Roles of local nonequilibrium free energy in the description of biomolecules

When a system is in equilibrium, external perturbations yield a time series of nonequilibrium distributions, and recent experimental techniques give access to the nonequilibrium data that may contain critical information. Jinwoo and Tanaka [Sci. Rep. 5, 7832 (2015)] have provided mathematical proof that such a process's nonequilibrium free energy profile over a system's substates has Jarzynski's work as content, which spontaneously dissipates while molecules perform their tasks. Here we numerically verify this fact and give a practical example where we analyze a computer simulation of RNA translocation by a ring-shaped ATPase motor. By interpreting the cyclic process of substrate translocation as a series of quenching, relaxation, and second quenching, the theory gives how the ATPase motor allocates the hydrolysis energy between individual substates until the end of the process. It turns out that the efficiency of RNA translocation is $48\%\text{–}60\%$ for most molecules, but $12\%$ of molecules achieve $80\%\text{–}100\%$ efficiency, which is consistent with the literature. This theory would be a valuable tool for extracting quantitative information about molecular nonequilibrium behavior from experimental observations.

Figure 1

Simulation results for a quenched process: (a), (b) The color-coded plus symbols indicate (negative logarithm of) the ensemble averages of Jarzynski's work on the left-hand side of Eq. (13) multiplied by $e^{-\beta F\left({\lambda }_0\right)}$. Each ensemble comprises paths that reach ${\mathrm{\Gamma }}_i$ at time $t$. The color-coded lines, the nonequilibrium free energy profiles, are drawn together for comparison. We show data for every five steps of $0\le t\le 1$ in (a) and every 200 steps of $1<t\le 40$ in (b) to prevent symbols from being overlapped (see Fig. 4 in the Appendix for all time data). Inset of (a) represents the color code for time steps. The same color code is used in (b). (c) Dashed blue line indicates the time series of the average over substates of nonequilibrium free energy, and dotted red line represents the time series of the average over substates of work content [negative logarithm of the left-hand side of Eq. (13)]. Inset of (c) shows the energy profile turned on at time $t=0$.

Figure 2

Simulation results for a process composed of quenching, continuous changing, second quenching, and relaxation: (a, b) The color-coded plus symbols indicate (negative logarithm of) the ensemble averages of Jarzynski's work on the left-hand side of Eq. (13) multiplied by $e^{-\beta F\left({\lambda }_0\right)}$. The color-coded lines, the nonequilibrium free energy profiles, are drawn together for comparison. We selected two time intervals that exhibit patterns not seen in the first experiment. We show data for every 50 steps of $1<t\le 19$ in (a) and every five steps of $19<t\le 21$ in (b) to prevent symbols from being overlapped (see Fig. 5 in the Appendix for all time data). Both panels used the same color code. (c) Dashed blue line indicates the time series of the average over substates of nonequilibrium free energy, and dotted red line represents the time series of the average over substates of work content. Inset of (c) shows the energy profiles $E(z;{\lambda }_t)$ for $0\le t\le 20$.

Figure 3

Analysis of the simulation result for RNA translocation by Rho hexameric helicase: (a) The rotary reaction from state $R$ (through $I$) to state $S$ is shown. Subunits' colors schematically represent subunits' conformations. Subprocesses (1), ATP binding, and (3), hydrolysis of another ATP (in the already occupied site), turn state $R$ into state $I$. Subprocess (2), the release of ATP-bound product ($\mathrm{ADP}+{\mathrm{P}}_i$), initiates the motor reaction phase during which subunits' conformations of state $I$ turn into those of state $S$. (b) Conformational free energies $G({\mathrm{\Gamma }}_i;E_{\mathrm{A}})-F\left(E_{\mathrm{A}}\right)$ in blue and $G({\mathrm{\Gamma }}_i;E_{\mathrm{B}})-F\left(E_{\mathrm{B}}\right)$ in orange are presented. (c) Initial probability distribution $p_0({\mathrm{\Gamma }}_i;E_{\mathrm{A}})$ in blue and final probability distribution $p_1({\mathrm{\Gamma }}_i;E_{\mathrm{B}})$ in orange are presented. The most populated substate in $p_0$ is state $I$, and that in $p_1$ is state $S$. (d) The nonequilibrium free energy profile $\mathcal{F}({\mathrm{\Gamma }}_i,\tau )-F\left(E_{\mathrm{A}}\right)$ is presented.

Figure 4

Simulation results for a quenched process: (a), (b) The color-coded lines indicate the nonequilibrium free energy profiles. Panel (a) shows data for $0<t\le 1$ for each step, and panel (b) shows data for $1<t\le 40$ in every 10 steps. (c), (d) The color-coded lines indicate (negative logarithm of) the ensemble averages of Jarzynski's work on the left-hand side of Eq. (13) multiplied by $e^{-\beta F\left({\lambda }_0\right)}$. Panel (a) shows data for $0<t\le 1$ for each step, and panel (b) shows data for $1<t\le 40$ in every 10 steps. In all panels, we used the same color code as depicted in the inset of panel (a) of Fig. 1.

Figure 5

Simulation results for a process composed of quenching, continuous changing, second quenching, and relaxation. (a), (b) The color-coded lines indicate the nonequilibrium free energy profiles. Panel (a) shows data for $0<t\le 1$ for each step, and panel (b) shows data for $1<t\le 40$ in every 10 steps. (c), (d) The color-coded lines indicate (negative logarithm of) the ensemble averages of Jarzynski's work on the left-hand side of Eq. (13) multiplied by $e^{-\beta F\left({\lambda }_0\right)}$. Panel (a) shows data for $0<t\le 1$ for each step, and panel (b) shows data for $1<t\le 40$ in every 10 steps. In all panels we used the same color code as depicted in the inset of panel (a) of Fig. 2.