, involving the dissociation Of A&) [Eq. (ll)]. The definitions of the fluxes corresponding to the two parts of the NTP cycle are: J, = (ff1c J2

= (ffz

ff-1)

- ff-zc)

+ ( P I C - p-1) + ( P 2 - P-ZC)

(36) (37)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

23

Under conditions when the polymer is either growing or shrinking and J1 # J2,the NTP flux, JT,is defined as the lesser of J1and J2,because it is only this amount of flux that refers to completed NTP cycles (i.e., part 1 and part 2). In the usual excellent approximation, we have

J1 = (al + pl)c

and

J2 = a2 + p2

(38)

The J1 flux is proportional to the free monomer concentration; the J2 flux is constant. These two lines are included in Fig. 6. Because the NTP flux, JT, is defined as requiring a complete NTP cycle [addition of A T ( s ) and removal of A,)(p)], JT increases with monomer concentration until c,. At this point subunits add and come off at the same rate. Above this concentration J T follows J2: subunits are being added faster than they come off, JT is independent of monomer concentration. The heavy portions of the lines J1 and J2 in Fig. 6 represent the two branches of JT. Explicitly,

J$-) = 51 = (a1 + p 1 ) C (c 5 c m ) (39) J$+) = J2 a2 + p2 (c 2 cm) The superscript (-) refers to the shortening case, whereas (+) refers to lengthening. The first branch of JT,.I$ is parallel -), to .lo,,.

a2 +82

J

-82 - a2

- (a2

+ (I21

FIG.6. Some fluxes taken from Fig. 5 together with additional fluxes defined in the text.

24

TERRELL L. HILL AND MARC W. KIRSCHNER

It is simple to verify from the definitions that J1 = J2 when J , that is, J1 = J2 at c = c,. At c = c,, JT is denoted JT: J? = J l ( c m )

=

= ff2 + P 2

+ J p = 0; (40)

J2(cm)

(42)

9

where we have used Eqs. (21), (30), and (35)-(37). If the two ends of the polymer are the same, JF is still positive (unlike Jf ,which is zero in this case). However J ; = 0 when XT = 0 (NTP hydrolysis equilibrium). In the typical example shown in Fig. 6, based on Fig. 5 , the flux of subunits through the polymer, J , , is significantly smaller than the number of complete cycles of assembly and disassembly, JT.The ratio .T=

Jf/JF

=

(alp2 -

azP1Ma1

+ P1Na2 + P 2 )

(43)

at c, is of particular interest. It should be noted that Eq. (43) follows from Eqs. (33) and (41) without the usual approximation that the reverse transitions can be neglected. A similar relation was obtained by Wegner (1976) without including reverse transitions, but was expressed in terms of c, . Because the terms alp2 and a2Pl appear in both numerator and denominator in Eq. (43), necessarily s < 1. One can regards as a kind of kinetic (not thermodynamic) efficiency: the treadmilling rate of the subunits (at c = c), relative to the total rate of NTP turnover at both ends of the polymer. The quantity s , above, was called r ) in Hill (1980a). Throughout this article we shall use r ) to represent a true thermodynamic efficiency, as in Hill (1977a). The definition o f s in Eq. (43), as J",JT, seems to us to be the most natural and logical choice. It should be noted that J? is nor quite the same as the total rate of association or dissociation events at the two ends at c = c,. This latter rate is given by (a1 +

p1

+ ff-2 + P-Z)C,

= CYZ

+ p2 + ff-1 + p-1

This quantity was used in the definition of s in Hill (1980a) but we abandon it here. Of course, for practical purposes (when reverse rate constants are negligible), (a1

+ Pl)Cm

=

a2

+ P2

=

JF

Thus, the distinction between the definitions is of more conceptual than practical interest. In the steady-state kinetics of independent enzyme molecules with multicycle kinetic diagrams (Hill, 1977a), a considerable conceptual simplification is realized if one regards the observable fluxes as being made up of

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

25

separate contributions from the various cycles of the diagrams. The same is true here (especially in Section V,B), at steady state (c = cm). We have so far referred to the two NTP cycles at the ends of the polymer. Actually, there arefour NTP cycles, as shown in Fig. 7. Cycles c and d make mixed use of the two ends. All of these cycles have the same force, XT.The relations of force to rate constants are (using the cycle designations in Fig. 7): (a) ala2/a-la--2 = eXT/kT (b) PlP2/P-lP-2 = eXTkT (44) (c) alP2/a-lp4 = eXTlkT (d) a2p1/a4p-1= eXdkT The flux J ; in Eq. (41) may now be regarded as a superposition of separate contributions from the four cycles: J ; = J a + Jb + J , + Jd (45) a-la-z)/D Jb = (pip2 - p-ip-z)/D J c = (alp2 - ~ - I P A ) / D Jd = (~2p1 - a-~p-i)/D (46) D + + + p-2 The general form of the connection between Eqs. (44)and (46) is conventional (Hill, 1977a); only the composition of D is different (because of enzyme aggregation here). Neglecting reverse rate constants, the relative contributions of the separate cycles a, b, c, and d to J ; are proportional to a1a2,PIP2, alp2,and a&, respectively. Furthermore, we see from Eq. (33) that J a = (%a2 -

JE

=

J,

-

(47)

Jd

In view of Fig. 7, this is just what we should expect: cycles a and b make no contribution to the treadmilling motion or rate; cycle c makes a positive contribution (a subunit is added at the a end and another is removed at the p end); and cycle d makes a negative (wrong-way) contribution. 3 . Steady-State Rate of Dissipation of Free Energy The rate of free energy dissipation, Td,S/dt (conventional notation), is important in bioenergetic kinetics because of its close relation to the effiPolymer

-a 82

a2 n 01

82

T w B1

n c

02

O&d

W

a1

B1

FIG.7 . Component cycles that contribute to the steady-state NTPase flux.

26

TERRELL L. HILL AND MARC W. KIRSCHNER

ciency of free energy transduction. Td,S/dt is equal to the rate of free energy consumed minus the rate of useful work or free energy produced. The efficiency is the ratio of the latter to the former. We ultimately would like to show (Sections IV,B and V,B) how the NTP hydrolyzed in polymer assembly can be made to do useful work. However, here we consider how much free energy is dissipated by a free treadmilling polymer in solution, at steady state (c = c,) . In this case NTP is consumed but no work or other form of free energy is generated. The rate of free energy dissipation should be the product of the flux and the force, or JFX, per polymer molecule. We can confirm this explicitly, as follows. There are four transition pairs in Fig. 3A, which we can designate as al, az,P1, and PZ.According to the second law of thermodynamics, the net flux in any transition pair must be in the same direction as the downhill free energy gradient for the process. The product of the flux and the free energy decrease gives the (never negative) contribution of the particular transition to the rate of free energy dissipation. In the present problem, at steady state, we have TdiSldt = J a l A ~ a l+

J ~ P P+~JP,AP.P, , + J P P P P ~ , (48)

where the net transition fluxes are Ja,

= Q~C, -

JP, =

a-1,

P i ~ m- P-1,

J,,

= CQ

JP, =

Pz

-

a

-

2

P-zc,

~

~

(49)

and the free energy differences (decreases) are kTln C,) - (PAD + PP) &al = A P P , = (P!T A ~ a 2= A P P ~= (P.\o + PP)- (P.XT + kTln c m - XT)

(50)

These are the free energy differences in Eqs. (18) and (20) except that here the standard chemical potential poITof the free monomers has been replaced by the complete chemical potential. The separate free energy levels, as in Fig. 4, from Eqs. (50), are analogous to the “gross free energy levels” (Hill, 1977a) in conventional (nonaggregation) systems. Note that

A P ~ ,+ A p a Z = APP, +

b u g , =

XT

( 5 1)

as in Fig. 4, and that, from Eqs. (18) and (20), Apal = Apol = kTln (alcm/a-l)= kTln (plc,/p-l) (52) APa2 = AP,, = kTln (azla-zc,) = kTln ( P z / P - z c m ) Comparison of Eqs. (49) and (52) shows that every term on the right-hand side of Eq. (48) is positive (i.e., .Ia, and A p a l , etc., have the same sign) as the second law demands. It is important to note that all of Eqs. (48)-(52)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

apply as well if c, is replaced by an arbitrary c. Finally, in the c on using

= c,

27 case,

and Eq. (51), Eq. (48) reduces to as was to be proved. We shall see in Sections IV,B and V,B that a steady-state polymer with one end anchored, or treadmilling between restraining barriers, can convert some of the NTP free energy of hydrolysis into mechanical work, if a resisting force is attached to the subunits of the treadmill. In this case, not all of JTXT is dissipated, as in Eq. (54), which applies to free treadmilling polymers in solution.

4. A New Method for the Measurement of Rate Constants by Exchange of Labeled Subunits Conventionally, individual rate constants are measured away from steady state, under conditions where it is assumed that certain transitions are negligible (see, for example, Johnson and Borisy, 1977). There are several problems with this analysis stemming from the possible concentration dependence of rate constants at concentrations other than the equilibrium concentration ce (as described in Section I1,A) and the difficulty of distinguishing nucleation from polymer elongation. An alternative to obtaining rate constants obtained away from steady state is to measure the exchange of labeled subunits as a function of monomer concentration in the regime just below c, to just above c p , as described below. Suppose, at the outset of an experiment ( t = 0), we start with radioactively labeled subunits in the polymer molecules but with unlabeled free monomer molecules, at concentration c. We consider here the mean rate of loss of label from the polymers at small enough times so that (a) c remains essentially constant, (b) the relative concentration of labeled free monomers is always negligible, and (c) no significant number of (short) polymer molecules completely disappear (if c < c,, the polymer molecules will shorten, as shown in Fig. 6). Thus, added unlabeled monomers will “cap” the original labeled subunits on a polymer end that lengthens, whereas labeled subunits will be lost to the solution (and measured) from an end that shortens. When c > cB (see Fig. 8, which is based on Fig. 6), both ends will lengthen and there will be virtually no loss of label to the solution (the heavy lines in Fig. 8 show the rate of loss of label per polymer molecule as a function of c ) . When c, < c < c B ,the (Y end grows but the p end recedes

28

TERRELL L. HILL AND MARC W. KIRSCHNER

a2 + 82

J

-82 - a2 - (a2 + 821

FIG.8. Rate of loss of label (heavy lines) for the illustrative system in Figs. 5 and 6.

and loses label at the rate it recedes. This is just what we have defined as the treadmilling rate, J,, above (Fig. 6). Thus, Rate of label loss

=

J,

=

-Jp

=

p2 - plc

(c, < c < cp) (55)

as shown (heavy line) in Fig. 8. At c = c,, this rate is J ; [Eq. (34)l. Both /I1 and p2 can be determined if this line [Eq. (55)] is established experimentally. Finally, when c < c,, label will be lost from both ends of the polymer: Rate of label loss

=

-.Io,,= (a2+

p2) - (ar + pl)c

( c < c,)

(56)

This is also included (heavy line) in Fig. 8. Experimentally, it should be possible to determine the four rate constants from the slopes of - J p and -Jon (which give p1 and a1 + &, respectively) and the two intercepts c, = p2/p1and a2 + p2. The break point at c, = a 2 / a lcould be used to confirm the results. Fluctuations are of considerable interest in this problem, especially at c = c,. They will be dealt with in Section VI.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

29

Gain of Label by Polymer. The opposite situation in which initially unlabeled polymer molecules pick up label from the solution, especially under treadmilling conditions, has been treated at length by Wegner (1976) and more recently by Cote and Borisy (1981). The treatment of this problem, which we shall not repeat here, is complicated by the fact that the label takes different lengths of time to pass through (by treadmilling) polymers of different sizes in a polydisperse sample. 111. Polymer with End or Ends Capped or Anchored

In this section we consider polymer molecules with one or both ends not free. This could arise, for example, if one end of the polymer is anchored at a nucleation site, or if one or both ends are partially or completely capped by foreign molecules or a foreign structure or structures, or if one end is anchored and the other capped. A polymer molecule with both ends capped can move about in solution, but we do not refer to these ends as ‘‘free.” We do not include in this section cases in which a polymer, in contact with structures of one kind or another, is under either an extensive or a compressive force, F . Here,F = 0. A cap, for example, will alter the on & off subunit kinetics at the capped end (see below), but the cap offers no mechanical resistance to lengthening or shortening the polymer. Examples in which F # 0 will be treated in Sections IV and V. A. EQUILIBRIUM POLYMER The discussion here is a continuation of that in Section I1,A. Consider a polymer end, which, when free, has rate constants a(on) and a’(of€).Now if the same end is in contact with a foreign structure or cap, these rate constants will in general be altered, say to k(on) and k‘(o@. We would naturally expect (though exceptions are conceivable) k < a and k‘ < a’ owing to the physical impediment placed in the way of free exchange between monomers in solution and monomers on the end of the polymer. In fact, the impediment might well amount to essentially complete blockage of monomer exchange, in which case k = 0, k‘ = 0. The term “cap” often refers to this situation only, but we use “cap” in a more general sense. We are interested here in cases in which the blockage of monomer exchange is not complete and k and k’ are nonzero. The monomer chemi. cal potential in the bulk of the long polymer molecule is still p O ( T )Any modification of an end (provided F = 0) has no influence on bulk ther-

30

TERRELL L. HILL AND MARC W. KIRSCHNER

modynamic properties. Addition of a monomer at the modified end actually adds a monomer to the bulk of the polymer and does not alter the equilibrium state of the modified end (see Section 11,A). Of course the chemical potential of free monomers in solution is unchanged: p sis given by Eq. (1) and p!(T) is the standard chemical potential. Thus p! - po and the equilibrium constant ce for the process A(po1ymer) + A(so1ution) are the same [Eq. (3)] whether the end is free or modified. That is, c, = a ' / a = k ' / k

(57)

In fact the relationship between a single end that is either free or modified is essentially the same as that between two different free ends of an equilibrium polymer [Eq. (4)]. Figure 9 shows the net rates of addition of monomer to the polymer end in question, as a function of c . This is analogous to Fig. 1B.Equation (57) tells us that the cap or structure at the polymer end alters a! and a' by the same factor: k / a = k ' / a ' . The kind of analysis in Eqs. (5)-(8) of the operational rate constants a and a', as composites of more microscopic rate constants (reflecting the variety of subunit configurations at the end of the polymer), can also be made for k and k ' . Of course the presence of a cap or structure at the polymer end will in general alter all of the microscopic rate constants as well as the relative free energies and probabilities of the various surface

J

- k'

- a'

FIG.9. Net aggregation flux at the a end of an equilibrium polymer when this end is free, modified by a cap or an attachment or a barrier, or completely blocked (zero flux).

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

3I

configurations. Interactions between end monomers and the foreign molecule or structure attached to the end, as well as the dependence of the structure's own free energy on the surface configuration of the end monomers of the polymer, have to be taken into account. We now supplement the above general discussion with an explicit treatment of a highly idealized model that will illustrate some of the above points, as well as Eqs. (5)-(8). This is perhaps the simplest possible model that can be used for this purpose; it is meant to be pedagogical rather than realistic. a. Simple Kinetic Model, without Cap. We begin by examining the free a end of a two-stranded polymer with a square lattice of square subunits (Fig. lOA), following Eqs. (5)-(8). We assume that only three end configurations are stable enough to take into account. These are denoted 0, 1, and 1' in Fig. 10A; 1 and 1' have the same properties, by symmetry. Configurations 1 and 1' are less stable than 0 because the top molecule in 1 and 1' has a missing horizontal neighbor. If w (negative) is the horizontal free energy of interaction between neighbors (Fig. IOB), 1 and 1' are less A

w

0 B

1

1'

C

FIG. 10. (A) Transitions at the free a end of a simple model of a two-stranded polymer. (B) Double-headed arrow indicated is a horizontal neighbor interaction with free energy 11' (negative) in the bulk polymer. (C) Addition of a hypothetical cap to this simple model.

32

TERRELL L. HILL AND MARC W. KIRSCHNER

stable than 0 by a free energy w / 2 (each subunit in Fig. 10B, in the bulk of the polymer, can be assigned a horizontal interaction free energy w/2). Here we are ignoring any differences in the vibrational motions of the three kinds of end subunits. Thus, at equilibrium, the probabilities of the three end configurations are related by [Eq. (7)] p; = p;,,

p;/p'o = ewl2kTsx

(58)

and thus p'o = 1/(1

+24,

p; = p;r = x/(l

+ 24,

(59)

wherex < 1 andp;>p$. Figure 10A shows all of the allowed on-off transitions and the corresponding microscopic rate constants. Detailed balance at equilibrium requires that on and off rates be the same for each elementary process [see Eq. Wl: p%eao

= pela;,

pfceal

= ~8aA

(60)

The first of these relations applies to both sites of configuration 0. The second applies to one site in each of 1 and 1'. Thus when we sum ovei sites and configurations, as in Eq. (6), we obtain a

=

2(p@o

+ pfal),

a' = Z ( p f 4 + pga;)

(6 1)

These express the operational rate constants a and a' as equilibrium averages over microscopic rate constants. We also have the microscopic equilibrium constants [Eq. (S)] a;/ao = c,x-1,

aA/a1 = c,x

(62)

Physically, these equations say that it is easier than average (c,x-l > c,) to remove a subunit from 1 or 1' and harder than average (c,x < c,) to remove one from 0. The average equilibrium constant is, from Eq. (61), using Eqs. (59) and (62), a ' / a = c,, as expected. One other topic is of importance. We have assumed above that the polymer is in equilibrium with the monomer at c,. However, at other monomer concentrations the actual distribution of configurations at the ends will not necessarily be the equilibrium distribution described above, but will be determined by the first-order rate constants that interconnect the three configurations, as shown in Fig. 11. Note that, in this figure, the free monomer concentration c is arbitrary, not necessarily c = c, . When c Z ce, there will be steady net addition or loss of subunits from the a end governed by J , = a c - a'. Our object is to deduce a and a' for arbitrary c.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

33

+

cv0c a; FIG.1 1 . Transitions among the three states 0, 1, 1’ in Fig. IOA.

The polymer end will attain a steady state among 0, 1, and l ’ , as determined by the rate constants in Fig. 11. From PdPO = Pl*/PO = (aoc + aA)/(.1c

+ a;),

(63)

where p o , p l , and p l t are the steady-state probabilities of the configurations, we can easily find Po

=

1/(1 + 2Y),

P1 = P1’

where y = (aoc + aA)/(a1c

= Y/(l

+ 2Y)

(64)

+ a;) (66) (67)

(59) for the probabilities. The averaging in Eqs. (6) is still appropriate for the separate calculation of a and a‘ at an arbitrary c , using the steady-state p s , but there is no longer detailed balance as in Eq. (5). Thus, in the present example, a

=

2(poao

+P

l 4 ,

a’ = 2 h a ;

+ pod)

(68)

as in Eqs. (61), but here the p s are given by Eq. (64). Because y is a function of c , the steady-state configuration probabilities pi are functions of c. Hence, a and a’ in Eqs. (68) depend on c. Thus, for this simple model, we would not expect J , = ac - a’ to be a straight line. Similar problems could arise if various associated proteins do not reach an equilibrium distribution at the ends of the filament. The above considerations could, of course, play a role in real polymers. An ameliorating circumstance, however, would be that on-off transitions among surface configurations, as in Fig. 1 1 , are not the only transitions involved. Subunits may diffuse on the surface, thus changing configurations without on or off transitions. If diffusion transitions are relatively

34

TERRELL L. HILL AND MARC W. KIRSCHNER

fast, the surface could maintain close to an internal equilibrium among configurations, even though the overall process is not at equilibrium (c # c e ) .In this case the equilibrium values of a and a' would be valid at any c. b. Simple Kinetic Model, with Cap. Now suppose that the a polymer end of Fig. 10A has a flexible cap that behaves as shown in Fig. 1OC. The cap bends in 1 and 1' to enhance interaction with the subunits, but at some cost in bending free energy. There is a corresponding new set of microscopic rate constants, b , etc., which are presumably significantly smaller than ao,etc. (We do not enter into this aspect of the problem.) Let wo (negative) be the interaction free energy between the cap and the end subunits in configuration 0, and let w1 be the analogous free energy for 1 or 1' plus the free energy required to distort the cap itself. We assume, where z because of the distortion, that w1 > wo, We define z = e(wo-wl)/kT, < 1. Thus, in the presence of the cap, 1 and 1' are destabilized further with respect to 0 (by a factor xz, compared to x without the cap). The equilibrium probabilities of the end configurations, with cap included, are

Equations (60)-(62) and the accompanying comments then all apply here, as well, but with the changes in notation: all at + ki ;x + xz ;and all pt + P I .The value of c, in these equations is unchanged [Eq. (57)l. Thus the cap adds some new kinetic and thermodynamic features, but no change in the fundamental approach is required. Although we can conclude from Eq. (62) that k ; / k o is larger than a ; / a o by a factor z-' and that kh/kl is smaller than a & / a lby a factor z , we can say nothing (from the above type of analysis) about ratios such as ko/ao, kh/a;, etc. (Figs. 10A and C).

