PCCP PAPER. Mesoscopic non-equilibrium thermodynamic analysis of molecular motors. 1 Introduction

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1 PAPER View Article Online View Journal View Issue Cite this: Phys. Chem. Chem. Phys., 13, 15, 1945 Received 5th June 13, Accepted 3th September 13 DOI: 1.139/c3cp5339j 1 Introduction The movement in space of a small-scale biological system is often well modeled as an activated process, in which the system proceeds when a certain amount of energy is available. 1 4 The binding of a ligand to a protein, the formation of loops in DNA segments, the stretching of a DNA molecule,,3 or the pumping of ions against their chemical potential 4 are all such activated processes, taking place under non-equilibrium conditions. The progress of these processes is usually analyzed in terms of chemical reaction kinetics or kinetic equations for probability distributions. 5 7 In particular, for molecular motors, a theoretical framework has been developed using a two-state model. This framework explains how the motor velocity emerges from transitions between the two internal states, introduced to characterize the structural changes of the molecular motor. 8,17 Traditionally, the principles of non-equilibrium thermodynamics have only been applied to the linear regime of fluxes and forces, 9 11 arguing that the assumption of local equilibrium can be justified only in the linear regime. In this article, we a Department of Chemistry, Norwegian University of Science and Technology, 4791 Trondheim, Norway. signe.kjelstrup@ntnu.no b Process and Energy Laboratory, Delft University of Technology, Leeghwaterstraat 44, 68CA Delft, The Netherlands c Departamento de Fisica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 88 Barcelona, Spain Electronic supplementary information (ESI) available. See DOI: 1.139/ c3cp5339j Mesoscopic non-equilibrium thermodynamic analysis of molecular motors S. Kjelstrup,* ab J. M. Rubi, c I. Pagonabarraga c and D. Bedeaux ab We show that the kinetics of a molecular motor fueled by ATP and operating between a deactivated and an activated state can be derived from the principles of non-equilibrium thermodynamics applied to the mesoscopic domain. The activation by ATP, the possible slip of the motor, as well as the forward stepping carrying a load are viewed as slow diffusion along a reaction coordinate. Local equilibrium is assumed in the reaction coordinate spaces, making it possible to derive the non-equilibrium thermodynamic description. Using this scheme, we find expressions for the velocity of the motor, in terms of the driving force along the spacial coordinate, and for the chemical reaction that brings about activation, in terms of the chemical potentials of the reactants and products which maintain the cycle. The second law efficiency is defined, and the velocity corresponding to maximum power is obtained for myosin movement on actin. Experimental results fitting with the description are reviewed, giving a maximum efficiency of.45 at a myosin headgroup velocity of ms 1. The formalism allows the introduction and test of meso-level models, which may be needed to explain experiments. analyze such activated processes using the principles of mesoscopic non-equilibrium thermodynamics. 1,13 Local equilibrium can in fact be expected at the shorter time and length scales relevant for activated processes. The mesoscopic description can be used to test proposals of molecular mechanisms, i.e. of the detailed sequencing of steps in the motor s cycle, when it is integrated into the level of macroscopic performance and compared to experiments. The aim of this work is to present one such mechanism for a progressing motor which can function or malfunction (slip), and find the corresponding mesoscopic thermodynamic description. As an example we shall take the movement of myosin headgroups along actin. We shall find flux equations from probability distributions on the meso level, integrate them out, relate them to experiments, and analyze the motor efficiency. While the first law efficiency has been used earlier to find limits for the maximum power, 14,4 we shall use a definition based on the second law as a basis, see e.g. ref. 11 and 16. The fuel that moves the molecular motor in a biological system is adenosine triphosphate (ATP). In the two-state model described by Jülicher et al., 17 thechangebetweenthedeactivated state and the activated state, which leads to motor rectification, is brought about by the hydrolysis reaction of ATP with adenosine diphosphate (ADP) and inorganic phosphate (Pi). The relationship between ATP hydrolysis and the molecular motor conformational changes is not described in detail. The model assumes implicitly that one forward step of the motor needs the hydrolysis of one ATP. Similar stoichiometric links are also taken for granted when post-albers schemes are used to describe analogous This journal is c the Owner Societies 13 Phys. Chem. Chem. Phys., 13, 15,

