Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles

Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the ${\text{F}}_1$-ATPase and the kinesin motor.

Figure 1

Schematic representation of a motor-bead model comprising a one-state motor (small blue sphere) attached via an elastic linker to the probe particle (large red sphere). An external force $f_{\text{ex}}$ is applied to the bead. The transition rates of the motor are denoted by $w^{+}(n,x)$ and $w^{-}(n,x)$. The load-sharing factors ${\theta }^{+}$ and ${\theta }^{-}$ indicate the position of an underlying unresolved potential barrier relative to the minimum of the free-energy landscape of the motor.

Figure 2

Coarse-grained rates ${\Omega }^{+}$ and ${\Omega }^{-}$ (top) and average velocity (bottom) as functions of $f_{\text{ex}}d$ for various friction coefficients $\gamma$ in the range $5\text{s}/d^2\ge \gamma \ge 5\times {10}^{-10}\text{s}/d^2$ (from bottom to top). With decreasing $\gamma$, the rates and the velocity approach the corresponding fast-bead limit (solid black lines). Parameters: $\kappa =40d^{-2}$, $c_T=c_D=2\times {10}^{-6}\text{M}$, $c_P={10}^{-3}\text{M}$, $\Delta \mu =19$, ${\theta }^{+}=0.1$, $w_{0}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=3\times {10}^7{\left(\text{Ms}\right)}^{-1}$.

Figure 3

Coarse-grained rates ${\Omega }^{+}$ and ${\Omega }^{-}$ as functions of $f_{\text{ex}}d$ for various $c_T$, $c_D$ in the range $2\times {10}^{-5}\text{M}\ge c_T,c_D\ge 2\times {10}^{-12}\text{M}$ (from top to bottom). With decreasing $c_T,c_D$, the rates approach the fast-bead limits ${\widehat{\Omega }}^{+}$ and ${\widehat{\Omega }}^{-}$ (straight lines). Parameters: $\kappa =40d^{-2}$, $\gamma =0.5\text{s}/d^2$, $c_P={10}^{-3}\text{M}$, $\Delta \mu =19$, ${\theta }^{+}=0.1$, $w_{0}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=3\times {10}^7{\left(\text{Ms}\right)}^{-1}$.

Figure 4

Top: Average rates $\langle w^{+}\rangle$ and $\langle w^{-}\rangle$ as functions of $f_{\text{ex}}d$ for various $\gamma$ in the range $5\text{s}/d^2\ge \gamma \ge 5\times {10}^{-9}\text{s}/d^2$. With decreasing $\gamma$, the rates approach ${\widehat{\Omega }}^{+}$, ${\widehat{\Omega }}^{-}$ (solid black lines). Bottom: Ratio of $+$ and $-$ rates. In contrast to ${\Omega }^{+}$, ${\Omega }^{-}$ (large red dots), the averaged motor rates do not fulfill the LDB condition (solid black line). The parameters are the same as in Fig. 2.

Figure 5

Coarse-grained rates ${\Omega }^{+}$ and ${\Omega }^{-}$ (top) and average velocity (bottom) for various $\gamma$ and $f_{\text{ex}}=0$ as functions of $c_T$. Since $c_D$ and $c_P$ are fixed, $\Delta \mu$ also increases with $c_T$. The rates and the velocity approach the fast-bead approximation (solid black lines). Parameters: $c_D=2\times {10}^{-6}\text{M}$, $c_P=1\times {10}^{-3}\text{M}$, $\kappa =40d^{-2}$, ${\theta }^{+}=0.1$, $w_{0}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=3\times {10}^7{\left(\text{Ms}\right)}^{-1}$, $\gamma$ in the range $5\text{s}/d^2\ge \gamma \ge 5\times {10}^{-9}\text{s}/d^2$ (from bottom to top).