B. STEADY-STATE POLYMER In this section we examine first the consequence of complete blockage of monomer exchange at the anchored end of microtubules or microfilaments. Some of the biological implications have been considered previously (Kirschner, 1980). We shall describe some explicit kinetic examples of such processes. Next we consider polymer molecules that have caps or anchors which inhibit but do not completely prevent monomer exchange. We then discuss how such capping of a steady-state polymer can lead to reversal of the direction of treadmilling and can also lead to a shift in the apparent critical concentration. The biological consequences for spatial control of assembly in cells will be discussed briefly.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

35

1. Some Consequences of Complete Blockage at One End Various dynamic changes in microfilament and microtubule assembly take place so quickly in cells that it is unlikely that appreciable changes in total subunit concentration take place on that time scale. For some considerations, therefore, the cell can be considered a closed system of constant volume. In such a system complete blockage of one end of a steadystate polymer results in specific stabilization of those polymers relative to free polymers, as shown below. Suppose that, in a small, closed volume (say a cell), there exist free monomers (tubulin or actin) at a concentration c > c p(see Fig. 5 ) as well as numerous polymers of three different types: (i) anchored (with monomer exchange blocked) at the /3 end, but with the a end free; (ii) anchored and blocked at the cy end, with the j3 end free; and (iii) both ends free. To avoid complications (special cases), suppose that there are about equal numbers of polymers of the three types. The rates of addition of monomers, per polymer, are shown in Fig. 5 as J,, Jp,and Jon,respectively. If c > c p initially, all of these rates are positive and all three types of polymers will grow (the free polymers will grow at both ends), at the expense of the monomers in solution. Consequently, because the volume is closed and finite, c will decrease. When c becomes less than c p , type (ii) polymers will begin to disassemble (Ja< 0) but the others will continue to grow ( J , > 0, Jon> 0). The concentration c will thus decrease further to c, and the type (ii) polymers may have disappeared (depending on their initial length). At c = c,, type (ii) polymers, if they still exist, will shrink until disappearance while type (i) polymers grow ( J , > 0). Meanwhile, type (iii) polymers maintain a constant length (treadmill). After disappearance of type (ii), c will again decrease because of growth of type (i). This decrease of c will cause type (iii) to shrink (Jon< 0) and finally disappear. Ultimately, type (i) will stop growing and c will stabilize at c = c,. The only surviving polymers are thus of type (i), with a or + end free. The specific physical-chemical implications of the above are that anchorage of polymers at the p end will ensure that those anchored polymers will persist and ultimately be the only polymers in the cell. A consequence of this would be that at steady state all polymers in the cell should be anchored, all should have the a end free, and the steady state should be governed by c,. As discussed previously (Kirschner, 1980; Cleveland and Kirschner, 1982), this has important biological implications. In order for the cytoskeleton to achieve specific cell morphology and directional intracellular transport, the filaments must have specific positions in the cell. Any specific arrangement is always potentially threatened by spontaneous as-

36

TERRELL L. HILL AND MARC W. KIRSCHNER

sembly. For anchored equilibrium polymers, at any concentration where they are stable (c,) the free polymers will also be stable since they are governed by ce as well. Thus spontaneous polymerization or breakage of existing filaments can always lead to randomization of the polymer distribution. What is more, free polymers have a kinetic advantage, since they can grow at both ends. Thus steady-state polymers have a unique ability to specifically stabilize anchored filaments and suppress spontaneous polymerization. Recently this concept has been tested, when spontaneous free microtubule assembly has been induced in living cells (DeBrabander et al., 1981a,b) or broken filaments have been artificially produced (Kirschner and Berns, unpublished). In these cases it could be shown that free filaments are unstable relative to the anchored filaments. As discussed in Cleveland and Kirschner (1982), polarity determination for microtubules and actin are also in general accord with the above theory. Kinetochores, though apparently nucleated with the opposite polarity, probably act as sites of insertion of the p end, while the other end is stabilized by the centrosomes. 2. Explicit Kinetic Example The sequence of events outlined at the beginning of the preceding subsection, with reference to Fig. 5 , can be followed explicitly, as a function of time, after solving several elementary differential equations and introducing values for the various parameters involved. We shall give the necessary theory here together with two quite arbitrary illustrative cases. The reader can easily generate other examples of his own choosing. The total concentration ct of subunits in the small, closed volume under consideration is ct = c

+ cp,N, + cp,N, + cp3N3,

where c is the free monomer concentration, as usual, cpl is the concentration of polymer molecules of type (i) (those blocked completely at the p end with a end free), N 1 is the number of subunits in each of these polymers, etc. For simplicity, we shall assume that the three polymer types are monodisperse; that is, all of type (i) have N1 subunits, etc. The quantities ct, cplrcpz,and cp3inEq. (70) are all constants, while c, N1,N,, and N3 change with time. On differentiating Eq. (70) with respect to t , and using dt

= a1c -

(Y2,

%

= plc -

p2,

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

37

we obtain the differential equation dc/dt

= K - - K+C

(72)

where K+ K-

= =

CPl(Y1 CPl(Y2

+ C P 2 P l + c*&1 + P1) + CP2P2 + C P h 2 + P 2 )

(73)

It will be seen that K + ( s - ~ ) and K-(Ms-')are total on and off rate constants including all polymer molecules. The solution of Eq. (72) is c(f) =

[Co - (K-/K+)]e-K+t

+ (K-/K+),

(74)

where c o is the value of c at t = 0. If we now substitute Eq. (74) for c in the first of Eqs. (71), we obtain after integration,

where NP is the value of Nl at t

=

0. In similar fashion, we get

Equations (74)-(77) allow us to follow c , N,, N,, and N3 as functions of time, if all the parameters are given. These equations are valid until the first polymer type disappears. This will usually be type (ii); disappearance occurs when N2(t)reaches the value zero, say at time t = t'. That is, N , ( t ' ) = 0. For the second stage of the calculation, with t > t ' , we can still employ Eqs. (75)-(77) but K+ and K - have to be recalculated using cm = 0. Also the values of c and the Niat t = t ', from the first stage, must be used as initial values in the second stage ( t = t' in the first stage is t = 0 for the second stage). The above procedure is then repeated, starting at t = t", when N3(t)reaches zero (in the third stage cp2= cp3= 0). In the third stage, K - / K + = a z / a l so that the linear term in N , ( t ) drops out: in this stage, N,(t) has a simple exponential behavior. The length of the first stage, in most examples, is determined roughly by the linear term (following an exponential transient). Thus, from Eq. (76), t' is given very approximately by t'

N$/@

-

(PlK-/K+)]

(78)

38

TERRELL L. HILL AND MARC W. KIRSCHNER

Similarly, the length of the second stage is determined roughly by the initial value N 3 ( f ' and ) the linear coefficient in Eq. (77) (with second stage K - and K+ values). In these linear regimes, c is constant with the value K-/K+ [Eq. (74)]. The ultimate value o f c (third stage) is c, = a z / a l . We turn now to two numerical examples that have not been adjusted at all in order to make the time scale biologically realistic. As can be seen from the above discussion, the time required for the system essentially to reach the final steady state at c = c , is a complicated function of many parameters. In the first example we use the following parameters: initial polymer sizes NP = 65,000, N,O = 8000, Ng = 3000; numbers of polymers 175,500, and 10, respectively, in a cell volume 2.6 x low9cm3;initial free monomer co = 3.4 pM; and rate constants a1 = 7.2 pM-' s-l, p1 = 2.25 p M - l s-l, a2 = 17 s-l, p2 = 7 s-l. Some derived concentrations of interest are then c, = 13.23 p M , c, = a 2 / a l= 2.36 p M , coo= 2.54pM, and cp = p2/p1= I

10 x

Ex

N, 6x

4x

2x

7 j 4 1 t

322x 103 .

2

lsecl

FIG.12. Arbitrary numerical example showing polymer sizes and free monomer concentration as functions of time, in a closed system. Nl refers to polymers with a or + end free, N2to polymers with p or - end free, and N3 to polymers with both ends free.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

39

3.11 p M . Also, K - / K + = 2.71 p M in the first stage and 2.37 p M in the second stage. Figure 12 shows the N,(t)and c(t)for this case. In the first stage Nz increases slightly until c passes cp, and then N2 decreases to zero. In the second stage, N3 increases until c passes c-, and then N3 decreases to zero. In the third stage there is a slight further increase in N1 as c drops to its final value c,. In the second example, all parameters are the same except that we start with 175, 500, and 10 small nuclei of the three types; that is, we take NB = N,O = N! = 0. The total monomer is kept the same; it is all free monomer at t = 0, ct = co = 13.23 p M . Figure 13 shows the behavior of the N,(t) and c(t). The initial linear increases in the N , , from zero, are proportional to al,pl, and a1 + pl, respectively [Eqs. (75)-(77)]. The maximum in Nz(t)is again, of course, at c = c p . The maximum value of Nzis 12,500. It is easy to prove from Eqs. (75)-(77) that N 1= N s at the end of the first stage 3.4

10

x 104 -

3.2

3.0

c (pM) 2.8

2.6

2.4

8 x 103

/'

16 x lo3

I

I

24 x 103

32 x 103

2.2

t (seci

FIG.13. Another example, as in Fig. 12, but starting ( 1 types.

=

0) with small nuclei of the three

40

TERRELL L. HILL AND MARC W. KIRSCHNER

( N z = 0). The linear regime in the second stage persists (not shown) until N3 reaches zero. This stage lasts longer than in Fig. 12 because N3(t’) is

much larger here. Then the third stage (not shown) is identical with that in Fig. 12. These kinetic examples demonstrate what the thermodynamic analysis has already revealed: whether one starts from preformed filaments or nuclei, the only stable polymers left at large times are those anchored at the p end. The exact time course is not important, since by changing the number of polymer nuclei or the individual rate constants, the example in Fig. 13 could take place in a much shorter time. It is interesting to note that polymers with two ends free, N3 in the examples, can be quasistable for some time, whereas those with only the p end free are quite unstable. We might therefore expect to see filaments or microtubules with two free ends transiently during processes such as mitosis or cell movement. 3. Polymer with Incomplete Blockage at One End To illustrate the consequences of Section II1,A for steady-state polymers, we discuss here the special case in which one end of a polymer is capped or anchored and the other end is free. To be specific, we suppose that the a end is capped (“cap” means cap or anchor in this paragraph). The rate constants a l , a,, a_, in the absence of the cap are designated k l , k-l, k,, k-,, respectively, in the presence of the cap. The considerations in Section III,A apply to both pairs of rate constants (1, - 1; 2, -2): the critical concentration for each of the two equilibrium reactions on one end (the a end in this example) must be the same in the presence and absence of a cap. However, as we demonstrate below, these caps will effect the steady state concentration for that end, c , , and hence the treadmilling concentration, c,. Thus we have [Eq. (57)l c e( ~=) ( ~ 2 / ( ~ -=2 k 2 / k _ 2 c(l)= a-l/al = k - , / k l , (79) We suppose here that all of the k s are somewhat reduced compared to the corresponding as. The rates of addition of monomers to the a end, as functions of c , with and without the cap, are illustrated in Fig. 14A. The a lines or rates are taken from Fig. 5 (as is the dotted J p line, needed below). The relation between the k lines (cap) and the a lines (no cap) in Fig. 14A is similar to that between the p lines and the a lines in Fig. 5 . This resemblance exists because, as already mentioned, we are dealing with two different polymer ends in both cases. The constants kWland k-, must be negligible if and a4 are negligible [Eq. (79)]. Consequently, the net rate of addition of monomers to the capped a end is J k = klc - k,, which is shown as a solid line in Fig. 14A. J k (cap) has a smaller slope (k, < a l )and

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

41

- k: - a:

FIG.14. (A) Illustration of the effect of capping the a end of the steady-state polymer of Fig. 5 . See text for details. (B) An alternative example of the same kind (described in the text).

42

TERRELL L. HILL AND MARC W. KIRSCHNER

a smaller intercept (k2 < az)than J , (no cap). J k = 0 at c = c k k2/k1. In the figure, we show c k > c a , but this is not necessarily the case, as illustrated in the next section. In fact, in the figure, we show c k > c p , as well. If this is the case, the direction of steady-state treadmilling is reversed by the cap on the a end: the p end is now the + (growth) end of the aggregate and the k end (cap on a end) is the - end. The new steady state ( J p = - J k ) occurs at c = ck (dotted vertical line). It is easy to see, from a diagram analogous to Fig. 14A, that a cap or anchor on the p end (with a end free) could also lead to a reversal of the + and - roles of the two ends. Thus, observed + and - ends of a steady-state polymer with a cap or anchor might not correspond to the intrinsic (free) + and - ends of the same polymer; some caution in this assignment may be in order. 4. Differential Effects of Capping on the Critical Concentration Capping of the steady state polymer can regulate sensitively the spatial distribution of filaments. In most cells the distribution of actin and microtubules is nonuniform. Examples are the polymerization of microtubules in developing neurites, where there is a preferential polymerization in the region of the neurite process (Spiegelmann et al., 1979), or the preferential dense localization of actin in the ruffling membrane (Heuser and Kirschner, 1980). Although many specific factors are probably responsible for this, the steady state polymer affords some unique ways of regulating the process. If we consider caps as basically inhibitory in nature, then, as shown in the previous section, it is possible to raise the critical concentration of an end by inhibiting both the A,(s) A&) + Pi reaction [Eq. (lo)] and the A&) + T G AT(s) + D reaction [Eq. (11)J to roughly the same extent. Such effects decrease the on and off rates for each reaction by the same factor; they do not effect either equilibrium constant [Eq. (79)l. However, the steady-state concentration was raised from c , to ck in Fig. 14A. On the other hand, since these are two chemically distinct processes, it is easy to conceive of effectors that compete with only one of these reactions or which affect them differentially. Despite the fact that the cap is inhibitory, it could then actually lower the critical concentration. This is demonstrated in Fig. 14B. We propose here a cap that competes with the complex process involving loss of A D from the polymer to the soluble phase and the exchange of NTP for NDP. This effector does not inhibit the other reaction (attachment of A, and hydrolysis on the polymer). We can imagine, for example, an associated protein that interacts strongly with the AD in the polymer and only weakly with AT in the polymer. Therefore, since the association However, the dissoof AT is unchanged, it is still governed by aIc ciation process changes from (Y-~C- a2to kac - kz. As in Eq. (79), c3’ =

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

43

C X ~ / C X - ~ = k2/k-,. Under the usual assumption, ignoring reverse rate constants, the effect of this is to leave the slope of the J , line unchanged so that Jk is parallel to J,. However, the intercept changes from -az to -kz. Thus the critical concentration on the a end is reduced from c, to c k , and there is a corresponding reduction in c, to ck, as shown in Fig. 14B. The ability of an inhibitory cap alone to decrease the critical concentration offers a convenient way to regulate polymer growth locally. Even if the monomer concentration is uniform in the cell, certain effector molecules that bind to the ends can raise or even lower the critical concentration and inhibit or stimulate polymerization.

IV. Polymer under a Moveable Force

Up to this point in our article, there has been no mechanism available for the conversion of any of the NTP free energy of hydrolysis (at the polymer ends) into mechanical work or into any other form of free energy. There is free energy dissipation but no free energy transfer. In this section and in Section V we consider polymers under a force. As a result, mechanical work and free energy transduction both enter the picture. (From this point on the term “force” might refer to either a thermodynamic or a mechanical force; the meaning will be obvious from the context.) In the systems studied in the present section there is a constant external mechanical force F, (x = external), generally at one end of the polymer, that is moveable and that tends to either extend or compress the polymer (Hill, 1981a). The force can be moved a significant distance if the polymer either shortens or lengthens by losing or gaining subunits, respectively. Examples are a shortening microtubule that pulls on a chromosome, or a lengthening bundle of microfilaments or HbS aggregates that push on and distort a cell membrane. In Section V, on the other hand, we consider polymers that grow up to and against rigid or almost rigid barriers at both polymer ends. In these cases polymer growth ceases after a sufficient internal compressive forceF has been built up in the polymer by monomer addition. However, if monomer exchange is still possible at the two polymer ends despite the barriers, treadmilling may occur. If, further, a source of external force F, is attached to the middle of the polymer while it is treadmilling between the barriers, mechanical work can be done. For simplicity, we do not take polymer bending into account in the presence of a compressive force on the ends. Microtubules are in fact relatively resistant to bending (Mizushima et al., 1982) whereas microfilaments (actin) bend easily (Oosawa and Asakura, 1975; Mizushima et al., 1982). However, bundles of microfilaments are presumably the significant system in vivo; these would, of course, have much greater flexural rigidity

44

TERRELL L. HILL AND MARC W. KIRSCHNER

than single polymers. Microtubules probably often work in groups, as well. Furthermore, the anchorage of microtubules at their ends will tend to stabilize the straight form relative to a bent form.

A. EQUILIBRIUM POLYMER 1. Basic Thermodynamics It is necessary to begin with a digression on the basic thermodynamics of a rod-shaped polymer under an external force F,. Figure 15A shows a polymer under an extending force F, (bottom), which we arbitrarily call a positive force. The shaded region above the polymer is a rigid anchor. The other three force arrows in Fig. 15A represent the balance of forces that will act essentially instantaneously in response to the imposed F, at the bottom of the figure. The force within the polymer itself is designated F. Because of the mechanical equilibrium that is set up, F = F,. Figure 15B shows the compressive case in which F, and F are negative. The forces in this case act in the same direction as in a gas under a positive pressurep = px imposed by a piston. The four pairs of arrows in Fig. 15A and B indicate that monomer exchange with a surrounding solution is possible. That is, the polymer is an open thermodynamic system. This system can be characterized thermodynamically by L (length), N (number ofmonomers), T (temperature), F (force), and p (chemical potential of monomers in the polymer), not all of which are independent, of course. The equilibrium with monomers in solution will be introduced later. For simplicity, we do not include the

A

B

FIG.15. (A) Equilibriumpolymer under an extending force F, (positive). The balance of forces is shown. The other arrows indicate possible off-on transitions at the two ends. (B) Same for a compressing force Fx (negative).

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

45

solvent explicitly (but see Hill, 1964) in the polymer thermodynamics, nor do we include pressure-volume effects. A basic equation for the polymer as a macroscopic thermodynamic system is then

dA = -SdT

+ FdL + p d N ,

(80)

where A is the Helmholtz free energy and S the entropy of the polymer. For a gas, FdL would be replaced by - p d V . Another standard equation, easy to derive is,

dp =

-

(S/N)dT - (L/N)dF

(81)

We define the length per subunit as I = L / N and let lo represent the value of I whenF = 0. The linear number density of the polymer is I-'. For a microtubule, lo = 80/13 = 6.15 A. A microtubule is a tube with 13 strands and HbS is a rod with 14 strands; microfilaments ( 2 strands each) occur in bundles of various sizes. All of these are long protein crystals, with presumably small linear compressibility. The compressibility will be discussed at the end of this section. In the meantime, we assume that it is small. The equation of state (relation betweenF and L ) can then be written

F

=

a(l - lo),

(82)

where a is a force constant. This is an empirical thermodynamic relationship; in fact, it is just Hooke's law for the polymer. Under compression, 1 < Id under extension, 1 > 1, (how large an extending force the anchoring of the polymer can withstand will be considered in Section IV,A,4). Equation (82) contains the first term in a power series; the next term is b(l etc. However, this quadratic term will not be used; indeed, for most purposes even the linear term is not needed (i.e., the polymer can be considered incompressible with 1 = lo, as a good first approximation-just as for a liquid or solid in conventional thermodynamics). The finite compressibility will play a significant role in Sections V and VI but it is not important in Section IV beyond the present subsection. If we now substitute lo + (Flu) [Eq. (82)]for 1 in d p = -1dF [Eq. (8l)l and integrate, we obtain

where p,, is the monomer chemical potential (in the polymer) at F = 0. This is the same po introduced in Section 11,A; a polymer molecule with free ends, as in Section II,A, necessarily has F = 0. We shall see below that the F 2 / 2 a term in Eq. (83) is usually negligible, as is the term Flu in

46

TERRELL L. HILL AND MARC W. KIRSCHNER

1 = 1, + ( F / a ) .Alternatively, using Eq. (82), p can be expressed as a function of 1 :

Compression ( F < 0) increases p (the monomers in the polymer are less stable than at F = 0); conversely, extension decreases p and makes the monomers in the polymer more stable. We now consider a polymer under an external force F,, and in contact with solution. The free monomers in solution have a chemical potential given by Eq. (1). Here we let c," (this is denoted c, in Section I1,A) be the free monomer concentration required in order for monomers in solution to be in equilibrium with polymer at F, = 0, and we let c, be the equilibrium concentration when the polymer is under the force F,. Then, at equilibrium, we equate chemical potentials and put F = F, (Fig. 15A and B): po = p: po - loFx = pS0

+ kTln c," + kTln c,

On combining these two equations, In c,

=

In c," - (IoF,/kT)

(89)

This equation shows that the critical concentration c, for polymer formation is increased when F, is negative (compression) and decreased when F, is positive (extension). a. Incompressible Special Case. In Hill (1981a), the polymer was assumed at the outset to be incompressible, with constant chemical potential p,,. To avoid possible confusion, we show here how the above more general treatment degenerates into that used in Hill (1981a). In the incompressible limit, for a finite F in Eq. (82), a += = and 1 - lo + 0. Thus Eq. (84) applies for p, and L + ION.As a consequence, L is no longer an independent variable: dL = 1,dN. When molecules are added to the polymer, the length necessarily increases. Equation (80) becomes, then, dA

= = -

SdT SdT

+

1oFdN

+ podN

+

(PO

-

1oF)dN

(90) The coefficient of d N in Eq. (90) is not a pure chemical potential; it is a hybrid coefficient from the terms FdL and p d N in Eq. (80). When polymer at F = F, is in equilibrium with free monomer at c = c, the equilibrium condition is p = (u, [Eq. (88)l. This is the same as Eq. (2) of Hill (1981a). The chemical potential poin Eq. (90) is a constant that takes no account of F, and cannot be set equal to E.LS at equilibrium unless F, = 0 as in Section

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

47

I1,A. If Eq. (90) is used as a starting point for the polymer, F, must be included as a separate part of the system (as in Hill, 1981a). b. Estimate of Compressibility. The compressibility here is K =

(l/L)(aL/dF)N,T= l/loa,

(91)

having used Eq. (82). Neither K nor a is known so we have to resort to a rough but reasonable estimate of these quantities. As an average for both torsional and center of mass vibrational motion of the molecules in HbS, using an Einstein model, Ferrone et al. (1980) chose a vibrational frequency of 3 x lo9 s-'. The center of mass motion, which is related to the radial pair potential for two protein molecules, is presumably more restricted than the torsional motion; we estimate the center of mass frequency to be somewhat higher than the average, say v = 8 x lo9s-l. If the intermolecular pair potential has a force constant 6 for displacements near its minimum, then the argument in Hill (1960, p. 293) can be used to show that the isotropic Einstein force constant for the motion of a molecule in the field of its 12 (close packed) neighbors, at optimal spacing, is 46. Then from 46 = 47r2v2m, the value of v above, and the molecular weight 64,500, we find 6 = 68 dyne cm-'. We now adopt this estimate of 6 for tubulin pair interactions in a microtubule of 13 strands. The same value (given as 0.7 x lo2 dyne cm-l) was estimated in a quite different way by Mizushimaet al. (1982). The relation between force and length for a single strand of subunits would be F1= 6(d - do), where do = 80 8, is the optimal nearest-neighbor distance between subunits along the strand and d is the actual nearest-neighbor distance. Obviously do= 131,and d = 131in the notation used above. The longitudinal force F for 13 strands is then

F = 13F1 = 1696(1 -

10)

Thus the force constant a above [Eq. (82)] is related to 6 by a = 1696. Hence our estimate for a is 1.1 x lo4 dyne cm-'. The compressibility K then follows from Eq. (91). A dimensionless equivalent of or substitute for the Hooke's law constant a will prove very useful, especially in Sections V and VI. This is y = IgalNkT

(92)

If we take the value of a as above, N = 2 x lo4, and lo = 6.15 A, then y = 0.05. We shall use this as a typical order of magnitude for y where needed. If a polymer of fixed length L contains No molecules at F = 0 and N molecules at an arbitrary F, and if we define this excess number of molecules as n = N - No,then, using/ = LINand lo = L / N o ,the Hooke's law

48 relation F

TERRELL L. HILL AND MARC W. KIRSCHNER =

a(l - lo) [Eq. (82)] can be rewritten in the form

loF/kT = - yn or n = KN~(-F) (93) Thus y is a dimensionless force constant related to number of molecules rather than to length. As a numerical example, if l0F/kT = - 1 (compression) and y = 0.05, thenn = 20. Thus, if No = 2 x 104(seeabove) a t F = 0, then under the compression loF/kT = - 1, if the length is kept constant, the polymer would contain 20 additional subunits. For most (but not all) purposes, this small number (one part in 1000) can be ignored. The second form in Eq. (93) exhibits, in a very concise way, the three ingredients on which the value of n depends. Incidentally, corresponding to Eq. (91), the three-dimensional compressibility (close-packed lattice) that can be deduced from the above pair potential, 6(d - d 0 ) 2 / 2 is , K = -(l/V)(aV/aph,T = 3 d 0 / 2 ~ ' ~ 6 (94) Finally, we want to show that the term F 2 / 2 a in Eq. (83) is usually negligible. This will be the case if 11;1/210aQ 1, or if

