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Stochastic action of actomyosin motor
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Physica A 336 (2004) 123 – 132
www.elsevier.com/locate/physa
Stochastic action of actomyosin motor Micha l Kurzy%nski∗ , Przemys law Che lminiak Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna!n, Poland Received 25 October 2003; received in revised form 15 December 2003
Abstract It is argued that the actomyosin motor can be e2ectively considered a common chemo-chemical enzymatic machine occurring, however, in multitude rather than a few conformational substates distinguished by the conventional kinetics. A technique was developed with the help of which relations were found between basic parameters of the machine’s 6ux–force dependences: the turnover number, the force stalling the motor as well as the degree of coupling between the ATPase and the mechanical cycles, and the mean :rst-passage times in a random movement between some distinguished conformational substates of the myosin head. The phenomenology proposed is consistent with all presently available experimental data including multiple stepping per one adenosine triphosphate molecule hydrolysed. c 2004 Elsevier B.V. All rights reserved. PACS: 87.15.He; 87.15.Rn; 87.16.Nn Keywords: Protein dynamics; First passage time; Biological free energy transduction; Molecular motors
1. Molecular versus macroscopic engines One of the nontrivial challenges for the contemporary statistical physics is :nding a conceptual apparatus proper to describe the action of molecular motors. Of these, the actomyosin motors that drive the animal muscles are the most studied [1]. Functionally, there is no essential di2erence between the muscle and, the steam engine. They both perform macroscopic work at the expense of certain chemical reaction, either adenosine triphosphate (ATP) hydrolysis or the fuel burning. However, the steam engine structure is macroscopic (Fig. 1a) whereas the muscle is organized on the microscopic or, more precisely, the mesoscopic level (Fig. 1b). ∗
Corresponding author. E-mail address: [email protected] (M. Kurzy%nski).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.01.017
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M. Kurzy!nski, P. Chelminiak / Physica A 336 (2004) 123 – 132
Fig. 1. (a) Newcomen’s heat engine from 1712. Steam from the boiler moves the piston upwards the cylinder. Then, after closing the valve, the cylinder is cooled, steam is precipitated, and the atmospheric pressure moves the piston downwards raising the load. (b) Muscle consists of many periodically repeated structures, the sarcomeres. They are composed of the myosin thick :laments along which the actin thin :laments can slide. After activation by calcium ions, the muscle contracts raising the load. This motion, being the result of many Brownian motions of the myosin heads, is directed due to ATP hydrolysis.
The muscle (a biological organ) consists of many :bres (cells), each composed of many myo:brils (organelles), being systems of periodically repeated sarcomeres. Fig. 1b presents schematically the structure of a sarcomere. It is composed of thick :laments, made of the myosin macromolecules, along which the thin actin :laments slide. The proper molecular motor is the myosin head, a protein composed of some 1200 amino acids, i.e., some 18,000 atoms [2]. Functionally, one can distinguish in it a catalytic subunit (630 amino acids) joined by a swivel with a regulatory subunit— a “lever arm” (570 amino acids). Details are shown in Fig. 2. The myosin head is ATPase, an enzyme that hydrolyses ATP to adenosine diphosphate (ADP) and Pi (the inorganic phosphate). As all ATPases, the myosin head does not perform practically its enzymatic function until the conditions appear for the process to be biologically useful. The enzyme activator is the actin :lament. Only after strong attachment to the :lament at two sites the myosin head is able to bind and rebind substrates and products of the catalysed reaction. The binding site is a cleft between the upper and the lower domains of the catalytic subunit (Fig. 2). This cleft can be in the open or closed state. Through a long -helix, referred to as the relay, the state of the cleft is transmitted onto the orientation of the lever-arm domain, unless another -helix, called the SH1– SH2 helix, is melted [3]. This swinging lever-arm picture [2] re:nes the classical H. E. Huxley’s swinging cross-bridge model that relates rotational motion of the regulatory subunit relative the catalytic subunit to the motor force generation. The still open question for molecular motors is how to combine chemistry and mechanics and describe the mechanism of chemo-mechanical coupling. Two causes of the motor motion are considered: either a conformational change consisting in the lever-arm rotation (“power strokes” alternating “recovery strokes”) [1] or a biased
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Fig. 2. Structure of the myosin head in the absence of nucleotite. The upper drawing, made using the program Rasmol on the basis of Protein Data Bank IDs 2MYS for the catalytic subunit and 1SCM for the regulatory subunit, shows the particular secondary structure elements, -helices and -pleated sheets. The lower drawing presents schematically the component domains of the catalytic subunit: the upper (U), the lower (L) and the amino end (N) one, as well as those of the regulatory subunit: the globular converter (C) becoming a single -helix stabilized by the essential light chain (ELC) and the regulatory light chain (RLC). The SH1–SH2 helix, the relay helix and the both binding sites to the actin :lament are also shown. The swivel is close to the hydrosul:de group SH2.
