A theory of linear and helical aggregations of macromolecules

J. Mol. Biol. (1962) 4, 10-21

A Theory of Linear and Helical Aggregations of Macromolecules FUMIO OOSAW A AND MICHIKI KASAl

Institute for Molecular Biology and Department of Physics, Faculty of Science, Nagoya University, Nagoya, Japan (Received 4 August 1961) A theory of the helical aggregation of macromolecules is presented in comparison with simple linear aggregation. The thermodynamic analysis of the equilibrium distribution of monomers and linear and helical aggregates shows that the transition from dispersed monomers to helical aggregates takes place as a condensation phenomenon. When the concentration of macromolecules is increased, the helical aggregates begin to appear at a critical concentration determined by the solvent conditions. Above this critical concentration very long helical aggregates coexist in equilibrium with a constant concentration of dispersed monomers (and a small amount of simple linear aggregates). Theoretical analysis of the process of the helical aggregation is also made, and the relation between the aggregation rate and the monomer concentration is investigated. The theoretical results are compared with experimental data obtained for various kinds of proteins. Particularly, the equilibrium and kinetic features of the globular-to-fibrous transformation of the muscle protein actin are found to be explained reasonably by assuming that this transformation is a helical aggregation. That is to say, F-actin can be regarded as a helical aggregate of G-actin. Finally, the possible functions of intermolecular superstructures, such as helical aggregates, are discussed.

1. Introduction Recent studies have shown that various biological macromolecules form intramolecular helices, which play an important role in their physiological functions (Doty, 1959). On the other hand, up to the present, nobody has discussed systematically the possibility of intermolecular helical structures. We can find, however, that experimental results suggest the formation of intermolecular helical aggregates by various biological macromolecules, in vitro and in vivo (Waugh, 1959). The purpose of this paper is to present a simple theory of the helical aggregation of macromolecules and to compare the theoretical prediction with experimental results. In linear polypeptides it has been found theoretically and experimentally that the transition from intramolecular helix to random coil takes place like a solid-liquid transition or like a melting phenomenon (Zimm, 1959). In the case of helical aggregation it is found that the transition from dispersed molecules to helical aggregates takes place like a gas-liquid transition or like a condensation phenomenon. The distribution of macromolecules in the equilibrium state of solution is determined by the total concentration of molecules and the condition of the solvent. Under suitable conditions of the interaction parameters of macromolecules it is seen that the helical aggregates appear only above a critical concentration of macromolecules which is 10

THEORY OF AGGREGATION OF MACROMOLECULES

11

determined by the solvent conditions. Above this critical concentration very long helical aggregates coexist in equilibrium with a constant concentration of dispersed macromolecules (and a small amount of linear aggregates). Therefore, this aggregation phenomenon can practically be regarded as a kind of condensation phenomenon. As a typical example of the fibrous aggregation of proteins we have the G-F transformation of the muscle protein actin (Szent-Gyorgyi, 1951; Straub, 1942; Mommaerts, 1952; Tsao, 1953). According to the experimental analysis made in the authors' laboratory (Oosawa, Asakura, Hotta, Imai & Ooi, 1957, 1959), the general features of the G-F transformation of actin fit very well the theoretical conclusion derived under the assumption of the helical aggregation of globular actin molecules. We do not have sufficient examples of the thermodynamic analysis of the association-dissociation equilibrium of proteins other than actin. We have, however, some examples in which kinetic analysis and crystallographic or ultrastructural analysis have shown the possibility of helical aggregation of proteins, if we apply a generalized definition of "helical aggregation". Similarly to the helix-coil transition of linear polypeptides, a transition of helical aggregates to linear aggregates can happen. Each macromolecule in the helical aggregate is bound to four neighboring macromolecules, usually by two kinds of bonds. By breaking either kind of bond between unit macromolecules in the helical aggregate we can form a linear aggregate. The transition between various kinds of the aggregates may possibly have some physiological meaning in biological systems, for example, in muscle (Szent-Oyorgyi, 1957; Oosawa, Asakura & Ooi, 1961).

