We investigate the dependence of the displacements of a molecular motor embedded inside a glassy material on its folding characteristic time τ f . We observe two different time regimes. For slow foldings (regime I) the diffusion evolves very slowly with τ f , while for rapid foldings (regime II) the diffusion increases strongly with τ f ( D ≈ τ −2 f ) suggesting two different physical mechanisms. We find that in regime I the motor's displacement during the folding process is counteracted by a reverse displacement during the unfolding, while in regime II this counteraction is much weaker. We notice that regime I behavior is reminiscent of the scallop theorem that holds for larger motors in a continuous medium. We find that the difference in the efficiency of the motor's motion explains most of the observed difference between the two regimes. For fast foldings the motor trajectories differ significantly from the opposite trajectories induced by the following unfolding process, resulting in a more efficient global motion than for slow foldings. This result agrees with the fluctuation theorems expectation for time reversal mechanisms. In agreement with the fluctuation theorems we find that the motors are unexpectedly more efficient when they are generating more entropy, a result that can be used to increase dramatically the motor's motion.

We use molecular dynamic simulations to investigate the motion of a folding molecular motor inside soft matter. Purcell’s scallop theorem forbids the displacement of the motor due to time symmetrical hydrodynamic laws at low Reynolds numbers whatever the asymmetry of the folding and unfolding rates. However, the fluctuation theorems imply a violation of the time symmetry of the motor’s trajectories due to the entropy generated by the motor, suggesting a breakdown of the scallop theorem at the nanoscale. To clarify this picture, we study the predicted violation of time reversibility of the motor’s trajectories, using two reverse asymmetric folding mechanisms. We actually observe this violation of time reversibility of the motor’s trajectories. We also observe the previously reported fluidization of the medium induced by the motor’s folding, but find that this induced diffusion is not enough to explain the increase of the motor’s displacement. As a result, the motor is not carried by the medium in our system but moves by its own, in violation of the scallop theorem. The observed violation of the scallop theorem opens a route to create very simple molecular motors moving in soft matter environments.

The motor's diffusion and Purcell's theorem violation depend on the folding speed in contrast to the medium's fluidization.

Using molecular dynamics simulations we compare the motion of a nano-swimmer based on Purcell's suggested motor with a time asymmetrical cycle with the motion of the same molecular motor with a time symmetrical cycle. We show that Purcell's theorem still holds at the nanoscale, despite the local structure and the medium's fluctuations. Then, with the purpose of both orienting the swimmer's displacement and increasing the breakdown of the theorem, we study the effect of an electric field on a polarized version of these swimmers. For small and large fields, the time asymmetrical swimmer is more efficient, as suggested by Purcell. However we find a field range for which Purcell's theorem is broken for the time symmetric motor. Results suggest that the breakdown of the theorem is arising from the competition of the orientation field and Brownian forces, while for larger fields the field destroys the effect of fluctuations restoring the theorem. arXiv:2001.03991v1 [cond-mat.soft]