Brownian walk (“thermal ratchet” models) [4–6]. Presumably, much of the super6uous discussion on this topic results from the fact that authors often do not clearly de:ne which notion of the force they have in mind: that on the micro-, meso- or macroscopic level. These are formally quite di2erent quantities. The force in the Newtonian sense can be de:ned only on the microscopic level of motion of individual atoms. This is the subject of molecular dynamics and will not be considered here. The forces exerted by a motor on a track and by a track on a motor, always balanced by the friction and the Brownian forces have a meaning on the mesoscopic level of stochastic dynamics of a
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Fig. 3. Lymn–Taylor–Eisenberg kinetic model of mechano-chemical cycle of the actomyosin motor. (a) The version of Ma and Taylor [10] with a few distinguished conformational states of myosin. A denotes the actin :lament, M the myosin head, T, D and Pi stand for ATP, ADP and inorganic phosphate, respectively. The original labelling of the reaction steps used by these authors is indicated. (b) The version with a quasi-continuum of conformational substates of myosin considered in the present paper. The multitudes of conformational transitions within E1 (the myosin–ADP complex strongly attached to the actin :lament), M (the myosin–ATP or ADP·Pi complex weakly attached) and E2 (the same complex detached from the actin :lament) are represented by shaded boxes. R1 = ATP, P1 = Pi whereas R2 and P2 denote the actin :lament before and after translation by a unit step, respectively. All bimolecular reactions are assumed to be gated, i.e., they take place only in certain conformational substates of the myosin head [13]. The distinguished conformational substates composing the gates are labelled as 1 , 1 , 2 and 2 .
single motor macromolecule. They are considered in various thermal ratchet models [4– 6] and observed in single molecule mechanical recordings [7,8]. Otherwise, the external load acts on a statistical ensemble of motor molecules composing the sarcomere or the whole muscle, and can be directly de:ned only on the macroscopic level of irreversible thermodynamics. Hill [9] stressed long ago that the force exerted by individual motor molecules is a strictly molecular property, not dependent on macroscopic external constraints such as the external load. Thus this load, which is simply subtracted from the mean force exerted by the ensemble of all motor molecules, can be only a property of the organization of the statistical ensemble, in particular the number of myosin heads bound to the actin :lament. In other words, external load attached to the myo:bril in6uences the energy of binding of the myosin head to actin :lament and not the energy of particular conformational states of the myosin. We adopt this point of view in the phenomenology proposed further on. 2. Eective chemo-chemical machine with multitude of conformational substates Fig. 3a shows the commonly accepted Lymn–Taylor–Eisenberg [1,10] kinetic scheme indicating how the ATPase cycle of myosin is related to a detached, weakly attached and strongly attached states of the myosin head to the actin :lament. Both the substrate and the products of the catalysed reaction bind to and rebind from the myosin in its
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strongly attached state, whereas the very reaction takes place either in the weakly attached or in the detached state. The changes in the binding free energy owing to the external load can be expressed as the changes in the e2ective rather than actual concentrations of the actin :lament A, which allows the motor to be treated as a usual chemo-chemical machine. However, the kinetic scheme in Fig. 3a is insuQcient for the proper description of the actomyosin mechanical cycle. To determine the force exerted by the myosin head on the actin :lament, one has to consider a quasi-continuum of conformational substates labelled with the help of a one-dimensional variable characterizing a position of a given fragment of the myosin head relative to a :xed point on the actin :lament. The force is a negative derivative of the free energy with respect to this variable [1,5,9]. In the weakly attached and detached states of the myosin head the stochastic dynamics of conformational transitions appears to be still more complex than one-dimensional di2usion [3]. Many studies performed in recent years indicate rich stochastic dynamics of transitions between a multitude of conformational substates in native proteins [11,12]. A slow character of this dynamics is the reason why the steady-state kinetics of biochemical processes involving protein enzymes cannot in general be described in terms of the conventional chemical kinetics, i.e., reaction rate constants. A more sophisticated language of mean :rst-passage times has to be used [12]. A technique was developed, enabling a calculation of the steady-state 6uxes for systems of enzymatic reactions controlled and gated by an arbitrary type stochastic dynamics of the enzymatic complex [13]. Here we present the most important results of application of this technique to the actomyosin motor and correct some far-reaching simpli:cations in Refs. [12,13]. Fig. 3b shows an extended version of the Lymn–Taylor–Eisenberg model we consider. Shaded boxes represent the multitudes of conformational substates and transitions within the three main states of the motor: E1 (the myosin–ADP complex strongly attached to the actin :lament), M (the myosin–ATP or ADP·Pi complex weakly attached) and E2 (the latter complex detached from the actin :lament). All binding–rebinding reactions are assumed to be gated, i.e., they take place only in certain distinguished conformational substates. Notation used is the same as in the paper [13] and di2ers slightly from that in the paper [12] (2 should be replaced by 2 and vice versa). In physiological and most of experimental conditions the ADP concentration is by a few orders of magnitude lower than the ATP concentration. Thus the ATP hydrolysis can be treated e2ectively as a unimolecular reaction R1 ↔ P1 , where R1 is ATP and P1 is an inorganic phosphate Pi. We have already argued that also physical motion itself can be treated as a unimolecular reaction R2 ↔ P2 , R2 and P2 denoting the actin :lament non-translated and translated by a step, respectively. 3. The ux–force relations As said above, the motor is formally considered a chemo-chemical machine that enzymatically couples the two unimolecular reactions: the free energy-donating reaction 1 and the free energy-accepting reaction 2. The input and output 6uxes Ji (i = 1 and
