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[EQUATION] Here, we see that Eq. ( 40 ) is the Michaelis-Menten constant defined as the ATP concentration at which the velocity [MATH] equals [MATH] |
HMSPR05 . This constant characterizes the crossover between the regime [MATH] where the rotation is limited by the slow arrival of ATP molecules and the saturation regime [MATH] |
where the velocity reaches its maximum value fixed by the finite rate of release of ADP and P . Accordingly, the constant ( 39 ) is the maximum angular velocity of the motor, which is reached in a regime where the ATP concentration is large enough with respect to the Michaelis-Menten constant: |
[EQUATION] the limit meaning that [MATH] The Michaelis-Menten constant itself can be determined at low ATP concentrations according to |
[EQUATION] this other limit meaning that [MATH] The third constant ( 41 ) characterizes the decrease of the velocity as the concentrations of the products ADP and P are increased. It turns out as we shall see in the following that this constant is larger by several orders of magnitude than the equilibrium constant: |
[EQUATION] Accordingly, the term involving the equilibrium constant in the numerator of Eq. ( 38 ) can be neglected. Hence, the constant [MATH] can be determined in the regime where |
[EQUATION] by taking the difference of the inverses of the velocities at two different concentrations of ADP, keeping fixed the other concentrations: |
[EQUATION] with the notation [MATH] Another consequence of the inequality ( 45 ) is obtained by using the definition ( 37 ) of the equilibrium constant: |
[EQUATION] whereupon the Michaelis-Menten constant ( 40 ) is essentially independent of [MATH] in the present case: [EQUATION] and thus directly determines the constant [MATH] of ATP binding once the constant [MATH] of product release is obtained thanks to the maximum velocity ( 39 ). Subsequently, the constant [MATH] ... |
[EQUATION] III Comparison with the continuous-state description In this section, we compare the aforementioned discrete-state description with a continuous-state description we previously reported on pgeg . The continuous-state description considers the rotation angle as a continuous random variable instead of supposin... |
III.1 Fokker-Planck equation description In the continuous-state model pgeg , the system is found at a given time [MATH] in one of the six chemical states [MATH] and the [MATH] -shaft at an angle [MATH] There are six chemical states because the three [MATH] -subunits can be either empty or occupied by a molecule of ATP... |
Consequently, the system is described by six probability densities [MATH] normalized according to [MATH] The time evolution of the probability densities is ruled by a set of six Fokker-Planck equations coupled together by the terms describing the random jumps between the chemical states [MATH] due to the two chemical r... |
[EQUATION] where the probability current densities are given by [EQUATION] The diffusion coefficient [MATH] can be expressed in terms of the friction coefficient [MATH] according to Einstein’s relation [MATH] . The friction coefficient [MATH] can be evaluated for a bead attached to the [MATH] -shaft, a bead duplex, or ... |
[EQUATION] with the water viscosity [MATH] pN s nm -2 and [MATH] pgeg KAI04 When the motor is in the chemical state [MATH] , the [MATH] -shaft is submitted to the external torque [MATH] and the internal torque [MATH] due to the free-energy potential [MATH] of the motor with its [MATH] -shaft at the angle [MATH] . Apply... |
III.2 Coarse graining into a discrete-state model The correspondence with the discrete-state description in terms of the master equation ( 25 ) can in principle be established by coarse graining the continuous angle into discrete states. These discrete states correspond to the angular intervals |
[MATH] where the [MATH] -shaft spends most of its time while in the chemical state [MATH] . Accordingly, the probabilities ruled by the master equation ( 25 ) are related to the probability densities of the continuous-state description ( 54 ) according to |
[EQUATION] where the angular integral is carried out over the aforementioned intervals. In this way, a fully discrete description could be inferred from the continuous-state description. In general, this reduction from one description to the other by the aforementioned coarse graining leads to non-Markovian equations. ... |