2 conformational changes with chemical reaction kinetics. It is known from experiments on Ca + -pumping by the Ca-ATPase, however, that there is only seldom a stoichiometric ratio between the moles of ions pumped and the moles of ATP used. 4 In a stochastic process, it is likely that not all events in the pumping are strictly coupled, but that some lead to slip. In other words that there are malfunctions and inefficient operation in the thermodynamic sense. This gives reasons to choose a non-equilibrium thermodynamic description that accounts for non-stoichiometry. 11 The understanding gained on how the Ca-ATPase works 4 shows the need to account for the fact that not all reacted ATP molecules lead to forward motion of the motor. Molecular motor efficiency should be a measure of this deficiency; a fact that has not been addressed systematically, despite the large number of previous works on molecular motor efficiency. 8,14 16,18,4 In the construction of our model, we have chosen to build on the well established two-state model. 17 We generalize the model so that the link to the energy source is made explicit, and introduce a mechanism of slip. We continue to describe the work performed against an external force, f ext, through controlled deactivation. This mechanism is probably relevant for kinesin motors, muscular contraction, and other motors. A macroscopic model with this mechanism was first described for muscular contraction by Førland, 19, arguing that such a mechanism of work production should give a higher efficiency than the famous Huxley model for muscle contraction. This work did not formulate a theory on the mesolevel, however, and did only use linear flux force relations. We shall see that we can arrive at non-linear flux force relations, using a mesoscopic approach, and derive expressions for the motor s efficiency at maximum power. Our formulation generalizes the early approach of Førland, and details the link between the meso and macro levels. We start in Section, giving a general basis of mesoscopic non-equilibrium thermodynamics. We formulate next the mechanism for the molecular motor function. Using in Section 3 that the transition state energy is much larger than the thermal energy for activation and uncontrolled and controlled deactivation, we are able to construct a solution for motion both along the path coordinates in state space, 1,1 associated with the conformational changes the molecular motor undertakes, as well as along the real space coordinate. In Section 4 we use the motor integrated equations of motion to describe its displacement along a filament. The integrated description contains Jülicher et al. s 17 description, but it is also able to give direct information on motor efficiency, and optimal performance; the latter is discussed in Section 5. A two-state model of slipping single motors The principles of mesoscopic non-equilibrium thermodynamics shall be applied to a two-state model of an isothermal single motor in this section. Consider therefore a motor moving along a filament with spatial coordinate x. The filament has a periodic structure with a repeat unit of length l. As the motor moves from the start to the end of a period, the bond to the filament varies in strength, visualized by an enthalpy profile that has a ratchet shape in space. At each position x, the motor can in principle be in two states, a deactivated state, d, characterized by an enthalpy h d = h d (x), as shown schematically in Fig. 1 or an activated state, a, characterized by an enthalpy, h a, assumed uniform for the sake of simplicity (following Jülicher 17 ), but that can be generalized without significant differences. The motor can pull against an external force f ext during controlled deactivation, i.e., when the motor moves down the ratchet with a decreasing value of an effective enthalpy profile h(x). Consumption of ATP makes the transition between the states possible and can also lead to forward movement along the x-axis. The three processes that characterize the motor motion can be described as chemical reactions ATP(x a )+H O(x a )+M d (x a ) " ADP(x a ) + Pi(x a )+M a (x a ) zm a (x a ) " zm d (x a +1) (1 z)m a (x a ) " (1 z)m d (x a ) (1) The first step, the reaction with ATP, takes place at position x a, and leads to activation of the motor M from state d to a, Fig. 1 Example of a molecular motor: the movement of a myosin headgroup along an actin filament due to ATP hydrolysis is shown in the top part (a). The myosin filament is able to sustain an external force f ext. The headgroup is binding to the filament in a progressively stronger manner, as it moves within one structural unit along the filament, enabling myosin to do work as it moves. The corresponding enthalpy profile for the headgroup is illustrated in part (b) at the bottom of the figure. The enthalpy of the activated myosin headgroup, h a, is considered constant, while the enthalpy of the deactivated headgroup is a periodic function of position, h d (x) Phys. Chem. Chem. Phys., 13, 15, This journal is c the Owner Societies 13