2. Theory (a) Equilibrium analysis Let us consider a solution of macromolecules which have the ability to make linear aggregates (linear polymers) by end-to-end association. In the equilibrium state the solution contains dispersed macromolecules (monomers) and linear polymers of various lengths (various degrees of polymerization). If we neglect the interaction between monomers and polymers, the following mass action law is established between the number concentration of monomers -\ and that of dimers '\2:

'\2 = a1'\~

(1)

where a1 is the equilibrium constant of dimerization. Similarly, between (i + I)-mer and i-mer: (2)

When a i is independent of i, or when the binding free energy of monomer to i-mer is independent of i, the equilibrium number concentration of i-mer is given by (3)

as a function of equilibrium constant a ( = ai) and monomer concentration ,\ ( = '\1)' At given total number concentration of macromolecules n, we have

o;

(4)

ia-1(aA)i = a-1(a'\)/(I-a,\)2

(5)

n= ~ i=l

when equation (3) can be applied,

n=

~ i=l

12

F. OOSA W A AND M. KASAl

By this equation we can determine the concentration of monomer at given n. From this value of >. the concentrations of i-mers are calculated by (3). In a linear polymer each monomer (except the end monomers) is bound to two other monomers and usually all bonds are of the same chemical nature. In the helix formed by a linear polypeptide each amino acid is bound with foar amino acids by two kinds of bonds, one being the primary bond with neighboring amino acids in the polypeptide chain and the other the hydrogen bond with the third preceding amino acid along the chain. Similarly to such an intramoleoular helix, imagine an aggregate in which each monomer macromolecule can make two kinds of bonds with other monomers. Then we can have various types of two- or three-dimensional

o:;:!: CO ~ CfX) ~CJ:XX):;:!:Cf:XXX) ~ ~t

H

&~6?J

~t

~OO

FIG. l(a). Schematic diagram of equilibria between monomer, linear polymer and helical polymers.

FIo. l(b). Helical polymer (left) and linear polymer (right).

polymers. As a special case, when on account of some steric effect each monomer in a linear polymer has the capacity to make a bond with another monomer at a regular distance along the linear chain, we have a helical polymer as shown in Fig. 1. For example, in the helical polymer having three monomers per turn each monomer is bound with neighboring monomers along the linear' chain and simultaneously with the third preoeding and succeeding monomers. In this oase the shortest polymer which can have a helical struoture is composed of four monomers. This shortest helix is made by attaching a monomer to a trimer having a speoial steric structure favoring the simultaneous formation of two bonds between the fourth monomer and both end monomers of the trimer. Such a trimer of a special conformation has a lower entropy or a higher free energy than the ordinary linear trimer. Therefore, if we consider an equilibrium solution containing both linear polymers and helioal polymers, the number concentration of the special trimer >'3h is related to that of the ordinary linear trimer >'3 by Aah = 8.\3 (6)

THEORY OF AGGREGATION OF MACROMOLECULES

13

where 8 = exp ( - of' j kT); of' is the free energy increment necessary for the special conformation. If we write the chemical constant of the equilibrium between the fourth monomer and the special trimer as b, the number concentration of the shortest helical polymer composed of four monomers A4I1 is given by (7)

Since the fourth monomer in the helix is bound with two monomers, b is usually larger than a, and ajb is expressed as exp ( - of jkT) by using the free energy of of the second bond characteristic of the helical structure. For the further growth of the helical polymer by attaching monomers to the helical nucleus we can assume the same chemical constant b. Thus the equilibrium number concentration \n of helical polymers composed of i monomers is given by (8)

where a = s(ajb)3. In the solution containing various linear and helical polymers the relation n = A+nl+nh n1 =

~

i=2

ia-l(aA)i,

~

=

~

ia-1a(bA)i

(9)