In this case, the transition rates [MATH] of the master equation ( 25 ) can be deduced from the solution of the Fokker-Planck equations ( 54 ). We emphasize that the transition rates [MATH] of the master equation ( 25 ) do not take the same values as the rates [MATH] appearing in the Fokker-Planck equations ( 54 ). Ind... |
We notice that the rates [MATH] appearing in the Fokker-Planck equations ( 54 ) only concern the chemical reactions and, therefore, do not depend on the friction coefficient [MATH] and the external torque [MATH] which only enter in the current densities ( 55 ) appearing in the left-hand side of the Fokker-Planck equati... |
III.3 Dependence of the reaction constants on friction and external torque The original and key feature of our work is the inclusion in the discrete-model reaction constants [MATH] [MATH] ) of the mathematical dependence on friction and external torque. This dependence expresses the coupling between the chemical and me... |
We determine the dependences of the reaction constants [MATH] on both friction and external troque by fitting them to the simulations of our continuous-angle model pgeg . Since this latter has been fitted to experimental data, the present fitting procedure is comparable to a fitting to the experimental data of Ref. N41... |
In this regard, an essential observation is that the reaction constants become independent of the friction coefficient [MATH] at low friction and decrease as the inverse of the friction at high friction: [MATH] . The motor is functioning in a reaction-limited regime at low friction and in a friction-limited regime at h... |
The crossover between the low- and high-friction regimes can be well described by giving the following analytical form to the reaction constants: |
[EQUATION] with [MATH] The coefficients of the function in the denominator are taken as exponentials in order to guarantee the positivity of the reaction constants, the friction coefficient [MATH] being always non-negative. The coefficients [MATH] can be determined in the low-friction regime and the coeffcients [MATH] ... |
The dependence on the external torque [MATH] appears in the functions [MATH] and [MATH] , which are taken as expansions in powers of [MATH] limited to the second order: |
[EQUATION] with [MATH] The coefficients of these expansions are fitted in intervals of values of the external torque which are typically [MATH] pN nm. The values of the coefficients of Eqs. ( 59 )-( 60 ) for the reaction constants [MATH] [MATH] , and [MATH] are given in Table |
The last constant for ATP unbinding is finally obtained by using Eq. ( 53 ) as [EQUATION] with [MATH] pN nm. IV Properties of the F motor |
The comparison between the discrete and continuous descriptions sheds a new light on the properties of the F motor. On the one hand, the continuous-angle model reproduces the experimental observations of Ref. N410 and its simulation can test the assumptions of the discrete-state model, in particular, the assumption of ... |
IV.1 Tight versus loose chemomechanical coupling In order to determine the regime of tight coupling between the chemistry and the mechanics of the F motor, both the angular velocity [MATH] and the ATP consumption rate [MATH] have been simulated with the continuous-angle model ( 54 ) for different values of the external... |
In the tight-coupling regime, the condition ( 21 ) should hold, which implies that the mechanical and chemical affinities [MATH] and [MATH] are no longer independent but should combined into the unique chemomechanical affinity ( 23 ). In this regime, the vanishing of the rates, [MATH] and [MATH] , should thus occur on ... |
[EQUATION] This is observed in Fig. for values of the external torque extending from zero down to about [MATH] pN nm and for chemical potential difference from zero up to [MATH] , which delimits the zone where the tight-coupling assumption is satisfied. Outside this zone for higher values of [MATH] or lower values of [... |
Since the coincidence of the curves [MATH] and [MATH] along the straight line ( 62 ) is a feature of the discrete-state model ( 27 ), we may expect it describes the F motor in the aforementioned zone of tight coupling. |
IV.2 Rotation rate versus ATP concentration Random trajectories of the discrete model can be simulated thanks to Gillespie’s numerical algorithm gillespie1 gillespie2 . Examples of random trajectories are depicted in Fig. for different values of ATP concentration, illustrating the Michaelis-Menten kinetics described by... |
The crossover between the regime at low ATP concentration and the saturation regime of the Michaelis-Menten kinetics is seen in Fig. where we directly compare Eq. ( 42 ) of the discrete model with experimental data from Ref. N410 . At low ATP concentrations, the rotation rate is proportional to the ATP concentration wh... |