3 denoted M d and M a, respectively. The probability for the motor to be activated depends on its position along the filament, x. It is sensible to assume that the activation takes place preferentially in a small domain which we restrict to the specific location x a ;we do not expect this restriction to have a relevant effect on the motor performance. During the process of activation and deactivation the motor can also move along the x-axis. How the motor rectifies its displacement and rotates or moves along a filament depends on the details of how the mobility of the motor is linked to the conformational changes it experiences. A fraction z of the activated motors can step forward to position x a + 1 doing work simultaneously, while the remaining fraction of motors (1 z)is unable to move forward and will rather dissipate their energy in the deactivation step. The overall velocity of the motor emerges as a result of the three processes, and means that a fraction, z, of motors moves along the filament at the expense of 1 ATP reacted, ATP(x a )+H O(x a )+M d (x a ) " ADP(x a ) + Pi(x a )+(1 z)m d (x a ) + zm d (x a + 1) () The deviation from ideality of the motor is then expressed through z, the ability of each ATP to produce a step forward. In order to describe the reaction kinetics in a consistent mesoscopic non-equilibrium thermodynamic framework, we need to account for the conformational changes that the motor undergoes. The chemical reaction with ATP, corresponding to the first process in eqn (1), can be described as an activated process along a reaction coordinate g 1. The chemical potential energy raise of the motor upon activation to h a can, however, be lost via a slip path, which is given here as coordinate g, corresponding to the third process of eqn (1). Both reaction coordinates have an activation energy barrier, making the overall processes slow, and allowing us to regard the reactions as diffusion processes along g 1 or g according to Kramers. 1 For convenience, and without loss of generality, the reaction coordinates take values between and 1. The limiting values of the g coordinate correspond to the conformations of the motor as a reactant or a product of the corresponding chemical reaction, respectively; intermediate values denote the conformational structures the molecular motor undertakes when moving from the initial to the final state. Specifically, when g 1 = and g =,org 1 = 1 and g = 1, the motor is in the deactivated state d, while for g 1 = 1 and g =, the motor is in the activated state a. We will also describe the movement along the x-coordinate as being continuous. The progression of the motor along the x-axis is measured in terms of the diffusion flux J x (x,g 1,g,t). We consider the filament period, l, as the reference unit of length, so x is dimensionless and varies between and 1. The state of the molecular motor at time t is given by a probability distribution P(x,g 1,g,t). The distribution is defined for an ensemble of single motors in a large landscape, where the variables are periodic along the x-coordinate with a period l. Since the landscape is large, the generated ensemble is sufficient for a thermodynamic analysis. The probability is conserved, meaning that the time derivative g 1 ; g ; g 1 ðx; g 1 ; g ; tþ g ðx; g 1 ; g ; xðx; g 1 ; g ; ¼ Thesecondandthirdtermsonthelefthandsidearethe divergence of the reaction rates J g1 (x,g 1,g,t) andj g (x,g 1,g,t) along the g 1 -andg -coordinates. The last term on the left hand side is the divergence of the diffusion flux J x (x,g 1,g,t). In order to find expressions for J x (x,g 1,g,t), J g1 (x,g 1,g,t) and J g (x,g 1,g,t) in terms of the corresponding driving forces, we need to determine the entropy production. To do this, we use the Gibbs entropy postulate for a motor dsðtþ ¼ 1 ð 1 T du mðx; g dx dg 1 dg 1 ; g ; tþ dpðx; g T 1 ; g ; tþ (4) The chemical potential of the molecular motor, m, isafunction of the degree of reaction with ATP, g 1, the degree of de-activation g, the position and the time. Assuming isothermal conditions and no supply of chemical reactants from the outside, the internal energy per unit length can change only when an external force is applied to the motor, du/dt = f ext J x where J x ðtþ ¼ ð 1 (3) dxdg 1 dg J x ðx; g 1 ; g ; tþ (5) is the average flux as the motor moves along the filament. The external force can for instance be due to an optical tweezer pulling molecular strands, to a load pulling on the myosin filament under muscular contraction or to a cargo being pulled by a kinesin motor. In these cases, the force is negative. The motor does not move in the absence of fuel (ATP), or when the external force is balanced by an internal driving force. Eqn (3) and (5) give the time derivative of the entropy (eqn 4) ð x ðx; g ¼ 1 dg 1 ; g ; tþ g 1 ðx; g 1 ; g 1 g ðx; g 1 ; g ; tþ mðx; g 1 ; g ; o þ J x ðx; g 1 ; g ; tþf ext We integrate by parts and obtain ð ¼ dx 1 dg J x ðx; g 1 ; g ; m ð x; g 1; g ; t þ J g1 ðx; g 1 ; g ; mðx; 1 ; g ; tþ 1 þ J g ðx; g 1 ; g ; mðx; 1 ; g ; tþ (6) Þ f ext (7) This journal is c the Owner Societies 13 Phys. Chem. Chem. Phys., 13, 15,

4 wherewehaveusedtheperiodicityoverthedistance1along the x-coordinate and set the reaction fluxes at the ends of the g-coordinate equal to zero. The boundary conditions for m will depend on the access of the motor to ATP, ADP and Pi. The entropy production (per unit of length l, and degrees of reaction) equals sðx; g 1 ; g ; tþ ¼ J g1 ðx; g 1 ; g ; tþ mðx; g 1 ; g ; tþ 1 J g ðx; g 1 ; g ; tþ mðx; g 1 ; g ; tþ J x ðx; g 1 ; g ; tþ m ð x; g 1; g ; t Þ f ext This local form of the entropy production is non-negative definite in accordance with the second law. From eqn (8) we obtain a product sum of three fluxes with their conjugate forces. The first products describe diffusion along reaction coordinates, g 1, g, while the last set describes diffusion along the filament. In the general description provided by eqn (8), the allowed states of the motor form a continuum with r g 1, g r 1 and r x r 1 and the motion along the spatial coordinate, x, is periodic (Fig. ). The mesoscopic approach put forward shows that ATP activation takes place along a straight line between (x a, g 1 =, g = ) and (x a, g 1 =1,g = ). The move forward accompanied by deactivation takes place between (x a, g 1 =1,g = ) and (x a +1, g 1 =1,g = 1), while the deactivation through slip takes place only between (x a, g 1 =1,g = ) and (x a, g 1 =1,g = 1). Since these three processes are activated, the probability distribution and the fluxes are therefore only unequal to zero along the 3 lines, which connect these points in state space, and where the fluxes are directed. We can therefore simplify the general expression (8) for the entropy production to give ð 1 ¼ 1 J g1 ðx a ;g 1 mðx a 1 ;;tþ 1 ð 1 dg J g ðx a ;1;g mðx a dg 3 J x ðx a þ g 3 ;1;g 3 ;tþ mðx a þ 3 ;1;g 3 ;t 3 ð 1 Þ f ext where we have introduced the coordinate g 3 to describe the displacement of the motor from (x a, g 1 =1,g =)to(x a +1,g 1 =, g = 1), and describe it as an effective activated process. Accordingly, we replace the spatial dependence of the motor by its displacement in coherence with the deactivation process. Hence, the diffusion flux J x (x,g 1,g,t) introduced earlier will be parametrized as J x = J x (x a + g 3,g 3,t) to express the dependence on g 3 ; the same applies to the spatial dependence of the diffusive fluxes that characterize motor hydrolysis and slipping. The derivative with respect to g 3 in the last term has contributions from the variation along the x-coordinate as well as the g - coordinate. The flux force relations that follow from this expression for the entropy production are J g1 ðx a ; g 1 ; ; tþ ¼ L g1 ðg 1 ; mðx a ; 1 ; ; tþ 1 J g ðx a ; 1; g ; tþ ¼ L g ðg ; mðx a ; 1; ; tþ J x ðx a þ g 3 ; 1; g 3 ; t Þ ¼ L x ðg 3 ; tþ mðx a þ 3 ; 1; g 3 ; tþ f ext 3 (1) Since we have identified the different configurations that characterize the motion of the molecular motor, these three reactions remain uncoupled. As discussed in ref. 1, three uncoupled reactions can be described by an equivalent formulation in terms of two independent coupled reactions. Hence, there is no loss of generality to take the reactions in eqn (1a and b) uncoupled. 1 The Onsager coefficients, L g1, L g, L x, have been introduced to describe the dependence of the fluxes on the forces. The boundary values m(x a,,,t) =m(x a + 1,1,1,t) = m(x a,1,1,t), m(x a,,1,t) and m(x a,1,,t) are given by the corresponding chemical potentials of the reactants and products, respectively, see reactions (1). Note that the state g 1 = g =1is the same as g 1 = g =, which we have used in the boundary values. The motor is only able to perform work when it moves along the x-coordinate, consistent with the fact that f ext only couples Fig. A sketch of the three activated processes that determine the motion of the molecular motor, according to the reaction scheme of eqn (1). The motor changes from the active to the deactivated state. All reactions are reversible, but in the deactivation process it can either change its conformation without displacing or can jump to the next attractive position along the spatial coordinate (and that can correspond either to a translation or rotation). We note that the choice of a linear dependence between the spatial displacement and the coordinate reaction g implies that the probability density of the activated state drops linearly to zero between these two points. For example, in the two state model for a molecular motor introduced in ref. 17, this corresponds to a choice of the fraction of molecular motors in the deactivated state that varies linearly along the filament. If one chooses a curved line, this corresponds to a nonlinear variation of the relative amount of activated and deactivated motors along the filament. We will come back to this in the ESI Phys. Chem. Chem. Phys., 13, 15, This journal is c the Owner Societies 13

5 to the flux along g 3. We introduce a more convenient, shorthand notation along the two reaction coordinates, J g1 ðg 1 ; tþ ¼ L g1 ðg 1 ; 1 ðg 1 ; tþ 1 J g ðg ; tþ ¼ L g ðg ; ðg ; tþ (11) By integrating over the coordinate g 1, we will obtain from eqn (1a) a description in terms of the reaction rate and the chemical driving force of the reaction eqn (1a). The term given by eqn (1b) will dissipate energy between states bounded by the chemical potential of the activated and the deactivated motor. From eqn (1c) we should find a contribution due to the measurable motor velocity J x and the net driving force on the motor in step eqn (1c), qm(x a + g 3,1,)/qg 3 f ext. Conservation of molecular motors gives J g1 (x a,1,)=j g (x a,1,)+j x (x a, 1, ) (1) Given the constant nature of the fluxes along the reaction coordinates, this implies that J g1 (x a,g 1,,t) =J g (x a,1,g,t) +J x (x a + g 3,1,g 3,t) (13) The parameter z introduced in eqn (1) is equal to z ¼ J xðx a ; 1; Þ J g1 ðx a ; 1; Þ ¼ 1 J g ðx a ; 1; Þ J g1 ðx a ; 1; Þ (14) The integration along the corresponding internal coordinates will eventually lead to flux equations on the macroscopic scale analogous to the ones given by Caplan and Essig, 11 as will become explicit later on. A formulation in terms of a mesoscopic description can be used to test how a model performs on the macroscopic level after integrating it. We shall now see how this can be done. 3 Reaction rates The flux along the reaction coordinates g 1 and g can be further described using Kramers theory 1,1 and its activated nature can be used to identify the magnitude of J g1 and J g. A full account of this theory applied to the present case is given as ESI. We