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2, respectively) and the conjugate thermodynamic forces Ai are de:ned as [12] d[Pi ]=dt Ji = ; [E]0 ≡ [E1 ] + [M] + [E2 ] [E]0 and Ai = ln Ki
[Ri ] ; [Pi ]
Ki ≡
[Pi ]eq : [Ri ]eq
(1)
(2)
Here, symbols of the chemical compounds in square brackets denote the molar concentrations in the steady state (no superscript) or in the equilibrium (the superscript eq), and is proportional to the reciprocal temperature, ≡ (kB T )−1 , where kB is the Boltzmann constant. Thermodynamic forces measure the distance from the equilibrium at which they vanish. The free energy transduction is released if the product J2 A2 , representing the output power, is negative [12]. Reaction 2 in the absence of reaction 1 proceeds from P2 to R2 and it can be driven against the conjugate force A2 provided reaction 1 occurs. The general expressions obtained in Ref. [13] for the system of two coupled enzymatic reactions are, unfortunately, very complex and not transparent. Serious simpli:cations result from the assumption that the machine is highly asymmetric and acts far from the chemical equilibrium: K1 eA1 1;
K2 eA2 6 1
(3)
which is well satis:ed for the actomyosin motor. Further simpli:cations are brought on by the assumptions that the binding–rebinding reactions are negligibly fast (cf. Fig. 3a) and that the mean :rst passage times within E1 are much shorter than those within M. In addition, it is assumed that E2 (2 → 2 ) E2 (2 → 2 ) ;
(4)
where E2 (2 → 2 ) and E2 (2 → 2 ) denote the mean :rst-passage times in the state E2 from the substate 2 to 2 and 2 to 2 , respectively. Two latter assumptions are the necessary condition for the J2 (A2 ) dependence being concave, which is in fact observed [1]. On the basis of the above assumptions the output 6ux can be approximated by a simple formula st k[R1 ] J2 (A2 ) = (eA2 − eA2 ) ; (5) K + [R1 ] where k≡
M (1 ↔ {1 ; 2 }) [E2 ]eq ; eq [M] M (1 → 1 ) E2 (2 → 2 )
K≡
[R1 ]eq [E1 ]eq M (1 ↔ 1 ) : [M]eq M (1 → 1 )
(6)
(note the Michaelis–Menten type dependence on the ATP concentration [R1 ]). Above, M (1 ↔ 1 ) denotes the mean :rst-passage time in the state M from the substate 1 to 1 and back and M (1 ↔ {1 ; 2 }) denotes the mean :rst-passage time in the state M from the substate 1 to 1 or 2 and back. The quantity Ast2 determines the (negative) value of the force A2 that stalls the machine: J2 (Ast2 ) = 0. The function (5) describes experimental behaviour equally well as the conventional hyperbolic dependence and
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1
J2(A2)/J2(0)
0
−1
st
−A2 /A2
0
Fig. 4. Fit of Eq. (5) to the data of He et al. [14], Fig. 3, for the sarcomere shortening velocity. The black circles correspond to the slow :bres and the white circles, to the fast 2A :bres. The :tted values of the negative stalling force in kB T units, −Ast 2 , were found to be equal 4.8 and 7.9 for the slow and the fast :bres, respectively. After Ref. [13].
in Fig. 4 we show how it :ts the data of He et al. [14]. The 6ux J2 is proportional to the mean velocity of the myosin head along the actin :lament and the force A2 is proportional to the load. Since our description of the motion is in terms of an e2ective reaction, we consider only the dimensionless quantities J2 (A2 )=J2 (0) and −A2 =Ast2 . The stalling force Ast2 found for an arbitrary value of A1 is given by the equation Ast2 = ln(e−A1 + c1 ) ; where c1 ≡
[M]eq =[E1 ]eq E1 (1 ↔ 1 ) + M (1 ↔ {1 ; 2 }) : M (1 ↔ {1 ; 2 })
(7) (8)
In Fig. 5 we show how a function of the form (7) :ts the experimental data of Pate et al. [15]. The maximum value of 10.0 of the negative stalling force in kB T units, −Ast2 , is comparable to the values of 4.8 and 7.9 determined for other samples from the 6ux–force dependence, Fig. 4. This, and the fact that in the linear range the negative stalling force −Ast2 is with a high accuracy equal to the force A1 mean that in any case the ratio c1 is a few orders of magnitude smaller than unity additionally justi:es our simpli:cations. All the experimental data in Fig. 5 are for A1 1, thus the assumption of reaction 1 being far from the equilibrium made in the derivation of Eq. (5) is also well justi:ed. Clearly, the interpretation of the experimental data in terms of Eq. (7) is more correct than in terms of the formulae derived by Pate et al. [15] under the assumption of a proximity to equilibrium. The eQciency of the machine is the ratio −J2 A2 =J1 A1 of the output power to the input power. In general, the 6ux J2 can di2er from the 6ux J1 because of a possible slippage (in either direction!) of the corresponding cycles when passing through the
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10
st
−βA2
8 6 4 2 0
0
2
4
6
8 βA1
10
12
14
Fig. 5. Fit of Eq. (7) to the data of Pate et al. [15], Fig. 6. We have assumed that the concentration of inorganic phosphate Pi determines directly the force A1 in kB T units. The maximum negative stalling force −Ast 2 in kB T units was :tted to be equal 10.0. After Ref. [13].