(IFllo/kT)/2yN Q 1 (95) The denominator is of order 2 x lo3. The numerator is usually less than 5 . For example, the estimated dragging force ( dyne) of a chromosome, with lo = 6.15 A, gives F,lo/kT = 0.015; also, c,/c," = 5 in Eq. (89) gives (F,l&/kT = 1.6. Thus the condition in Eq. (95) is well satisfied. In the expression 1 = lo + ( F / a ) ,F l u is negligible and 1 = lo if Iq/loaQ 1. Except for a factor of 2, this is the same condition as in Eq. (95). Because IFllo/kTis usually not more than order unity, the condition (95) is equivalent to N B l/y. In view of Eq. (93), it is also eauivalent to N B Inl. 2. Effect of Force on Rate Constants In Fig. 16, F, may have either sign and monomers A exchange at either polymer end, or at both ends. The assignment of aand @ t othe two ends is arbitrary. We shall allow for both ends to exchangz. but we can merely put a = 0, a' = 0 or p = 0, p' = 0 if one end or the other does not exchange. The rate constants a, a',p, p' refer to arbitrary or whereas we use ao, ab, Po, F0for the special case F, = 0. When the system is at equilibrium with F, = 0 and c = c,", or with arbitrary F, and c = c,, Eqs. (87) and (88) apply, respectively, along with the detailed balance relations c," = a;/ao = pb/po, c, = ab/a = p ' / p , (96) just as in Eq. (4). The F, = 0 rate constants correspond to those in Section II,A, but they would not have the same values for the same polymer

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

49

FIG.16. Rate constant notation for Fig. 15.

because here the ends are anchored and there the ends are free. However c,O here would be the same as c, in Section II,A because, for an equilibrium polymer, the anchors do not alter the equilibrium constant. The notation is changed here to allow for greater generality. From Eqs. (87)-(89) and (96), we have kTln (ao/ah Of POIP8 = P: - Po kTln ( a / a ' or p / p ' ) = ko- p0 + loFx

(97) (98)

and cy/(y'

= (ao/(y6)eWxF.kT =

p/p'

=

(p0/p6)&oFx/kT

(99)

The last equation shows the influence of F, on the ratios a/a' and P I P ' . These ratios are increased (favoring aggregation) if F, is positive (extension) and decreased if F, is negative (compression). The separate rate constants a and a' (the following remarks also apply to p and p', of course) also depend on F, but this is a kinetic, not a thermodynamic matter. The effect e l o F x / k T [Eq. (99)] on a/a' is necessarily divided between a and a'. Whatever the division, it can be expressed conveniently in terms of a dimensionless parameter fu, as follows: a

= aoe&loFx/kT,

a' =

a6e(fa-1)10Fx/kT

(100)

50

TERRELL L. HILL AND MARC W. KIRSCHNER

For /3 and j3’,f, is replaced by fp . An explicit molecular model would be required to predict f, (or, in principle, f, can be determined from measured values of ao,a,and F,). Also, we have to expect that f, is itself a function of F, (see below). Figure 17 illustrates the physical significance off in the case of attachment under compression (F, < 0). The full curve in Fig. 17 shows the hypothetical free energy of interaction of a monomer with the end of the polymer, when F, = 0, as a function of the distance of the monomer from its attachment site on the polymer. When the polymer is under compression, the attached state is less stable than when F, = 0 [Eq. (84)], and the attachment free energy well is raised an amount - loF, to curve C. At the same time, the rate constants change from a. and ab to a and a’.The new (F,) free energy curve from “off’ to C might have various levels at the position of the transition state (maximum). If f = 0, the transition state level is unaffected by F,, and a = a@ In this case, the full effect of F, is in a’,which is increased over ah by a factor e-l$x’”(because the free energy barrier to escape, on + off, is reduced). If fu = 1, the transition state level is increased by the full - I&,. In this case, there is no effect of F, on a6 (a’= a;)but a is smaller than a. by a factor e loFx’kT because the barrier to attachment is higher). If f = 1/2, a is smaller than a. and a’is larger than ah, but both by less severe factors. Incidentally, in principle f is not confined to values between 0 and 1, but the range 0 If 5 1 is no doubt most likely. When the polymer is under significant compression, it seems intuitively reasonable that it will be more difficult for an oncoming monomer to

On

FIG.17. Physical significance off,; free energy barrier that determines the on-off rate

constants in the case of compression.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

51

squeeze between the polymer end and the anchor and also that the compression will tend to push end monomers out of the polymer. That is, a will decrease and a' increase. Hence we might expect an intermediate fa, say fa = 1/2, when F, is large and negative. On the other hand, extension (F, > 0) should make it easier for monomers to attach, though a at large F, cannot exceed the diffusion-controlled limit. Thus, for large F,, we might expect fa UF,, so that faFxbecomes constant. Figure 18 shows, qualitatively, the above described F, dependence o f f . This should be considered as merely a plausible possibility. At the detailed molecular level, f at any F, is actually an averaged quantity, similar to a in Eq. (6).

-

3. Kinetics of Monomer Exchange The net rates of addition of monomers to the two ends, at arbitrary F,, are

J, =

(YC -

a',

Jp =

P c - p',

(101)

where the F, dependence of the rate constants is given in Eq. (100). We assume throughout that mechanical equilibrium, F = F,, is maintained virtually instantaneously even in nonequilibrium circumstances ( c # c,, here). The total rate of addition is Jon= J, + Jp. The dependence of these rates on force is illustrated for J, in Fig. 19. The three solid lines are for loF,/kT = + 1, 0, - 1, taking fa = 1/2 in the + 1 and - 1 cases. Both slopes (a)and intercepts (- a') change with force. The broken line represents the loF,/kT = + 1 case if we take fa = 0 for this F,. In this case the on rate is unaffected by F,: a = a@J, = 0 for each line at c = c,; c, itself depends on l,F,/kTaccording to Eq. (89). This latter dependence is shown in Fig. 20 where the heavy line is the logarithm of the critical concentration, In c,, as a function of the extending or compressing force loF,/kT.

I

Compress 0

Extend

Fx+

FIG.18. Illustration of howf, might depend on the external force F,.

52

TERRELL L. HILL AND MARC W. KIRSCHNER

F ,I,

/kT =

+1

Jo

FIG.19. Illustration of the effect of F, on J , for an equilibrium polymer.

If we were to include Jp lines in Fig. 19, they would cross the corresponding J, lines on the abscissa [Eqs. (96)], as in Fig. 1, but the slopes would generally be different (also as in Fig. 1). We can rewrite Eqs. (101) as

where [see Eqs. (l), (87), and (89)l

with A p defined by Ap

= ps - po = p! + kTln c

- po =

kTln (c/c,")

(104)

The expressions in Eq. (103) are alternative ways of writing eXlkT,where

X = A p + loFx

(105)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

53

is the total thermodynamic force (the same at a and p ends) driving monomers from the solution onto the polymer ends. When free monomers in solution are in equilibrium with monomers in the polymer, c = c,, X = 0, and J, = Jp = 0. When X > 0, that is, when c > c, for a given F,, J, and Jp are both positive. In this case the polymer grows at the total rate

Jon = J,

+

Jp = (a'

+ p')( eXlkT-

1)

(106)

Thus the polymer grows for all points in the plane of Fig. 20 above the heavy line c = c,. Conversely, the polymer shortens when c < c, and X < 0 (points in Fig. 20 below the heavy line). Free Energy Transduction. The two separate contributions to X are the intrinsic (or pure) monomer or subunit aggregation thermodynamic force A p and the mechanical force term loFx. A p is the thermodynamic force driving monomers onto the polymer in the absence of a mechanical force F,. A p [Eq. (104)] is positive above the horizontal line c = c," in Fig. 20 and negative below this line. For any choice of c and F, in region I of the plane of Fig. 20, the polymer is under compression ( loFx < 0) yet it grows ( X > 0, Jon > 0) against the compressive force because the intrinsic subunit thermodynamic force Ap is large enough ( c > c,) to counteract the opposing effect of ZoFx.An illustration would be the growth of a bundle of actin

I

t

x>o

Compressing

In c

0

Extending

IoFx-

FIG. 20. The In c versus /,F, plane for an equilibrium polymer. On the line c polymer is in equilibrium with monomer. See text for details.

= ce,

54

TERRELL L. HILL AND MARC W. KIRSCHNER

microfilaments (ignoring ATPase activity for the moment), or HbS aggregates, against a cell membrane that resists the growth, thus pushing the membrane out and distorting the cell shape. This is an example of free energy transduction: mechanical work is done against the force F, at the expense of the intrinsic subunit aggregation thermodynamic force Ap. In the aggregation (growth) process, some subunit free energy is converted into mechanical work. The efficiency of the conversion is r) = - l o F x / A k If - loFxis close to A p ( c = ce), r)+ 1. The rate of free energy dissipation is where Jon[Eq. (106)] and Xare both positive. Note that this free energy transduction process does not involve any NTPase activity; rather, part of the subunit aggregation free energy is converted into mechanical work. Incidentally, in order to trigger a process such as the above, in vivo,it is not necessary that the subunit concentration itself should change from c 5 c, to c > c,. More likely would be an increase in the value of p; - poin Eq. (104) owing to phosphorylation (or dephosphorylation) of free subunits or of molecules attached to free subunits, to Ca2+binding to or release from the subunits, to changes in intracellular pH, etc. An increase in pj - pocorresponds to greater stability of the polymer relative to free subunits. After the triggering process on the free monomers, the preexisting polymer would serve as a seed for the new growth that would occur spontaneously if the new A p exceeds - loFx. The other case of primary interest in Fig. 20 corresponds to region 11. Here the subunit concentration is low enough ( c < c,) so that the polymer shortens ( X <0, Jon< 0) despite the extending force loFx> 0 that opposes the shortening. The intrinsic subunit force - A p favoring depolymerization exceeds loFx.Hence some of the subunit free energy - Ap is used to do work against the resisting mechanical force F,. The efficiency of free energy transfer is r) = loF,/(- Ap). The rate of free energy dissipation is again given by Eq. (107), but in the present case both Jon and X are negative. An example would be the shortening of a group of microtubules (ignoring the GTPase activity until Section IV,B) that are attached to a chromosome which offers viscous resistance to its movement (caused by the shortening). Aside from a lowering of c from c 2 c, to c < c,, triggering (see above) of the depolymerization could occur by a change in state (see above) of the subunits of the polymer such that the polymer is destabilized relative to free subunits 0.: = po decreases). Some further details and a discussion of other cases in Fig. 20 are included in Hill (1981a).

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

55

F=

F= FIG.21. Equilibrium polymer anchored at the p end, and with a source of external force F , (the ring) attached to the central region of the polymer. F , is positive in the downward direction.

4. Polymer with a Centralized Attached Force Figure 2 1 presents a variant on the above type of system. We consider a

polymer with p end anchored, a end free or capped, and a source of external force F, (the ring) attached to the polymer somewhere in its central region. F, is positive when directed toward the a end. Between anchor and ring, the force is F = F,; between the ring and the a end, the force is F = 0. Because of different forces in the two regions, the chemical potential of the monomer in the polymer is also different. Hence the on + off equilibrium constant is different. In Fig. 22, J," = aoc - dofor the a end ( F = 0). If the external force F, on the ring is zero, the Jp line is J$ = poc - /36.This crosses the J,Oline on the abscissa at c,O = a6/ao= FO/po (the two equilibrium constants c,O are equal in this case). But if F, # 0, the .Ipline ( pc - p') shifts as illustrated by the two examples in the figure, one for F, > 0 and one for F, < 0; however, J," is unaffected. The three J p lines in Fig. 22 are analogous to the three J, lines in Fig. 19. When F, > 0 (extention), there is on-off equilibrium at the p end when c = ce(+) = p' / p . This is the on -+ off equilibrium constant for the pend. If c > ce(+), monomers add to the p end and the ring moves in the direction of the force F, (i.e., away from the p end). If c < ce(+), monomers are lost from the Bend ( Jp < 0); the ring is pulled against F, and moves toward the p end. In summary, the p end behaves just as already described in the preceding section. Meanwhile, the activity at the a end, where F = 0, is

56

TERRELL L. HILL AND MARC W. KIRSCHNER

FIG.22. Illustration of the effect of F, onJDin the system shown in Fig. 21. Treadmilling is possible (lines A and B) without NTPase activity.

independent of that at the p end, and has nothing to do with moving F,, and is therefore wasteful. The free energy transduction situation when F, > 0 and Jp < 0, is the following. The rate of free energy dissipation [Eq. (48)] is

Td,S/dt = J , " A k + JpApp = J,"Ap + Jp(Ap + &FX) = JmAP + JpbFx, (108) where we have used Jon = J," + Jo. This differs from Eq. (107), which applies to Fig. 16, because only p transitions move F, in the Fig. 21 system. The efficiency of free energy transfer is then = (- J d M x / ( - J o n ) ( -

(109)

Because of the wasteful a end activity (i.e., -Jon > -J,>, this is smaller than 7) = l,,E,/(-Ap), in the preceding subsection, for region I1 of Fig. 20. A novel feature of the present system, with F, > 0, is the possibility of treadmilling without the involvement of NTP. This is indicated by the vertical line A in Fig. 22, drawn so that Jp = - J,".Thus, at this concentration, the polymer maintains a constant length, shortening at the a end and lengthening at the p end, while the ring moves in the direction of F,

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

57

(downward). The extending force F, is large enough to induce aggregation at the p end ( J p > 0) despite the fact that A p is negative ( c < c,").This behavior corresponds to region I11 of Fig. 20 (see also Hill, 1981a). When F, < 0 (compression), equilibrium occurs at the pend ifc = ce(-) in Fig. 22. When c > ce(-), the situation at the p end is analogous to that for region I of Fig. 20. There is growth at the p end ( J p > 0) despite the opposing force F, < 0. Again Eq. (108) applies; the efficiency of free energy transduction is 77

=

Jd- 1oFJ /Jon A p

(1 10)

This is less than 77 = - f,,Fx/Ap(region I, Fig. 20) because of, as before, the wasteful activity at the a end. Whenc < ce(-), treadmilling can again occur at line B of Fig. 22, where J," = - Jp. In this case compression ( F , < 0) induces loss of subunits from the p end (.Ip< 0) despite the fact that A p is positive ( c > c!). This corresponds to region IV of Fig. 20 (see also Hill, 1981a).

5 . Stability of Anchoring u Microtubule against CI Force For many schemes where microtubules or actin filaments push or pull objects, they must be anchored at their ends and still be able to exchange subunits. This puts unusual constraints on the capping or anchoring structures, for example, they must bind tightly but not block exchange. In this section we consider some general examples of such structures, which illustrate some of the properties they must possess. We consider a microtubule that (a) is anchored at one end, (b) is under an extending force F, > 0, and (c) exchanges subunits at the anchored end. We give here a very brief and qualitative discussion of the question of anchoring stability under these conditions. That is, what determines whether the anchor can withstand the force F, and hold the microtubule? The simplest possible case is illustrated in Fig. 23A. The anchor is a flat but somewhat flexible molecular surface, perpendicular to the microtubule. Such a structure may in fact correspond to the outer layer of the kinetochore of metaphase chromosomes (Ris and Witt, 1981). Subunit exchange is possible radially, as indicated by the double-headed arrow. As discussed in Sections II,A and III,A, subunit exchange and surface diffusion of subunits will produce fluctuations in the end or surface configuration of the microtubule. The free energy curve in Fig. 23A with maximum slope Fb represents the free energy of molecular interaction between the end of the microtubule and the flat anchor, as a function of the distance between the anchor and the microtubule end, when the microtubule has as smooth a molecular surface configuration as possible. (Note that the distance scale is very different in the two parts of Fig. 23A.) Even in

58

TERRELL L. HILL AND MARC W. KIRSCHNER

I

1

Free Dista,nce Energy

I

A

0

23. Analysis of the stability of two simple kinds of anchor for a microtubule. See text for explanation. FIG.

the smoothest configurations perhaps only 8 or 9 of the 13 end subunits could interact effectively with an essentially flat anchor (Amos, 1979). Configurations that optimize the number of good tubulin-anchor contacts will of course be favored in the equilibrium distribution (Section 111,A). That is, the presence of the anchor biases the distribution of the end configurations. The free energy curve in Fig. 23A with maximum slope Fa represents the microtubule-anchor interaction when the surface configuration is relatively jagged, say in a fluctuation resulting from on-off transitions, with only 4 or 5 effective interactions of end subunits with the anchor. Both the depth and the maximum slope of a free energy curve will be more or less proportional to the number of “good” subunit-anchor interactions. The slope of one of these free energy curves at any point is equal to the attractive force between microtubule and anchor. Thus Fb is the maximum possible attractive force in this example. If F, > Fb,the anchoring is definitely unstable and the microtubule will pull away. If Fa < F, < Fb, the anchoring will be stable for smooth surface configurations ( Fb) but will pull apart if a fluctuation reaches a surface configuration corresponding to the curve with maximum slope Fa. In order for the anchor to be stable indefinitely under a force F,, we must have F, < Fa, where Fa is the smallest maximum slope reached in a significant surface fluctuation.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

59

Figure 24 presents the point just made in a different way. In this figure we plot the slope of the free energy curve just mentioned (with “smallest maximum slope”), that is, we plot the attractive force against distance. The maximum on this curve is Fa. A pulling force F, < Fa cannot surmount the force barrier. Retraction of the microtubule from the anchor, with force F,, will stop at the arrow in the figure; that is, this is the equilibrium position of the microtubule relative to the anchor when the extending force is F,. When F, > Fa,the anchor is unstable. Incidentally, we are using a completely mechanical argument here and ignoring Boltzmann fluctuations within any given free energy curve because the structures under discussion are very large. The anchor in Fig. 23B is idealized but still much more realistic than the flat plate in Fig. 23A. In Fig. 23B, the anchor is a cylindrical sleeve with an annular lid. Subunit exchange (double-headed arrow) must take place through the hole in the lid. As the end of the microtubule enters the sleeve, the (attractive) free energy of interaction between the sleeve and the outside surface of the microtubule decreases linearly, with slope F,; the free energy is proportional to the amount of overlap. When the end of the microtubule reaches the lid, the lid-tubulin interactions (taken to be the same as in Fig. 23A) are added to the sleeve-tubulin interactions. Thus the maximum attractive forces (Faand Fb)in the two examples in Fig. 23A become here Fa + F, and Fb+ F,. The sleeve adds an “insurance” contribution F,. If F, < F,, the anchor is stable no matter what kind of fluctuation in surface configuration occurs (i.e., no matter how small Fa). F, is the (negative) free energy of interaction of the sleeve material

tI

Attractive Force

F,, unstable

-

-

Fx, stable

’ \EquilibriumDistance

FIG.24. Attractive force between microtubule and anchor as a function of distance in the Fig. 23A case.

60

TERRELL L. HILL AND MARC W. KIRSCHNER

with the microtubule outer surface, per unit length of overlap. This cannot be estimated without molecular information about the hypothetical sleeve. B. STEADY-STATE POLYMER 1. Basic Thermodynamic and Kinetic Equations

We have seen above, for an equilibrium polymer under a moveable force (Fig. 16), that mechanical work can be obtained from the force F, if the polymer lengthens (and F, < 0) or shortens (and F, > 0). The driving thermodynamic force for the length change is a subunit chemical potential difference between polymer and solution. NTPase activity is not needed and is not involved. In this section we introduce NTPase activity into this type of system (Fig. 25). As will be seen below, we again find that, in order to accomplish mechanical work against a moveable outside force F,, polymer length changes are imperative. The NTPase activity is essentially wasteful and the basic properties of the system are not very different from those of the equilibrium polymer. All transitions are of course related to NTP (Fig. 3B) but there is no way to convert NTP cyclic activity into mechanical work. The basic reason for this is that, for the system in Fig. 25, every possible

t Fx

FIG.25. Steady-state polymer under a moveable force F , (positive in the downward direction).

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

61

NTP cycle (Fig. 7) first adds and then subtracts a monomer from the polymer; hence, the polymer does not change in length and F, does not move. However, the situation is different if the external force is attached to the central region of the polymer, as in Fig. 21 (see Section IV,B,3 below). Figure 25 is a generalization of Fig. 16, to include NTP activity as introduced in Fig. 3. The polymer is under a force F = F,. The rate constants al,etc., in Fig. 25 apply at F,; when F, = 0, the constants are designated a!, etc. The basic thermokinetic equations are immediate consequences of Eqs. (18), (20), and (98): kT1n (a!/a!l or pp/ptl) = P X T - (CLAD + P P ) kT1n (aI/a-lor plIp-1) = PP\T - (CLAD + P P ) + l a x kTln (d/a!2 or pZ"/pE2) = (CLAD + P P ) - (P!T - XT) kTln ( a 2 / a - 2 or p2Ip-2) = ( P A D + P P ) - ( P X T - XT) -

l a x

(111) (1 12) (113) 7114)

The loF, terms originate from the polymer chemical potential pAD - l0F,, as in Eq. (84). Equations (44)were originally written for a steady-state polymer with free ends ( F= 0). They also apply, because no equilibrium constants are changed, if the polymer ends are capped' or anchored, with F = 0. Thus Eqs. (44) apply here to the rate constants a!, etc. (anchored, F = 0), as can easily be verified from Eqs. (1 11) and (1 13). In fact, Eqs. (44) also apply here to the set of rate constants al,etc. (F, # 0) in Eqs. (1 12) and (114), because the loF, terms always cancel in a complete cycle (in a complete cycle no subunit is added to or subtracted from the polymer, as already mentioned above). On combining Eqs. ( 1 11) and ( 1 12), and (1 13) and ( 1 14), we obtain the analogues of Eqs. (99): al/a-l= (a!/&)r a2/a-2

=

(ap/a")r-'

= pl/p-l= =

(P!/Ptl)r

p2/p-2= (pp/p!2)r-',

(1 15) (116)

where at this point we introduce the shorthand notation eloFx/kT

(1 17)

This will be used throughout the remainder of the article to save space. The rate constants a2and p2refer to "off" transitions; hencer -'occurs in Eqs. (1 16) rather than r itself. Note that under extension (F, > O ) , we have r > 1, and that under compression (F, < 0), we have r < 1 . The F, = 0 case is r = 1 . Equations ( 1 15) and (1 16) show how ratios of inverse rate constants, or equilibrium constants, depend on F,.