In order to appreciate the nonequilibrium thermodynamics of the molecular motor, it is interesting to depict the rotation rate as a function of the affinity ( 23 ) instead of the ATP concentration. Indeed, the former is a substitute of the latter, as shown by expressing the concentrations in terms of the chemical poten... |
[EQUATION] In this way, we recover the equilibrium relation ( 36 ) between the concentrations with Eq. ( 37 ) in the thermodynamic equilibrium state [MATH] corresponding to given concentrations of ADP and P . Substituting Eq. ( 63 ) into Eq. ( 38 ), we obtain the following expression for the rotation rate: |
[EQUATION] with the constant [EQUATION] This coefficient controls the linear response of the molecular motor because [EQUATION] The analytic form ( 64 ) shows that the rotation rate depends on the thermodynamic force [MATH] in a highly nonlinear way, in contrast to what is often supposed. The nonlinear dependence is ve... |
IV.3 Rotation rate versus friction In Fig. , we show the effect of friction on the motor’s velocity in the absence of ADP or P . At low friction, the velocity saturates exhibing the reaction-limited regime. At high friction, we see the rapid decrease of the velocity with increasing friction in the friction-limited regi... |
IV.4 Rotation in the presence of ATP hydrolysis products In the presence of ADP and [MATH] in the environment of the motor, Eq. ( 38 ) shows that the rotation rate decreases, as expected since these products tend to counteract ATP hydrolysis that is powering the motor. This phenomenon is known as ADP inhibition N410 KA... |
As observed in Fig. which compares the continuous and discrete models, this effect manifests itself above millimolar concentrations of ADP if inorganic phosphate is in millimolar concentration. We notice that the decrease of the rotation rate goes as the inverse of the concentrations of ADP and P [MATH] as described by... |
IV.5 Dependence on the external torque Figure depicts the dependence of the rotation and ATP consumption rates on the external torque [MATH] for both the continuous-angle and discrete-state models, showing their agreement in the range of validity of tight coupling. This range of validity is observed in Fig. to extend d... |
Figure also shows the remarkable feature of the discrete model to reproduce the phenomenon of stalling torque that the mean rotation rate can be stopped at a negative critical value of the external torque opposing the rotational motion. Indeed, the value of the external torque where both the rotation and ATP consumptio... |
[EQUATION] in the tight-coupling regime. For [MATH] , the chemical potential difference takes the values [MATH] pN nm and [MATH] pN nm, for respectively [MATH] M and [MATH] M. Hence, Eq. ( 67 ) gives respectively the stalling torques [MATH] pN nm and |
[MATH] pN nm, as indeed observed in Figs. a and b. IV.6 Chemical and mechanical efficiencies Under a negative external torque [MATH] , the F motor can synthesize ATP, in which case the ATP consumption rate as well as the rotation rate are negative, [MATH] and [MATH] In this regime of ATP synthesis, the chemical efficie... |
[EQUATION] such that [MATH] rmp A mechanical efficiency can similarly be defined in the regime where the rotation is powered by ATP as the inverse of the chemical efficiency rmp |
[EQUATION] The mechanical efficiency satisfies [MATH] in the regime where the external torque is still non-positive while both the rotation rate and the ATP consumption rates are positive, [MATH] and [MATH] |
It is known rmp that the chemical and mechanical efficiencies reach their maximum values under different conditions as shown in the Appendix. In the tight-coupling regime where [MATH] , these conditions coincide and the chemical and mechanical efficiencies ( 68 ) and ( 69 ) become |
[EQUATION] in agreement with Eq. ( 97 ) derived in the Appendix from the assumption of linear response. In the tight-coupling regime, the chemical and mechanical efficiencies can reach the maximal unit value at the stalling torque where Eq. ( 67 ) holds. This remarkable result is observed in Fig. depicting the chemical... |
In the interval [MATH] pN nm [MATH] , the external torque is opposed to the mean rotation rate but the motor consumes ATP, so that the mechanical efficiency ( 69 ) is postive. In this interval, the coupling is tight so that the mechanical efficiency computed with the continuous-angle model (squares) agrees with the pre... |