substates in M (cf. Fig. 3b). The measure of the slippage is a deviation from the unity of the ratio of both 6uxes, which in the same approximation as Eq. (5) and on st neglecting the small quantity e−A2 , is given by the equation M (2 ↔ {1 ; 2 }) J1 (1 + c2 e−A2 ) ; = (9) J2 M (1 ↔ {1 ; 2 }) where c2 ≡
[M]eq =[E2 ]eq E2 (2 ↔ 2 ) + M (2 ↔ {1 ; 2 }) : M (2 ↔ {1 ; 2 })
(10)
We do not discuss the conditions for the maximum eQciency of the machine as even in the linear approximation of the 6ux–force relations (which is certainly a bad approximation) the formulae for the values of forces maximizing the eQciency are very complex. Anyhow, the conditions for the maximum eQciency and those for the maximum output power contradict each other. The machine is the more eQcient when free energy dissipation is lower, i.e., it works slower. But the slower it works the lower is its output power. Moreover, not always the maximum eQciency or the maximum output power are the optimum from the point of view of living organism. Very often the power output of biological machines equals simply to zero, i.e., the output forces stall the machines. Muscles of a man sustaining a big load do not perform any work but, of course, ATP is consumed in some amounts. st On multiplying Eq. (9) by 6ux J2 , which on neglecting the quantity e−A2 is proA2 portional to e , we get an approximately linear dependence between the rate of ATP utilization and the rate of the muscle shortening. This is indeed observed experimentally [14] and allows evaluation of the ratio c2 to be of the order of 0.5 [13]. As expression (9) does not depend on the ATP concentration [R1 ], both J1 and J2 should
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show the same Michaelis–Menten type dependence on this concentration, cf. Eq. (5). Indeed, such a dependence was recorded experimentally [16,17] with similar values of the Michaelis constant K of the order of 10−5 –10−4 M. The maximum turnover number found for the ATPase is of the order of 20 s−1 whereas the maximum velocity per half sarcomere of the unloaded :lament is of the order of 5 m s−1 [10,16,17]. Thus the maximum distance travelled per one ATP molecule hydrolysed equals some 250 nm. The reciprocal of the ratio (9) represents the number of steps travelled per one ATP molecule hydrolysed. The size of the step depends on the system we consider. In the myo:bril, when a given myosin head is in the detached state, the actin :lament is submitted to the action of the external load as well as the force exerted by all remaining myosin heads (our theory assumes in fact some mean-:eld approximation) and the size of the step equals the actin :lament period 36 nm [1]. On the other hand, in the appropriately organized single molecule mechanical recording the size of the step can equal directly the single actin molecule diameter 5:5 nm [8]. During one ATPase cycle the myosin head transits only once through the state E1 where a power stroke takes place. Thus, the very ratio (9) is proportional to the power stroke distance divided by the distance travelled per one hydrolytic cycle which is refereed to as a duty ratio [1]. In the conventional approach, ratio (9) is assumed to equal unity (no slippage) [1]. However, some recent single-molecule recordings clearly demonstrate multiple steps produced by the myosin heads during a single ATPase turnover [8]. There is no reason to doubt that the actin heads behave in a similar manner in the assembly which they form in the myo:brils. The maximum value of J2 =J1 equals some 5–7, both in the case of single molecule recordings [8] and in the case of whole myo:brils (see data above). In order to :t this value, the ratio M (1 ↔ {1 ; 2 })= M (2 ↔ {1 ; 2 }) must be of the order of 10. The long mean :rst-passage time M (1 ↔ {1 ; 2 }) can be explained by the necessity of melting and recrystallization of the SH1–SH2 helix [3] during a transition in M from the substate 1 and back, cf. Fig. 3b. The relatively short mean :rst-passage time M (2 ↔ {1 ; 2 }) is the reason why, before coming back to the strongly attached state E1 , the myosin head can stochastically undergo several mechanical cycles through the detached state E2 . The direction of that motion is determined by a large disproportion between the times E2 (2 → 2 ) and E2 (2 → 2 ). It is the latter time that determines the motor velocity, cf. Eq. (5). It is worth noting here that, as opposed to the reaction rate constants, the mean :rst passage times do not obey in general the detailed balance condition. 4. Summary Besides taking into account the power stroke [1], the model considered is also a generalization of the thermal ratchet models [4–6] in the sense it assumes that the conformational substates of the motor need not form a one-dimensional quasi-continuum. On the contrary, the interiors of the shaded boxes symbolise arbitrary lattices of conformational substates, in particular fractal lattices, di2usion on which has been
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demonstrated to be a reasonable model of the protein’s intramolecular dynamics [11,18]. The phenomenology proposed is consistent with all presently available experimental data including multiple stepping per one ATP molecule hydrolysed [8]. The power stroke, being the source of the (mesoscopic) force the myosin head exerts on the actin :lament, takes place in the strongly attached state E1 . However, details of the power stroke mechanism are not important for the action of the actomyosin motor. The fundamental parameters of the theory presented are several :rst-passage times between the distinguished conformational states in M and E2 but not E1 . A present challenge for experimentalists is to determine their values precisely and for theorists to calculate them for simple but adequate models of conformational transition dynamics of the myosin head. Acknowledgements The study has been supported in part by the Polish State Committee for Scienti:c Research (project 2 P03B 056 18) and by the MITACS-MMPD project funded by NSCRC (Canada). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, Sunderland, 2001. M.A. Geeves, K.C. Holmes, Annu. Rev. Biochem. 68 (1999) 687–728. N. Volkmann, D. Hanein, Curr. Opin. Cell Biol. 12 (2000) 26–34. R.D. Astumian, Science 276 (1997) 917–922. F. JVulicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69 (1997) 1269–1281. B.J. Gabry%s, K. Pesz, S.J. Bartkiewicz, Physica A 336 (2004) 112–122 [this issue]. J.T. Finer, R.M. Simmons, J.A. Spudich, Nature 368 (1994) 113–119. K. Kitamura, M. Tokunaga, A.H. Iwane, T. Yanagida, Nature 397 (1999) 129–134. T.L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics, Springer, New York, 1989. Y.Z. Ma, E.W. Taylor, Biophys. J. 66 (1994) 1542–1553. M. Kurzy%nski, Prog. Biophys. Mol. Biol. 69 (1998) 23–82. M. Kurzy%nski, Physica A 285 (2000) 29–47. M. Kurzy%nski, P. Chelminiak, J. Stat. Phys. 110 (2003) 137–181. Z-H. He, R. Bottinelli, M.A. Pellegrino, M.A. Ferenczi, C. Reggiani, Biophys. J. 79 (2000) 945–961. E. Pate, K. Franks-Skiba, R. Cooke, Biophys. J. 74 (1998) 369–380. C. Lionne, F. Travers, T. Barman, Biophys. J. 70 (1996) 887–895. I. Amitani, T. Sakamoto, T. Ando, Biophys. J. 80 (2001) 379–397. M. Kurzy%nski, K. Palacz, P. Chelminiak, Proc. Natl. Acad. Sci. USA 95 (1998) 11685–11690.