62

TERRELL L. HILL AND MARC W. KIRSCHNER

We represent the dependence of individual rate constants on F, just as in Eq. (100): (yl

=

lr

( y o fa ‘ 7

p1 = pPrf’, (ye

= apr-e,

pz = ppr-@,

p-l

p-2

=

= = =

aylr(f?-l)

pyly(f@-l) a!zyl-f:

pyzr’-fP

(118) (1 19) ( 120) (121)

Here, for generality, we have included four differentfs. Each of the four is a function of F, (Fig. 18 is an example). The convention we have adopted here (see Hill, 1977b) is that, in both pairs of transitions of a two-state cycle, the parameterfis used for the dominant transition direction (a1,pl, a2,p2) and then, necessarily,f - 1 is used for the usually unimportant a+, p-,). The overall chemical processes at the two inverses (adl, different polymer ends are the same for a1and p1[Eq. ( lo)], and for a2and pZ[Eq. (1 l)]. If the kinetic mechanisms (see Fig. 2 for examples) are also the same at the two ends, as is probably but not certainly the case, then f P = fQ = f l and fg = fP = fz.With this simplification, there are only two different fs. If, say, the a1 (on) and az (off) processes were elementary physical attachment and detachment processes, with no chemistry involved, we and the two would expect that the two “on” rate constants a1and “off’ rate constants a2and would have the same dependences on F,. That is, from Eqs. (118) and (120), we would expectf? + fg = 1. Unfortunately, the two-state cycles in Fig. 3 are reduced from chemical cycles that are at least as complicated as those in Fig. 2, so that the two-state rate constants in Fig. 3 are composites of many, more elementary, rate constants (Hill, 1980a, 1981b). Thus there is no simple connection between the a1 and processes, nor between at and a+ Hence there is no reason to anticipate that f P + fg = 1 and fQ + fP = 1. With the understanding that all eight rate constants a l , etc. (Fig. 25) depend here on F,, the various flux and other definitions and expressions in Eqs. (24)-(43) hold without formal change, and will not be repeated. The explicit F, dependence can be seen, in any given relation, by substituting Eqs. (1 18)-(121); an example appears below. Equations (24)-(43) also all hold here in the special case F, = 0, with the rate constant set a!, etc. For quantities such as J,, c,, etc., at F, = 0, we use the notation J,”, cO,, etc., to distinguish from J,, c,, etc., at arbitrary F,. One difference here, compared to Section II,B,2, is that we are not particularly interested in the treadmilling case, Jon= 0, c = c,, because no mechanical work is done on F, when Jon= 0: the polymer maintains a constant mean length and F, does not move. To do work on F,, we require Jon# 0. In view of Eqs. (118)-(121) and the fact that we would usually expect

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

63

thefs to have values between 0 and 1 , al and p1 (“on” rate constants) will tend to increase with increasing F, while atand p2 (“off” rate constants) will tend to decrease as F, increases. Figure 26 illustrates this. The solid lines are for F, = 0, the broken lines for F, > 0 (extension), and the dotted lines for F, < 0 (compression). The value of C, (where Jon= 0) decreases as F, increases, which is qualitatively the same behavior as in Fig. 20 for an equilibrium polymer. But the dependence of In c, on F, is in general not linear at steady state. Thus, from Eqs. (31) and (118)-(121),

=

( A= ff,A = A )

C;r-(fl+R)

(123)

where the second form is the plausible special case mentioned above. Equation (123) can be rewritten In

Cm =

In ~ 0 -,

(fi + h)(&FX/W,

( 124)

/

J

-

a; 0

- a2

FIG.26. Illustration of the effect of F , on J , and Jp in the Fig. 25 system. Vertical lines are drawn at treadmilling concentrations. r < 1 corresponds to F , < 0 (compression), etc.

64

TERRELL L. HILL AND MARC W. KIRSCHNER

which is analogous to Eq. (89). But we would expect fl and fi to be functions of F,, and also fl + fi # 1; hence, Eq. (124) is not so simple as Eq. (89). Figure 27 shows a hypothetical plot of In cmagainst IoF,, from Eq. (122) or (124). For points above the curve in the figure, c > coo,Jon> 0, and the polymer grows. The situation is reversed below the curve. The total rate of addition of subunits to the polymer is Jon =

(a1+

= (a2

+

- (a2 + P2) P2)Kc/cm) - 11, P1)c

(125)

( 126)

where a2and p2 are given in Eqs. (120) and (121). Equation (126) resembles Eq. (106) but the latter gives Jonwith reference to an equilibrium point (X= 0, c = CJ whereas the former gives Jonrelative to a steady-state point (c = cm). 2. Rate of Free Energy Dissipation The simplest way to obtain an intuitive feeling for the complete activity taking place in the system depicted in Fig. 25 is to examine the rate of free energy dissipation and the efficiency of production of mechanical work. Specifically, we shall compare the efficiency of an actin filament or microtubule with NTPase activity (steady-state polymer) pushing or pulling against a force with the corresponding efficiency for an equilibrium polymer (just considered). The analysis shows that, for pulling and pushing, the NTPase activity only adds to the ineficiency of the process. We start with the very general dissipation Eq. (48), which applies at arbitrary c. For the present system, in view of Eqs. (50), (112), and (114), we have for the free energy changes in Eq. (48), A k l = A h , = AP+ + I O F X (127) A k 2 = ApOP= - Ap- - /OF,, (128) where we have defined Ap+ and Ap- by analogy with A p in Eq. (104), as follows: AP+ ~ 0 +1kTln ~ c - (p.m + PP) (129) Ap- = pp\T+ kTln c - X, - (p.,,, + pp) (130) The f subscripts refer to growth and shortening, as in Eq. (39). Their appropriateness here will appear below. In Eq. (104), Ap is the thermodynamic force driving subunits from the solution onto the equilibrium polymer when F, = 0. Correspondingly, Ap+ is the force (when F, = 0) driving subunits onto the steady-state polymer using the al, PI process [Eq. (lo)]. Similarly, Ap-is the force driving subunits onto the steadystate polymer using the a-2, P-zprocess [the reverse of Eq. (1 l)]. Because XT=

A p + - Ap-,

(131)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

65

and X, is of order 12- 14 kcal mole -*,we might expect each of Ap+ and - Ap- to be roughly half this size, say 5-8 kcal mole-' (the division depends on c). That is, the spontaneously operating NTP two-state cycle (Fig. 3), at either end of the polymer, with total free energy drop XT,is comprised of two successive spontaneous steps with free energy drops Ap+and - Ap-,presumably of similar magnitude. Thus Ap+ is fairly large and positive whereas Ap- is fairly large and negative. By contrast, A p in Eq. (104), which is equal to kTln (c/c$), can be positive or negative (Fig. 20) and would usually be of order one to three kT, or only 0.5-2 kcal mole -'. The transition flux expressions in the dissipation Eq. (48), for the four transition pairs in Fig. 25, are defined in Eq. (49), but here c, is to be replaced by c . All of the fluxes A,, etc., and free energy changes A k l , etc., in Eq. (48) are positive, except in very unusual circumstances. The second law of thermodynamics requires that each product in Eq. (48) is always positive (or zero). From the definitions in Section II,B,2, there are a number of useful flux relations that we collect here: Ja

= Jal -

= Jon = J1

J$-)

=

Ja1

Ja2,

Jp = Jp,

+ JPI,

Ja + Jb

= Ja,

J2

- Jp2

+

JP2

J p = J1 - J 2 J$+)

= J2

(132) (133) (134) (135)

We now consider two separate cases, marked I and I1 in Fig. 27. These are of most interest from the point of view of free energy transduction. For a point in region I, the polymer is under a compressing force F, < 0 but c is large enough, c > c,, so that the polymer grows (Jon > 0), thus pushing back the force F, and doing mechanical work. For a point in region I1 of Fig. 27, the polymer is under an extending force F, > 0 but c is small enough, c < c, so that the polymer shortens (Jon < O), thus pulling F, along and doing work. In either case, from Eqs. (127), (128), (133), and (134), the dissipation Eq. (48) can be rewritten as TdtSldt =

J1

Ap+ -

J2

Ap-

+

Jon

IOFX

(136)

This division of the rate of free energy dissipation into contributions from the al,p1process, the ag pzprocess, and mechanical work is interesting but the separate contribution of cyclic NTPase activity is not obvious. To rectify this shortcoming, we introduce Eq. (135). Turning first to the region I case, it will be recalled that, when the > J2and hence the rate of completing NTP cycles is polymer lengthens, .I1 Then the excess of Jlover J2, that is, the lesser of these, namely, J2= J$+). J1- J2= Jon , represents addition of subunits to the polymer via the al,Dl

66

TERRELL L. HILL AND MARC W. KIRSCHNER

It" t c

0

Compressing

FIG.27. Illustrative plot of In c, (Jon= 0 at c figure is the steady-state analog of Fig. 20.

toFx-

Extending =

c,) versus loFxin the Fig. 25 system. This

process but it does not represent complete (cyclic) NTP activity. Thus, in Eq. (136), for the region I case, we replace J,by J 2 + Jonand J2by J4+! We then obtain, using Eq. (131),

TdtS/dt

= J$+'XT

+

Jon

Ap+

+

JonloFx

(C

>

C-,

Fx < 0) (137)

All factors on the right are positive except F,. The first product (positive) is due to NTP cyclic activity, the second (positive) is due to spontaneous addition of subunits to the polymer via the al,PI process, and the third (negative) represents mechanical work that is done on F,. In other words, not all of the NTP and subunit free energy dissipation (first two products) is wasted; some of this is used to push back the compressing force F, at a velocity Jonlo.The efficiency of the free energy transduction is then 77 =

Jon(-

LFx) /(J~+'XT + Jon &+I ( C >

C-7

F, < 0)

(138)

The + subscript on Ap,, as well as the + subscript on J$+),is used to refer to the fact that the polymer is growing in this case. The efficiency in Eq. (138) should be compared with the corresponding expression, above Eq. (107), for the equilibrium polymer, that is, 77 = - loFx/ A k The former efficiency is much smaller than the latter. The NTP term in Eq. (138) is excess baggage; it simply adds to the dissipation and reduces 7. Even without the NTP term, the remaining efficiency, -lOF,/Ap+, is generally significantly smaller than -ld;;/Ap because Ap+

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

67

is probably of order 5-8 kcal mole - l whereas A p is of order 1 kcal mole -l (see above). In summary, the use of NTP activity in the aggregation of actin and/or tubulin, for the purpose of pushing membranes, distorting cell shapes, etc., seems to be an example of‘ considerable thermodynamic overkill. In the region I1 case in Fig. 27, the polymer shortens ( c < c,, J,, < 0) despite the extending force F, > 0. Here .Iz > J1;hence the rate of NTP Theexcess ). of J2over Jb that is, Jz - J1= cycle completions is J1 = .I$ - J,,, represents loss of subunits from the polymer by the a2,p2process. In Eq. (136), we put J 2 = J 1 - J o n and J1 = J $ ) :

TdiS/dt

=

J$-’XT+ JonAp-

+ Jonlolc,

(C

< c, F, > 0) (139)

In this equation J,, and Ap-are negative but the other factors are positive. Again the first two products are positive and the third negative. Mechanical work is produced (the polymer shortens at velocity - Anlo against the resisting force F, > 0) with efficiency 7) = (- J o n ) l & x / [ J $ - ) X T

+ (-

Jon)(-

&-)I

(C

<

c-9

Fx

> 0) (140)

As in the discussion of region I, this efficiency is much smaller than the corresponding efficiency for an equilibrium polymer, 7 = loFx/(- Ap). In the treadmilling special case, c = cm,

J1 = Jz = J T ,

and

Jon =

0

(141)

No work is done on the force F,, whatever its sign, because the polymer maintains a constant mean length ( J n = 0). Equation (136) simplifies to

Td,S/dt = JFX,

(142)

All of this NTP free energy is dissipated, as in Eq. (54).

3. Polymer with a Centralized Attached Force In this section we treat the system shown in Fig. 28 (compare Figs. 21 and 25). There is NTPase activity at the anchored @end and also at the free or capped (Y end. In addition, an external source of force F, (shown as a ring) is attached to the polymer in its central region. This force might be positive (extending) or negative (compressing). Thus the force on the polymer itself is F = F, between anchor and ring but it is F = 0 between the ring and the a end. The essential feature here is that monomer exchange (with the concomitant NTP activity) at the @ end moves F,, and can do work, but monomer exchange at the a end has no effect on F,. Transitions at the (Y end contribute to free energy dissipation but not to production of work. In the Fig. 25 system, on the other hand, monomer exchange at both ends can move F,.

68

TERRELL L. HILL AND MARC W. KIRSCHNER

0

"-2

0

a- 1

FIG. 28. Anchored (at p end) steady-state polymer with centralized attached force F , (positive downward).

The four P rate constants depend on F, as in Eqs. (1 19) and (12 1) but the four a rate constants (Fig. 28), a!,aOl, etc., are independent of F, . Equations (1l l) and (1 13) (for F = 0) are still valid but Eqs. (1 12) and (1 14) (for F = FA now apply only to P1/p-l and Pz/p-2, respectively. Thus the symmetry between the two ends in the equations of Sections IV,B ,1 and 2 is lost. In place of Eqs. (44), we have here from Eqs. (111)-(114), as just amended, (b) PlPZIP-lP4 = (a) ayai/aYlaY2= eXTlkT (c) aYl,Pz/aYl,P-z= exTIkTr--l (d) = e*IkTr (143) (el aYIPIIaPP-l = r (f) aiP-z/ff%Pz = r These equations show the thermodynamic forces in six different cycles. Cycles c and d in Fig. 7 now move F, as well as hydrolyze NTP. Hence the forces in these cycles are, from Eqs. (143c) and (143d), XT - loFxand X, + loFx,respectively. There are also two new cycles, Eqs. (143e) and (143f) (see also Fig. 36), that move F, but do not hydrolyze NTP. These are relatively unimportant because of the appearance of inverse (negative subscript) rate constants in both numerator and denominator (i.e., in both cycle directions); these cycles will be discussed further in Section V.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

69

The various flux expressions, Eqs. (132)-(135), have to be altered because

J,", = agc- a!,

and

J:,

=

a$- at2c

are independent of F,. Hence, we have J , O = J Oa1 - J Oa, - agc -- az" J2 = J,"* + Jp, J1 = J,", + Jp,, J , n = J," + Jp = J1- J2

(145) (146) (147)

%

From 4,

=

0 (polymer length is constant) at c

= c,

( 144)

we find for c,

This cannot be put in the simple form of Eq. (124). To examine the rate of free energy dissipation, we need to revise Eqs. (127) and (128): = AP+ + loFx = b+, (149) AM = - Ap- Ap,, = - A p - , These again reflect the asymmetry between the two ends of the polymer ( F = 0 at the aend). We can again obtain the dissipation Eqs. (136), (137) (for c > c-), and (139) (for c < cm) except that J,, loFxin all three equations is replaced by JploFx,and, of course, J1, J% and J,, in the other terms of these equations are redefined as in Eqs. (146) and (147) (a?,etc., in place of al,etc.). We should expect Jon to be replaced by Jp in the work term because J," in J,, makes no contribution to the work. The classification according to Eq. (137) or (139) depends on whether Jon> 0, c > c,(polymer growing) or Jon < 0, c < c, (polymer shrinking), respectively. In either case, in order for there to be free energy transduction, the work term JploFxmust be negative. This can be accomplished in two ways, in either case: Jp > 0, F, < 0; or Jp < 0, F, > 0. Thus there are four possible free energy transducing cases here: polymer as a whole growing, with p end growing against a compressing force; polymer as a whole growing, with p end shortening against an extending force; etc. Obviously the four cases are possible (instead of two, as in the preceding subsection) because of the independent, nonproductive, and variable contribution of the a end. The efficiency expressions, when Jp and F, have opposite signs, are &a,

( 150) ( C > c,, Jon > 0) 7 = - JpIoFx/(J4+)XT+ Jon Ap+) 7 = - J ~ ~ o F ~ / [ J+ ~(-Jon)(-Ap-.)] -)~T ( C < c,, Jon < 0) (151)

70

TERRELL L. HILL AND MARC W ,KIRSCHNER

The transitions at the a end contribute to both fluxes in both denominators, but not at all to the numerators. Hence a transitions reduce the efficiency, as expected. An interesting but probably not very important special case arises when either Jp > 0, F, < 0 or Jp < 0, F, > 0 (so that work is being done on F,) and at the same time J," = - Jp so that Jon= 0 and c = c, In other words, there is treadmilling (polymer length is constant), with work being done on F,. Two examples are shown in Fig. 29, with Fig. 29A the more likely

I

B

FIG.29. Two examples in which, for the Fig. 28 system, work is accomplished against F , by the NTPase activity under treadmilling conditions (vertical lines). In the dashed-line case in Fig. 29A ( r I), however, no work is done (F, and Jp have the same sign).

*

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

71

possibility (the free or a end is the + end, as in Section III,B,2). Treadmilling is the boundary case between Eqs. (150) and (151): 7 = - Jp(c,)lOF,

(C =

/J$XT

c,, 4,

=

0)

(152)

where Cm =

JTr

=

JP(C,)

(4+ Pz)/(aP + a: + pz = p1cm -

pz =

P1)

( h a : - P2aQ)AaP+

(153) P1)

Of course p1and pz here are functions of F, [Eqs. (1 19) and (121)l. In this special case NTP hydrolysis is used as the sole free energy source for the work done on F,. This possibility arises again, and is more significant, in Section V. Free energy transduction (work production), while treadmilling, was not possible in the previous subsection (Fig. 25) because F, does not move, in that system, under treadmilling conditions [Eq. (142)]. But F, can move in Fig. 28 when the polymer has a constant length.

V. Polymer between Two Barriers In this section we consider cases in which the free monomer concentrationc is high enough to cause a polymer with free ends to grow until, in the course of its growth, the polymer encounters obstacles or barriers at both ends (Hill and Kirschner, 1982). In another case the polymer may be anchored at one end to begin with and grows until the other end reaches a barrier. Even if, as we assume in this section, monomer exchange is still possible at both ends after the polymer has made contact with both barriers, net growth of the polymer will quickly cease. The physical reason for this is that addition of further monomers to the polymer (between barriers) will induce a rising compressive force F (negative) within the polymer that, in turn, will increase the critical concentration for growth of the polymer. When the critical concentration reaches c , the polymer will stop growing. Because these polymers are rather incompressible, not many additional monomers will be required to raise the compressive force enough to turn off the growth. We shall not be concerned explicitly with the transient process just described (see, however, Section VI,A,2). After the transient, there is no net growth, as just explained. Therefore, in this section, our concern will be with polymers either at equilibrium or at steady state (e.g., if there is NTP activity). Consequently, treadmilling will be particularly important here. In contrast, treadmilling played a minor role in Section IV, where polymer length changes were crucial. Furthermore, if an external force F,

72

TERRELL L. HILL AND MARC W. KIRSCHNER

is attached somewhere in the central region of a polymer that is treadmilling between barriers, some NTP free energy of hydrolysis can be converted into mechanical work. There is consequently a formal thermodynamic resemblance between such a system and the actin-myosinATP system in muscle contraction. A. EQUILIBRIUM POLYMER 1. Polymer between Rigid Barriers In Fig. 30, there are rigid barriers a fixed distance L apart. Somewhat elastic barriers are treated in the next section. The solution contains free monomers at a concentration c that is arbitrary except that it is larger than the critical concentration c," for growth of the free polymer. In Fig. 31, which presents an illustrative case, the dotted lines labeled s;t' and JZ represent the growth rates of the two ends of thefree polymer. The points A and B on these lines are the rates of addition of monomers to the two ends of the free polymer at the particular monomer concentration c. When the growing polymer first reaches the barriers, the force on the polymer is stillF = 0 but the growth rates change to points C and D on the lines J,O and J J , with rate constants ao,a6,Po, Po.These rate constants refer to polymer in contact with barriers but with F = 0 (Section 111). The critical concentration is still c," because the barrier merely acts as an inhibitory cap (Section 111,A). At this point ( F = 0), let Nobe the number

@ F L

A at c F

FIG.30. Equilibrium polymer that has grown up against rigid barriers a distance L apart.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

73

of monomers in the polymer of length L . Then lo = L/N,. Because c > c,9 a few more monomers (n) will add to the polymer until the compression (F < 0, N = No + n, I = L / N ) ,is sufficient for the lines J," and J j in Fig. 3 1 to shift (arrows) to the lines J , and J , with rate constants, a ,(Y ', /3 ,/3 '. Here the rate constants refer to a force F (negative) just sufficient to increase the critical concentration (or equilibrium constant for on + off) from c,"to c, or just sufficient to move points C and D in Fig. 3 1 to point E on the abscissa. Of course in the aggregation process at concentration c , just described, only point E on the lines J, and J, is actually realized. The lines themselves represent hypothetical growth rates at the same F but at variable monomer concentrations. We now examine the quantitative aspects of the above discussion. The chemical potential of free monomers is [Eq. (l)] p s = p!

+ kTln c

( 154)

For the polymer just in contact with the barriers, at F = 0, and in equilibrium with monomers at c,U, the chemical potential of monomers in the

FIG.31. Flux changes when an equilibrium polymer, with monomer concentration c , encounters barriers at the two ends. See text for details.

74

TERRELL L. HILL AND MARC W. KIRSCHNER

polymer is [Eq. (2), but using c:! instead of c,] po = p,"

+ kTln c:!

(155)

Because c > c,"and hence p, > po,monomers at c will add to the polymer until F becomes sufficiently negative [Eq. (84)] to raise p for monomers in the polymer up to the value of p, in Eq. (154). At this point, the polymer will be in equilibrium with monomers at c: p = po - I,,F =

&=

p!

+ kTln c

( 156)

This equation determines F as a function of c . The independent thermodynamic variables here are T, L , and c (or E.LS = p in place of c). In Section IV, on the other hand, F was determined by the external force F, (i.e., F = F,); F, was an independent variable. On combining Eqs. (155) and (156), the explicit expression for F(c) is

-IoF/kT

= In (c/c,") =

yn

(157) We have included y n here from Eq. (93). If y is known [Eq. (92)], n as well as F can be calculated from c. However, the much more important relation is F(c); this does not depend on knowledge, or an estimate, of y (provided, however, that the inequality ( 9 9 , or N 9 l/y , is satisfied). Because c/cg would usually be of order 10 or less, the three expressions in Eq. (157) are usually of order 2 or less. For example, if yn = 1 and y = 0.05 [Eq. (93)], n = 20. From the detailed balance relations at F = 0 and F, respectively,

c:! = .;/a0 = Pb/Po, and using Eq. (157), we have

dff' = bo/ffb)E

c =

=

PIP'

a ' l f f= P ' / P ,

= (PO/Pb)E,

(158) (159)

where we introduce the shorthand notation ,+l.'/kT

=

c:/c 5 1 (160) Equation (160) is similar to Eq. ( 1 17), but F is not an independent variable, whereas F, is. Equations (99) and (159) show the same effect of F on

equilibrium constants, but in different contexts. From the split of E between forward and backward rate constants, we have, as in Eq. (loo),

where fa and& themselves depend on F, in general. Because, throughout this section, we are dealing with compression, it may be a rather good approximation to take fa and & as constants (see Fig. 18).

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

75

Except for the brief transient referred to above (F = 0 -+ F , n = 0 + n), the state of this system is equilibrium (point E in Fig. 31); there are no monomer fluxes to consider. 2. Polymer between Slightly Elastic Barriers The polymer may grow, at monomer concentration c > c,O, against an elastic barrier or barriers, instead of against rigid barriers. Examples of this may be the plasma membrane or a cortical array of actin. In this case, as the compressive force F (negative) is built up in the polymer, the barriers are pushed back in accordance with some macroscopic law of force appropriate to the particular barrier material or materials. In Fig. 32, the barriers are a distance Lo apart (lo = Lo/No)when the polymer first contacts the barriers (F = 0). After growth has ceased, the barriers are a distance L apart (L > Lo. I = L / N ) , and the forceF in the polymer is again determined only byc, from Eq. (157). Because of the mechanical equilibrium, this same force F acts on the barriers. To keep the remaining discussion simple, we now assume that the response of the combined barriers (when pushed by the polymer) follows another Hooke’s law relation, F = A(Lo - L ) . and that the length change, L - Lo, is small compared to Lo (e.g., L - Lo is several hundred A whereas Lo is of order 1Oj A). In the final equilibrium state, c , F, and L here have the same significance as in the preceding section, so most of the discussion of the equilib-

F F

gA g A at

L

C

F F

mE

FIG.32. Encounter of a growing equilibrium polymer with slightly elastic barriers that yield to the extent Lo + L .