On the other hand, the motor synthesizes ATP under the action of the external torque for [MATH] pN nm, where the velocity becomes negative and the chemical efficiency ( 68 ) positive. As explained above, the motor is no longer in a regime of tight coupling below the stalling torque so that the chemical efficiency compu... |
Conclusions In the present paper, we have studied one of the simplest possible stochastic processes describing the stochastic chemomechanics of the F -ATPase molecular motor, as observed in Ref. N410 . This description considers the two discrete states corresponding to the steps and substeps observed in the rotary moti... |
N410 . This two-state discrete model is based on the master equation ruling the time evolution of the probabilities of the two discrete states. The model is analytically solvable, which provides us with an interesting insight in our understanding of the behavior of the F motor. The discrete model is set up by fitting i... |
The comparison between the discrete-state and continuous-angle descriptions reveals important properties of the F motor. Indeed, the discrete-state model presupposes that the mechanical motion and the chemical reactions powering the motor are tightly coupled, although the continuous-angle model does not. Accordingly, t... |
An important difference between the continuous and discrete models is that mechanical properties such as the friction of the object attached to the [MATH] -shaft in the liquid surrounding the motor or the external torque acting on the shaft explicitly appear in the Fokker-Planck equations defining the continuous model,... |
In the discrete-state model, the tight coupling between mechanics and chemistry implies that the motor is driven out of equilibrium by the unique chemomechanical affinity ( 23 ) combing the external torque with the chemical potential difference of ATP hydrolysis. Thermodynamic considerations shows that the chemical equ... |
Thanks to the solvability of the two-state model, the stationary solutions of the master equation can be exactly deduced, allowing us to show analytically that the mean rotation rate obeys a Michaelis-Menten kinetics with respect to the ATP concentration. Moreover, the analytical formula ( 38 ) is obtained for the rota... |
The highly nonlinear dependence of the mean rotation rate of the [MATH] -shaft ( 64 ) on the chemomechanical affinity ( 23 ) shows that the F motor is not functioning in the linear-response regime defined by Onsager’s linear-response coefficents, but instead typically runs in a nonlinear-response regime which is more t... |
Furthermore, the crossover between the reaction-limited and friction-limited regimes is well described by the two-state model as the friction coefficient [MATH] is increased. Although, the mean rotation rate is nealry independent of the friction coefficient in the reaction-limited regime at low friction, it decreases a... |
We have also investigated the behavior of the F motor in a surrounding filled with ADP and inorganic phosphate. As shown by Eq. ( 38 ), the mean rotation rate decreases as the concentrations of ADP and P exceed a crossover value. The reason is that the release of the products of ATP hydrolysis in the motor is counterac... |
The dependence of the rotation and ATP consumption rates on the external torque is also of great interest because it reveals the interval of values of the external torque where the tight-coupling condition holds. It is in this regime that the discrete-state model provides a good description of the motor. This compariso... |
Moreover, the nonequilibrium thermodynamics of the F motor has been developed, allowing us to study the chemical and mechanical efficiencies defined in Ref. rmp . In the tight-coupling regime, these efficiencies can reach their maximal unit value near the stalling torque. The coupling between mechanics and chemistry is... |
In conclusion, the two-state model and its comparison with the continuous-angle model of our previous paper pgeg is providing a powerful method to study the kinetic and thermodynamic properties of the F motor and, especially, the coupling between its mechanics and chemistry. |
Acknowledgments. This research is financially supported by the “Communauté française de Belgique” (contract “Actions de Recherche Concertées” No. 04/09-312), by the “Fonds pour la Formation à la Recherche dans l’Industrie et l’Agriculture” (F. R. I. A. Belgium), and the National Fund for Scientific Research (F. N. R. S... |
Appendix A Thermodynamic relations between affinities and currents In this appendix, the nonequilibrium thermodynamics of the F motor is presented in the linear regime very close to the equilibrium. |