Physica A 336 (2004) 123 – 132
www.elsevier.com/locate/physa
Stochastic action of actomyosin motor Micha l Kurzy%nski∗ , Przemys law Che lminiak Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna!n, Poland Received 25 October 2003; received in revised form 15 December 2003
Abstract It is argued that the actomyosin motor can be e2ectively considered a common chemo-chemical enzymatic machine occurring, however, in multitude rather than a few conformational substates distinguished by the conventional kinetics. A technique was developed with the help of which relations were found between basic parameters of the machine’s 6ux–force dependences: the turnover number, the force stalling the motor as well as the degree of coupling between the ATPase and the mechanical cycles, and the mean :rst-passage times in a random movement between some distinguished conformational substates of the myosin head. The phenomenology proposed is consistent with all presently available experimental data including multiple stepping per one adenosine triphosphate molecule hydrolysed. c 2004 Elsevier B.V. All rights reserved. PACS: 87.15.He; 87.15.Rn; 87.16.Nn Keywords: Protein dynamics; First passage time; Biological free energy transduction; Molecular motors
1. Molecular versus macroscopic engines One of the nontrivial challenges for the contemporary statistical physics is :nding a conceptual apparatus proper to describe the action of molecular motors. Of these, the actomyosin motors that drive the animal muscles are the most studied [1]. Functionally, there is no essential di2erence between the muscle and, the steam engine. They both perform macroscopic work at the expense of certain chemical reaction, either adenosine triphosphate (ATP) hydrolysis or the fuel burning. However, the steam engine structure is macroscopic (Fig. 1a) whereas the muscle is organized on the microscopic or, more precisely, the mesoscopic level (Fig. 1b). ∗
Corresponding author. E-mail address: [email protected] (M. Kurzy%nski).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.01.017
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M. Kurzy!nski, P. Chelminiak / Physica A 336 (2004) 123 – 132
Fig. 1. (a) Newcomen’s heat engine from 1712. Steam from the boiler moves the piston upwards the cylinder. Then, after closing the valve, the cylinder is cooled, steam is precipitated, and the atmospheric pressure moves the piston downwards raising the load. (b) Muscle consists of many periodically repeated structures, the sarcomeres. They are composed of the myosin thick :laments along which the actin thin :laments can slide. After activation by calcium ions, the muscle contracts raising the load. This motion, being the result of many Brownian motions of the myosin heads, is directed due to ATP hydrolysis.
The muscle (a biological organ) consists of many :bres (cells), each composed of many myo:brils (organelles), being systems of periodically repeated sarcomeres. Fig. 1b presents schematically the structure of a sarcomere. It is composed of thick :laments, made of the myosin macromolecules, along which the thin actin :laments slide. The proper molecular motor is the myosin head, a protein composed of some 1200 amino acids, i.e., some 18,000 atoms [2]. Functionally, one can distinguish in it a catalytic subunit (630 amino acids) joined by a swivel with a regulatory subunit— a “lever arm” (570 amino acids). Details are shown in Fig. 2. The myosin head is ATPase, an enzyme that hydrolyses ATP to adenosine diphosphate (ADP) and Pi (the inorganic phosphate). As all ATPases, the myosin head does not perform practically its enzymatic function until the conditions appear for the process to be biologically useful. The enzyme activator is the actin :lament. Only after strong attachment to the :lament at two sites the myosin head is able to bind and rebind substrates and products of the catalysed reaction. The binding site is a cleft between the upper and the lower domains of the catalytic subunit (Fig. 2). This cleft can be in the open or closed state. Through a long -helix, referred to as the relay, the state of the cleft is transmitted onto the orientation of the lever-arm domain, unless another -helix, called the SH1– SH2 helix, is melted [3]. This swinging lever-arm picture [2] re:nes the classical H. E. Huxley’s swinging cross-bridge model that relates rotational motion of the regulatory subunit relative the catalytic subunit to the motor force generation. The still open question for molecular motors is how to combine chemistry and mechanics and describe the mechanism of chemo-mechanical coupling. Two causes of the motor motion are considered: either a conformational change consisting in the lever-arm rotation (“power strokes” alternating “recovery strokes”) [1] or a biased
M. Kurzy!nski, P. Chelminiak / Physica A 336 (2004) 123 – 132
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Fig. 2. Structure of the myosin head in the absence of nucleotite. The upper drawing, made using the program Rasmol on the basis of Protein Data Bank IDs 2MYS for the catalytic subunit and 1SCM for the regulatory subunit, shows the particular secondary structure elements, -helices and -pleated sheets. The lower drawing presents schematically the component domains of the catalytic subunit: the upper (U), the lower (L) and the amino end (N) one, as well as those of the regulatory subunit: the globular converter (C) becoming a single -helix stabilized by the essential light chain (ELC) and the regulatory light chain (RLC). The SH1–SH2 helix, the relay helix and the both binding sites to the actin :lament are also shown. The swivel is close to the hydrosul:de group SH2.