i=3

or an = aA/(l-aA)2+ a[bA/(1-bA)2-(bA+2b 2A2)] must be established. Constants a, a, and b are functions of temperature T and other conditions of the solvent. By this equation we can calculate the concentration of dispersed monomers A at given values of total concentration n. The first term on the right-hand side of the equation gives the total number A+nl of dispersed monomers and monomers forming part of linear polymers. The second term gives the total number ~ of monomers forming part of helical polymers. Before analysing this equation we must remark that the value of constant a is usually very much smaller than unity. If we assume that the free energy of of the bond for forming the helical structure is of the order of 5 kcaljmol, the value of ajb becomes 10-2 • If the free energy increment for forming the special trimer of' is of the same order, the value of s is also of the order of 10-2 • Then a becomes 10-8 • Such a small value of a accounts for the rarity of nucleation of helical polymers. Now, we follow the change A with increasing concentration n of macromolecules. For very small values of concentration n, monomer concentration A increases in parallel with n and then a very small number of linear polymers oflow polymerization degrees (dimers, trimers, etc.) begins to appear in the solution. If the second term of equation (9) is not taken into account, with increasing total concentration n the concentration of linear polymers increases continuously and A has to tend to a-I. Actually, the second term of (9) is negligible because of the very small value of a as long as A does not approach b-l . Since, however, b is larger than a, A must reach b-l before it tends to a-I. When A approaches b-l very closely, the second term of (9) cannot be neglected however small a may be. In other words, helical polymers begin to appear when the total concentration n approaches very closely the value given by (10)

14

F. OOSAWA AND M. KASAl

When n exceeds this value nc' the value of ,\ must approach b-1 unlimitedly and the second term of (9) increases with increasing n. On the other hand, the first term of (9) is kept nearly constant because the increment of ,\ is very small. That is to say, above the critical concentration given by (10), with increasing total concentration only helical polymers increase under constant concentration of dispersed monomers and small linear polymers, as shown in Fig. 2.

6 1.0

__ -,-

-------_....

5

4 3

2

o

5

10

15

an

FIG. 2(a). Physical features of linear polymerization. Ordinate: monomer concentration in equilibrium with linear polymers (continuous line) and number average degree of polymerization (dotted line). Abscissa: total concentration of macromolecules.

FIG. 2(b). Physical features of helical polymerization. Ordinate: monomer concentration (and linear polymer concentration) ,\ and helical polymer concentration nh, and number average of degree of polymerization
Such a phenomenon resembles gas-liquid condensation. Dispersed monomers (and linear polymers) correspond to gas molecules and helical polymers correspond to liquid. When the number of gas molecules (density of gas) is increased gradually, at a critical density a liquid phase begins to appear and with further increase of the total number of molecules the amount of liquid phase increases under a constant density of gas giving the saturated vapor pressure. In the present case the condensation of monomers into helical polymers is not completely discontinuous because CT takes a finite value; but in practice for very small values of CT it can be regarded as a kind of phase transition.

THEORY OF AGGREGATION OF MACROMOLECULES

15

To understand the sharpness of the transition, we estimate the average degree of polymerization of helical polymers. It is given by (11)

Near the critical point n c we obtain from (10) (~>1

= (Ela)l (alb)! where E = (n-nc)/nc' When alb = 10-2 and a = 10-8 , (ih > 1 = 300 for

<

(12) E

= 10-1 and

10-2 .

i h > 1 = 100 even for E = Very near the critical point the total number of helical polymers is very small, but even under such a condition the average degree of polymerization of helical polymers is very large. Dispersed monomers and short linear polymers coexist in equilibrium with very long helical polymers. With increasing values of the constant a the transition becomes gradual and the average degree of polymerization of helical polymers becomes small. In Fig. 2, the relations of various quantities to the total concentration of macromolecules in helical polymerization are compared with those in simple linear polymerization. So far we have considered only completely linear polymers and completely helical polymers. When, however, a is not so much smaller than b and a is not so small, various polymers containing both linear and helical parts in themselves can be formed. The general treatment of these various polymers can be developed according to the method employed in the theory of helix-coil transition of linear polypeptides. If only completely linear and completely helical polymers are taken into account, the total concentration of i-mers, summing up linear and helical polymers, is given by (13) where Ai means the average activity of monomer in the i-mer, and depends on the polymerization degree i. Taking into account the polymers having helical and linear parts in themselves, we can derive the generalized expression of Ai for large values of i by the matrix method and when a is smaller than 10-1 we can use the approximate formula: (14) where constants a', b', and a' are functions of a, b, and a. This formula is equivalent to that given by Zimm & Bragg (1959). The first term of (14) gives the concentration oflinear-rich i-mers and the second term gives the concentration of helical-rich i-mers. Since (14) has a form similar to (13), the polymerization equilibrium may still be described in all its features as a condensation phenomenon. (b) Kinetic analysis Here, on the basis of the same model as above, a kinetic analysis is made of the process of helical polymerization caused by a sudden change in the solvent condition of a monomer solution. Let us denote the number concentrations of monomers, linear i-mers, and helical i-mers at time t as A(t), ,\(t), and "-ih(t) , respectively. The rate of growth of helical i-mers to helical (i+l)-mers can be expressed as k+ A(t) Aih(t) (for i ~ 3); and the rate of detachment of monomers from helical i-mers can be expressed as k_ \ll(t) (for i ~ 4). Corresponding to the previous assumption that the binding constant b is independent of i, it can be assumed that kinetic constants k; and k: are independent of i because k+lk_ = b. The transformation