76

TERRELL L. HILL AND MARC W. KIRSCHNER

rium state there is still valid. However, there are now two contributions to n = N - No, which we consider below [yn in Eq. (157) needs modification]. Fluctuations in N and L are also different here; these will be treated in Section VI. The concentration c determines the equilibrium F , and F then determines L from F = X(Lo - L ) :

L = Lo + (kT/hlo)ln (c/c!),

( 162)

where the last term is much smaller than Lo. With L available, N can then be found from the polymer equation

If we put

N-I z N6'[1

-

(n/No)]

because No 9 n , and use Eq. (162) for L , we then deduce from Eq. (163),

n

=

[(kT/h/;)

+ y-'I

In (c/c!)

( 164)

From Eq. (162), we can see that the new contribution ton here is equal to (L which is of order 30 to 40. This is of the same magnitude as the term in y-*. The new term in n is obviously due to the extra space made available to subunits when the barriers are pushed back (Lo + L ) . Aside from the extra contribution to n , elastic barriers do not introduce any really new features. L and F refer to the final state with stretched barriers. For this reason, we need not consider elastic barriers again until Section VI (fluctuations).

3. Polymer with a Centralized Attached Force We discuss in this section an equilibrium polymer (no NTPase activity) that has grown to a length L between rigid barriers, in the presence of an arbitrary free monomer concentration c > c;, and to which is attached an external force F, located at a distance BL from the p end (8 is a fraction, 0 5 8 I1). The principal parameters of the system are thus L ,c , F,, and 8. This system is illustrated in Fig. 33. Because of the asymmetry introduced by F,, instead of a uniform force F in the polymer there are now two different forces Fa and Fp in the two sections of the polymer (Fig. 33B). Qualitatively, it is obvious that if the polymer is compressed to begin with (F negative) and then, say, a positive F, is introduced (pulling the ring in the Q direction), the compression in the p end of the polymer will be reduced while that in the a end will be increased. Fp might even become positive; in this case the stability of the anchoring at the p end would be

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

A

77

B

FIG. 33. Equilibrium polymer, grown against two barriers, with an external attached force F , at position 0 (a fraction of L ) . F , is positive toward the a end.

tested. Because the two ends are now under different forces, the on + off equilibrium constants are different. Consequently the two ends of the polymer cannot be in equilibrium with free monomers at the same concentration; the final state of this polymer, in contact with monomer at c, will therefore be a steady state rather than an equilibrium state. Our object here is to examine the properties of this steady state. Even though there is no NTPase activity, treadmilling is induced by F, # 0. We now perform a sequence of thought experiments in order to relate a, a‘,p, p‘ in Fig. 33A to arbitrary values o f F and F,. First, with F, = 0, we select the proper monomer concentration [Eq. (157)l to produce a preselected, arbitrary compressive force F (negative) in the polymer, at equilibrium. Second, with F thus established, we inhibit or block monomer exchanges at both ends. Third, with exchanges blocked, we impose an arbitrary external force F, on the ring. The ends of the polymer are fixed at the barriers ( L is constant) and the polymer is only slightly compressible. Exploiting this compressibility, the ring will make a very small adjustment in position (9 changes, say, to 9’)until a balance of forces on the ring is achieved. From this balance, we can find F, and Fo in terms of F, F,, and 9. If, for concreteness, we think of F, as positive, as shown in Fig. 33B, then the two downward (Fig. 33B) forces on the ring, F, and - F p , must be balanced by the upward force -Fa. That is, F, - Fp = - F,. If 9 += 9’ (an increase, if F, > 0), then in the p (top) part of the polymer1 += l 9 ‘ / 9 while in the a (bottom) part I + l(1 - O f ) / ( 1 - 9). From F = a ( / - lo) for each part of the polymer and the force balance equation (above), we then have

F,

-

o[(w/e) - lo] = -a{[i(i

-

o f ) / ( i - e)]

-

lo)

(165)

78

TERRELL'L. HILL AND MARC W. KIRSCHNER

From this we find = 1 + [ ~ , ( i- e ) / ~ r ] ( 1 - 8')/(l - 8) = 1 - [ F X 8 / d ] .

e'/e

(166)

It is easy to show, as expected, that [ ] << 1 in these equations if N o + l / y [Eq. (95)l. That is, the change in 8 is very slight. If we now use Eqs. (166) in Fp and F, [Eq. (165)], we obtain the required relations

Fp = F Fa = F

+ (1

-

- 8)FX =

OFx

Fa

+ F,

The fact that 8 appears here rather than the final 8' is of no consequence because the terms in Eq. (167) are all very large compared to (8' - 8)F,. An alternative, more intuitive way, to deduce Eqs. (167) is the following. With F and F, given, the mechanical work necessary to introduce a subunit at the p end is -Flo against the force F (negative) and -Fxlo(1 - 8) against F,, because the position of F, moves a distance lo(l - 8) toward the a end (utilizing the compressibility) if a subunit is added to the p end. Similarly, the work necessary for a subunit insertion at the a end is - Flo against F and F,lo8 against F,. The total work expressions here suffice to define the effective forces (Fa and Fp). These totals agree with Eqs. (167) (except for the factor -lo). In the fourth and final step of our hypothetical sequence of events, we adjust c to a value cP, such that free monomers will be in equilibrium with the p end of the polymer at Fp (above). The p end transitions are then turned on (a end still inhibited); the rate constants at Fp are p and p ' . Then, from [compare Eq. (88)l the equilibrium relations

+

+

p o - lo[F ( 1 - @F,] = p," kTln cP, ( F = 0, F, = 0 ) p o = p,"+ kTln c,"

and pcP,

=

( 168) ( 169)

p', pot," = PA, we find p/p'

(po/p&wl-e =

(170) where E = e'oFlkT< 1 andr = erifJkT, as before. Alternatively, if only the a end is unblocked, we can derive in the same way =

a/a'

= (a0/a&re=

(Po/ph)e'iifhlkT,

(ao/aA)e'i~p~/kT

(171) Recall also that p0/pA = a0/a;)( F = 0 , F, = 0). Equations (170) and (171) show the dependence of PIP' and a / a' (equilibrium constants) on arbitrary F, F,, and 8. These relations were derived, as usual, using an equilibrium arrangement. But they are also valid at any F in a transient or steady state because of our assumption that the mechanical equilibrium (balance of forces) is readjusted virtually instantaneously as monomers

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

79

enter or leave the polymer. The equilibrium constants a/a' and p/p' at the two ends differ because Fa # Fp. a. Steady-State Kinetics. As in Eqs. (161), the individual rate constants are written in terms of fs as a

p

a' =

= ao(&r+))/.,

p'

= pO(&r'-B)'D,

cyb(&r+))tY-'

= pb(&r'+))'@-'

(172)

In these equations, f u is in general a function of Fa and fa is a function of FD(=Fa+ F,). Even if& and fa are the same functions of the force they will differ in value, in general, because Fa # Fp. Operationally, the free monomer concentration has a given value c , the external force is fixed at F,, and is attached to the polymer at a position defined by 8. The polymer, in growing against the two barriers under these condition,s, adjusts the value of F (by changing n ) until steady state is reached, that is, until growth ceases:

This is an equilibrium relationship [Eq. (I%)] only if F, = 0. If Eqs. (172) are substituted into Eq. (173), we then have a relation that determines the steady-state value of E (i.e., F) as a function of c , F,, and 8 (keeping in mind that f u andfs, in general, also depend on F). Actually, we note that E and 8 always occur (also in f, and f p ) in the combination &r-e(=elflJkq. Consequently, it is simpler to regard Eq. (173) as determining as a function ofc and F,. Once &r-O is so determined, numerically if necessary, it can be introduced into Eqs. (172) to give the values of the rate constants that actually obtain at steady state. The steady-state rate constants are functions of c and F,, but not 8. We shall illustrate this procedure in a special case below. There is treadmilling at steady state. If F, is positive, monomers will enter the polymer at the p end and leave at the a end. The reverse is true if F, is negative. To confirm this, let J i be the steady-state rate of addition at the p end. Then, from Eq. (173),

Ji; = -J,"

=

PC - P'

= p'a(r -

l)/(a + p),

(174)

where we have used ).

=

l,)l.',/kT

=

Pa' I P 'a,

(175)

which follows from Eqs. (170) and (171). There is a steady-state cycle operating here: a monomer adds to the polymer at the p end and another leaves at the (Y end. This is similar to cycle e in Eq. (143). The thermodynamic force driving the cycle is loFx. This is the only force in the

80

TERRELL L. HILL AND MARC W. KIRSCHNER

cycle. Hence free energy transduction is not possible, only dissipation of free energy at the rate

TdiSldt = Jp"lJ,

2

0

(176)

Note in Eq. (174) that J ; > 0 if F, > 0 (r > 1) and J ; < 0 if F, < 0 (the cycle runs in reverse). In summary: because of the presence of the two barriers, the equilibrium polymer in this system cannot move F, and do work by means of net growth or shortening driven by a monomer chemical potential difference, as in Section IV. Instead, the polymer here necessarily has a constant length; F, can move only through treadmilling; no other thermodynamic force is present in this system. b. Number of Subunit Insertions. The polymer is only slightly compressible. Hence, even though the force F, introduces asymmetry (Fa and Fp),the linear number density ( 1 / 1 ) has essentially the same value in the a and /3 regions. However, this very small difference has an effect on the number of subunit insertions, n . According to Eq. (93), n = K N ~ ( - F )In . the /3 region (top, in Fig. 33) of the polymer, F = Fp and the number of monomers is essentially ONo. In the a region, F = Fa and the number of monomers is (1 -@No. Thus the numbers of subunit insertions, owing to compressibility, in the two separate regions are np =

na = K(1 - 8)No(-F,)

K8No(-Fp).

(177)

These numbers differ significantly. If we now use Eqs. (167) for Fp and Fa, we find n = n,

+ np = K N ~ ( - F )

(178)

This shows that, after averaging over a and /3 regions, Eq. (93) still holds for this more complicated system. Equation (178) also establishes the operational significance o f F , as the mechanical force that determines n for the present system. At steady state, F is given by Eq. (173), as already explained. Were it not for this simple result [Eq. (178)], it would have been easier to work, from the outset, with F, and F, as independent variables, rather than with F, F,, and 8, as we have done. c. Simple Special Case. To illustrate some of the above relations, we examine the special case fa = fp = f = constant. In view of Fig. 18, this could well be a realistic approximation because we are concerned here with compression. In this case, we can solve Eq. (173) explicitly for F. We find er-B

= eI,,FalkT =

(ah

+ /3&rf-')/(a0+ Porf)c

(179)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

81

and -In

E =

-I,,F/kT

=

In ( c l c ' ) = yn,

(180)

where

+ p;rf-')re/(ao + perf)

c ' = (a(,

(18 1)

Here, c' is an effective concentration that depends on r and 8. Hence, F andn dependonc,r,and& Wenotethatifr = 1(FX=O)inEq.(181),c'= a;/ao= p;/po = c,". In this case, Eq. (180) reduces to Eq. (157). The fact that c ' f c,"for arbitrary r and 8 confirms, as expected, that, for the same monomer concentration c, the steady-state F in Eq. (180) differs from the equilibrium F (when F, = 0) given by Eq. (157). The difference depends on c'/c,".Consequently, the n values [Eqs. (157) and (180)l also differ. The explicit expression for JpZ, using Eqs. (172), (174), and (179), is not very simple:

JpZ

=

ao&,rf-'(ab

+ p&rf-l)f-l(r-

l)/(ao+ /30rf)fcf-1

(182)

This gives J ; as a function of c and r ; 8 is not involved. POLYMER B. STEADY-STATE 1. Polymer between Rigid Barriers In this section we extend the treatment in Section V,A,l to a steadystate polymer, that is, to a polymer with NTPase activity. There is free monomer AT at a concentration c large enough to cause the polymer to grow up against both barriers until compression halts the growth. No external force F, is attached to the polymer here. The compressive force F (negative) is uniform through the polymer. At the steady-state (no net growth), treadmilling occurs (we assume, as usual, that subunits can exchange at both ends of the polymer despite the presence of the barriers). Except for the fact that the polymer is under compression, the treadmilling resembles closely that of a free polymer. In particular, NTP is used to maintain the treadmilling, but no outside work is done; all of the NTP free energy of hydrolysis is dissipated. Because of the formal resemblance to the treadmilling of a free polymer, much of Section II,B,2 can be taken over here without change. In the next section we shall add to this system an external force F, attached to the central region of the polymer. In this case, free energy transduction (NTP + work) is possible. We turn directly to the steady state, as this is the only condition of interest. The independent thermodynamic variables are L , T, and c. The compressive force sufficient to stop growth is F. We again use the notation

82

TERRELL L. HILL AND MARC W. KIRSCHNER

= eCIFIkT. The rate constant notation is a l ,etc., at F and at,etc., at F = 0 (polymer just in contact with barriers). Figure 34 shows the transitions to be considered. The dependences of the individual rate constants on E are the same as in Eqs. (118)-(121) except that Y is to be replaced by E . The steady state is defined by Jon= J, + Jp = 0 [Eq. (28)l. The steadystate value of F or E , as a function of c , is found by substituting Eqs. (1 18)-( 121) (with E for Y) into

E

+ Pz +

+ P-d/bl +

(183) + P-2) and solving (numerically if necessary) for E . Equation (183) is analogous to Eq. (30). Because F is negative, E < 1 . This value of E is then put into the c =

(a2

a-1

P1

+

a-2

eight separate rate constant expressions to give the steady-state values of the rate constants. These are then used in the flux expressions of interest (below). We adopt the notation c , to denote the monomer concentration at which the steady-state polymer is just in contact with the barriers but without any compression (i.e., at F = 0). In this case, Jon = 0 with the rate constants a!, etc. Hence C, = (a!

+ P! + a!!,+ P!!l)/(ay+

+ a t z + P!!J

(184)

The concentration c , above, is of course larger than c,. This larger concentration results in some compression and n extra subunits being added to the steady-state polymer at c , where n is given by Eq. (93) [F having been found from Eq. (183)l.

FIG.34. Steady-state polymer grown against two barriers.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

83

The steady-state Eqs. (30)-(34) and (40)-(54) of Section II,B,2 (free treadmilling polymer) all apply here provided that the following changes are made: (a) C, is replaced by c ; (b) in Eqs. (50), P . , ~is replaced by P.\D - /OF;and (c) all eight rate constants al,etc., are understood in these equations to be the steady-state rate constants evaluated as described following Eq. (183). Simple Special Case. We now consider an explicit example that is simple but probably quite realistic. We assume that

f?

=

ff = fl

=

const.,

& = f [ =f2

=

const.

(185)

As mentioned before, fP = ff is likely because the mechanism in part 1 of the NTP cycle (see Figs. 2 and 3) is probably the same at the two ends of the polymer. Also,fl = const. is plausible because we are concerned here with compression only (see Fig. 18). Of course similar comments apply to part 2 of the NTP cycle. We also assume, in this example, that inverse transitions (negative subscripts) can be neglected. Then we have C Y ~=

( ~ f ~ f l ,

p1 = P ~ E ~ I C, Y ~ = CYSE-~’,

p2

=

P ~ E - ~(186) ’

Because E < 1 (see below) the effect of E (compression) is to reduce the “on” rate constants a1and p1and increase the “off” constants a2and p2. When these expressions are substituted into Eq. (183), we find for E , &

(187)

= (C,/C)Il(fi+h)

and for F ,

-IOF/kT = [l/Vl

+ fi)lln

(c/c,)

= yn

(188)

The relation between F and n is an elastic (mechanical) property of the polymer that holds at steady state as well as at equilibrium. Equation (188) is the analog of the equilibrium Eq. (157). In Eq. (188), c > c,, F is negative, and n is positive. The sumfl + fi is of order unity. For example, iffl + fi = 1, c/c, = 5, y = 0.05, and lo = 6.15 A (microtubule), then dyne. -/,FIAT = 1.61 and n = 32. At 25”C, - F = 1 . 1 x Having found the compressive force F necessary to stop growth of the polymer between barriers, when the monomer concentration is c , we can now obtain the steady-state treadmilling rate J z and NTPase rate J?. We substitute E from Eq. (187) into Eqs. (186), and then put these rate constant expressions into Eqs. (34) and (42). The simple result is

where

84

and

TERRELL L. HILL AND MARC W. KIRSCHNER

JF(0) = a; + pp

are the two fluxes at c = c,, F = 0. It will be noted that both the treadmilling and NTPase rates are increased modestly when c > cm,and by the . the value of s [Eq. (43)l is same factor [for example, ( C / C , ) ~ / ~ ] Thus independent of c (in this special case). It should, perhaps, be mentioned again that J;(O) and JF(0) are fluxes for the treadmilling polymer in contact ( F = 0) with barriers at both ends; these are not the fluxes for the free treadmilling polymer (with no caps, anchors, or barriers). 2. Polymer with a Centralized Attached Force This important system is shown schematically in Fig. 35. The polymer, in the presence of monomer (AT) at concentration c , has NTPase activity at both ends, despite the presence of barriers (or anchors). In addition, there is an external force F,, attached at position 8 (Fig. 3 3 , which is taken as positive in the a direction. This could be any kind of a structure that pushes or pulls on the surface of the polymer, or that offers resistance to the treadmilling of the polymer. Because of the barriers, steady-state treadmilling is the only polymer motion possible. Because of the attached external force, some of the NTP free energy of hydrolysis can be converted into mechanical work. In this respect, the present system resembles the actin-myosin-ATP system of muscle, or the axonemal-ATP sys-

FIG.35. Steady-state polymer, with external force attached, grown against two barriers.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

85

tem of eukaryotic cilia and flagella. In fact, in a limiting hypothetical case (below), the efficiency of conversion of NTP free energy into work can be 100%. Under realistic conditions, however, the efficiency is probably very low. The essential theoretical ingredients for this subsection have already been worked out in Sections V,A,3 and V,B, 1. We merely have to combine these approaches. The rate constants here are influenced by both the compressive force F and the external force F,, as in Eqs. (172). In place of Eqs. (118)-(121), we have here a!(~r-@)f', p1 = p ! ( E r ' - @ ) f f ,

p-l

= p!l(Erl-e)ff-l

= (I!g(Er-e)f',

(I!-2

=

a0_2(&r-@)1-fB

p2 = p$(&r'-@)-fg, p-2

=

pz(&+@y-fB

= ~~,(Er-~)fB-1

( ~ = 1

a2

(193) (194) (195) (196)

These follow from Eqs. ( 1 18)-( 121) on replacing, at the (I! end, r by w-@ (i.e., by ewJwT) and, at the /3 end, r by ~ r l - @ (Le., by ewOlkT).The rate constants a!, etc., refer to polymer in contact with barriers but with F = 0, F, = 0. The parametersff andff are in general functions of F,;ff andfl are functions of F , . The quantities c , F,, and 8 are independent variables of the system. The polymer stops growing (.lo"= 0) when F becomes sufficiently negative (or positive, in some unusual cases). This force is found by substituting Eqs. (193)-(196) into Eq. (183) and solving for E . Again, as with Eq. (173), E and 8 occur (including thefs) only in the combination EP. Hence, Eq. (183) may be regarded as determining E r 4 as afunction ofc and F,. When 8r4 has been found (numerically, if necessary) for a given c and F,, this value is substituted into Eqs. (193)-(196) to give the steady-state rate constants that are to be used in the following discussion of fluxes, cycles, etc. We shall use the regular notation al,etc., below with the understanding that these are steady-state rate constants, actually determined by the values of c and F,. Because E T - @ is a function of c and F,, F (or E ) is a function of c , F,, and 8. The same is true of n , the number of subunit insertions, in view of Eqs. (93) and (178), which hold here as well. There are six steady-state cycles possible for this system, with various thermodynamic forces [compare Eqs. (44), (143), and Fig. 71. The cycles and forces are shown schematically in Fig. 36. Cycles a and b hydrolyze a molecule of NTP; the force is X , . These cycles use NTP without accomplishing anything. Cycle c hydrolyzes NTP and also moves the attachment a distance lo against F, (if F, is positive); the force is XT- loF,. This, in

86

TERRELL L. HILL AND MARC W. KIRSCHNER

FIG.36. Component cycles for the steady-state system in Fig. 35, with thermodynamic forces given.

fact, is the only cycle that contributes to free energy transduction (NTP + work), if F, is positive. Cycle d hydrolyzes NTP and moves the attschment a distance lo in the direction of F, (if positive); the force is XT+ &F,. This cycle is counterproductive if F, is positive and if the objective is to do work against F,. Of course, if F, is negative, cycles c and d exchange roles. Cycles e and f move F, only; the thermodynamic force (in the direction chosen for these cycles) is loFx.Cycles e and f are generally unimportant (have a negligible cycle flux; see below) because a negative subscript (inverse) transition is used in traversing either of these cycles in either direction. In the other four cycles, on the other hand, one cycle direction uses principal (positive subscript) rate constants only, hence producing a significant cycle flux in this direction. This is the direction chosen in Fig. 36a-d. It should be noted that the positive direction chosen for cycles e and f is the opposite of that for cycles E and F in Hill and Kirschner (1982). Unlike cycles E and F, cycles e and f have positive cycle fluxes if F, is positive. There are thermodynamic relations between the rate constants a ! , etc., that we will need (Section 11,B): apag/a!lla!L2 = p$3$/p?1p?2 = eXTlkT ap/ao_,=

p:/p?l,

pg/p0_2

ag/CY!L*=

(197) (198)

On combining rate constants in Eqs. (193)-( 196) according to the cycles in Fig. 36, we find, on using Eqs. (197) and (198), = (a) a1a2/a-la-2

(b)

PlP2IP-lP-2=

(199)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

87

(c) al&/ ~ r - , p -=~ e *TlkTr-l , (d) azPl/a-zP-l= eXTlkTr (200) (el a-lPJalP-l= r , (f) a~P-zIa-zPz= r (20 1) All the other factors cancel. Note that Eqs. (198) no longer hold in the presence of F , (the two polymer ends are under different forces, Fa and F p ) . Instead, we have Eqs. (201). Equations (200) follow from Eqs. (199) and (201); they are not independent. The steady-state treadmilling rate J R and NTPase rate J? can be obtained from [Eqs. (24) and (36)] J"in

-J,

= Ja =

J;=

= (a1

J1= J 2 =

+ a - 2 ) ~- (at +

(a1+ P 1 ) C -

(202) (203)

(a_,+& I ) ,

if we substitute Eq. (183) for c . In each case, after the substitution, half of the terms in the numerator cancel and the remaining terms can be paired off by cycles. For example, in Eq. (203), the numerator terms a1a2- a-la-z= alaz(1 - e-xT'kT)

refer to cycle a. Thus J ; and JTm can be expressed as a sum of separate contributions from four cycles each, as follows:

where the separate cycle fluxes are defined by ( Y ~ ( Y Z (1 - e-XT/kT)/D, J, = a1P2(1- e-*TlkTr)/D, J, = (Y-~PI( 1 - r-l)/D,

Ja =

1 - e-*TIkT)/D (206) azPl(l- e-*TIkTr-l)/D (207) a&z( 1 - r-l)/D (208)