Brownian walk (“thermal ratchet” models) [4–6]. Presumably, much of the super6uous discussion on this topic results from the fact that authors often do not clearly de:ne which notion of the force they have in mind: that on the micro-, meso- or macroscopic level. These are formally quite di2erent quantities. The force in the Newtonian sense can be de:ned only on the microscopic level of motion of individual atoms. This is the subject of molecular dynamics and will not be considered here. The forces exerted by a motor on a track and by a track on a motor, always balanced by the friction and the Brownian forces have a meaning on the mesoscopic level of stochastic dynamics of a
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Fig. 3. Lymn–Taylor–Eisenberg kinetic model of mechano-chemical cycle of the actomyosin motor. (a) The version of Ma and Taylor [10] with a few distinguished conformational states of myosin. A denotes the actin :lament, M the myosin head, T, D and Pi stand for ATP, ADP and inorganic phosphate, respectively. The original labelling of the reaction steps used by these authors is indicated. (b) The version with a quasi-continuum of conformational substates of myosin considered in the present paper. The multitudes of conformational transitions within E1 (the myosin–ADP complex strongly attached to the actin :lament), M (the myosin–ATP or ADP·Pi complex weakly attached) and E2 (the same complex detached from the actin :lament) are represented by shaded boxes. R1 = ATP, P1 = Pi whereas R2 and P2 denote the actin :lament before and after translation by a unit step, respectively. All bimolecular reactions are assumed to be gated, i.e., they take place only in certain conformational substates of the myosin head [13]. The distinguished conformational substates composing the gates are labelled as 1 , 1 , 2 and 2 .
single motor macromolecule. They are considered in various thermal ratchet models [4– 6] and observed in single molecule mechanical recordings [7,8]. Otherwise, the external load acts on a statistical ensemble of motor molecules composing the sarcomere or the whole muscle, and can be directly de:ned only on the macroscopic level of irreversible thermodynamics. Hill [9] stressed long ago that the force exerted by individual motor molecules is a strictly molecular property, not dependent on macroscopic external constraints such as the external load. Thus this load, which is simply subtracted from the mean force exerted by the ensemble of all motor molecules, can be only a property of the organization of the statistical ensemble, in particular the number of myosin heads bound to the actin :lament. In other words, external load attached to the myo:bril in6uences the energy of binding of the myosin head to actin :lament and not the energy of particular conformational states of the myosin. We adopt this point of view in the phenomenology proposed further on. 2. Eective chemo-chemical machine with multitude of conformational substates Fig. 3a shows the commonly accepted Lymn–Taylor–Eisenberg [1,10] kinetic scheme indicating how the ATPase cycle of myosin is related to a detached, weakly attached and strongly attached states of the myosin head to the actin :lament. Both the substrate and the products of the catalysed reaction bind to and rebind from the myosin in its
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strongly attached state, whereas the very reaction takes place either in the weakly attached or in the detached state. The changes in the binding free energy owing to the external load can be expressed as the changes in the e2ective rather than actual concentrations of the actin :lament A, which allows the motor to be treated as a usual chemo-chemical machine. However, the kinetic scheme in Fig. 3a is insuQcient for the proper description of the actomyosin mechanical cycle. To determine the force exerted by the myosin head on the actin :lament, one has to consider a quasi-continuum of conformational substates labelled with the help of a one-dimensional variable characterizing a position of a given fragment of the myosin head relative to a :xed point on the actin :lament. The force is a negative derivative of the free energy with respect to this variable [1,5,9]. In the weakly attached and detached states of the myosin head the stochastic dynamics of conformational transitions appears to be still more complex than one-dimensional di2usion [3]. Many studies performed in recent years indicate rich stochastic dynamics of transitions between a multitude of conformational substates in native proteins [11,12]. A slow character of this dynamics is the reason why the steady-state kinetics of biochemical processes involving protein enzymes cannot in general be described in terms of the conventional chemical kinetics, i.e., reaction rate constants. A more sophisticated language of mean :rst-passage times has to be used [12]. A technique was developed, enabling a calculation of the steady-state 6uxes for systems of enzymatic reactions controlled and gated by an arbitrary type stochastic dynamics of the enzymatic complex [13]. Here we present the most important results of application of this technique to the actomyosin motor and correct some far-reaching simpli:cations in Refs. [12,13]. Fig. 3b shows an extended version of the Lymn–Taylor–Eisenberg model we consider. Shaded boxes represent the multitudes of conformational substates and transitions within the three main states of the motor: E1 (the myosin–ADP complex strongly attached to the actin :lament), M (the myosin–ATP or ADP·Pi complex weakly attached) and E2 (the latter complex detached from the actin :lament). All binding–rebinding reactions are assumed to be gated, i.e., they take place only in certain distinguished conformational substates. Notation used is the same as in the paper [13] and di2ers slightly from that in the paper [12] (2 should be replaced by 2 and vice versa). In physiological and most of experimental conditions the ADP concentration is by a few orders of magnitude lower than the ATP concentration. Thus the ATP hydrolysis can be treated e2ectively as a unimolecular reaction R1 ↔ P1 , where R1 is ATP and P1 is an inorganic phosphate Pi. We have already argued that also physical motion itself can be treated as a unimolecular reaction R2 ↔ P2 , R2 and P2 denoting the actin :lament non-translated and translated by a step, respectively. 3. The ux–force relations As said above, the motor is formally considered a chemo-chemical machine that enzymatically couples the two unimolecular reactions: the free energy-donating reaction 1 and the free energy-accepting reaction 2. The input and output 6uxes Ji (i = 1 and