report here the important premises of the derivations and the resulting expressions for the flux equations. In the mesoscopic description, the chemical potential of a molecular motor is given in terms of the motor enthalpy and the probability distributions P 1 (g 1,t) orp (g,t) (where we have introduced the short-hand notation, P 1 (g 1,t) P(x a,g 1,,t) and P (g,t) P(x a,1,g,t)) along the two reaction coordinates by m j (g j,t) =h j (g j )+k B T ln[p j (g j,t)] (15) Here h j (g j ) are the enthalpies 16 per molecular motor along the reaction paths. We have also introduced the short-hand notation, m 1() (g 1 (g ),t) = m(x a,g 1 (1),(g ),t) and h 1() (g 1 (g ),t) = h(x a,g 1 (1),(g ),t). A good approximation to h j (g j ) may be derived from the potential of mean force of the motor in its activated or deactivated conformations. In equilibrium the chemical potential, m eq, is constant (independent of x,g 1,g,t). The motor probability distribution becomes " # m eq h j g j P j;eq g j ¼ exp : (16) k B T Along the first reaction path, the molecular motor goes from state d to state a and one ATP molecule is hydrolyzed to ADP and Pi. Therefore, h 1 () = h d + h ATP + h H O and h 1 (1) = h a + h ADP + h Pi. Along the second reaction path the molecular motor goes from state a to state d. Using the same reference value we have h (1) = h d +h ADP + h Pi and h () = h 1 (1). When a fraction of the motors first moves from state d to a using the first reaction, and then from a to d using the second reaction (both on the same position x a ) the reaction enthalpy h ATP h ADP h Pi is lost as heat in the surroundings. The enthalpy profiles associated with both reaction paths are illustrated in Fig. 3 and 4. The reaction coordinates are characterized by high enthalpy barriers, larger than the thermal energy k B T,atg 1, and g,, which determine the corresponding transition states, as a consequence, the probability of finding molecular motors in the transition states is very small, both in equilibrium, eqn (16), and away from equilibrium. Therefore, the enthalpies at the transition states can be referred to as the transition state energies, h 1 (g 1, ) h 1, and h (g, ) h,, see Fig. 3 and 4. With large transition state energies, the reactions become slow and therefore quasi-stationary. In a quasi-stationary state, the reaction rates are independent of the reaction coordinate. It is well-known that the Onsager kinetic coefficients, L gj (g j,t), Fig. 3 A sketch of the enthalpy profile along the reaction coordinate for ATP hydrolysis. Boundary values for reactants and products are shown. The deactivated motor is seen as a reactant with enthalpy h d. Through the hydrolysis of ATP it is brought to an activated state, with enthalpy h a. The height of the activation enthalpy barrier, h 1,, is found for g 1,. The chemical potential difference of the reaction is also shown, in terms of reactant and product states. This journal is c the Owner Societies 13 Phys. Chem. Chem. Phys., 13, 15,

6 Fig. 4 A sketch of the enthalpy profile for the motor slip reaction. Boundary values for reactants and products are shown. The motor deactivation starts at an enthalpy h a and ends with h d ; both states are referred to the enthalpy sum of ADP and Pi. The height of the activation enthalpy barrier, h,, is found for g,. The chemical potential difference of the slip reaction is also shown, in terms of reactant and product states. are usually in good approximation proportional to P j (g j,t), and that we can define a diffusion coefficient for each reaction D gj k B T L g j g j ; t (17) g j ; t which can be regarded as a constant. The factor k B T gives D gj the dimensionality s 1 which is appropriate along a dimensionless reaction coordinate. Using these assumptions, we show in the ESI that the fluxes are equal to DG DmðtÞ J g1 ðtþ ¼ J g1 ;þðtþ 1 exp k B T P j J g ðtþ ¼J g ;þðtþ 1 exp DmðtÞ k B T (18) where J g1,+(t) and J g,+(t) are unidirectional forward reaction rates, Dm(t) m d (t) m a (t) and DG m ADP + m Pi m ATP m H O. The signs of Dm(t) and DG have been chosen so that they are both negative when the reaction proceeds from reactants to products. The absolute value of both is generally much larger than the thermal energy; in the system considered, the motor usually displaces in the regime Dm(t) DG c k B T. It is possible to extract the backward rate, when information is available on equilibrium exchange rates. Both forward reaction rates are usually much larger than the backward reaction rates, so that in good approximation J gj (t) C J gj,+(t). When Dm(t) and DG aresmallcomparedtothethermal energy k B T,onecanlinearizeeqn(18)andobtainexpressions used in classical non-equilibrium thermodynamics. For example, in this regime, the reaction rate of eqn (1a), J g1,is proportional to a linear combination of DG for the ATP hydrolysis and the chemical potential difference of the activation step. The large values of Dm and DG imply that the fluxes become nonlinear in the driving forces. The expressions of these fluxes can be simplified, taking advantage of the large magnitude of the transition state enthalpies, h j,. We obtain the forward fluxes J g1 ;þðtþ ¼ D g 1 exp m dðtþþm ATP þ m H O h 1; C 1; k B T J g ;þðtþ ¼ D g exp m (19) aðtþþm ADP þ m Pi h ; C ; k B T where the C j, are constants of order one (see ESI ). Once the fluxes have been identified, it is possible to quantify the slipping of the molecular motor, 