Jb = P1&(

Jd= Jf =

D = a 1+ P1+

+ P-z

The treadmilling rate J ; is defined as the rate of adding monomers at the a end (or removing monomers at the P end). Hence it is clear, merely by inspection of Fig. 36, that Jf should be related to cycle fluxes as in Eq. (204). Similarly, the relation of J ; to cycle fluxes [Eq. (205)l is also obvious from Fig. 36 (cycles a, b, c, and d all hydrolyze NTP). If we neglect inverse transitions and drop the terms in e-XT'kT(which is or 10-lo), we have the simpler relations usually of order

JE= JcJa=

Jd,

alaZ/(al+

J , = alPz/(al+

Pl), Pl),

J y = Ja+ J,+ Jc+ Jb= Jd

PlPZ/(al+ P1)

= azP1/(a1+

P1)

Jd

(209) (210)

The relative contributions of the separate cycles to the steady-state kinetic activity are proportional to a I a z ,PIPz,alp2,and azPl,for cycles a, b, c, and d, respectively. Substitution of Eqs. (210) in Eqs. (209) gives sim-

88

TERRELL L. HILL AND MARC W. KIRSCHNER

ply Eq. (34) for J ; and Eq. (42) for J : (but the rate constants here are steady-state rate constants, functions of c and F,). We return now to the general treatment in Eqs. (199)-(208). The rate of free energy dissipation is given by the general Eq. (48). However, in Eqs. (50), we have to replace p by p ,,D - loFa for A p a l and A p a 2 ,and replace p,\Dby p,lDor hpp,and App2.Then, if we use

lap

JY = Jz = J,, Jm ” = - J --J F,

=

Fa

,

+ Jp, 02

+ F,,

- J

PI

= J a = J a1 - J

a2

Eq. (48) simplifies. to 9

Td$/dt

=

JTXT

-

JGloF,

(211)

This is just what we should expect for this free energy transducing system. This equation tells us that not all of the NTP free energy production, at the rate J;XT, is dissipated; some of it is converted into mechanical work, at the rate J;IoF,, if J ; and F, are both positive (or if J ; and F, are both negative). The attachment to the polymer is moved against the positive force F, (i.e., moved in the direction a p ) , by the treadmilling, at a velocity J;I0. The efficiency of the free energy transfer is

-

q = J;IOF,/J,”xT

(212)

It can be seen from Eq. (204) that cycles d, e, and f, to the extent that they are used, reduce J ; and hence reduce the efficiency ( J d ,J,, and J f are all positive, if F, is positive). These cycles move F, in the wrong direction. Furthermore, from Eq. (205), we see that J ; is increased by cycles a, b, and d without any contribution to the work production (in fact cycle d moves F, the wrong way, as just mentioned). Thus, to the extent that these cycles contribute to J ; , they reduce efficiency. Indeed, from the point of view of free energy transduction, only cycle c is a “good” cycle (if F, is positive). In this cycle, NTPase activity and motion of the attachment against F, are completely (or “tightly”) coupled. In the strictly hypothetical, tight coupling, case in which p f , PO1, a!, and are all zero, only cycle c remains. Molecules enter the polymer at the (Y end by process 1 (Fig. 3) and leave at the p end by process 2. In this case, J ; = J ; = J,= aiPz(l- e-*T’”r)/(a,+ q = IOFXIXT

/3-2)

(213) (214)

In principle, ifF, is made large enough,loF, - X T andq + 1 (equilibrium, with JG = JY + 0 as well). If XT is 13 kcal mole-’ and lo = 6.15 A, the required F, for equilibrium is 1 . 5 ~ dyne. [At such a large force, Eq.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

89

(84) is no longer adequate.] For comparison, the average force exerted by a myosin cross-bridge, while attached to actin in an isometric contraction, is about dyne (Eisenberget al., 1980). Also, we found lop6dyne in the example following Eq. (188). The drag of a chromosome in anaphase has been estimated at about dyne (Taylor, 1965; Nicklas, 1965). Returning to the free energy dissipation Eq. (21 l), if we substitute Eqs. (204) and (205), we obtain

Td,S/dt = JaXT

+ JbXT + J,(XT

-

lax)

+ Jd(XT + loFJ + JeIoF, + JJax

(215)

This exhibits the free energy production by separate cycle contributions (Hill, 1977a). Each of the six terms on the right is positive or zero: for each cycle, flux and force necessarily have the same sign [Eqs. (199)(201) and (206)-(208)l. a. Simple Special Case. 'We return to the simple but probably important special case introduced starting with Eq. (185). We takefi= f!= fi = const.,fg= f$-f2= const., and neglect all inverse transitions. When Eqs. (193)-( 196) are substituted into c=

(a2

+ Pd/((.l+ P A

we can deduce an explicit formula for er-e as a function of c and F,: e y - 8 = eloFJkT = ( 1 / R )I/(fl+fZ) (216) where R

-

(a?+

P?rfi)c/

(a:

+ P8rfz)

-

(217)

R is defined in this way as a convenience below. Note that R c and that when r = 1(F, = 0 ) , R = c/cm [Eq. (184)l. From Eq. (216), we then find for F : -1,FIkT = [l/(fl + f2) ]In ( c / c '1

= yn

(218)

O1rf1)

(219)

where, again for convenience, we have defined c' = (a! + /3!r-fz)Y(fi+f2)'3/

(a! + P

Equation (218) is very similar to Eq. (188). When r = 1, c ' = cmand Eq. (218) reduces to Eq. (188). Note that F and n are functions of c , F x ,and 8. The physical significance of c ' , which is a function of F , and 8 , is that if c happened to have the value c ' at a given F , and 8 , then the steady state of the polymer (J,,, = 0) would occur at F = 0 and n = 0. No subunit insertions or deletions would be necessary. It is possible forc' to be either larger or smaller than cm, If c ' > cmand c is chosen between cm and c ' , then F would be positive and n negative. But we are primarily interested

TERRELL L. HILL AND MARC W. KIRSCHNER

90

in cases in which c is fairly large, say several times larger than either crn or c’. Then F is negative (compression) and n positive. If we use Eq. (216) in the rate constant expressions (193)-(196) and these, in turn, in Eqs. (34) and (42) for JP;;and J ? , we deduce JP;;

=

J; =

(QPPgr-fz - agPprfi)Rfi/(fi+f~)/(,p+ PPrfi)

(a:+ pgr-fz)Rfzlvl+fz)

(220) (221)

These flux expressions depend on c and F, but not on 8. Both JG and J ? are proportional to Whenr= 1 (F,= 0), J z and J? reduce to Eqs. (189) and (190), as expected. By analogy with the force-velocity curve in muscle, we expect the “velocity” JP;; as a function of F,, with c held constant, to have a maximum at F,= 0 [Eq. (189)] and to become equal to zero at a sufficiently large F, (the “isometric” force). We shall calculate a force-velocity curve Jg(F,) below, in a numerical example. But it is easy to see from the numerator in Eq. (220) that, in general, the value of F,, call it F $ (the isometric force), necessary to make J ; = 0 is U ’ $ I k T = [l/(fl + fi)lln (aPPBIagP3 The ratio aP@/a$PSis related to JP;;(O)[Eq. (191)l. The individual cycle fluxes are J , = spa$ Y, J, = o@$r-f2Y,

Jb= PPfl!jrfi-fz Y Ja = a!#frfiY,

(222)

(223)

where y c Rfz/(fi+fZ) /(a! + PPrf1)

All of these cycle fluxes are proportional to cfzlcfi+fz)and independent of 8. b. Numerical Example. We now illustrate the above special case with a rather arbitrary numerical example in which, however, the rate constants have realistic magnitudes. For the rate constants at F = 0, F, = 0 (i.e., at c = G), we take a;= 7 p J 4 - 1 s-1,

pp= 1 p J 4 - 1

s-1,

a$= 7

pg= 7

s-1 s-1

(224)

These rate constants then give [Eqs. (31), (34), and (42)] Crn=

1.75 / L M ,

Jz(O)= 5.25 s-’,

J?(O)= 14 S-’

(225)

These are reference quantities. We now suppose that c = 5.25 p M (i.e., c/crn= 3), and we takefl= fi= 1/2. We shall allow F, to vary from F, = 0 to F, = F ; , where, from Eq.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

91

(222), l,,F:/kT= In 7 = 1.95. At F,= 0 and c = 5.25 p M , we have [Eqs. (188)-( 192)] J ; = 9.1 s-l,

J ; = 24.2 s-l,

loF/kT= -In 3 = - 1.10 (226)

These are the conditions at C/Cm= 3, before F, is imposed. Both J ; and J ; are increased over the c = cm values by a factor of 3lI2. Figure 37 shows the effect on the various fluxes of varying F, while keeping c = 5.25 p M . J , , J b , and J : are not very sensitive to F,. J ; reaches zero when J, = Jd (at F, = F ; ) . As already mentioned, J;(F,) is the “force-velocity” curve for this system. It is practically linear in this example, unlike the force-velocity curve in muscle (Eisenberg et al., 1980). Despite this linearity, this system is very far from equilibrium. To calculate the mechanical forces, and related quantities, we assume further that 8 = 1/2. While 1z.JkT ranges from 0 to 1.95, l,,F/kT varies from -1.10to -0.68,loF,/kTfrom -1.10to -1.66, loF,/kTfrom -1.10to +0.29 (extension rather than compression), and c ’ from 1.75 to 2.65 pM. From y n = -loF/kT, we find, if we take y = 0.05, that n varies from 22 to 14. The NTP free energy of hydrolysis, XT,is not specified explicitly in this example because we are using one-way cycles. But if we take X T / k T = 23

*O

t

10

5

0.5

1.0

1.5

loFX/kT FIG.37. Dependence of various fluxes on F, in a numerical example for the system in Fig. 35. The J”,(F,) curve is the “force-velocity” curve for this system. See text for details.

92

TERRELL L. HILL AND MARC W. KIRSCHNER

(13.6 kcal mole-l at 25"C), we can calculate the efficiency of free energy transduction from [Eq. (212)l 7 = (/oF,/xT)(J;/JTm)

(227)

The efficiency is zero at F, = 0 and F, = F,* (because J ; = 0 ) , and has a maximum of only 0.81% at IoFx/kT= 0.982. The two factors in Eq. (227), at this point, are IoF,/XT= 0.043 and s = J g / J ? = 0.19. The first factor is primarily responsible for the low efficiency. The NTP free energy of hydrolysis is much larger than needed for the mechanical work obtained. In muscle contraction, on the other hand, the corresponding maximum efficiency is about 50% (Eisenberg el a / . , 1980). Incidentally, l,,F,/kT= 0.982 corresponds to 6.6 x lo-' dyne (at 25°C; lo = 6.15 A); also, F; = 1.3 x dyne. c. More Than One Attached External Force. There is no essential complication if several external forces act on the surface of the polymer. For example, if there are two forces F1 and Fz located at el and &, then it is not difficult to show that, at the two ends, Fa= F - 81F1- &Fz FO= F + (1- Bi)Fl+ (1- 6,)Fz = Fa+ F l + Fz

(228)

Consequently, we merely have to replace in the various kinetic equations 8r-O byEr;Olr;OZ

and r by rlrz,

(229)

where rl'

el#l/kT

and

rzE el#zlkT

One can also show that Eq. (178) for n still holds in this more complicated case. VI. Fluctuations and Stochastics

This section is essentially a supplement or appendix. For readers not interested in fluctuations and stochastics, the first five sections present a self-contained thermokinetic analysis of the problem addressed in this article; the present section may be omitted. The treatment here is selective and illustrative rather than exhaustive. No attempt is made to analyze every type of system mentioned in the first five sections.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

93

A. EQUILIBRIUM POLYMER 1. Completely Open Systems

The essential first step in studying fluctuations in an equilibrium polymer is to adopt the particular statistical mechanical partition function (Hill, 1960) that is appropriate for the actual “environment” of the polymer. The environment is characterized by the independent thermodynamic variables. An example is the polymer between rigid barriers in Fig. 30. The independent thermodynamic variables of the polymer in this case are the chemical potential p , determined by the concentration c of free monomers in solution, and p = &(c) ; the length of the polymer L (distance between barriers); and the temperature T . The extensive variableN, conjugate to p , fluctuates;N is not an environmental variable. The appropriate partition function for these independent variables is the grand partition function E ( p , L , T ) . The fluctuations in N are the main point of interest. This case will be considered in the next subsection. A polymer with free ends, as discussed in Section II,A, has environmental variables p and T as above. However, L is not fixed; instead, the variable conjugate to L , namely F , is fixed at the value F = 0. The environmental variables are thus p , T , and F = 0. The unusual feature of this set of variables (Hill, 1964) is that none of them is an extensive variable and hence the size of the system is not prescribed. In more typical environmental variable sets, for example, p , L , T , or N , L , T , or N,F,T, at least one extensive variable is included; thus the size of the system is specified. When no extensive variables are given, as in the set p , T , F = 0, the system is said to be “completely open.” That is, it has fluctuations in all of the conjugate extensive variables, for example, in N , E (internal energy), and L in the case p , T , F = 0. The fluctuations in a cbmpletely open system are abnormally large (Hill, 1964). If, as a simplification, the polymer is assumed to be incompressible, then L is not an independent variable but is always exactly proportional to N . Hence, the pair of variables F , L drop out of the discussion. The environmental variable set for an incompressible polymer with free ends becomes simply p , T . Fluctuations in N in a completely open, incompressible polymer require special and detailed treatment that we shall not include here, though such treatment is available elsewhere (Hill, 1980a). The main results are, as c approaches c, (critical concentration, Section II,A), the mean polymer size N approaches infinity; and for values ofc close to but less than c, ,the probability PB that the polymer contains N subunits is proportional to N ” ( c / c J N ,where n is probably between 4 and 6 ( n depends on the trans-

94

TERRELL L. HILL AND MARC W. KIRSCHNER

lational, rotational, and internal vibrational motion of the polymer molecule). This n is, of course, not related to the n introduced in Eq. (93). It is usually assumed in the literature, incorrectly, that n = 0. This value of n would be appropriate only if the above mentioned degrees of freedom are absent (e.g., if the polymer is immobilized on a surface). Actual observation of the true equilibrium distribution is a problem: it may take a very long time to reach the final distribution; and some methods may introduce artifactual results. Fluctuations in such an equilibrium distribution are considered in Appendix 2. From the above mentioned probability distribution in N (or L , because of proportionality), one can deduce for the variance in N , relative to the square of the mean value,

(N"

- @2)/N2=

=

(1'1.ave./no.

l/(n

where no. ave. = N=

2N P k N ,

+

1)

ave.) - 1

(230)

- -

wt. ave. = N 2 / N

The normalized variance in Eq. (230) is of order unity, which is relatively large (Hill, 1964). In systems with normal fluctuations, e.g., in the next subsection, the variance itself is of order N and the normalized variance is of order l / k (Hill, 1960). An equilibrium polymer with one end anchored or with one or both ends capped is also a completely open system with fluctuations of the above type. The macroscopic thermodynamics of such a polymer is the same as for a polymer with free ends, butfinite polymers are of interest here (c just less than c,). Because the translational, rotational, and internal vibrational motions of the finite polymer molecules will be altered by the anchor or caps, we would expect n (but not c, , a macroscopic property) to be different than in the case of free ends. An equilibrium polymer under an external force F,, as in Figs. 16 and 20, is another example of a completely open system with fluctuations of the above type. The environmental variables are p , F = F,, T. To have finite but sizable polymers, the concentration of free monomers c must be close to but less than c,(F,), as shown in Fig. 20 and Eq. (89). We would expect n to be small because translational and rotational motion is not possible. 2. Polymer between Rigid Barriers We return now to the system shown in Fig. 30 and treated in Section V,A, 1 . The independent variables are p , L , and T, with p determined by

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

95

p = &(c) [Eq. (156)l. This system is open with respect to the number of

subunits N in the polymer; that is, N fluctuates. The appropriate partition function for these variables is the grand partition function E(p,L/T) =

2 Q(N,L,T)eNPIkTT,

(232)

N

where Q is the canonical partition function (given below). This system has normal (small) fluctuations in N . The size of the system is set by L . It suffices to use macroscopic thermodynamic functions below ( N o is of order 2 x lo4). The finite aspect of the polymers dealt with in this article is important only for completely open systems. We can obtain an explicit expression for Q using the well-known relationA = -kTln Q, and alsoA = FL + p N , which follows on integrating Eq. (80). We needA and Q as functions of the variables N, L , and T . Hence, in the second expression forA , we put a(l - lo) in place o f F [Eq. (82)], where I = L / N . Also, in place of p , we introduce Eq. (85). Then we find Q(N,L,T)

= e--NPo/kTp-WlkT

(233)

where W = (Na/2)(1- 1,)’

(234)

W can be interpreted as the work or free energy required to insert n subunits, starting with No = L/lo and ending with N = L / l (recall that n = N - No). This is more obvious if we use Eqs. (92) and (93) to show that

W/kT = yn2/2

(235)

Note that W is the potential of the force F [Eq. (157)l. That is, F = -dW/d(lon). Incidentally, the work per subunit added is W/n

=

ynkT/ 2

=

(kT/2) In (c/cO,),

(236)

having used Eq. (157). It is interesting that this quantity depends only o n c and not on the value of the force constant y . An important property of E is that the probability PX;that the polymer contains N molecules is proportional t o the summand in Eq. (232), that is, to Q ( N , L , T ) e N ” k TFrom . this it follows (Hill, 1960) that the mean value of N is =

and the variance in N is

-

-

kT(3 In

E / dp)L,T

N 2 - N 2 = a;= k T d ( N / a p ) L . T ,

(237) (238)

96

TERRELL L. HILL AND MARC W. KIRSCHNER

where uN is the standard deviation of the PX;distribution. An alternative expression for the variance (Hill, 1960) follows from Eq. (81):

N2 - N 2 = N 2 k T K / L ,

(239)

where K is the compressibility. Using Eqs. (91), (92), and (239), we then obtain -

N2

-

-

N2

=

US

=

l/y

(240)

This same simple result is also obtained from Eqs. (86) (in which we put = L / N ) and (238). This is another reason why y is an important parameter. Using n = N - No, where No = L/lo is a constant, it is easy to see that - - N2 - N 2 = n 2 - n - 2 = u; = l/y (241)

I

In switching from the N distribution to then distribution, we have merely shifted the origin. The mean value of n is given by -loF/kT

=

In ( C / C ~ = y Z

(242)

This is just Eq. (157) rewritten, but here we recognize explicitly that N and n fluctuate and that fi should strictly be used in this thermodynamic equation in place of n . As a numerical example, if c / c $ = 2.72 and y = 0.05, thenyii = 1, E = 2 0 , ~ ;= 20, andu, = 4.5, whereN, is of order2 x lo4.The fluctuations in N are quite small because the polymer is not very compressible. The probability distribution P," is essentially a Gaussian function about the mean value ii with standard deviation u, = l/y1'2.The width of the Gaussian distribution is roughly ?3u,. Thus the P: distribution will be substantially confined to positive values of n if ii 3 3u,, or if In ( c / c z ) 2 3y1/2

(243)

This will usually be the case. In the above example, the Gaussian in n ranges from about n = 7 to n = 33. Stochastic Approach. The above discussion recognizes the fact that n (or N ) fluctuates at equilibrium. Of course n also fluctuates when the system is not at equilibrium, that is, in a transient. An example of a transient is the further addition of subunits after the polymer just comes in contact with the second barrier (Fig. 30). Let P,(t) be the probability that the polymer has n extra subunits at I. The probability distribution P , changes with time and approaches P," as t +. m. P, represents an average over an ensemble of identical systems, or an average for a single system if the same experiment is repeated a large number of times. Figure 38 (see also Fig. 30) shows the rate constant notation we use in following transitions between different values ofn (i.e., we follow the gain

-

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

0.0

n- 1

n

an'

n+l a;+ 1 +PIS+ 1

+Pi

97

...

FIG.38. Kinetic scheme for stochastic study of fluctuations in an equilibrium polymer between rigid barriers.

and loss of individual subunits). The rate constants depend on n because the work of inserting or removing a subunit depends on n . The differential equation in Pn(t)is, from Fig. 38,

+ (an-lCPn-1 - aAPn) + (Pn-lcPn-1 - Pt/Pn)

(244)

There is an equation like this for each value of n . At equilibrium (which is the only case we consider here), each pair of terms in parentheses must vanish separately because of detailed balance. We now consider the rate constants a, and a;+l for the process n F?: n + 1 at the a end of the polymer. If there were no work or force involved in the addition or subtraction of a subunit, these rate constants would be a0 and ad, as in Section V,A,I. We need to correct these rate constants for the work of insertion. The work AW necessary to add one subunit to a polymer already with n excess subunits is given by [Eq. (235)l

AW/kT

+ 1)2/2]- ( y n 2 / 2 ) + l)y/2

(245)

= (ffo/ab)x2n+'

(246)

= [ ~ ( n =

(2n

This is also the value of -loF/kT in Eq. (93) at n + 1/2. Thus ffn/(Y;+1

where y,

e-712

<1

Equation (246) is the discrete analog of Eq. (159). Note that when we deal with insertions explicitly, y becomes a key parameter. For the separate rate constants we write, as in Eq. (161), ffn = ffo,y("+l)f,

=

a;x.(2n+l)(f,-l)

(247)

Similarly, at t h e p end,

p,

=

pox(2n+l)fB

P ; + ~= ~ ; ~ ( l n + l ) ( f ~ - - l )

(248)

These are the rate constants one would use in Eq. (244) in studying a particular transient.

98

TERRELL L. HILL AND MARC W. KIRSCHNER

At equilibrium, from Eq. (244), for n

en +

cr;+lP,'+l = ancP,' PZtl/P,' = a n c / a ; + , =

1 at the a end,

((UoC/(U~)X2""

(249)

= (C/C2X2n+l,

where we have made use of Eq. (158) in the last form. This same result would be found on considering detailed balance at the /3 end. As a check on the above kinetic argument, let us also derive Eq. (249) directly from the equilibrium grand partition function. Because P& is proportional to QeNNkT, we have from Eq. (233) (we put N = No + n and omit the N o factors) P,' -nP,,/kTe-yn2/2,nNkT (250)

-

For

we replace n by n + 1. Then we find p;+,/p,' = e ( w - ~ ) l k T x 2 n + l = (c/c,)x 0

2n+l

9

(25 1)

having used Eqs. (155) and (156).

