Tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor

We present a tight-binding derivation of a discrete-continuous description of mechanochemical coupling in a molecular motor. Our derivation is based on the continuous diffusion equation for overdamped Brownian motion on a time-independent tilted periodic potential in two dimensions. The free-energy potential is nonseparable to allow coupling between the chemical and mechanical degrees of freedom. We formally discretize the chemical coordinate by expanding in Wannier states that are localized along the chemical coordinate and parametrized along the mechanical coordinate. A discrete-continuous equation is derived that is valid for anisotropic systems with weak mechanochemical coupling and deep potential wells along the chemical coordinate. The discrete-continuous description is consistent with established theoretical models of molecular motors with discrete chemical states but is constrained by the underlying continuous two-dimensional potential. In particular, we derive analytic expressions for the effective potential along the mechanical coordinate and for the rate of thermal hopping between chemical states. We determine the thermodynamic efficiency of energy conversion and find that, for a molecular motor with one chemical state per cycle, the derived discrete-continuous equation can accurately describe mechanochemical coupling but cannot describe energy conversion.

Figure 1

Periodic potential Eq. (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, and $C=6\mathrm{\Theta }$.

Figure 2

Steady-state probability density for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, and $C=6\mathrm{\Theta }$. Other parameters are $f_x=0$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.

Figure 3

Steady-state current density ${\gamma }_x(q_x{J}_x,q_y{J}_y)/\mathrm{\Theta }{q}_x^3{q}_y$ for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, and $C=6\mathrm{\Theta }$. Other parameters are $f_x=0$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.

Figure 4

The hopping rates (black) ${\kappa }_1\left(x\right)$ and (gray) ${\kappa }_{-1}\left(x\right)$ for the potential (62) with $B=20\mathrm{\Theta }$ and $C=2\mathrm{\Theta }$, and with force ($\times$) $f_y=0$, ($\circ$) $f_y=5\mathrm{\Theta }/a_y$, and ($\square$) $f_y=10\mathrm{\Theta }/a_y$. The Kramers escape rate Eq. (47) is shown as the solid, dashed, and dotted curves, respectively.

Figure 5

The Wannier states for potential (62) with $B=20\mathrm{\Theta }$, $C=2\mathrm{\Theta }$, and $f_y=0$. The white dashed curve shows the minimum of the potential $V^c\left(y\right)+W(x,y)$ for each $x$.

Figure 6

The onsite additional terms ($\times$) ${\mathcal{K}}_0^{\left(1\right)}$ and ($\circ$) ${\mathcal{K}}_0^{\left(2\right)}$ defined according to Eqs. (36) and (64) for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=2\mathrm{\Theta }$, $f_x=-2\mathrm{\Theta }/a_x$, and $f_y=5\mathrm{\Theta }/a_y$. The curves are (solid) ${\mathcal{K}}_0^{\left(1\right)}$ and (dashed) ${\mathcal{K}}_0^{\left(2\right)}$ for the approximate expression (50) with $W(x,y)$ evaluated at (black) $y=y_0\left(x\right)$ and (gray) $y=0$.

Figure 7

Drift (a) $v_x$ and (b) $v_y$ according to ($\times$) Eq. (52) and (solid) Eq. (1) for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=2\mathrm{\Theta }$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.

Figure 8

Steady-state current density ${\gamma }_x(q_x{J}_x,q_y{J}_y)/\mathrm{\Theta }{q}_x^3{q}_y$ for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=6\mathrm{\Theta }$, $f_x=0$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=1$.

Figure 9

Drift (a) $v_x$ and (b) $v_y$ according to ($\times$) Eq. (52) and (solid) Eq. (1) for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=2\mathrm{\Theta }$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=1$.

Figure 10

Steady-state current density ${\gamma }_x(q_x{J}_x,q_y{J}_y)/\mathrm{\Theta }{q}_x^3{q}_y$ for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=30\mathrm{\Theta }$, $f_x=0$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.

Figure 11

Drift [(a) and (b)] $v_x$ and [(c) and (d)] $v_y$ according to ($\times$) Eq. (52) and (solid) Eq. (1) for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=30\mathrm{\Theta }$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.

Figure 12

Drift (a) $v_x$ and (b) $v_y$ according to ($\times$) Eq. (52) and (solid) Eq. (1) for potential (62) with $A=-5\mathrm{\Theta }$, $C=2\mathrm{\Theta }$, $f_x=-2\mathrm{\Theta }/a_x$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$. The vertical axis on subplot (b) is scaled logarithmically.

Figure 13

Thermodynamic efficiency for potential (62) with $A=-\mathrm{\Theta }$, $B=20\mathrm{\Theta }$, $C=30\mathrm{\Theta }$, $f_y=5\mathrm{\Theta }/a_y$, and $\mathrm{\Gamma }=10$.