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2, respectively) and the conjugate thermodynamic forces Ai are de:ned as [12] d[Pi ]=dt Ji = ; [E]0 ≡ [E1 ] + [M] + [E2 ] [E]0 and Ai = ln Ki
[Ri ] ; [Pi ]
Ki ≡
[Pi ]eq : [Ri ]eq
(1)
(2)
Here, symbols of the chemical compounds in square brackets denote the molar concentrations in the steady state (no superscript) or in the equilibrium (the superscript eq), and is proportional to the reciprocal temperature, ≡ (kB T )−1 , where kB is the Boltzmann constant. Thermodynamic forces measure the distance from the equilibrium at which they vanish. The free energy transduction is released if the product J2 A2 , representing the output power, is negative [12]. Reaction 2 in the absence of reaction 1 proceeds from P2 to R2 and it can be driven against the conjugate force A2 provided reaction 1 occurs. The general expressions obtained in Ref. [13] for the system of two coupled enzymatic reactions are, unfortunately, very complex and not transparent. Serious simpli:cations result from the assumption that the machine is highly asymmetric and acts far from the chemical equilibrium: K1 eA1 1;
K2 eA2 6 1
(3)
which is well satis:ed for the actomyosin motor. Further simpli:cations are brought on by the assumptions that the binding–rebinding reactions are negligibly fast (cf. Fig. 3a) and that the mean :rst passage times within E1 are much shorter than those within M. In addition, it is assumed that E2 (2 → 2 ) E2 (2 → 2 ) ;
(4)
where E2 (2 → 2 ) and E2 (2 → 2 ) denote the mean :rst-passage times in the state E2 from the substate 2 to 2 and 2 to 2 , respectively. Two latter assumptions are the necessary condition for the J2 (A2 ) dependence being concave, which is in fact observed [1]. On the basis of the above assumptions the output 6ux can be approximated by a simple formula st k[R1 ] J2 (A2 ) = (eA2 − eA2 ) ; (5) K + [R1 ] where k≡
M (1 ↔ {1 ; 2 }) [E2 ]eq ; eq [M] M (1 → 1 ) E2 (2 → 2 )
K≡
[R1 ]eq [E1 ]eq M (1 ↔ 1 ) : [M]eq M (1 → 1 )
(6)
(note the Michaelis–Menten type dependence on the ATP concentration [R1 ]). Above, M (1 ↔ 1 ) denotes the mean :rst-passage time in the state M from the substate 1 to 1 and back and M (1 ↔ {1 ; 2 }) denotes the mean :rst-passage time in the state M from the substate 1 to 1 or 2 and back. The quantity Ast2 determines the (negative) value of the force A2 that stalls the machine: J2 (Ast2 ) = 0. The function (5) describes experimental behaviour equally well as the conventional hyperbolic dependence and
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1
J2(A2)/J2(0)
0
−1
st
−A2 /A2
0
Fig. 4. Fit of Eq. (5) to the data of He et al. [14], Fig. 3, for the sarcomere shortening velocity. The black circles correspond to the slow :bres and the white circles, to the fast 2A :bres. The :tted values of the negative stalling force in kB T units, −Ast 2 , were found to be equal 4.8 and 7.9 for the slow and the fast :bres, respectively. After Ref. [13].
in Fig. 4 we show how it :ts the data of He et al. [14]. The 6ux J2 is proportional to the mean velocity of the myosin head along the actin :lament and the force A2 is proportional to the load. Since our description of the motion is in terms of an e2ective reaction, we consider only the dimensionless quantities J2 (A2 )=J2 (0) and −A2 =Ast2 . The stalling force Ast2 found for an arbitrary value of A1 is given by the equation Ast2 = ln(e−A1 + c1 ) ; where c1 ≡
[M]eq =[E1 ]eq E1 (1 ↔ 1 ) + M (1 ↔ {1 ; 2 }) : M (1 ↔ {1 ; 2 })
(7) (8)
In Fig. 5 we show how a function of the form (7) :ts the experimental data of Pate et al. [15]. The maximum value of 10.0 of the negative stalling force in kB T units, −Ast2 , is comparable to the values of 4.8 and 7.9 determined for other samples from the 6ux–force dependence, Fig. 4. This, and the fact that in the linear range the negative stalling force −Ast2 is with a high accuracy equal to the force A1 mean that in any case the ratio c1 is a few orders of magnitude smaller than unity additionally justi:es our simpli:cations. All the experimental data in Fig. 5 are for A1 1, thus the assumption of reaction 1 being far from the equilibrium made in the derivation of Eq. (5) is also well justi:ed. Clearly, the interpretation of the experimental data in terms of Eq. (7) is more correct than in terms of the formulae derived by Pate et al. [15] under the assumption of a proximity to equilibrium. The eQciency of the machine is the ratio −J2 A2 =J1 A1 of the output power to the input power. In general, the 6ux J2 can di2er from the 6ux J1 because of a possible slippage (in either direction!) of the corresponding cycles when passing through the
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10
st
−βA2
8 6 4 2 0
0
2
4
6
8 βA1
10
12
14
Fig. 5. Fit of Eq. (7) to the data of Pate et al. [15], Fig. 6. We have assumed that the concentration of inorganic phosphate Pi determines directly the force A1 in kB T units. The maximum negative stalling force −Ast 2 in kB T units was :tted to be equal 10.0. After Ref. [13].