16

F. OOSAWA AND M. KASAl

rate of ordinary trimers to the special trimers (nuclei of helical polymers) is given by k~ A3(t) and the reverse transformation rate is given by k_ A3h(t), where k~/k'- = 8-1. If we neglect the direct transformation of linear i-mers of i ~ 4 to helical i-mers of i ~ 4 and also the reverse transformation, we obtain: (d/dt) [i~3 Aih(t)] = and

r; A3(t) - v: A3h(t)

(15)

(d/dt) [i~3 i\n(t)] = [k+ A(t)- k_] [i~ Aih(t)] + 3[k~ A3(t) - k'- A3h(t)] + k_ A3h(t)

(16)

If, for instance, the direct transformation of a linear tetramer to a helical tetramer is taken into consideration, the terms k~ \(t) - k:' A4h(t) and 4[k~ A4(t) - k:' A4h(t)] must be added to the right-hand sides of the above two equations, respectively, where k~ and k:' are kinetic constants of winding and unwinding of tetramers. Such terms, however, are small in usual cases because A4(t) < A3(t) and k~ < k~ ( ~ k+ A). The winding of larger linear polymers makes a much smaller contribution. Thus the increasing rate of the total number concentration n" of monomers participating in helical polymers, or the decreasing rate of the total number concentration A+ n 1 of dispersed monomers and monomers participating in linear polymers, is approximately given by -[d(A+n1)/dt] = (dn,,/dt) =

[k+A(t)-k_]r[k~A3(t)-k'-A3h(t)]dt + 3[k~ A3(t) - k'- A3h(t)] + k_ A3h(t)

(17)

In the process of helical polymerization where the condition [i~3\h(t) ~ As!l(t)] is satisfied, the first term of the right-hand side of this equation is predominant as compared with other terms. Moreover, it is allowable to assume that A3 (and Ash) is proportional to A3 because the linear polymerization-depolymerization reaction is more rapid than helix formation, and to assume that concentration of linear polymers is very much lower than the monomer concentration (n1 ~ A). Then we have (18)

where c is a constant. By solving this equation we can follow the decrease in monomers or the increase in helical polymers. It is difficult to obtain a simple expression for the solution of this equation. Fortunately, however, if the solvent condition is optimal for helical polymerization, i.e. if k; A~ k_, we have the solution: 1 [l + (1-x3)l][l- (l-xg) ]l _ 3 n [1- (1-x3)l][1 + (l-xg)]l - OI.t

(19)

where x = f3(A/Ao)3, xg = f3, 01. = ylf3-l, f3-1 = l+k~h~/y, y = ik+cAg, Ao is the initial concentration of monomers, and hois the initial value of (.~ \n) or the initial number t=3

concentration of helical nuclei. If all nuclei are formed after the change of the solvent condition at time t, i.e, if ho = 0, then (20)

THEORY OF AGGREGATION OF MACROMOLECULES

17

Fig. 3 gives the A or ~ versWi t relations under various conditions of solvent or at various values of kinetic constants. One of the remarkable results derived from these equations is that at the initial stage the helical polymerization rate (d~/dt)t=o is approximately proportional to At whether ho is zero or not. The second point is that when AI Ao = constant, ~ t = constant; therefore, for example, the half polymerization time tj defined by A(tj ) = (1/2) Ao becomes proportional to "0•. If it is assumed that the helix has p monomers per turn, we have In [1 + (1- AP 1"8)t] = p(2lpk c)t "8/2 t [l-(l-API"8)t] +

(21)

instead of (20), and, therefore, the initial rate OC AK+l and the half polymerization time tt OC "oP/2. These theoretical predictions are useful for comparison with experimental data. 1.5

1.0

0.5

10

3(2{3k+ c)t

15

,\! t

FIG. 3. Kinetic features of helical polymerization. a,

Ao = 0·5;

b,

Ao = 1·0;

c,

Ao = 1·5.