3. Polymer between Slightly Elastic Barriers Here we examine the fluctuations in N and L in the system already discussed thermodynamically in Section V,A,2 (see Fig. 32). The barriers, it will be recalled, are assumed to obey the Hooke's law relation F = A(Lo - L). The associated barrier free energy function is then A(L - Lo)2/2. The combined partition function of the systempolymer plus barriers, for given N and L , is then [Eq. (233)] -N%/kTe -W/kTe-ML-Lg)Z/2kT

(252)

with

w

=

( N a / 2)[(L/N) -

kI2,

lo = Lo/No

The hybrid canonical partition function, a function of N , Lo, and T, is the sum, or integral, of (252) over L , because L can fluctuate for given N. The size of the system is essentially established by L o , The hybrid grand partition function is then found, as usual, by multiplying by eNMkT and summing over N: z(p,Lo,T) =

N

L

eN(Ir,rro)/kre-W/kTe-A(L--L0)2/2kT

(253)

The probability of the system having particular values of N and L, for given p (determined by c ) , Lo, and T, is proportional to the summand in Eq. (253). We denote this summand by R . In contrast with our use of thermodynamic relationships in Eqs. (237)(240), we shall work here entirely with the probability function a ( N , L )

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

99

(the results have been checked by the other method). We shall treat R as a continuous Gaussian distribution - - in two variables. If we expand In R about N, L, where N and L are the values of N and L that satisfy dln R/dN = 0, then we have, to quadratic terms,

dln R/dL = 0,

(254)

Using In R from Eq. (253), Eqs. (254) give I*- Po=

--

-(a/2)KL/N)2-

(256)

and A(L0-

Z)= U[((Z/R - lo]

(257) Equation (256) is the same as Eq. (85) and Eq. (257) is simply the condi- thermodynamic results tion for mechanical equilibrium. That is, these are - L / N as a function of p. already encountered. Equation (256) provides Equation (257)-gives as a function of L / N and hence as a function of p . - I Finally, and L / N determineN(p). We shall not write out these functions explicitly. To simplify notation, we define a;, miL, and a2 by

Then from In R ( N , L ) [Eq. (253)], we deduce

kTN3/az2, &= kTN2/aZ, at= k T N / ( a + AN)

a; =

(259)

Equation (255) can now be rewritten in Gaussian form as

R ( N , L ) = R(N,Z)exp{-[(N - N)'//2aA] + [ ( N - N ) ( L - L)/&] - [ ( L - E)'/~u:]}

(260)

By completing the square in the quadratic form, the cross term can be eliminated. The necessary integrals are then easy to carry out. We find for the desired variances

100

TERRELL L. HILL AND MARC W. KIRSCHNER

z)

(N - N)(L = NkT/At ( L - L)2 = k T / A P

These fluctuations are all small in magnitude. In Eq. (265), k T y / h f Z is of order unity [Eq. (164)l; the fluctuations in N are about twice as large as in the rigid barrier case [Eq. (240)], but they are still small. Finally, we note from Eqs. (163) and (164) that

-IoF/kT= In (c/cz)= y i i / [ l + ( k T y / h l i ) ] ,

(268)

which differs from Eq. (157). However, both of these equations can be put in the form Fi = (N2 - N 2 ) ( - l o F / k T ) ,

(269)

which is related to the second of Eqs. (93). B. STEADY-STATE POLYMER

1. Completely Operi Systems In studying fluctuations in any polymer system at a nonequilibrium steady state, the only approach available is the stochastic or “master equation” method [as in Eq. (244)l. Partition functions cannot be called upon because they do not exist at steady state. Completely open systems at steady state are those already mentioned in Section VI,A,l: polymers at F = 0 (both ends free or capped, or with one end anchored), or with a moveable force F = F, at one end of the polymer. The stochastic treatment of those polymers is relatively elaborate because finite size effects on rate constants have to be taken into account. The interested reader should refer to Hill (1981b). We shall merely mention here that the final results on polymer size distribution resemble in a formal way those for the equilibrium system (Section VI,A,l). However, as might be expected, some of the parameters have to be redefined to take care of the steady-state conditions. The steady-state treatment has also been extended (Hill, 1981b) to the three-state kinetic cycle case (A, in solution is the third state; see Fig. 2). Polymer breakage would alter the steady-state distribution but not the equilibrium distribution.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

101

2. Polymer between Rigid Burriers We return to the system shown in Fig. 34 and treated in Section V,B, I , which should be reviewed. Here, we consider this system stochastically. The transitions to be taken into account are

. . .n - l a n @n+ I a n + 2 . . . where the total rate constant for n

-+

n

+

(270)

1 is

+ a - z ( n ) + P-2.n) 3 = k+(n)

cbl(n> + P d n )

+ 1 + n is a2(n+ 1 ) + P2(n + 1) + c c l ( n + 1) + P-l(n + 1 )

(271)

and that for n

k-(n

+

1)

The individual rate constants follow from Eqs. (1 18)-( 121) (with (247), and (248): al(n)

=

4

(Y$p+l)f

P1(n) = p y 2 n + l ) f

a 2 ( n + 1) = (&-‘2n+”ff p2(n

+

+

,

Q,

1)

P-l(n + 1)

=

aO_,x‘2n+l)(f Y1)

for v), (273)

= pO_1x‘2n+1)(ff--1) (274)

a - 2 ( n ) = aO_2x(2n+1)(1-f3

P&)

1) = p g x - ( 2 n + 1 ) f g

E

(272)

== pO_2x‘2n+l)(l-f $5

(275) (276)

The definition of x is e - y / 2 , as before [Eq. (246)l. In general, thef’s are all functions of F , and therefore of n , through the relation -loF/kT = y n . More explicitly, for the transitions n d n + 1 via a1and for example,fy(F) would be evaluated using the value n + 1/2 in the above F(n) relation. This is appropriate becausefp is a property of the intermediate transition state between n and n + 1. See also the comment following Eq. (245). Rate constants analogous to (271) and (272) apply to all transitions in the kinetic scheme (270): the value of n is shifted as appropriate. Even in the most general case (i.e., without any simplications in the approach just mentioned), it is easy to obtain the numerical steady-state solution for P, from the scheme (270). This is a consequence of the fact that (270) is a linear kinetic diagram. Because of this, the steady-state solution of the dPn/dt set of equations is formally of the detailed balance type. Thus, the set of equations dP,/dr

=

0

=

+ l)Pn+l - k+(n)P,] + [k+(n - l)P,-1 k-(n)P,]

[k-(n

-

has the “detailed balance” solution

k-(n

+

l)P,+1 = k+(n)P,

(277)

102

TERRELL L. HILL AND MARC W. KIRSCHNER

Because k+(n) and k-(n + 1) are provided at the outset by the details of the particular model used, repeated (seriatim) application of Eq. (278) over the range ofn in which P, is not negligible, plus normalization (Z,P,= l), will determine all the P,,values. With P, available, various averages of interest may then be calculated. Simple Special Case. In general, Eq. (278) cannot be put in a quasiequilibrium form analogous to Eqs. (249) or (251), i.e., in the form P,+l/P, cyZ%+l, where y is some quantity independent ofn . However, in the important special case already used in Eqs. (185)-(192) and (216)(223), Eq. (278) does simplify in this way. In this case

-

k-(n

k+(n) = c ( a f + pp)x(zn+l)fl + 1) = (a&+pp)x-(zn+I)fz

(279)

so that

f'n+1/Pn= (c/cm)yZn+l,

(280)

where Cm =

+ pp),

(a& + P&)/(aQ

y =

XfIff2

(281)

On comparing Eqs. (250) and (251), we see that here

Pn

- (c/cm) ny

nz

(282)

This is the steady-state solution of the set of Eqs. (277) in this special case. If we put this in Gaussian form, using the procedure in Eqs. (254) and (255) (but for only one variable), we find p, e - ( n - ~ ) r / 2 ~ f i (283)

-

where the parameters of the distribution are

[r(h +

= -

u; = nz -

n2 =

In (c/cm)

llr(f1 + fz)

(284) (285)

Equation (284) is the same as Eq. (188) (with E in place ofn) but Eq. (285) is new; the stochastic approach is needed to obtain this result. The contribution from state n to the steady-state NTP flux .I? can be written in two ways [see Eq. (279)l: n +n + ,

+ 1: - 1:

c(aQ+

pp)x(zn+l)fl

~;)~-[Z(n-l)+llf~

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

103

If we multiply either of these by the normalized P , from Eq. (283), and integrate over n (after completing the square), we derive for the mean NTP flux per polymer molecule,

J-F

=

(cup + p g ) ( C / C , ) f z / ( f l + f z ~ e

-YfIf2/2(fI+fd

(286)

This is the same as Eq. (190) except for the last factor, which is usually a small (< 1%) correction arising from the distribution in n. Similarly, if we calculate 7; as the mean rate of loss of monomers from the p end, the contribution from polymers in state n is ppx-[2(n-1)+llf2

-

0 (zn+l)fl C P 1X

If we multiply this by the normalized Pn and integrate over n , we obtain

7;

= J,” (0)(C/Cm)R/(R+k)eYR6/2(fi+R)

(287)

This is the same as Eq. (189) except for the final small correction factor, which is the same as that in Eq. (286).

3. Polymer with a Centralized Attached Force We now extend the discussion in the previous section by including an external force F,, as in Fig. 35. Insertion of a monomer at either end compresses both ends slightly. Equation (178) shows that the relation between n and F is the usual one [Eq. (93)] despite the asymmetry introduced by F,. Consequently, Eqs. (235) and (245) still apply for the work of insertion. The generalization of Eqs. (273)-(276) to the present case is then obtained by replacing E in Eqs. (193)-( 196) by P + l . The general discussion following Eqs. (273)-(276) also applies here except that the new parameters F, and 8 are involved. In the most general case,fl andfg are functions of Fa andff andfl are functions of F, [Eqs. (167)l. For n e n + 1 transitions, F is determined by n + 112, as above, and then Fa and Fp follow from F . Simple Special Case. The usual special case again exhibits quasiequilibrium behavior. We have, instead of Eqs. (279), L + ( ~ )= c(cu;+ p 0 ~ f ~ ) ~ ( 2 n + i ) f i ~ - e f ~ (288) k-(n + 1 ) = (a! + p~r-fi)X-(2n+l)fzy~f2 Consequently, Pn,l/Pn = (C/Cf)y2nf’ Pn (c/c’)nyn2,

-

where c ’ is defined in Eq. (219). The Gaussian form, Eq. (283), then has parameters

n=

[r(f1

+

In ( c / c ’ >

(291)

104

TERRELL L. HILL AND MARC W. KIRSCHNER

and CT; as in Eq. (285). Equation (291) agrees with Eq. (218), with fi in place of n . The small correction factors in y , in 5,”and 52 [see Eqs. (220) and (221)], are found to be the same as in Eqs. (286) and (287). 4. Rate of Label Loss from Polymer As a rather different example of fluctuation behavior, we return to the problem discussed in Section II,B,4: rate of label loss from steady-state polymer molecules with free ends. At t = 0, the subunits of the polymer molecules are labeled whereas the free subunits in solution are unlabeled. As time passes, the number of labeled subunits entering the solution by disaggregation is assumed to be negligible compared to the number of free unlabelled subunits. Consequently, a subunit that adds to a polymer end is always unlabeled, but one that leaves a polymer end may be labeled or unlabeled (when t > 0). Let us consider first the concentration range c, < c < cp in Fig. 8. In this range, the a end is growing and the p end receding. Thus, the a end is “capped” by unlabeled monomers while the p end loses labeled mono= Pz - Plc [Eq. (5511. mers to the solution at the net mean rate -.I, In the following discussion, we consider an idealized single-strand polymer. The detailed stochastic analysis of this system would be applicable to real multistranded polymers if the strands were independent of each other. This, however, is certainly not the case because of intersubunit interactions between strands that tend to smooth out the polymer end “surfaces” and also because of possible diffusion of subunits among ends of strands (Section 11,A). In the treatment below the rate constants used should be considered to be averaged rate constants. This prior averaging is not a strictly correct procedure and may introduce some second-order error. Figure 39 shows a single-stranded polymer at an arbitrary time at which it has N labeled subunits and q unlabeled subunits (at the p end). We can ignore the a end. At t = 0 there are No labeled subunits and no unlabeled subunits ( q = 0). Some of these labeled subunits are lost from the p end as time passes; thus N 5 No. No is a large number (say lo4) whereas q is zero or a positive integer, usually small. N can decrease to N - 1 only when q = 0 ; if q 2 I , the p end is temporarily “capped.” This system can be treated in detail stochastically but here we give a condensed and somewhat intuitive discussion. The activity at the p end can be summarized as follows: plc is the rate of adding unlabeled monomers; p2 is the total rate of losing monomers (of which plc are unlabeled); and pz - plc is the rate of losing labeled monomers. Labeled monomers never return to the polymer, once lost. In effect, then, the kinetics of label loss can be treated as a unidirectional random walk on the integers N with rate constant PZ - Plc for each step N + N - 1 . The

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

o=

0

105

unlabelled

= labelled

N

q

FIG.39. p end of a labeled polymer that is losing monomers to a solution of unlabeled monomers.

temporary cap of the P end by unlabelled monomers (excursions in q with 2 1) is responsible for the effective reduction of Pz to Pz - &c as the rate constant for N + N - 1). Let P ( N , t )be the probability of a polymer having N labeled monomers at r . Because N is large, we treat it as a continuous variable. Then, as is well known for random walks of this type (Hill, 1977a, p. 131), P ( N , t ) satisfies the differential equation

q

This equation has the Gaussian solution

P(N,r) =

1 [2 7ra'( t )]

[

exp -

"- mI2} 2(+2(t)

'

where mean and variance change with time according to

N(t) = No - ( P Z - P1c)r a'(t)= (P2 - &c)t = No - H ( t )

(294)

The Gaussian distribution in N moves toward smaller N and spreads as it moves. Equations (294) apply to one molecule. For M molecules, multiply Pz - P I C by M . The excursions in q to q 2 1 are very limited in extent so q must be treated discretely. Figure 40 shows the rate constant scheme that governs these excursions. This applies to every value of N . Shifting of N with t does not perturb the distribution in q . Consequently, after a brief transient, a steady state is set up among the q values. Because the scheme in Fig. 40 is linear, the steady state has a quasiequilibrium form. Let ps be the steady-state probability of q , irrespective of N value. Then, from Fig. 40, we find

z)z* ( q = 0, 1, . . .) z = PIC/PZ < 1

pq = (1 -

(295)

106

TERRELL L. HILL AND MARC W. KIRSCHNER

F I G .40. Kinetic scheme for Fig. 39 in following fluctuations in q .

The factor 1 deduce

-

z is a consequence of normalization of the p s . We then

In the interval 0 < c < c, in Fig. 8, there would be label loss from both ends of the polymer. The small excursions in q just treated would now occur independently at each end (at the a end, the rate constants a1and a2 replace p1and p2 above). Also, in place of Eq. (294), we have (Hill, 1977a, p. 132)

The Case p2 = picB. At c = cp in Fig. 8, pica = p2.That is, the on and off rates are the same. This case has to be treated separately. Figure 41 shows a sampling of the states passed through, by the p end of the polymer in a hypothetical sequence, as time passes. The mean position of the end of the polymer remains at s = 0, because on and off rate constants are the same, but labeled monomers are gradually lost as a result of fluctuations that

FIG.41. Hypothetical sequence of polymer states at the p end in the special case of the system in Fig. 39 when the mean gain or loss of monomers at this end is zero. See text for details.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

107

temporarily remove all of the unlabeled monomers. These fluctuations became less frequent, because they must be larger, as more labeled monomers are lost. This problem is a random walk on the integerss (Fig. 41), starting a t s = 0 (with completely labeled polymer), and with rate constant Pz for steps in either direction. We are interested in cases with a large number of steps. The mean time between steps is 1/2p2. The number of steps by time t is 2p2t.This is the maximum possible number of labeled monomers lost by t . Chandrasekhar (1943, Eq. 24) practically gives the solution, but in different notation. The probability that the walk has never gone beyond (to the right) a particular positive s value, after time t , is

After a change of variable in the second integral, this can be simplified to

It then follows that the integrand, Qm(f)

= e-m2/4P1/(.rrPzt) 112,

(300)

is the probability (density) that m is the largest value of s that has been reached in a walk lasting a time f . Thus, Qm(f)is the probability that m labeled monomers have been lost to the solution between t = 0 and t . The mean number of labeled monomers lost by t is then -

m

=

lo'"" mQm(t)dm

= 2(p2f/7r)1'2

= 2(P,t/~)~/~ -( el - V

(r

9

pi11

(30 1)

The rate of loss of label, after the transient, is dZ/dt

=

( & / r f'" )

(302)

This approaches zero slowly as t + m. The a end behaves in the same way at c = c, (Fig. 8), but with a2 replacing p2. However, this loss of label to the solution at the a end would be swamped by the much larger steady loss at the p end, at this concentration. As a numerical example of Eq. (301), suppose p2 = 10 s-'. Then the mean number of labeled monomers lost from the p end, per polymer molecule, is 3.6 at t = 1 s, 11.3 at t = 10 s, 35.7 at t = 100 s, and 112.8 at t = 1000 s (16.7 minutes).

108

TERRELL L. HILL AND MARC W. KIRSCHNER

5 . Gain of Label by Polymer from Solution

At the end of Section I1 we mentioned very briefly the case in which an initially unlabeled polymer picks up label from solution under treadmilling conditions. Here we comment on the special case of this kind of system in which, at the polymer end of interest (say the /3 end), the on and off rates are the same. That is, we are referring here to precisely the reverse problem of the one we have just considered: the designation of the two types of monomers in Fig. 41 is simply exchanged. We are interested, then, in the rate at which label from solution penetrates into an end of an initially unlabeled polymer when the end has .zero mean growth. Clearly, the mathematical argument just given applies and Eq. (301) is again the result. However, K? in Eq. (301) now represents the mean penetration of label, in monomer units, by time t . Essentially the same problem is encountered in the more complicated pulse and chase experiments analyzed by Zeeberg et al. (1980). Their Eq. (13), derived in a different way, is the same as our Eq. (301).

VII. Afterword

The recent era of molecular biology has witnessed far-reaching biological implications drawn from the detailed molecular study of important macromolecules. In some of the most notable successes, such as the explanation of genetic mechanisms from a detailed study of the structure of DNA, or the explanations of feedback regulation through a detailed study of allosteric enzymes, important biological properties of complex systems have been deduced from the study of purified components. The cytoskeleton is presently one of the most complex systems under study. However, what looks like hopelessly intricate and individualistic structures in different cells has been reduced recently to the study of a few major structural elements common to all eukaryotic cells, which differ little from organism to organism and presumably are under strong selective pressures. We can identify important biological questions, such as: What regulates filament assembly? What determines the organization of the cytoskeletal filaments? How do the cytoskeletal filaments interact with other organelles and other materials in cells? What role does the cytoskeleton have in intracellular transport and cell motility? How are all these properties integrated into the cell cycle and the programs of development and cell differentiation? Unfortunately the known biochemical properties of actin filaments and microtubules fall short, as yet, of answering all of these

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

109

biological questions. Yet it seems possible that a detailed study of the unique and highly conserved properties of these filaments themselves will hold the key for understanding many of their cellular properties. In this article we have concentrated on one important property of actin filaments and microtubules: their utilization of the free energy of ATP or GTP hydrolysis. To interpret the function of nucleoside triphosphate hydrolysis we have had to consider a number of other properties: the structural and kinetic polarity of the filaments; their interaction at the ends with specific components; and the action of forces on the filaments. To take all of these into account we have reformulated the theory of polymerization and treadmilling on a thermokinetic basis. The major result of this theoretical analysis is to provide some answers to the question: “How might the free energy of ATP or GTP hydrolysis be utilized in actin and tubulin assembly?” We feel that the hydrolysis free energy can be utilized in two general ways. In the first, under the proper circumstances, it can be used to do work. This requires, in general, that the filaments be polymerized between special barriers, which still allow exchange of subunits. Treadmilling, under these conditions, can transport material against a resisting force. In a bioenergetic sense this is similar to the conversion of ATP to work in muscle contraction. When we considered, in addition, the related effect of compression of the filament against barriers on the rate constants for monomer assembly, we found that the assembly reactions are very sensitive to compression. This might be an important method of regulating the length of polymers in the cell. We have also shown that under nonsteady-state conditions polymerization or depolymerization can also move materials, providing that the ends of the filaments can both exchange subunits and retain their attachment. The second way in which ATP or GTP hydrolysis is “used” is to allow the filament at steady state to have different critical concentrations at the two ends. By utilizing two different modes of binding and detachment of the monomer (one in the NTP form, the other in the NDP form), it is possible to make the effective affinity of the two ends at steady state differ. This is an example of the use of nucleoside triphosphate hydrolysis to avoid a thermodynamic requirement at equilibrium. We have discussed how this disparity in the critical concentration can promote the selective stabilization of nucleated filaments in the cell. It can also allow local control over filament assembly. Selective capping of the ends can effectively decrease or increase the effective affinity of the ends for monomer, thereby locally promoting or stopping growth. This use of ATP or GTP free energy is novel and it is likely that it is biologically important. The free energy is being used here to make certain inverse reactions essentially

110

TERRELL L. HILL AND MARC W. KIRSCHNER

irreversible. It is reminiscent of the use of nucleoside triphosphate hydrolysis free energy to increase fidelity of translation or DNA replication by kinetic proofreading (Hopfield, 1974). In addition to the broad biological implications of the polymerization kinetics and nucleoside triphosphate hydrolysis for microtubules and actin, our analysis raises a number of interesting theoretical problems on mechanism of assembly. These arise from the detailed nature in which subunits are added to the ends of the filaments, the heterogeneity in lattice sites, the possible persistence of GTP or ATP deep into the polymer, and fluctuations and stochastic problems. The simplifying assumption that the polymers may be considered to be composed of single strands of subunits may be incorrect, and we have tried to indicate how rejecting this assumption might affect some of the results of our analysis. The more subtle properties of these polymeric systems will be revealed when there is more experimental data. It is a major function of a review of this type to focus on the needs for specific areas of understanding that can come only from experiments. The details of the polymerization reaction, taking into account all possible states and interactions, can be taken into account by Monte Carlo calculations if some realistic values for rate constants and free energies are known. It would then be useful to attempt to discuss the polymerization reaction more precisely. There is also a great need for more steady-state data, under the same experimental conditions, for rate constants, nucleoside triphosphate fluxes, and monomer fluxes. We need to know if there are demonstrable nonlinearities with respect to monomer concentration that might indicate the existence of some potential problems raised here. Much of the discussion has centered on the interaction of structures at the ends of filaments. In some cases they may be nucleating structures, tight caps, or loose caps. In other cases they may be materials that bind to an end of the filament and still allow subunit exchange. The study of nucleation and attachment structures is a very important and active one in the cytoskeleton field. We need to know more of how these structures interact with the filament end, whether they allow for subunit exchange, and whether they interact preferentially with NTP-containing, or NDPcontaining, subunits and hence influence the critical concentration. Ultimately the assessment of function for specific properties of the cytoskeletal proteins will depend on evaluation in vivo. For example, we would like to know what effect will a shift in the treadmilling or nucleotide hydrolysis have on the growth and stability of microfilaments and microtubules in the cell. The answers to questions like these will probably ultimately come from genetics. The answers will require, however, detailed kinetic analysis in v i m and parallel physiological studies in vivo.