A.1 The general case The molecular motor can be driven out of equilibrium by the two independent affinities which are the mechanical and the chemical affinities proportional respectively to the torque [MATH] and the chemical potential difference [MATH] . These affinities are the corresponding fluxes or currents can be ... |
[EQUATION] in which case the thermodynamic entropy production ( 20 ) takes the following form: [EQUATION] In general, the currents are nonlinear functions of the affinities |
[EQUATION] which vanish at the thermodynamic equilibrium where the affinities vanish, [MATH] Close to equilibrium, the currents can be expanded in powers of the affinities, which defines the linear response coefficients [MATH] as |
[EQUATION] The microreversibility implies the Onsager reciprocity relation: [EQUATION] and the non-negativity of the entropy production ( 73 ) the inequality |
[EQUATION] The coupling between the mechanics and the chemistry is here possible because the motor is attached to a solid support and keeps a fixed orientation. The chemistry thus becomes vectorial as well as mechanical. Therefore, there is no contradiction with the Curie symmetry principle according to which scalar an... |
In the plane of the affinities [MATH] , the curve where the velocity vanishes behaves around the origin as [EQUATION] while the curves where the ATP consumption rate vanishes is given by |
[EQUATION] Near the origin, the curve ( 80 ) has therefore a more negative slope than the curve ( 81 as the consequence of the inequality ( 79 ) resulting from the second law of thermodynamics rmp |
A.2 The case of tight coupling In the case of a tight coupling between the mechanics and the chemistry of the molecular motor, these two curves have the same slope around the thermodynamic equilibrium point. Indeed, the tight-coupling condition ( 21 ) reads |
[EQUATION] so that the entropy production becomes [EQUATION] with the chemomechanical affinity ( 23 ), which here reads [EQUATION] |
Therefore, the mechanical and chemical affinities are no longer independent in the tight-coupling regime where the chemomechanical affinity ( 84 ) becomes the unique nonequilibrium driving force. Accordingly, the unique current ( 82 ) is a function of this unique affinity: |
[EQUATION] and all the linear response coefficients become related to each other by [EQUATION] In this case, the inequality ( 79 ) reaches the equality: |
[EQUATION] which is also characteristic of tight coupling. A.3 The efficiencies The chemical and mechanical efficiencies ( 68 )-( 69 ) can be expressed as follows in terms of the affinities and the currents: |
[EQUATION] In the regime of linear response where the currents are the linear functions ( 76 )-( 77 ) of the affinities, the efficiencies can be written in the form |
[EQUATION] in terms of the coefficient [EQUATION] and the constant [EQUATION] such that [MATH] by the inequality ( 79 ). Equations ( 89 ) and ( 90 ) show that the efficiencies only depend on the ratio of the affinities rmp . In their respective domains of variation, the chemical and mechanical efficiencies can reach th... |
[EQUATION] This happens along different curves in the plane [MATH] of the affinities with respect to the curves where the rotation and ATP consumption rates vanish since rmp |
[EQUATION] In the tight-coupling limit, the coefficient ( 91 ) vanishes because of Eq. ( 87 ), [MATH] , so that the chemical and mechanical efficiencies take the value |
[EQUATION] according to Eqs. ( 89 ) and ( 86 ). In this case, the four conditions ( 93 )-( 96 ) coincide in [MATH] where the chemical and mechanical efficiencies reach their maximal value which is equal to unity, [MATH] . Therefore, tight coupling favors the optimization of the efficiencies. |
# Source: arxiv 0905.2145 # Title: Predictions from a stochastic polymer model for the MinDE dynamics in E.coli # Sections: all # Downloaded: 2026-03-03T05:14:52.254262+00:00 |
Predictions from a stochastic polymer model for the MinDE dynamics in E.coli Abstract The spatiotemporal oscillations of the Min proteins in the bacterium Escherichia coli play an important role in cell division. A number of different models have been proposed to explain the dynamics from the underlying biochemistry. H... |
Min proteins, polymer dynamics, stochastic oscillations, hybrid dynamical systems pacs: 87.16.A- Theory, modeling, and simulations 87.17.Ee Growth and division 87.10.Mn Stochastic modeling |