+

The equation (19) is not valid near the final equilibrium state because the condition k+A}>k_ cannot be satisfied. It is notable that if at all times t~3 \h}>A3 A at the final equilibrium state becomes nearly equal to k_lk+ (= b-1 ) . The equilibrium analysis in the previous section showed that the condition (~Aih}> A3h) is always i=3

satisfied under the solvent condition where the helical polymers are stably formed. Therefore, it is reasonable that, similarly to the equilibrium analysis, the kinetic analysis also gives the result that the concentration of monomers coexisting with helical polymers at the final equilibrium is determined by the solvent condition, independently of the amount of helical polymers.

3. Comparison with Experimental Results Many kinds of protein molecules make aggregates under suitable conditions of the medium. Concerning the dissociation-association equilibrium, however, we have no adequate data. An example in which thermodynamic analyses were made is the 2

18

F. OOSAWA AND M. KASAl

fibrinogen-fibrin conversion (Scheraga & Laskowski, 1957). As another example we have the G-F transformation of actin. Here we compare the theoretical results with the experimental data obtained on this transformation. A most remarkable experimental finding was that this G-F transformation can be regarded as a condensation phenomenon (Oosawa et al., 1957, 1959; Asakura & Oosawa, 196Q; Asakura, Kasai & Oosawa, 1960; Kasai, Kawashima & Oosawa, 1960; Ooi, 1960). As is well known, actin is in the dispersed state (G-actin) in the salt-free solvent and is polymerized into the fibrous form (F-actin) by the addition of neutral salts (Straub, 1942; etc.). In the intermediate range of salt concentration we can obtain solutions which contain both G-actin and F-actin in equilibrium. The equilibrium state is determined by the condition of the solvent, independently of the initial state of the solution. In such equilibrium solutions it was confirmed that each actin molecule undergoes the cyclic change from G-actin to F -actin and again to G-actin, and the equilibrium state is established when the transformation rate from G to F is equal to the rate from F to G (Asakura & Oosawa, 1960). The concentration of F-actin in the solution can be determined by measurements of flow birefringence, viscosity, rigidity, light scattering and sedimentation. Summarizing all the results of these measurements it was found that F-actin can be formed only above the critical concentration of actin which is determined by the solvent conditions. Above this critical concentration the F-actin concentration increases linearly with the increase of the total actin concentration (Oosawa et al., 1959). Each G-actin molecule was found to have one ATP per molecular weight about 57,000 and this ATP is dephosphorylated into ADP and inorganic phosphate during the transformation into F-actin (Straub & Feuer, 1950; Mommaerts, 1952). Therefore, the concentration of G-actin coexisting in equilibrium with F-actin can be determined by measurements of the concentration of ATP bound to G-actin. The experimental results showed that below the critical actin concentration almost all the actin is in the state of G-actin, and above the critical concentration the G-actin concentration is kept nearly constant independently of the F -actin concentration. This G-actin concentration coexisting with F-actin is equal to the critical actin concentration, as shown in Fig. 2 (Asakura et al., 1960). Moreover, the length of F-actin estimated by measurements of flow birefringence, rigidity, and light scattering was found to be very large. Even near the critical concentration where a very small amount of F-actin is formed, the average degree of polymerization of F-actin reaches one hundred (Ooi, 1960; Kasai et al., 1960). All of these results agree with the conclusion derived in the theory of helical polymerization. Hence it is very likely that F-actin is a helical polymer of G-actin. In addition, we have kinetic data which support this interpretation of the G-F transformation of actin as a helical polymerization. The process of polymerization of G-actin induced by addition of salt was followed by measuring flow birefringence, viscosity and dephosphorylation of ATP. The result revealed a remarkable dependence of the polymerization rate on the actin concentration. At the optimal salt condition, the initial polymerization rate was found to be nearly proportional to the third to fourth power of the initial G-actin concentration, i.e. (dntldt)t~o o: ,\3'5, and the half polymerization time tt was found to be nearly proportional to ..\;1-5 (Kasai, Asakura & Oosawa, 1962). These results can be consistently explained by putting p = 3 into equation (21) in the kinetic theory of the previous section, namely by assuming that the helical polymer F-actin has three monomers per turn. Furthermore, the polymerization rate is increased