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

11 1

Perhaps we can look forward to a similar physiological and structural study for microtubules as has been so successful with hemoglobin. In such a study one might be able to evaluate how physicochemical differences affecting, for example, treadmilling flux, will affect processes in the cell, such as chromosome movement. The study of the filamentous proteins of the cytoskeleton has already yielded important information and uncovered novel biochemical principles. It is an area where theory, in vitro experiments, and in vivo observation can each make an important contribution to what is now still an imperfect picture of how the cell regulates its own morphology. VIII. Appendix 1. Comparison of Rate Constant Notations

We have used the present rate constant notation in earlier papers (Hill 1980a, 1981a,b; Hill and Kirschner, 1982). Some alternative choices of notation are given in Table I. This table should be of use in converting from one system to another. IX. Appendix 2. Fluctuations in Polymer Length Distribution

The expected form of the polymer length distribution PN at a true equilibrium or true steady state has already. been discussed briefly in Sections VI,A,I and VI,B,l. The distribution is of the form PN N " ( c / c , ) rather ~ than the usually assumed form ( c / c , ) ~ Attainment . of the final distribution is very slow (Engel et al., 1977); this in itself leads to uncertainty about the validity of many published distributions. Artifacts

-

TABLE I COMPARISON OF RATECONSTANT NOTATIONS Present article ff1 a-I

ff2 ff -2

P1

P -1 P2

P3

Wegner (1976)

Bergen and Borisy (1980)

Kirschner (1980)

Pollard and Mooseker (1981) @.T

k FT k FD kFD k 7T k P_.T kP,D kyD

112

TERRELL L. HILL AND MARC W. KIRSCHNER

of electron microscopy are another possible problem with many observed distributions. A further complication is that a measured distribution is necessarily based on a finite sample of polymer molecules. The observed distribution, from the finite sample, will in general differ from the true distribution (based on an infinite sample). In this appendix we give the basic equations that allow us to estimate the magnitude of this discrepancy as a function of sample size. Proofs are given elsewhere (Hill, 1981b). Suppose PN is the normalized probability of observing N for a given polymer or system in which N fluctuates. For an infinitely large collection or ensemble of systems, PNwould be the fraction of systems in the ensemble - with N . The mean of the distribution PN is W and the variance is @ N 2 . Now in a particular experiment, suppose only a finite sample M of these polymers or systems is observed, say M = 200 or 300 rather than M = =. In this sample, let the number of systems with k be M k(we use k as the index for the finite sample M , and N as the same index for the “true” or M = x distribution). For the set of numbers Mk (&Mk = M ) , the mean and variance are denoted k and u. Practical questions of interest are: how respectively? much are F and u likely to differ from W and W 2- p, For the first question, the answer is quite simple. The mean square deviation of k from N is This approaches zero as M to above, this becomes

-+m.

(E -

With the polymer distribution PNreferred

N)2

= l/(n

+ I)M

Thus, if M = 200 and n = 5 , the root-mean-square deviation of k from @ is 0.029@. The mean variance U for the finite ensemble is

v = [ ( M - l)/M](N3 -

N2

1

(304)

The correction factor [ ] approaches unity as M + a. The variance in the variance is found after a long calculation, to be

= M-’{(N This approaches zero as M + we find

m.

W)4

-

[ ( N - N)2]2}

(306)

Using the polymer PN distribution above,

(2- U2)/ij2 = 2M-’(n

+ 4)/(n +

1)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

113

For example, ifM = 200 and n = 5 , the root-mean-square deviation from V is 0.122U = 0.122 (Fz - N2) (this corrects the numerical factor in Hill, 1981b). A slightly different quantity is [v -

(F-

P ) ] 2

(307)

This is the mean-square deviation of u from the true variance. This is easily shown to be the same as 2 - U 2 in Eq. (305) except for a small added term

M-"(N

-

N)2]2,

(308)

which makes no contribution to Eq. (306). X. Appendix 3. Persistence of NTP at Polymer Ends

We have been using Fig. 2A as the basis for the biochemistry occurring at the polymer ends. In the case of microtubules, at least, it is possible that AT(p), as well as AD(p), is a significant species at and near the polymer ends (Carlier and Pantaloni, 1981). We have been assuming, as is conventional, that the rate constant for AT(p) + AD(p) + P is fast compared to the rate constants for AT(s) + AT(p) and for AD(p) + T +AT(s) + D. If this is not the case, the species AT@) must be included in the analysis. To treat this problem exactly would no doubt require Monte Carlo calculations. These calculations would be straightforward but a very detailed model would have to be specified as a prerequisite. Such calculations may prove to be worthwhile in the near future but they do not seem to us to be justified at present, because of lack of experimental information about which model to choose. As an alternative, our plan in this appendix is to adopt an explicit but simplified model for microtubules, based to a considerable extent on that proposed by Carlier (1982), and to formulate an analytical solution in this case. This model is sufficiently complicated to bring out the main features of the problem. It is also complicated enough so that our analysis will have to be based on two approximations in the treatment of the model. It seems likely that even for this simplified model a Monte Carlo approach would be necessary in order to obtain exact results. a. The Simplijied Model. We assume, as shown in Fig. 42, that for aggregation-disaggregation purposes the 13-strand microtubule is comprised of five helices, labeled I, 11, . . . V. The two extra subunits (tubulin dimers) outside the numbered 13 strands are repeats, to show the periodic boundary conditions. We assume that new subunits can add to the particu-

114

TERRELL L. HILL AND MARC W.KIRSCHNER

/

a end

E /

‘5

FIG.42. Model of microtubule end, of 13 strands, considered as a 5-start helix with no vacancies or overhangs. See text for details and notation.

lar illustrative configuration of the a end, shown in Fig. 42, only at positions I, 11, IV, and V, where two new nearest-neighbor interactions will be formed. Addition at I11 is not allowed (only one neighbor). Also, departure of subunits from this configuration can occur only from positions A, B, C, and E, where two neighbor interactions are broken. Departure from D is not allowed (three neighbors). Addition is not allowed in the dotted position (one neighbor) nor is departure possible for the subunit below the dotted position (three neighbors). With these simple rules, all helices are intact (no vacancies) and there are no overhangs (as there would be if, for example, another subunit were at position 111). The two-state kinetic cycle in Fig. 2A now becomes a three-state cycle (Fig. 43). We assume that the on and off rate constants in Fig. 43 may have different values, depending on the state of the immediate neighbor in the same helix. We need, therefore, to introduce more general rate constant notation. For concreteness, let us refer to the sequence B , B’, B, I1 in Fig. 42. For the process A,(s) z 3 AT@): the on rate constant at I1 is designated aITif B is AT and am if B is AD;and the off rate constant for B is if B’ is AD. Similarly, for the process A D @ ) if B’ is AT and AT(s): the off rate constant for B is aZT if B’ is A T and aZD if B’ is AD; and the on rate constant at I1 is a-zTif B is A T and if B is AD. Finally, we assume that the rate constants for AT@) e AD(p) + P are

*

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

115

Solution AT(s)

Polymer FIG. 43. Three-state NTPase cycle needed if the hydrolysis step on the polymer, &(p) + &,(p), is not fast compared to other transitions.

K’(f0rWaI-d)and K’_(backward)for an end subunit of a helix (e.g., B) and are K(f0rWard) and K-(backward) for any interior subunit of a helix (e.g., B’ and B”). In this case, the state of neighbor or neighbors in the helix does not matter, only the number of neighbors in the helix (one neighbor for B, two for B’ and B”), These K rate constants have no connection with K+ and K- in Eq. (72). To summarize: the a and K’ transitions (details above) occur only at the very end of a helix, but the K transitions occur in the interior. With the above assumptions, the rate constants are not all independent. Corresponding to Eq. (21), we have the following relations involving the thermodynamic force:

/

a 1 T K ’ ffw ff -~TK’_(Y-~T = fflDK’ffzD/a-1DKh!-2D

=

eXTlkT eXTikT

(309) (3 10)

In the first equation, for example, a subunit goes through an NTPase cycle at position I1 with B in state A,; in the second equation, B is in state A D . From Eqs. (309) and (310), we then have the necessary relation ( a l T / a-lT)(ffZT

/ a-Zr> = ( a l D /

ff-lD)(a2D/

Q-ZD)

(311)

The quantities in parentheses are equilibrium constants [see Eq. (18), for example]. If we consider the two transition sequences, at the end of a helix, *

*

*

AT

+

. . . AT+

*

. . ATAT + . . ADAT . A D + . ADAT

* .

*

*

*

116

TERRELL L. HILL AND MARC W. KIRSCHNER

which have the same initial and final states, we can derive a second relation among rate constants: (CX~T/CX-~T)(K/

K-)

= (~ID/~-ID)(K'/KL)

(312)

= (cu~r/a-n)(~' /

(3 13)

Then, from Eq. (311), ( a m 1(

Y - ~ ( K /K - )

This relation, which is not independent, also follows on comparing the sequences "'A~AD+"*ADAD+'*'AD . . . ATAD+ . ' . AT+ ' . . AD As an example, if we assume as in Carlier (1982) that ~ ~ D / c x - ~isD practically zero, this has implications for the other equilibrium constants in Eqs. (311) and (312) if self-consistency is to be maintained. b. Approximations to Be Used. With the above assumptions, the five helices are almost but not quite independent of each other in their kinetic activity. The remaining type of interaction between successive helices is illustrated in Fig. 42: helix I11 cannot add a subunit because it would then overhang helix IV; and, for the same reason, helix IV cannot lose a subunit. Thus, in the configuration shown in Fig. 42, helices I11 and IV "interact" with each other. Despite this kind of interaction, as an approximation we treat the five helices as independent. Thus, because in any case the helices are equivalent, it suffices to study the kinetics of a single helix. In a Monte Carlo treatment, the complete 5-helix microtubule could be handled, without making this approximation. Actually, if the subunits have only one state in the polymer (as in Sections 1-VI), the above complication can be corrected for exactly and no approximation is involved in using a single helix for a microtubule with no vacancies and no overhangs. In the Sections I-VI case, with only AD in the polymer, every allowed surface configuration (as in Fig. 42) has the same free energy and is to be given equal weight in averaging. This follows because, in this simplified model, every surface configuration is missing 5 horizontal neighbors (I, * , V) and 13 vertical neighbors. Different configurations then influence the kinetics only through variations in the number of on and off sites present. All sites present are equivalent (for each type, on or off). Thus the configuration in Fig. 42, which can be represented by I( 1)3(1)6(2)3(1), (3 14) has four on sites and four off sites, already enumerated above. The numbers in parentheses here give the vertical steps (in subunits) and the ther

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

1 17

numbers give the horizontal steps (left to right). Here, 13 (horizontal) is partitioned into 4 positive integers (1,3,6,3) and 5 (vertical) is also partitioned into 4 positive integers (1,1,2,1). By considering all possible partitions of 13 and 5 into not more than 5 integers, and all permutations of these partitions that lead to distinguishable configurations, we have found a total of 482 surface configurations. Of these, 1 has 1 site (of each type), 24 have 2 sites, 150 have 3 sites, 208 have 4 sites, and 99 have 5 sites. The average number of sites per configuration is 3.79. Thus, if the kinetics of a single isolated helix is used to represent one-fifth of a 5-helix microtubule, each end subunit is actually available for an on or an off transition only a fraction of the time: 3.7915 = 0.758. If this model (no vacancies, no overhangs) were correct, this factor would have to be used to convert an operational rate constant into a microscopic rate constant for an actually available site. The above considerations apply strictly only to the polymer with onestate subunits (Sections I-VI). But when we use on and off rate constants below for a single isolated helix with two polymer states, the factor 0.76 may be considered to be imbedded in these rate constants as an approximate way of correcting for missing sites when we replace one-fifth of the full microtubule by a single helix. So far we have considered the approximation of representing the 5-helix microtubule by a single helix (or, to be more precise, by five independent helices). Even in treating a single helix analytically, we have to introduce a further slight approximation. We number the subunits of the single helix, 1 , 2, . . . , n , . . . , starting with the end subunit (thus B = 1, B’ = 2, etc., in Fig. 42). We let pn be the probability that thenth subunit is in state AT. Thus 1 - p n is the probability that the nth subunit is in state A,. In Sections I-VI we took pn = 0 for all n . We shall assume, as an approximation, that probabilities for neighboring subunits in a helix are uncorrelated. For example, we shall assume that the joint probability that subunit n is in state AT and subunit n + 1 is in state & is given by p,(l - p n + J . This would be true at equilibrium, but not at steady state (Hill, 1977e, p. 552) or in a transient. We are, of course, primarily interested in steady states, where the approximation is probably a very good one. A Monte Carlo treatment would avoid this approximation. c. Analytical Treatment. We shall write, below, equations for dp,/dt (n = 1 , 2, . . .) for a single helix. We are interested here in the steadystate case only, where dp,/dr = 0. The same equations apply to a full microtubule with rate constants that are five times as large. Because subunits go on and off of the helix end, we shall be following the state of particularpositions relative to the end rather than particular subunits whose positions change stochastically.

118

TERRELL L. HILL AND MARC W. KIRSCHNER

For position 1 in the helix, dp,/dr

=

(1 - Pi)aiDC - pla-2l.c + P2(1 - Pl)azT - (1 - P2)Pla-lD -IKL(1 - PI) - K’PI

(315)

The terms on the right correspond to the different ways in which p1 can change in value. The first term refers to the addition of a A T onto a AD in position 1 and the second term to the addition of a AD onto a A T in position 1. The sign is positive in the first case and negative in the second because in the first case dpl increases (AD + AT at position 1) and in the second case dpl decreases (AT + A D at position 1). The third term refers to the loss of A D from * . .ATAD(thus bringing the A T to position 1) and the fourth to the loss of A T from . -ADA,. The remaining two terms are obvious. There is no contribution to dpl when, for example, AT adds onto A T or A D departs from . *ADAD, etc. For position 2,

-

dP2/dr

=

(1 - PZ)J+T - P2J+D + P3(1 - P2)J-D - (1 - P ~ ) P ~ J - T K-(l - P2) - KP2,

(316)

where J+T

J-T

~ ~ ( a +i QI-ZTC), ~ c

Pla-lT + (1 - Pl)a2T,

(1 - Pl)(alDc + ~ - z D C ) Pla-lD + ( l - Pl)%D

J+D J-D

(317)

*AT,J+D refers to The quantity J+T is the rate of adding A T or AD onto adding AT or AD onto . . . AD, J-T refers to removing AT or AD from * * *ATX, and J-Drefers to removing A T or AD from . . .A&, where X is A T or A D . The first four terms on the right of Eq. (316) arise, respectively, from the following processes: *

*

‘ADAT+-, *

1

*

ATAD+,

*

*

*

ATADX+, .

*

*

ADATX+

where +- refers to on and + refers to off. For position n , with n 2 3, dP?l/dr

+ J+D) + b n + l - PdLP2J-T P2)J-DI + K-(l - Pn) - KPn

= (Pn-1 - Pn)(J+T

+ (1

-

(318)

In this equation, J+T+ J + D is the total rate of adding subunits; when a subunit adds, positionn - 1 becomes position n . Similarly, [ ] in Eq. (318) is the total rate of losing subunits; loss of a subunit converts positionn + 1 into position n . Equations (319, (316), and (318) determine the kinetics of the helix (actually, of the mean behavior of a large ensemble of such helices). At steady state, we put dpn/dt = 0 (n = 1, 2, . . .). These equations can be solved numerically, at steady state, as follows. With all parameters given,

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

1 19

we guess a value of p1 and then calculate p 2 from Eq. (315). With p1and p2 available, p 3 is calculated from Eq. (316). We can then find successively, p4, p5, . . . in the same way from Eq. (318). The values of p n should converge, for large n , to K - / ( K + K - ) . Convergence is extremely sensitive to the choice of p l ; the true value of p 1 is found by adjusting the initial guess until convergence is obtained. To establish contact with our previous notation, summarized in Eqs. (13214 1351, we note that here Jai

Jo,

+ (1

= plalTc = pZ(1 -

-(I Ja = Jai

-

-

Pl)aZT

-

p1)alDC

+ (l

Plb-2DC

Ja, = (J+T

-

+ J-D)

- (1 - P 2 ) P l a - l D - Pla-ZTC

Pflla-lT

p2)(1

-Plb2D

-k (1 -

- [PEJ-T

pZ)J-D]

(319) (320) (321)

There would of course be analogous expressions at the /3 end of the helix. Jp would not be linear in c because p 1 and p2 would in general be functions of c‘. Equations (133)-( 135) still apply. Treadmilling at constant length would occur at a value c = c , such that

J, and

J,,,,

=

0,

J,

= -Jp

=

J1 =

JI”,,

J2 =

JY

(322)

There is no simple expression for c, but it can be located graphically as before (Figs. 5 and 6). If we add Eqs. (315), (316), and (318) (all n 2 3), at steady state, we obtain after cancellations,

+ n=2 m

Jp’

K’PI

=

- KL(1 - pi)

[K/(K

-k

K-)]J,,

-k

2

[K P n -

[K-/(K

+

K-(1

-

p,)]

(323) (324)

K-)]Jaz

The quantity J‘,*)is the net total rate of the hydrolysis step (Fig. 43), R T ( p ) &(p) + P, at the a end, at steady state. The first term on the right of Eq. (324) is the net rate of adding AT(p) from solution, at the a end, multiplied by the fraction of this that ends up as A D @ ) and thus contributes to Jp’. The second term has an analogous interpretation if we write J,, = -(-.I In, the ,) special . case J, = 0, J,, = J,, , Eq. (324) becomes J p ) = JU1 = J,*. In this case each step in the GTP cycle at the a end (Fig. 43) has the same flux, which is equal to the rate of GTP hydrolysis at t h e a end. At the p end, where K and K - are the same as at the a end (this is not true, in general, of the other rate constants), we have +

JL” =

[K/(K

-k K - ) ] J p ,

+ [K-/(K

-k K-)]J,,

(325)

On adding Eqs. (324) and (325): J‘,*’

+ JL’’

J,

= [ K / ( K -k

K-)]J1

-k

[K-/(K

+ K-)]Jz

(326)

120

TERRELL L. HILL AND MARC W. KIRSCHNER

At c = cm,we have from Eq. (322), J,

= J1 = J z =

J;,

(327)

as we should expect. The three steps in the GTP cycle, including activity at both ends, have the same flux. d. Special Cases. If we consider KI- and K- to be negligible (that is, that the reverse of the hydrolysis step does not take place), which is no doubt an excellent approximation, Eqs. (315), (316), and (318) at steady state have a relatively simple solution of the formp, = p z a n - 2 ,for n 2 2, where a is a constant whose value is to be determined. With K- = 0, we have here p , + 0 for n + m. Thus a < 1, to achieve pn + 0 for large n . If we substitute the abovep, in Eq. (318), we obtain, for any n 2 3 ,

0

=

(1 - ~ ) ( J + T+-

J+D> -

a(1

-

a ) b i , Pzl -

Ka,

(328)

where [ I is given in Eq. (318). Because J+T and J + D depend on p l , this provides us with one relation between pl, p z , and a . Similarly, introduction of p3 = pza in Eq. (316) gives a second relation between pl, p z , and a . Finally, Eq. (315) connects p1 and p z . Thus, these three equations can be solved for pl, p z , and a . Instead of pursuing this relatively general case further, we turn to a much simpler example that readily yields closed expressions and numerical results. Here we use a one-way cycle (all reverse rate constants are assumed negligible), and we take ff1T

= (Y1D

ffZT

ff1,

= ffZD

(Yz,

K'

= K

(329)

That is, the rate constants do not exhibit the differential nearest-neighbor effects that we included at the outset. Equations ( 3 1 9 , (316), and (318) then become, at steady state, 0 0 0

(1 - Pi)aiC + Pz(1 - Pi)% - K P I (330) (pi - pz)aic (P3 - Pz)(l - Pi)% - K P z (33 1) = h - 1 - Pn)alC + (pn+1 - p,)(l - ~ J a z - KPn ( n 2 3) (332)

=

=

In this case, Eq. (331) can be included in Eq. (332), if we take n z 2. Equations (330) and (332) then have a solution of the form p, = plan-1 for n 2 1. If we put p z = p,a in Eq. (330) and solve for a , we find a = [KPl - ( 1

- Pl)alcl/Pl(l - PlbZ

(333)

Equation (332), on substituting p1an-l for p,, yields a second relation between a and pl: pi = 1

{[.a

- (1

- a)ff1C]/U(1

- U)O!z}

(334)

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

12 1

Elimination of a between Eqs. (333) and (334) then gives p1as a root of the cubic equation

For very large is

K,

p1 =

ff,C/K.

The next higher approximation, for large

K,

After p1is found from Eq. ( 3 3 9 , a follows from Eq. (333). Then p n can be obtained for any n from plan-l. Incidentally, the corresponding cubic equation in a is +

a2( K

- 2ff1C) ff

1c

+

a

(ff1Cff~

-

K2

-

ff1CK

-

(YlCff2

)+

f f ~ K

=

0

(337)

For very large K , a = f f l c / K . Thus p n = ( f f l c / K ) R , which approaches zero rapidly with n . The quadratic approximation, for large K , is

e. Numerical Example. In the special case just discussed, we have calculated J,(c) for a2 = 1 and K = 1, 10, and x (i.e., using dimensionless rate constants). Thus, we are considering cases in which the hydrolysis rate is equal to the rate of subunit loss ( K = I), substantially larger ( K = lo), or infinitely fast ( K = m) so that all subunits on the polymer are in state AD. These J,(c) curves are presented in Fig. 44. For this example, J,

= f f 1 C - (1 - p1)az = ff1C -

1

+ p1(c)

(339)

When K = =, we have p1= 0 and J , = alc - 1 . This is our usual two-state cycle flux, with a2 set equal to unity. This flux is shown as the lower straight line in the figure. When K = 10, J, is practically a straight line with a slightly increased slope but with the same intercept. J, is increased when K = 10, compared to K = m, because the off rate is reduced by the factor 1 - p1 (i.e., the probability that the end subunit is in state AD). But because p1 c [Eq. (336)], this effect appears in the slope of J, rather than in the intercept. The difference between the two lines in Fig. 44 is p l ( c ) [Eq. (33911. The effect of the new polymer state A T on J , is much larger when K has the same magnitude as a2 and ale: A&) survives long enough to be as important at and near the helix end as R,(p). This enhanced effect can be seen in the K = 1 case in Fig. 44. Again the difference between J,(K = 1)

-

122

TERRELL L. HILL AND MARC W. KIRSCHNER

1

J,

0

1

L FIG.44. Numerical illustration of the effect of AT persistence at and near the polymer a end on Ja, for different values of the hydrolysis rate constant K . See text for further explanation.

TABLE I1 PENETRATION OF & INTO ff ENDOF MICROTUBULE AT J , = 0

P1

PZ P3 PI

PI

PS P7

PS

0.4142 0.1213 0.0355 0.0104 0.003 1

0.0009 0.0003 0.0001

0.0844 0.0067 0.0005

MICROTUBULE AND ACTIN FILAMENT ASSEMBLY-DISASSEMBLY

123

and J,(K = x) is pl(c). In this case Eq. (336) is inadequate and p 1 is not proportional to c. We see from the figure that penetration of AT into the polymer end can also contribute to the nonlinearity of J, and J p . A few further numerical details may be of interest. In the case K = 10, we find J, = 0 at a I c = 0.9156 where p1 = 0.0844 and a = 0.0791. In the case K = 1 , we find J , = 0 at a1c = 0.5858 where p 1 = 0.4142 and a = 0.2929. Successive pn values in these two cases are given in Table 11. The penetration of A, into the end of the helix is more extensive when K = 1 , as expected.

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Erickson, H. P. (1974). J . Cell Biol. 60, 153-167. Ferrone, F. A., Hofrichter, J., Sunshine, H. R., and Eaton, W. A. (1980). Biophys. J . 32, 361-380. Heidemann, S. R., and McIntosh, J. R. (1980). Nature (London) 286, 517-519. Heuser, J., and Kirschner, M. (1980). J. Cell Biol. 86, 212-234.

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