1 z, related to the conformational changes it experiences without ATP hydrolysis. In the usual regime at which the motor operates, using eqn (14) and J gj (t) E J gj,+(t) one obtains for z z 1 J g ;þ J g1 ;þ () ¼ 1 D g C 1; exp DmðtÞ DGþh 1; h ; D g1 C ; k B T 4 Molecular motor displacement Once we have identified the relevant thermodynamic expressions for the fluxes associated with the conformational changes that the molecular motor experiences, when it consumes ATP on a filament, we can analyze, within the same framework, how these changes lead to a displacement of the motor. We have seen, eqn (1c), that this process in the steady state is characterized by a constant flux along the g 3 J x ðx a þ g 3 ; 1; g 3 Þ ¼ L x ðg 3 Þ mðx a þ 3 ; 1; 1Þ f ext (1) 3 The chemical potential associated with the displacement of the motor can be expressed in terms of the motor probability density as m(x a + g 3,1,1) = h(x a + g 3 )+k B T ln P(x a + g 3 ) () where h is an effective enthalpy profile that characterizes the effective energetic changes the molecular motor experiences as it moves along the filament. As shown in Fig. 5, this effective enthalpy, h(x a + g 3 ), associated with the displacement, is not periodic out of equilibrium. Its departure from a periodic function describes the departure from equilibrium of the activated processes; it is therefore an implicit function of the control parameters that determine the velocity of the molecular motor. If we keep the assumption that the Onsager kinetic coefficient is proportional to the probability density, as in eqn (17), we can introduce the constant diffusion coefficient L x (g 3 )=DP(x a + g 3 ) (3) 1941 Phys. Chem. Chem. Phys., 13, 15, This journal is c the Owner Societies 13

7 Fig. 5 The activated molecular motor motion along the effectively sinking enthalpy profile, h(x a + g 3 ). The motor is pulling against an external force f ext. After ATP-hydrolysis, the motor slips or moves past barriers at x a + N, where N = 1,,3... that characterizes the spatial diffusion of both activated and deactivated molecular motors. By substituting eqn () and (3) into eqn (1) we obtain J x ¼ D k B ð a þ g 3 Þ þ Px ð a þ 3 ð a þ g 3 Þ f ext 3 (4) which agrees with the two-state model. 17 We can proceed with our analysis without the need of this expression. In the third process described in eqn (1), activated motors deactivate along the g 3 -coordinate and work is done, if the motors displace under the action of an applied external force. During deactivation, the sum of the activated plus the deactivated motors remains the same and as a consequence the probability distribution is uniform, P(x a + g 3 )=P a (x a + g 3 )+ P d (x a + g 3 ) = 1 if properly normalized. Eqn (4) therefore simplifies to J x ¼ ð a þ g 3 Þ f ext 3 and, using quasi-stationarity and the fact that the flux is homogeneous, integrating over g 3 we identify the magnitude of the motor velocity along the x-axis J x = D[h(x a +1) h(x a ) f ext ] (6) which shows the central role played by the effective enthalpy profile. At equilibrium h(x a +1)=h(x a ) and no load can be carried. Away from equilibrium, the imbalance h(x a +1)a h(x a ) depends on Dm and DG (the driving force for activation and ATP hydrolysis), see eqn (8) and (9). This allows the molecular motor to carry a load f ext, up to the value of the stall force fext M = h(x a +1) h(x a ). This identity quantifies how the stall force will depend on ATP concentration. We show in the ESI that the enthalpy difference depends on the position where the reaction takes place, x a. With a linear variation in h d we obtain h(x a +1) h(x a )=db( 1 x a), where the product db is a constant. Due to the choice of units, these diffusion coefficients have also units of S 1. Far from equilibrium, mass conservation, J x = J g1 J g (eqn (13)), reduces to J x C J g1,+ J g,+ (7) showing how slipping is decremental for motor displacement. Close to equilibrium, displacement will obey eqn (13), and will vanish in equilibrium, when J g1 = J g =. Combining eqn (19) and (7) J x ¼ D g 1 exp m d þ m ATP þ m H O h 1; C 1; k B T D g exp m a þ m ADP þ m Pi h ; C ; k B T (8) provides an explicit expression of how the molecular motor translocation depends on the chemical reaction in the absence of any applied force on the motor. This expression provides a means to identify how h(x a +1) h(x a ) depends on the thermodynamic forces driving the conformational changes of the molecular motor, because h is independent of external forces. Hence, in the usual regime when the motor is far from equilibrium hx ð a þ 1Þ hx ð a Þ ¼ 1 D g1 exp m d þ m ATP þ m H O h 1; DC 1; k B T 1 D g exp m a þ m ADP þ m Pi h ; DC ; k B T (9) Under the action of an external force, f ext, the molecular motor will be able to phosphorylate ADP or hydrolyze ATP depending on whether it acts against or favors a motor displacement, and the values of DG and Dm. 5 The entropy production and the motor efficiency The formalism put forward allows us to determine the entropy production for a moving molecular motor. When we multiply the entropy production with the temperature of the surroundings we obtain the lost work, or the energy dissipated as heat in systems and surroundings. 16 In most biological systems, it is sufficient to look at the system as being isothermal. In an analysis of experimental data from the Ca-ATPase, Lervik et al. 4 found a second law efficiency near.13. The contribution from the heat flux to the entropy production was negligible. We shall therefore use the system temperature as the temperature of the surroundings, in the calculation of the dissipated energy at the stationary state. In that case eqn (9) applies. All fluxes are constant along the integration coordinate, and we find, similar to ref. 17 T dsðtþ irr dt ¼ J g1 ðdg DmÞ J g Dm J x ðdm f ext Þ (3) ¼ J g1 DG þ J x f ext where the second identity follows from mass conservation, and indicates that the molecular motor motion is characterized by This journal is c the Owner Societies 13 Phys. Chem. Chem. Phys., 13, 15,

