Single-polymer Brownian motor: A simulation study

Numerical simulation is used to study a single polymer chain in a flashing ratchet potential to determine how the mechanism of this Brownian motor system is affected by the presence of internal degrees of freedom. The polymer is modeled by a freely jointed chain with $N$ monomers in which the monomers interact via a repulsive Lennard-Jones potential and neighboring monomers on the chain are connected by finite extensible nonlinear elastic bonds. Each monomer is acted upon by a 1D asymmetric, piecewise linear potential of spatial period $L$ comparable to the radius of gyration of the polymer. This potential is also characterized by a localization time, $t_{\text{on}}$, and by a free diffusion time, $t_{\text{off}}$. We characterize the average motor velocity as a function of $L$, $t_{\text{off}}$, and $N$ to determine optimal parameter ranges, and we evaluate motor performance in terms of finite dispersion, Peclet number, rectification efficiency, stall force, and transportation of a load against a viscous drag. We find that the polymer motor performs qualitatively better than a single particle in a flashing ratchet: with increasing $N$, the polymer loses velocity much more slowly than expected in the absence of internal degrees of freedom, and the motor stall force increases linearly with $N$. To understand these cooperative aspects of motor operation, we analyze relevant Rouse modes. The experimental feasibility is analyzed and the parameters of the model are scaled to those of $\lambda$-DNA. Finally, in the context of experimental realization, we present initial modeling results for a 2D flashing ratchet constructed using an electrode array, and find good agreement with the results of 1D simulations although the polymer in the 2D potential sometimes briefly “detaches” from the electrode surface.

Figure 1

Schematic diagram of the applied ratchet potential characterized by an asymmetry parameter, $\alpha$, height, $V$, and period, $L$. The potential acts on each monomer of a freely jointed polymer with $N$ monomers, each monomer having approximate dimension, $\sigma$.

Figure 2

Velocity of the polymer center of mass as a function of time after the ratchet potential is switched on, averaged over repeated cycles of the potential ($N=50$; from left to right: $L=2.5\sigma ,5.5\sigma ,12\sigma$).

Figure 3

Polymer center-of-mass velocity averaged over many cycles as a function of $L$ for $t_{\text{off}}=2.5\tau$, $5\tau$, $10\tau$, $20\tau$, $30\tau$, $40\tau$, $80\tau$ ($t_{\text{on}}=20\tau$; $N=50$).

Figure 4

Average velocity of polymers of contour length, $50\sigma$, and total drag coefficient, $\gamma =50$, for different values of $V$ against $L$ in units of ratchet periods per time cycle (main figure) and in absolute units (inset). $t_{\text{on}}=t_{\text{off}}=20\tau$. Thick solid line: analytical calculations for a single particle with drag coefficient $\gamma =50$ assuming perfect confinement during $t_{\text{on}}$.

Figure 5

Average velocity of the polymer for $t_{\text{on}}=t_{\text{off}}=20\tau$ for values of $N$ varying by more than one order of magnitude. Although the velocity decreases slightly with increasing $N$, the position of the peak remains at approximately $L=5\sigma$.

Figure 6

(a)–(c) Trace of the polymer position with time along with corresponding information on the polymer conformation ($L=12\sigma$; $t_{\text{on}}=t_{\text{off}}=20\tau$).

Figure 7

(a) Trace of the polymer position with time and the corresponding values of the first and second Rouse modes [(b) and (c)] ($L=5\sigma$; $t_{\text{on}}=t_{\text{off}}=20\tau$).

Figure 8

Second Rouse mode versus absolute value of the first Rouse mode at the end of the localization period of the external ratchet potential for different values of $L$.

Figure 9

Average velocity of polymers with drag coefficient, $N\gamma$, as a function of $N$ for different ratchet periods, compared to a point particle with the same drag coefficient ($t_{\text{on}}=t_{\text{off}}=20\tau$, $V=4ϵ$).

Figure 10

Dispersion and Peclet number for $t_{\text{off}}=2.5$, 5, 10, 20, 30, 40, and $80\tau$ for $N=50$ and $t_{\text{on}}=20\tau$. (a) Dispersion, $D_{\mathrm{eff}}$, as a function of $L$. The diffusion constant $D=0.02{\sigma }^2∕\tau$ of a free polymer is indicated as a horizontal line. (b) Pe, calculated for transport over a distance $l=1000\sigma$.

Figure 11

Force-velocity response curves for a flashing ratchet motor in an external field for different ratchet lengths ($t_{\text{on}}=20\tau$; $t_{\text{off}}=10\tau$; $N=50$). Inset: Stall force of the polymer as a function of $L$ (circles) and single particle with drag coefficient, $50\gamma$ (squares).

Figure 12

(a) Comparison of the stall forces calculated from load velocity curves using a constant force (dots) and using a harmonic potential (squares). For the harmonic potential method the value is found by averaging over ten independent simulations of polymers translating against a spring with constant $\kappa =0.01ϵ∕{\sigma }^2$. (b) Stall force as a function of $N$ calculated using the harmonic-potential method ($t_{\text{on}}=20\tau$; $t_{\text{off}}=10\tau$; $L=5\sigma$).

Figure 13

(a) Rectification efficiency calculated using Eq. (7) for zero external force. (b) Efficiency as a function of force ($N=50$; $t_{\text{on}}=20\tau$).

Figure 14

Average velocity of the polymer for different drag coefficients, ${\gamma }_{\mathrm{b}}$, of a bead attached to one end of the polymer ($t_{\text{on}}=t_{\text{off}}=20\tau$; $N=50$).

Figure 15

Characteristics of the potential formed by an electrode array. Solid line; left axis: Decrease in the potential strength with distance from the surface; $V_0$ is an arbitrary scaling factor. Dashed line; right axis: Effective asymmetry measured by the separation of the maxima and minima of the potential as a fraction of $L$. Inset: The $x$ dependence of the potential at a distance $0.5\sigma$ from the surface $(L=5\sigma )$.

Figure 16

(a) Velocity of a polymer in a ratchet formed by an array of electrodes as a function of ratchet spatial period. (b) Trace of the polymer position along the $x$ axis (top) and $z$ axis (bottom) for $L=5\sigma$ ($N=50$; $t_{\text{on}}=t_{\text{off}}=20\tau$).