substates in M (cf. Fig. 3b). The measure of the slippage is a deviation from the unity of the ratio of both 6uxes, which in the same approximation as Eq. (5) and on st neglecting the small quantity e−A2 , is given by the equation M (2 ↔ {1 ; 2 }) J1 (1 + c2 e−A2 ) ; = (9) J2 M (1 ↔ {1 ; 2 }) where c2 ≡
[M]eq =[E2 ]eq E2 (2 ↔ 2 ) + M (2 ↔ {1 ; 2 }) : M (2 ↔ {1 ; 2 })
(10)
We do not discuss the conditions for the maximum eQciency of the machine as even in the linear approximation of the 6ux–force relations (which is certainly a bad approximation) the formulae for the values of forces maximizing the eQciency are very complex. Anyhow, the conditions for the maximum eQciency and those for the maximum output power contradict each other. The machine is the more eQcient when free energy dissipation is lower, i.e., it works slower. But the slower it works the lower is its output power. Moreover, not always the maximum eQciency or the maximum output power are the optimum from the point of view of living organism. Very often the power output of biological machines equals simply to zero, i.e., the output forces stall the machines. Muscles of a man sustaining a big load do not perform any work but, of course, ATP is consumed in some amounts. st On multiplying Eq. (9) by 6ux J2 , which on neglecting the quantity e−A2 is proA2 portional to e , we get an approximately linear dependence between the rate of ATP utilization and the rate of the muscle shortening. This is indeed observed experimentally [14] and allows evaluation of the ratio c2 to be of the order of 0.5 [13]. As expression (9) does not depend on the ATP concentration [R1 ], both J1 and J2 should
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show the same Michaelis–Menten type dependence on this concentration, cf. Eq. (5). Indeed, such a dependence was recorded experimentally [16,17] with similar values of the Michaelis constant K of the order of 10−5 –10−4 M. The maximum turnover number found for the ATPase is of the order of 20 s−1 whereas the maximum velocity per half sarcomere of the unloaded :lament is of the order of 5 m s−1 [10,16,17]. Thus the maximum distance travelled per one ATP molecule hydrolysed equals some 250 nm. The reciprocal of the ratio (9) represents the number of steps travelled per one ATP molecule hydrolysed. The size of the step depends on the system we consider. In the myo:bril, when a given myosin head is in the detached state, the actin :lament is submitted to the action of the external load as well as the force exerted by all remaining myosin heads (our theory assumes in fact some mean-:eld approximation) and the size of the step equals the actin :lament period 36 nm [1]. On the other hand, in the appropriately organized single molecule mechanical recording the size of the step can equal directly the single actin molecule diameter 5:5 nm [8]. During one ATPase cycle the myosin head transits only once through the state E1 where a power stroke takes place. Thus, the very ratio (9) is proportional to the power stroke distance divided by the distance travelled per one hydrolytic cycle which is refereed to as a duty ratio [1]. In the conventional approach, ratio (9) is assumed to equal unity (no slippage) [1]. However, some recent single-molecule recordings clearly demonstrate multiple steps produced by the myosin heads during a single ATPase turnover [8]. There is no reason to doubt that the actin heads behave in a similar manner in the assembly which they form in the myo:brils. The maximum value of J2 =J1 equals some 5–7, both in the case of single molecule recordings [8] and in the case of whole myo:brils (see data above). In order to :t this value, the ratio M (1 ↔ {1 ; 2 })= M (2 ↔ {1 ; 2 }) must be of the order of 10. The long mean :rst-passage time M (1 ↔ {1 ; 2 }) can be explained by the necessity of melting and recrystallization of the SH1–SH2 helix [3] during a transition in M from the substate 1 and back, cf. Fig. 3b. The relatively short mean :rst-passage time M (2 ↔ {1 ; 2 }) is the reason why, before coming back to the strongly attached state E1 , the myosin head can stochastically undergo several mechanical cycles through the detached state E2 . The direction of that motion is determined by a large disproportion between the times E2 (2 → 2 ) and E2 (2 → 2 ). It is the latter time that determines the motor velocity, cf. Eq. (5). It is worth noting here that, as opposed to the reaction rate constants, the mean :rst passage times do not obey in general the detailed balance condition. 4. Summary Besides taking into account the power stroke [1], the model considered is also a generalization of the thermal ratchet models [4–6] in the sense it assumes that the conformational substates of the motor need not form a one-dimensional quasi-continuum. On the contrary, the interiors of the shaded boxes symbolise arbitrary lattices of conformational substates, in particular fractal lattices, di2usion on which has been
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demonstrated to be a reasonable model of the protein’s intramolecular dynamics [11,18]. The phenomenology proposed is consistent with all presently available experimental data including multiple stepping per one ATP molecule hydrolysed [8]. The power stroke, being the source of the (mesoscopic) force the myosin head exerts on the actin :lament, takes place in the strongly attached state E1 . However, details of the power stroke mechanism are not important for the action of the actomyosin motor. The fundamental parameters of the theory presented are several :rst-passage times between the distinguished conformational states in M and E2 but not E1 . A present challenge for experimentalists is to determine their values precisely and for theorists to calculate them for simple but adequate models of conformational transition dynamics of the myosin head. Acknowledgements The study has been supported in part by the Polish State Committee for Scienti:c Research (project 2 P03B 056 18) and by the MITACS-MMPD project funded by NSCRC (Canada). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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