Figure 6

Trajectories of the one-state model for the ${\text{F}}_1$-ATPase for several parameter sets obtained from simulations. The trajectory of the detailed model (motor: steplike blue lines; probe: fluctuating red lines) is shown together with a trajectory of its corresponding coarse-grained model [green (light gray)]. Parameters: $\kappa =40d^{-2}$, ${\theta }^{+}=0.1$, $w_{0}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=3\times {10}^7{\left(\text{Ms}\right)}^{-1}$, $\gamma =0.5\text{s}/d^2$, $f_{\text{ex}}=0$, $c_T=c_D=2\times {10}^{-6}$ M, $c_P=0.001$ M (top left); $\gamma =0.5\text{s}/d^2$, $f_{\text{ex}}=40d^{-1}$, $c_T=c_D=2\times {10}^{-6}$ M, $c_P=0.001$ M (top right); $\gamma =0.005\text{s}/d^2$, $f_{\text{ex}}=0$, $c_T=c_D=2\times {10}^{-6}$ M, $c_P=0.001$ M (bottom left); $\gamma =0.5\text{s}/d^2$, $f_{\text{ex}}=0$, $c_T=0.001$ M, $c_D=2\times {10}^{-6}$ M, $c_P=0.001$ M (bottom right).

Figure 7

Network representation of a motor-bead model with four internal motor states and discretized state space of the probe particle (left). Each row of black dots represents one motor state while the dots themselves represent specific distances $y$ accessible to the probe particle (via the vertical red lines) within the same motor state. Transitions between motor states either leave $y$ the same (horizontal green lines) or can advance the motor by $d_{ij}^{\alpha }$ and change $y$ (diagonal blue lines). The top view of this network corresponds to the coarse-grained version of this model (right).

Figure 8

Schematic representation of a motor-bead model for the ${\text{F}}_1$-ATPase with two internal states of the motor, 1 [blue (dark gray)] and 2 [pale green (light gray)]. Transition between states 1 to 2 corresponding to the ${90}^{\circ }$ (${30}^{\circ }$) substep are labeled with superscript 90 (30). The transition rates are chosen accordingly from Eqs. (31) and (32).

Figure 9

Coarse-grained rates for the ${90}^{\circ }$ (top) and the ${30}^{\circ }$ (center) substep and average velocity (bottom) as functions of $f_{\text{ex}}d$ for various $\gamma$ in the range $5\text{s}/d^2\ge \gamma \ge 5\times {10}^{-10}\text{s}/d^2$ (from bottom to top). With decreasing $\gamma$, the rates and the velocity approach their corresponding fast-bead limit (solid black lines). Parameters: $\kappa =40d^{-2}$, $c_T=c_D=2\times {10}^{-6}\text{M}$, $c_P={10}^{-3}\text{M}$, ${\theta }_{90,30}^{+}=0.1$, $k_{12}^{90}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=3\times {10}^7{\left(\text{Ms}\right)}^{-1}$, $k_{21}^{90}exp\left[{\mu }_D^{\text{eq}}\right]/c_D^{\text{eq}}=3667.5{\left(\text{Ms}\right)}^{-1}$, $k_{21}^{30}=1000{\text{s}}^{-1}$, $k_{12}^{30}exp\left[{\mu }_P^{\text{eq}}\right]/c_P^{\text{eq}}=40{\left(\text{Ms}\right)}^{-1}$. The attempt frequencies are chosen on the basis of Refs. [53, 54] where very small probe particles have been used.

Figure 10

Coarse-grained rates for the ${90}^{\circ }$ (top) and the ${30}^{\circ }$ (center) substep and average velocity (bottom) for various $\gamma$ and $f_{\text{ex}}=0$ as functions of $c_T$. Since $c_D$ and $c_P$ are fixed, $\Delta \mu$ also increases with $c_T$. The rates and the velocity approach the fast-bead approximation (solid black lines). Parameters: $c_D=2\times {10}^{-6}\text{M}$, $c_P=1\times {10}^{-3}\text{M}$, $\kappa =40d^{-2}$, ${\theta }_{90,30}^{+}=0.1$, $\gamma$ in the range $5\text{s}/d^2\ge \gamma \ge 5\times {10}^{-10}\text{s}/d^2$ (from bottom to top).

Figure 11

Coarse-grained forward rates for various $\gamma =2.1\text{s}/d^2$, $0.26\text{s}/d^2$, $0.14\text{s}/d^2$, $0.017\text{s}/d^2$, $0.003\text{s}/d^2$, $0.001\text{s}/d^2$, $3.8\times {10}^{-4}\text{s}/d^2$ (from bottom to top) for the parameter sets I: $c_T=430$ nM, $c_P=1$ nM; II: $c_T=1$ mM, $c_P=1$ nM; III: $c_T=1$ mM, $c_P=200$ mM as used in [52]. Parameters that are the same for all sets I–III: $c_D=1$ nM, $\kappa =40d^{-2}$, ${\theta }_{90,30}^{+}=0.1$ and the attempt frequencies $k_{ij}^{\alpha }$ as given in Fig. 9. The values of $c_D=c_P=1$ nM are a rough estimate because there is no information about these concentrations in Ref. [52].