THEORY OF AGGREGATION OF MACROMOLECULES

19

effectively by addition of F-actin to a G-actin solution (Kasai, Asakura & Oosawa, 1962). This cooperative nature of the G-F transformation of actin is also a natural consequence of the helical polymerization assumption. As described above, actin has one nucleotide (ATP in G-actin and ADP in F-actin) per molecular weight of about 57,000. On the other hand, recent studies on the kinetic unit of G-actin by the light scattering made in the authors' laboratory gave a molecular weight of 117,000 which is twice the chemical unit having one nucleotide (Ooi, 1961). The presence of such a physical unit has also been deduced by fluorescence depolarization (Tsao, 1953). Since all ATPs bound to G-actin are dephosphorylated during polymerization, it is very probable that both the two nucleotide molecules bound to each monomer G-actin of molecular weight 117,000 play a role in making bonds between monomers in F-actin. Therefore, each monomer has the ability of making two kinds of bonds. This means that each monomer can be bound with four monomers just like the monomer in the helical polymer, unless it is the case that two nucleotides take part in one bond. If ADP in F-actin directly participates in the bond between the G-actin monomers, F-actin is gradually destroyed with the removal of ADP by prolonged dialysis. During prolonged dialysis the ADP content of F-actin was compared with the total amount of surviving F-actin estimated from the degree of flow birefringence (Kasai & Oosawa, to be published). The amount of surviving Factin was not proportional to the ADP content, but first kept nearly constant and then decreased rapidly with decreasing ADP. Such an experimental relation is found to be explained not by a simple end-to-end polymer model but by a helical polymer model in which each monomer G-actin is bound with four monomers through four bonds each of which contains ADP. Thus, the thermodynamic and kinetic analysis on the G-F transformation of actin suggests that F-actin is a helical polymer of G-actin. Such a helical structure of F-actin was also proposed from the X-ray analysis of actin gel by Bear & Selby (1956), although the detailed picture of the helix is very different from that presented here. Electron microscope pictures of F-actin also show the possibility of helical structure (Rosza, Szent-Gyorgyi & Wyckoff, 1949). In various kinds of proteins other than actin we also often meet experimental data suggesting helical aggregation. The polymerization process of insulin was well analysed by Waugh (1957), who showed that this polymerization consists of two steps: the formation of nuclei and their growth. The rate of nucleus formation was found to be proportional to the third to fourth power of concentration of insulin monomer. From this result he drew a model of the nucleus, which is just the same as the nucleus of the helical polymer of Fig. 1. Although the polymerization of insulin is not limited to one-dimensional growth but takes place two-dimensionally, its whole aspect resembles the G-F transformation of actin. From some physical measurements on the polymerization of fibrin monomer Ferry (1957) presented a model of the fibrin polymer consisting of the side-by-side association of two rod-like linear polymers accompanied by a staggered overlapping. This model is equivalent to a helical polymer having two monomers per turn. In such a helix, the polymerization takes place sharply but continuously because constant u cannot be so small. Scheraga & Laskowski (1957) analysed the equilibrium state of fibrin polymerization on the assumption of linear polymerization and concluded that the chemical constant at depends on i in such a way that at slightly increases

