Force generation in small ensembles of Brownian motors

The motility of certain gram-negative bacteria is mediated by retraction of type IV pili surface filaments, which are essential for infectivity. The retraction is powered by a strong molecular motor protein, PilT, producing very high forces that can exceed $150\mathrm{pN}$. The molecular details of the motor mechanism are still largely unknown, while other features have been identified, such as the ring-shaped protein structure of the PilT motor. The surprisingly high forces generated by the PilT system motivate a model investigation of the generation of large forces in molecular motors. We propose a simple model, involving a small ensemble of motor subunits interacting through the deformations on a circular backbone with finite stiffness. The model describes the motor subunits in terms of diffusing particles in an asymmetric, time-dependent binding potential (flashing ratchet potential), roughly corresponding to the ATP hydrolysis cycle. We compute force-velocity relations in a subset of the parameter space and explore how the maximum force (stall force) is determined by stiffness, binding strength, ensemble size, and degree of asymmetry. We identify two qualitatively different regimes of operation depending on the relation between ensemble size and asymmetry. In the transition between these two regimes, the stall force depends nonlinearly on the number of motor subunits. Compared to its constituents without interactions, we find higher efficiency and qualitatively different force-velocity relations. The model captures several of the qualitative features obtained in experiments on pilus retraction forces, such as roughly constant velocity at low applied forces and insensitivity in the stall force to changes in the ATP concentration.

Figure 1

(Color online) (a) The elements of the retraction motor model consist of a flexible ring of motor subunits that interact with a moving helical filament. (b) Equivalent geometry after the change of variables $y_i=x_i-id∕M$, which places the binding potentials of the filament monomers on top of each other. The binding potential is assumed to be an asymmetric ratchet potential with amplitude $U$ and asymmetry $a$. The undeformed state of the motor protein complex is described by the equilibrium positions $y_i^0$ of the motor subunits, and the subunits are elastically confined to their equilibrium positions. Due to the helical structure of the filament, the equilibrium positions become evenly spread over one period. The motor subunits at positions $y_i$ (black circles) interact with the filament potential. The open circle represents an unbound subunit. During a successful retraction process, a motor subunit detaches from the filament, relaxes in its confinement potential, and rebinds near the next binding site along the filament. (c) Distribution of the unbound subunit in (b). The shaded area represents the probability for the subunit to bind to the left of the potential maximum and produce a failed step. As the applied force increases, the filament is pulled forward relative to the distribution and the probability of a failed step increases. (d) Qualitatively new behavior emerges in the limit of high stiffness (strong confinement) and many motor subunits. In this limit, all bound motor subunits will not relax to the minima in the binding potential. Instead, some subunits will interact with the shaded part of the potential and oppose force production. In this regime, one expects that the stall force depends nonlinearly on the number of motor subunits and nonmonotonically on the stiffness.

Figure 2

(Color online) Force-velocity relations for high (triangles), intermediate (squares), and low stiffness (circles) calculated for $M=5$. The $y$ axis is velocity normalized by $\lambda d∕M$, which corresponds to the velocity for $M$ motor subunits if all steps were successful.

Figure 3

(Color online) (a) Stall force as a function of $k∕U$ in the range $5<k<250\mathrm{pN}∕\mathrm{nm}$ and $80<U<200\mathrm{pN}\mathrm{nm}$. The normalization $F_{\inf }$ is chosen according to Eq. (3). Lines are guides to the eye. The case of many subunits $(M=12)$ is qualitatively different from the other cases. Here, the stall force depends nonmonotonically on $k∕U$. (b) Stall force per bound motor subunit in the stiff limit $k\to \infty$ as a function of $U-U_M$. For $M⩽6$ the simulations are consistent with $f_M\left(\infty \right)=1$ in Eq. (3) (solid line). $M=12$ gives a lower stall force that does not fit Eq. (3), which again illustrates the nonlinear behavior of the stall force on $M$. Error bars in both graphs are smaller than the symbols.

Figure 4

(Color online) Force-velocity characteristics from simulation and experiment. Triangles: average experimental velocity from Fig. 3(b) of Ref. 10. Thick solid curve: $M=5$ model with $U=200\mathrm{pN}\mathrm{nm}$, $k=25\mathrm{pN}∕\mathrm{nm}$, $F_{\mathrm{stall}}=123\mathrm{pN}$, and $\lambda =1180{\mathrm{s}}^{-1}$. Diamonds: a single retraction event from Ref. 8. Dashed curve: $M=5$ model with $U=200\mathrm{pN}\mathrm{nm}$, $k=60\mathrm{pN}∕\mathrm{nm}$, $F_{\mathrm{stall}}=160\mathrm{pN}$, and $\lambda =1800{\mathrm{s}}^{-1}$. Isolated motor subunits—i.e., simple flashing ratchet models (black symbols)—are qualitatively different, as illustrated for $M=1$, $U=200\mathrm{pN}\mathrm{nm}$, $\lambda =2000{\mathrm{s}}^{-1}$, and $t_{\mathrm{off}}=25\mathrm{ms}$ $(\ast )$ or $2.5\mathrm{ms}$ $(\ast )$.