Introduction The spatiotemporal oscillations of the Min proteins in the bacterium Escherichia coli have been well studied both experimentally and theoretically (see Lutkenhaus ( 2008 for a recent review). Whereas many of the proposed models Meinhardt and de Boer ( 2001 ); Meacci and Kruse ( 2005 ); Huang et al. ( 2003 ... |
Our stochastic model consists of a set of four interacting linear polymers, a pair of MinD and MinE polymers at either pole of the cell, each of which can be in either a growing or a shrinking state. For any fixed combination of states for the four polymers, the dynamics are deterministic and described by a system of o... |
Cytrynbaum and Marshall Cytrynbaum and Marshall ( 2007 analysed a deterministic limit of infinitely high cooperativity, the solutions of which can be expressed easily in the form of fixed points of a one-dimensional map which we describe in the first subsection of Sec. IV . Formulating the model in this deterministic l... |
II Biology It is well established experimentally that two main processes control the position at which the rod-shaped bacterium E.coli divides. Nucleoid occlusion Yu and Margolin ( 1999 prevents division at sites within close proximity of the nucleoid, which – at the relevant time after DNA replication – is everywhere ... |
MinD is an ATPase that, in its ATP-bound form, binds to the inner cell membrane de Boer et al. ( 1991 . MinE is found to activate the ATPase activity of membrane-bound MinD and thereby removes MinD from the membrane Hu and Lutkenhaus ( 2001 . MinC co-localises to membrane-bound MinD Raskin and de Boer ( 1999a and is kn... |
The interaction of MinD and MinE in the presence of ATP and lipid membranes has been studied in vitro and MinD was found to accumulate into polymers and fibre bundles above certain concentrations Suefuji et al. ( 2002 . In vivo, the fluorescently labelled Min proteins appear to be organised into helical structures on t... |
III Model – a hybrid dynamical system We assume that both MinD and MinE aggregate only in the form of polymers on the inner side of the cell’s membrane. The Min oscillation results from an interplay between two MinD- and two MinE-polymers, one pair on each side of the cell. |
MinD monomers from the cytosol can start a polymer (i.e. nucleate) at one of the nucleation sites that are assumed to be positioned at each pole of the cell, an idea supported by recent experiments Mileykovskaya et al. ( 2009 ); Mazor et al. ( 2008 ); Touhami et al. ( 2006 . The probability of nucleation on an empty si... |
The geometry of our model cell is that of a cylinder with fixed length [MATH] and diameter [MATH] . Reported diffusion coefficients for the Min proteins are around [MATH] |
Meacci et al. ( 2006 so in a cell of length [MATH] and with characteristic reaction times on the order of seconds, the cytosolic concentrations of MinD and MinE are essentially uniform throughout the cell. Also, the time scale of ADP-ATP exchange in cytosolic MinD is assumed to be fast compared to the oscillatory dynam... |
The equations governing the dynamics of the polymers are distinct for the different discrete states between which the system jumps. The system describing the behaviour therefore is a hybrid dynamical system – a combination of continuous and discrete dynamics Champneys and di Bernardo ( 2008 . In our case, the continuou... |
III.1 Polymer dynamics For each of the four polymers (two MinD, two MinE), we track the projection of the polymers onto the long axis ( [MATH] ) of the cell (Fig. ). The four variables [MATH] [MATH] [MATH] and [MATH] describe these projected lengths for the polymers attached to the left and right pole of the cell, resp... |
III.1.1 MinD MinD polymerisation can start at either of two nucleation sites located at the two poles of the cell, i.e. at the positions [MATH] and [MATH] . We assume that one end of the MinD-polymers is fixed to one of these stationary nucleation sites, whereas at the other end (the ‘tip’), the polymer can elongate or... |
[EQUATION] [MATH] represents either constant disassembly or first order assembly and depends on the discrete state variable [MATH] of the polymer in question: |
[EQUATION] The dynamics of switching for the discrete state variables [MATH] , between the three states [MATH] , 0 and 1, will be explained in the next subsection. |
The cytosolic concentration [MATH] of MinD is determined by conservation of monomers: [EQUATION] where [MATH] is the volume of the cell (in [MATH] ), [MATH] converts between particles per [MATH] and [MATH] , and [MATH] is the total concentration of MinD monomers. |
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