Fluctuation theorem and large deviation function for a solvable model of a molecular motor

We study a discrete stochastic model of a molecular motor. This discrete model can be viewed as a minimal ratchet model. We extend our previous work on this model, by further investigating the constraints imposed by the fluctuation theorem on the operation of a molecular motor far from equilibrium. In this work, we show the connections between different formulations of the fluctuation theorem. One formulation concerns the generating function of the currents while another one concerns the corresponding large deviation function, which we have calculated exactly for this model. A third formulation concerns the ratio of the probability of observing a velocity $v$ to the same probability of observing a velocity $-v$. Finally, we show that all the formulations of the fluctuation theorem can be understood from the notion of entropy production.

Figure 1

A schematic of the rates for this two-state stochastic model of a single processive molecular motor. The position of the motor is $n$ and $y$ is the number of ATP consumed. The even and odd sites are denoted by $a$ and $b$, respectively. In the case of two headed kinesin, site $a$ represents a state where both heads are bound to the filament, whereas site $b$ represents a state with only one head bound. Note that the lattice of $\mathit{a}$ and $\mathit{b}$ sites extend indefinitely in both directions along the $n$ and $y$ axis. All the possible transitions are represented with arrows on this particular section of the lattice.

Figure 2

Cycles associated with the evolution of the motor of Fig. 1. Left: cycle for the position variable $n$; the average length $n$ undergone by the motor corresponds to half the number of full turns of the clock (the factor half is due to the period being equal to two lattice units) and with the rates as shown. Right: cycle for the chemical variable $y$, the average number of ATP units corresponds to the number of full turns with the rates shown. The affinities associated with these cycles are given by Eq. (11) for the mechanical variable and by Eq. (62) for the chemical variable.

Figure 3

Four modes of operation of a molecular motor, as delimited by $\overline{v}=0$ and $r=0$ [10]. The lines are generated with parameters that we have extracted by fitting the data for kinesin in Ref. [17] to our model, and this fit is shown in Fig. 4.

Figure 4

Kinesin velocity vs ATP concentration under an external force [30]. The solid curves are the fits of our model to data from Ref. [17]. From the top down, the plots are for $F_e=-1.05,-3.59$ and $-5.63\text{pN}$, respectively. Inset: Kinesin velocity vs force under a fixed ATP concentration. The solid curves are fits to the data of Ref. [17]. From the top down, the plots are for $\left[\text{ATP}\right]=2\text{mM}$ and $5\mu \text{M}$. From this fit, we obtained the following parameters for our model: $ϵ=10.81$, $k_0=1.4\times {10}^5\mu {\text{M}}^{-1}$, $\alpha =0.57{\text{s}}^{-1}$, ${\alpha }^{\prime }=1.3\times {10}^{-6}{\text{s}}^{-1}$, $\omega =3.5{\text{s}}^{-1}$, ${\omega }^{\prime }=108.15{\text{s}}^{-1}$, ${\theta }_a^{+}=0.25$, ${\theta }_a^{-}=1.83$, ${\theta }_b^{+}=0.08$, and ${\theta }_b^{-}=-0.16$.

Figure 5

Curves of equal efficiency $\eta$ within region A (which is delimited by the solid line and by the $y$ axis). The parameters are those used in Fig. 3 and obtained from the fit of Fig. 4. From the outside to the inside the curves correspond to $\eta =0.2$, $\eta =0.3$, $\eta =0.4$, $\eta =0.5$, and $\eta =0.58$. The absolute maximum efficiency for these parameters is about 59% and is located at $\Delta \mu \simeq 14$ and $f\simeq -4.9$.

Figure 6

Graphical illustration of the symmetry of the long-time fluctuation theorem (for simplicity a model without chemical variable has been used). The largest eigenvalue $\vartheta$ is shown as function of $\xi =\lambda /f$ for different values of the normalized force $f$. The symmetry of the long time fluctuation theorem corresponds to the symmetry of this curve with respect to $\xi =1/2$.

Figure 7

Graphical illustration of the symmetry of the finite-time fluctuation theorem for a simplified model without chemical variable. The left hand side of Eq. (39) is shown as function of $\xi =\lambda /f$ for different times (arbitrary units). The symmetry of the finite time fluctuation theorem amounts to the symmetry of this curve with respect to $\xi =1/2$.

Figure 8

Large deviation function $G\left(v\right)$, the solid line is the exact expression using Eqs. (49, 50) and the points are the numerical evaluation of the Legendre transform using Eq. (48). For the values of the rates used here, the average velocity, as given by Eq. (18), is $\overline{v}\simeq 80$: thus, the system is far from equilibrium. Note that $G\left(v\right)$ is minimum at $\overline{v}$ and that $G\left(\overline{v}\right)=0$.