Analytical theory of the stochastic dynamics of the power stroke in nonprocessive motor proteins

Statistical distributions of the structural states of individual molecules of nonprocessive motor complexes such as actomyosins are examined theoretically by considering a two-state stochastic model coupled by chemical reactions along the reaction coordinate representing the internal conformational states of the motor. The use of a conformational reaction coordinate allows for the approximation of taking the rate constants as local in their dependence on the reaction coordinate, and yields a simple analytic solution of the stationary states. The approximation is also tested against numerical solutions with a nonlocal form of rate constants. The theory is well-suited for computational treatments based on atomic structures of protein constituents using free energy molecular dynamics simulations. With empirical sets of free energy functions, stationary distributions of forces exerted by a motor head compare well with known experimental data.

Figure 1

(Color online) The Brownian ratchet and the power stroke mechanisms as projections of two-dimensional free energy landscapes $G_i(x,y)$ into one-dimensional axes. (a) In the Brownian ratchet models, the stochastic variable $x$ is defined as the linear displacement of the motor protein center-of-mass along the track. (b) The conformational changes of the motor protein occur via the Brownian motion of the conformational reaction coordinate $y$. See Fig. 3 for the interpretation of the arrows in (a) within the conventional ratchet perspective.

Figure 2

(Color online) Swinging lever arm mechanism of muscle contraction. The myosin heads in strained states ${\mathrm{E}}_1^{*}$ and ${\mathrm{E}}_2^{*}$ are shown as shaded. Binding (unbinding) of S1 to an actin filament facilitates unbinding (binding) of phosphate (P), ATP (T), and ADP (D), and vice versa. The conformational reaction coordinate $y$ is defined here as the lever arm angle, and the distance $r$ between the center of the “hinge” and the actin binding site is also shown.

Figure 3

(Color online) The equivalence of the Brownian ratchet and the power stroke perspectives. $P_1^{*}$ and $P_2^{*}$ are the quasiequilibrium distributions of the motor head displacement $x$ under the PMF $G_1\left(x\right)$ and $G_2\left(x\right)$, respectively [Fig. 1a]. After the potential has been switched on to become $G_1$, the near-uniform distribution (dotted line at the bottom) has to evolve toward $P_1^{*}$, generating a net flux equivalent to the power stroke depicted as the downhill drift in Fig. 1a. Note that the true stationary distributions $P_1\left(x\right)$ and $P_2\left(x\right)$ will differ from quasiequilibrium ones due to the contributions of transient distributions indicated by arrows.

Figure 4

(Color online) The stationary population ${\overline{P}}_1\left(y\right)$ (solid lines) and ${\overline{P}}_2\left(y\right)$ (dotted lines) at various different ATP concentrations: (a) $c=0.01$, (b) $c=0.1$, (c) $c=1\mu \mathrm{M}$. Other parameter values are $\beta {\kappa }_1=\beta {\kappa }_2=2$, $y_0=\pi ∕4$, $D_1=D_2=0.2{\mathrm{s}}^{-1}$, and $F_{\mathrm{ex}}=0$.

Figure 5

(Color online) Stationary distributions of $P_1$ and $P_2$ with different width of rate constants $\sigma$ in Eqs. (24). The solid, dotted, and dashed lines are for $\sigma =0$ from Eqs. (20), and the Monte Carlo results for $\sigma =0.1$ and 0.3, respectively. Parameter values are the same as in Fig. 4 with $c=0.1\mu \mathrm{M}$ and $k_{-h}=0$.

Figure 6

(Color online) The mean value ${⟨y⟩}_1$ averaged over the distribution $P_1$ as a function of $c$ with different values of $\sigma$. The solid line is for $\sigma =0$ from Eqs. (20a), and the circles, squares, and triangles are the Monte Carlo data for 0.01, 0.2, and 0.3, respectively. Parameter values are the same as in Fig. 4.

Figure 7

(Color online) Distribution $P\left(F_l\right)$ of forces exerted by a myosin head to the actin filament while bound. The dotted and solid lines are from Eqs. (20a, 26) with $c=0$ and $0.1\mu \mathrm{M}$, respectively. The rectangles and circles are the experimental data in the given conditions from Ref. 33. Parameter values are the same as in Fig. 4 except $D_1=D_2=0.22{\mathrm{s}}^{-1}$.