20

F. OOSAWA AND M. KASAl

with i from i = 2 to 7 and then begins to increase rapidly with increasing i. The same result can be derived from the model of a helical polymer by giving suitable values (which are not so small) to two constants o and a/b. Tropocollagen forms various types of aggregates, one of which can be regarded as a modified helical aggregate (Schmitt, 1959). The protein of tobaeeo mosaic virus is considered to be a helical polymer of monomer proteins in which one turn contains many monomers. We have described various examples of protein aggregation which can be interpreted as helical polymerizations. We can also find, of course, examples of simple linear polymerization of proteins. Tropomyosin in the salt-free solvent forms polymers which are depolymerized by addition of salts (Bailey, 1948). According to light scattering and electric birefringence measurements in the authors' laboratory, the experimental relation between the average length of tropomyosin polymers and the tropomyosin concentration fits well the theoretical relation for simple linear polymerization, i.e. the curve in Fig. 2(a) (Asai, 1961; Ooi, Kobayashi & Mihashi, to be published). In the case of actin, also, simple linear polymers often. appear in concentrated G-actin solutions under salt-free conditions. The relation between the average length and the actin concentration becomes of the type shown in Fig. 2(a) (Asai, 1961). Therefore, the same protein can make different kinds of polymers under different solvent conditions. Different kinds of polymers may have different properties, for instance, different length, birefringence, dipole moments, etc. Actually, the preliminary experiments on linear and helical polymers of actin showed that these two kinds of polymers have opposite signs of birefringence (Asai, unpublished results). A helical polymer can be constructed by the binding of each monomer with four neighboring monomers through two kinds of bonds. The possibility of two kinds of bonds means that two kinds of linear polymers can be formed. Changes of the solvent conditions may cause a transition between various kinds of polymers. The possible role of such a transition in biological phenomena, especially in muscle contraction, is discussed in another paper (Oosawa et al., 1961). REFERENCES Asai, H. (1961). J. Biochem., Tokyo, 50, 182. Asakura, S., Kasai, M. & Oosawa, F. (1960). J. Polym. Sci. 44, 40. Asakura, S. & Oosawa, F. (1960). Arch. Biochem, Biophys. 87, 273. Bailey, K. (1948). Biochem, J. 43, 271. Bear, R. & Selby, C. (1956). J. Biophys. Biochem, Cytol. 2, 71. Doty, P. (1959). Rev. Mod. Phys. 31, 107. Ferry, J. (1957). J. cell. comp. Physiol. 49, 185. Kasai, M., Asakura, S. & Oosawa, F. (1962). Biochim. biophys. Acta, in the press. Kasai, M., Kawashima, H. & Oosawa, F. (1960). J. Polym. Sci. 44, 51. Mommaerts, W. (1952). J. biol. Ohem, 198, 467. Ooi, T. (1960). J. Phys. Ohem, 64, 984. Ooi, T. (1961). J. Biochem., Tokyo, 50, 128. Oosawa, F., Asakura, S., Hotta, K., Imai, N. & Ooi, T. (1957). Proc. Conj. Ohem, Muscular Contraction, Lqakushoin, Tokyo, p. 57. Oosawa, F., Asakura, S., Hotta, K., Imai, N. & Ooi, T. (1959). J. Polym. Sci. 37, 323. Oosawa, F., Asakura, S. & Ooi, T. (1961). Progress in Theoretical Physics, SuppI. no. 17. Rosza, A., Szent-Gyorgyi, A. & Wyckoff, R. (1949). Biochim. biophys. Acta, 3, 561. Scheraga, H. & Laskowski, M. (1957). Advanc. Protein Chem. 14, 1.

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Schmitt, F. (1959). Rev. Mod. Phys. 31, 349. Straub, F. (1942). Studies Inst. Med, Ohem, 2, 3. Straub, F. & Feuer, G. (1950). Biochim. biophys. Acta, 4, 455. Szent-Gyorgyi, A. (1951). Chemistry of Muscular Contraction. New York: Academic Press. Szent-Gyorgyi, A. (1957). Bioenergetics. New York: Academic Press. Tsao, T. (1953). Biochim. biophys. Acta, 11, 227. Waugh, D. (1957). J. cell. comp, Physiol. 49, Suppl, 1, 145. Waugh, D. (1959). Rev. Mod. Phys. 31, 84. Zimm, B. (1959). Rev. Mod. Phys. 31, 123. Zimm, B. & Bragg, J. K. (1959). J. Ohern; Phys. 31, 526.