Figure 1

Scheme of the dynamics of the $F_1$-ATPase. The shaft (circle) advances by the force introduced from the flashing between the relaxed potential (solid line) and the excited potential (dashed line). Solid thick arrows show the advance of the shaft along each substep under the corresponding torques ${\tau }_1$ and ${\tau }_2$, while dashed thick arrows point out the different flashings occurring along the step and correspond to conformational changes of the molecule. The grayscale used for coloring the shaft indicates the time evolution of each step (from darker to lighter tones). The scheme is presented for (a) the pure dissipative case without external torque and (b) an external conservative torque ${\tau }_c>0$ assisting the motion of the shaft.Reuse & Permissions

Figure 2

Effect of a conservative torque in the equilibrium probability distribution of the shaft (gray shade) in the minimum of a piecewise well (solid line). When no external torque is considered (top case) the profile is symmetric and centered in the minimum of the potential, while an external applied force (bottom case) shifts the profile so the average position of the shaft does not coincide with the minimum of the potential.Reuse & Permissions

Figure 3

Experimental values for the excited potential (circles) to the left of its minimum. The potential introduces a torque of ${\tau }_1=35\pm 2\mathrm{pN}\mathrm{nm}$. Data are extracted from Ref. [16].Reuse & Permissions

Figure 4

Experimental results for the ATP binding constant. The linear approximation in Eq. (9) for the binding fits well with the experimental values extracted from Ref. [17].Reuse & Permissions

Figure 5

Angular mean velocity versus conservative torques for three ATP concentrations. Solid lines are the theoretical prediction in Eq. (10), dashed lines correspond to numerical simulations, and symbols are from experimental data [17]. The parameters used are those of Table .Reuse & Permissions

Figure 6

Comparison between the velocities obtained from Eq. (10) for hindering conservative torques (solid line) and dissipative torques in the same range of values (dashed line). The dissipative torque is obtained through the variation of ${\gamma }_L$. The parameters used are those of Table and [ATP]$=50\mu$M.Reuse & Permissions

Figure 7

Different behaviors for high values of hindering torques compared with experimental results (crosses). Usually the motor stays still around $\omega =0$; however, slippage events (crosses inside circles) allow backward motion. The velocity during the slippage is the one fixed by the excited potential (dashed line), while the measured velocity must be the result of a mixture of both behaviors, fixed and slippage (shaded area). Experimental data are extracted from Ref. [17].Reuse & Permissions