8 two independent fluxes, as discussed in Section. The bilinear form of this equation does not mean that the flux force relations are necessarily linear. As pointed out by Ross and Mazur, the entropy production of non-linear chemical reactions maintains this form. So, the form is not reserved for near-equilibrium conditions, but applies to the model described here. We observe the interesting feature that, by going from the first to the second line in eqn (3), the terms containing Dm cancel. This is because the present model specifies work done on the cost of the internal energy storage capacity in h a, not depending on the boundary conditions. The internal reservoir is a property of the model, and disappears by integrating into the macroscopic level at the stationary state. As a consequence, we have three fluxes on the mesoscopic level, which do not couple, while there are two independent fluxes on the macroscopic level, which have a non-zero coupling coefficient. The mechanism behind this coupling coefficient can be understood by the mesoscopic model, or another model compatible with the one presented here. 1 The macroscopic thermodynamic description does in principle not give information on the mechanism. In classical irreversible thermodynamics the entropy production (eqn 3) leads to the famous linear relations between fluxes and forces, discussed at length by Caplan and Essig. 11 For muscular contraction they give, using our terminology: Jg 1 = L rr DG + L rx f ext J x = L xr DG + L xx f ext (31) with L rx = L xr. The coefficients in eqn (31) characterize the macroscopic performance of a system described by the entropy production in (3). The coefficient can be traced back to the explicit expressions for the fluxes that we have derived at the mesoscopic level. They can in this manner be related to relevant thermodynamic variables and transport coefficients that characterize the conformational changes that the molecular motor experiences, given by eqn (18) and (8). Processes can be characterized by efficiencies based on the first or on the second law. The first law efficiency is defined by the work output per unit of heat added to the system. Z I ¼ W Q (3) Subscript I refers to the first law. This efficiency is used to compare combustion engines. The heat input is then the enthalpy change of the fuel. In the absence of any entropy production (a reversible engine) it gives the Carnot efficiency for a motor working between two temperature reservoirs. This efficiency does not provide a measure of the energy quality and cannot be used to compare different types of engines. In a saline power plant, for instance, the enthalpy difference is very small (Q = DH), making Z I c 1. This plant runs on the entropy of mixing of salt and fresh water. An efficiency based on the second law will measure the deviation of motor operation from the reversible limit (unit efficiency), and is therefore well suited for performance comparisons of all motor types. The definition is 16 Z II ¼ W ideal T dsðtþ irr =dt ¼ W (33) W ideal W ideal where T is the temperature of the surroundings. Subscript II refers to the second law. Here W ideal is the maximum energy available for work per unit of time, equal to J g1 DG. The form is useful when the energy source is a chemical reaction. The energy available is then minus the Gibbs energy change of the fuel, as not only the enthalpy, but also the entropy (heat) can be transformed into work in a reversible process. This is well known from electrochemical cells. Chemical energy sources are most relevant in biology. The real work per unit of time W (power) done by the present motor becomes W = J x f ext (34) By introducing the expression for the entropy production, we reduce the expression to Z II ¼ J x f ext J g1 DG (35) which agrees with the definition proposed previously by many authors, see e.g. ref. 11. The ratio is always a number between and 1. The value is positive as DG o. In our case, the expression for the external force from eqn (6) is f ext ¼ 1 D J x þ ½hx ð a þ 1Þ hx ð a ÞŠ In the absence of an external force, the efficiency is zero. In that case (equilibrium), h(x a +1)=h(x a ). Clearly J x is a function of the enthalpy difference, or vice versa. By introducing the expression for the external force in W, the power is a parabolic function of the motor velocity for a constant enthalpy difference: W ¼ 1 D J x þ ½hx ð a þ 1Þ hx ð a ÞŠJ x (36) This dependence of the power W on the motor velocity is illustrated in Fig. 6. The parabolic work function in Fig. 6 is bounded by the maximum work available in the system, DG. The reaction Gibbs energy for ATP hydrolysis is commonly near 46 kj mol 1 ATP reacted. 