Figure 12

Six-state-model representing a kinesin motor adapted from Ref. [5]. The transition between states 2 and 5 is purely mechanical and corresponds to a step of length $d$ whereas all other transitions are pure chemical transitions. The motor model includes three cycles: $\mathcal{F}$, which, in the $+$ direction, includes ATP hydrolysis and forward stepping; $\mathcal{B}$, which includes ATP hydrolysis and backward stepping in its $+$ direction; and a pure chemical cycle (around the circle) that includes hydrolysis or synthesis of two ATP.

Figure 13

Coarse-grained rates for the mechanical transitions (with $y$ dependence) for various $\gamma$ in the range $0.077\text{s}/d^2\ge \gamma \ge 7.7\times {10}^{-10}\text{s}/d^2$ (from bottom to top). The rates approach the one-particle rates from Ref. [5] (solid black lines). Parameters: $\kappa =10d^{-2}$ [55], $c_T=0.001\text{M}$, $c_D=c_P={10}^{-9}\text{M}$ (estimated), ${\theta }^{+}=0.65$, ${\chi }_{ij}=0.25,0.15$, $k_{12}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=k_{45}exp\left[{\mu }_T^{\text{eq}}\right]/c_T^{\text{eq}}=2\times {10}^6{\left(\text{Ms}\right)}^{-1}$, $k_{21}=k_{23}=k_{34}=k_{56}=k_{61}=100{\left(\text{s}\right)}^{-1}$, $k_{32}exp\left[{\mu }_D^{\text{eq}}\right]/c_D^{\text{eq}}=k_{65}exp\left[{\mu }_D^{\text{eq}}\right]/c_D^{\text{eq}}=2\times {10}^4{\left(\text{Ms}\right)}^{-1}$, $k_{43}exp\left[{\mu }_P^{\text{eq}}\right]/c_P^{\text{eq}}=k_{16}exp\left[{\mu }_P^{\text{eq}}\right]/c_P^{\text{eq}}=2\times {10}^4{\left(\text{Ms}\right)}^{-1}$, $k_{25}=3\times {10}^5{\left(\text{s}\right)}^{-1}$, $k_{52}=0.24{\left(\text{s}\right)}^{-1}$, $k_{54}={(k_{52}/k_{25})}^2{k}_{21}$.

Figure 14

Left: Average velocity for the model with probe particle (colored lines with symbols) an the one-particle model from Ref. [5] (solid black line). Right: Coarse-grained rates for chemical transitions (with $y$ dependence) for $\gamma =0.077\text{s}/d^2$. Other parameters as given in Fig. 13.

Figure 15

Comparison of the simulated trajectory of motor and probe with a trajectory of the probe and the estimated motor position that was reconstructed using the simulated trajectory of the probe. The trajectories are shifted for better visibility. Parameters: $\kappa =40d^{-2}$, $\gamma =0.005\mathrm{s}/d^2$, $c_T=c_D=2\times {10}^{-6}\text{M}$, $c_P={10}^{-3}\text{M}$, $c_T^{\text{eq}}=3.33\times {10}^{-7}\text{M}$, $c_D^{\text{eq}}=0.0682\text{M}$, $c_P^{\text{eq}}=1\text{M}$, $f_{\text{ex}}=0$, lower boundary to set $i=2$: $x-\lfloor x\rfloor =0.375d$, upper boundary to set $i=2$: $x-\lfloor x\rfloor =0.89d$.

Figure 16

Average velocity of the kinesin model as a function of the external force. Depending on the size of the probe, stall conditions are reached for different values of the external force. Parameters as in Fig. 13.

Figure 17

Detail of the coarse-grained mechanical transition rates of the kinesin model as shown in Fig. 13. Near the stall force at $f_{\text{ex}}d\simeq 14$ these rates exhibit a pole. In the gray shaded range, they should not be interpreted as physical transition rates.