3 The power is zero when the velocity is zero, and when the whole energy is dissipated as work, when the velocity is maximum. The situation at maximum velocity can be understood as an event where there is no time to form the bond that will start controlled deactivation and work. All myosin heads are slipping, 1 z =,andallenergyis dissipated as heat. There is an optimal velocity of operation, where the highest efficiency is reached. We find this velocity, J x,opt, by differentiating, setting qw/qj x =. The velocity depends on the diffusion constant, and the shape of the potential barrier according to, J x;opt ¼ D ½ hx ð a þ 1Þ hx ð a ÞŠ (37) We can use the expression to find D once the enthalpy difference is known from the initial slope of the curve. The maximum 1941 Phys. Chem. Chem. 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9 Fig. 6 The power of the molecular motor W in kj per mol ATP reacted, as a function of the motor velocity in m s 1 and the lost work. The maximum work outcome possible is given by DG (46 kj mol 1 ). The area under the maximum work-line is divided between work W and lost work T ds irr /dt. The work per unit of time is J x f ext. The lost work, or the entropy production times the temperature of the surroundings, can be measured by isothermal calorimetry. 3 The slope of the work curve is determined from eqn (38). efficiency, Z II,opt, valid for the particular non-equilibrium situation, follows from Z II;opt ¼ 1 D ½ J g1 DG hx ð a þ 1Þ hx ð a ÞŠ þ 1 D J x;opt (38) In his review of experimental results, Førland 19 suggested that a constant fraction of motors 1 z was always slipping (deactivating without stepping forward). He estimated the loss due to slip equal to 14 kj mol 1 per mol ATP reacted. The distance between the line for the maximum theoretical available work in the figure and the maximum in the expression of the work is then 14 kj mol 1.The difference in the two numbers, 3 kj mol 1 per mol ATP reacted, can then be associated with the maximum allowed value of the chemical potential change in the myosin heads (the enthalpy change), Dm = h(x a +1) h(x a )= 3 kj mol 1. The lost work is at any time a function of the velocity measured in muscle unit lengths per second. Førland 19 gave a nearly parabolic function for W = W( J x ), similar to the picture in Fig. 6. Linear flux force equations were used in the part of the cycle where work was performed, the second step in eqn (1). A constant slip leading to a loss of 14 kj mol 1 corresponds to a reduction in the second law efficiency from 1 (the reversible limit) to.7. Frictional losses lead to further reductions. The optimum of the curve in ref. 19 gives a maximum velocity of J x,opt =5 1 7 ms 1 and a value of Z II,max =.45. An efficiency based on the second law measures the degree of reversibility of the actual motor, contrary to the first law efficiency, Z I. 14,4 Most biological systems are nearly isothermal, even if they are large heat producers (heat can be produced reversibly). A comparison with a heat engine is therefore less relevant. 8 We suggest that the second law efficiency be systematically used, as it can be used to compare all types of motors, smallorlarge.thevalue.45compareswellwithapolymer electrolyte fuel cell efficiency. Clearly, also the movement of filaments and of cargo implies energy dissipation in terms of viscous losses. We have drawn the system boundaries around the motor to make clear that the force f ext is controlled externally. Viscous losses in the surroundings, as, e.g., if the molecular motor carries and displaces a cargo, do then not enter the definition of the motor efficiency in eqn (33). Internal viscous losses can also play a role, giving an explicit term in the entropy production. Førland estimated internal viscous losses in a qualitative manner. Such losses are not described in our model. Following eqn (33), a system that includes dissipation due to movement of filaments, will obtain an extra viscous term in the denominator of eqn (35). 6 Conclusions In this article, we have shown that mesoscopic non-equilibrium thermodynamics can be used in a systematic way to find nonlinear flux force relations for a slipping molecular motor. The movement of myosin along an actine filament is taken as an example. The motor becomes activated upon reaction with ATP far from a linear response regime.wehavechosenasimple mechanism proposed earlier for this case 19 to find the entropy production and conjugate flux force relations on the mesoscopic level. The mesoscale formulation was integrated to give the classical macroscale form of the entropy production. 11 An expression was found for the maximum velocity of myosin during muscular contraction. Literature data 19 indicate that the efficiency of muscular contraction under these conditions is.45, a value comparable to that of a polymer electrolyte fuel cell. The particular example could be solved, and we could clarify how our results are related to the description used by Jülicher et al. 17 Acknowledgements DB and SK are grateful for the financial support during a sabbatical stay at University of Barcelona. IP and JMR acknowledge the Dirección General de Investigación (Spain) and the DURSI project for the financial support under projects FIS and 9SGR-634, respectively. JMR acknowledges the financial support from Generalitat de Catalunya under program Icrea Academia. References 1 T. L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics, Dover, New York, S. Kjelstrup, J. M. 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