Statistical mechanics of a single active slider on a fluctuating interface

We study the statistical mechanics of a single active slider on a fluctuating interface, by means of numerical simulations and theoretical arguments. The slider, which moves by definition towards the interface minima, is active as it also stimulates growth of the interface. Even though such a particle has no counterpart in thermodynamic systems, active sliders may provide a simple model for ATP-dependent membrane proteins that activate cytoskeletal growth. We find a wide range of dynamical regimes according to the ratio between the timescales associated with the slider motion and the interface relaxation. If the interface dynamics is slow, the slider behaves like a random walker in a random environment, which, furthermore, is able to escape environmental troughs by making them grow. This results in different dynamic exponents to the interface and the particle: the former behaves as an Edward-Wilkinson surface with dynamic exponent 2, whereas the latter has dynamic exponent $3/2$. When the interface is fast, we get sustained ballistic motion with the particle surfing a membrane wave created by itself. However, if the interface relaxes immediately (i.e., it is infinitely fast), particle motion becomes symmetric and goes back to diffusive. Due to such a rich phenomenology, we propose the active slider as a toy model of fundamental interest in the field of active membranes and, generally, whenever the system constituent can alter the environment by spending energy.

Figure 1

Pictorial representation of the system dynamics for $\omega \le 1$, where $\omega$ is the interface-to-slider timescale ratio mentioned in the text. In the top panel, the particle has reached a local interface minimum. After some time (bottom panel), this location has become a local maximum due to local growth stimulation by the particle. The particle can now diffuse to the next local minimum, as suggested by the yellow arrow.

Figure 2

Cartoon depicting the system behavior for intermediate $\omega$. The particle is surfing a membrane wave that is both pushing and being pulled by the particle.

Figure 3

Typical interface shape for $\omega \to \infty$. As the interface attains a tentlike profile between two subsequent particle jumps, there is no longer a preferred direction for the particle to move along. Hence, it diffuses around as if it was not coupled to the interface.

Figure 4

Schematics of our model active interface. The particle (red circle) presence favors the move that makes the interface grow, as denoted by the black arrows. The particle, in turn, will jump with left-right symmetric rates when on a hill or in a trough, as the leftmost and rightmost particles in the figure; whereas, when sitting on a slope as the middle particle of the figure, it will be more likely to jump towards the lower height.

Figure 5

Scaling of the averaged width for $\omega =1$; values of $L$ are given in the key. The best collapse is achieved by setting the exponents $\alpha$ and $z_1$ to the EW class values, even if the width oscillations cause a slight departure from the scaling hypothesis, Eq. (3). A power law $x^{\alpha /z_1}$ is shown as a guide to the eye (black dashed line). The interface width has been averaged over at least 1000 realizations of the stochastic dynamics for $L$ up to 8000, and more than 100 realizations for $L=16000$. The number of realizations used for averages is the same in all the following figures, unless stated otherwise.

Figure 6

Scaling of the particle MSD for $\omega =1, L$ as in the key. The data are collapsed using values $\chi =1/2$ and $z_2=3/2$ consistent with the second class particle scaling discussed in the text. The black dashed line is a guide to the eye suggesting $〈X_t^2〉\sim t^{4/3}$. In this figure, and all the following figures, the MSD is computed in steady state, meaning that time starts running after the interface has reached its saturation width.

Figure 7

Correlations spreading from a fixed active particle which catalyzes growth in the interface. The averages here are performed over 10 000 different realizations of the interface dynamics. The slope correlation function defined in the text is plotted against $j/t^{1/2}$ in the top panel and $j/t^{2/3}$ in the bottom panel. The overlap of the functions is much better in the latter case, suggesting that correlations spread around the particle as $t^{1/z_2}$ where $z_2=3/2$.

Figure 8

Width (top) and MSD (bottom) vs $\omega t$, for $L=8000$ and $\omega$ as in the key. The width of a passive, EW interface is also shown for comparison. While rescaling time by $\omega$ renders the interface dynamics independent of this parameter, the particle displays an early-time, subdiffusive regime, the extent of which scales as $1/\omega$, as pointed out in the text.

Figure 9

Scaling plot of the particle MSD for $\omega ={10}^{-2}$ (left) and ${10}^{-1}$ (right). The black solid lines in both plots are guides to the eye for the superdiffusive law $〈X_t〉\sim t^{4/3}$.

Figure 10

Tentlike shape of the interface, which is obtained in our model in the limit $\omega \to \infty$. The dotted line plots the analytic prediction (18), with $\mathrm{\Lambda }/\nu =2$ and $L=4000$, while the blue dots represent a simulated $L=4000$ interface.

Figure 11

Width scaling in the $\omega \to \infty$ limit.

Figure 12

Surfing regime snapshots. The interface profiles are ordered in time according to their color, from lighter to darker, while the particle is represented by a yellow dot of a significantly larger size, to ease understanding of the picture. The earliest snapshot (light blue) depicts the initial growth, whose dynamics is analogous to that of the $\omega \to \infty$ regime. The second (azure) is taken some moments after the particle has started moving: the wave is broken together with the left-right symmetry of the system. The last (dark blue) is the latest, and it shows that the particle keeps moving while “ironing out” the interface: this is, in fact, the regime with the smallest roughness exponent. Notice how, due to the system finite size, the particle will soon reach the back of the wave: at this point it could stochastically revert his motion, so that the long-time dynamics is still diffusive (see discussion in the text).

Figure 13

MSD at $\omega =10$ (left) and 100 (right), system size as in the key. The dashed lines are guides to the eye, the black corresponds to the ballistic law $〈X_t^2〉\sim t^2$, and the blue corresponds to $〈X_t^2〉\sim t^{4/3}$. Though both left and right panels display superdiffusive behavior, true ballistic behavior is achieved for $\omega =100$ only (right panel).

Figure 14

MSD scaling at $\omega =100$. If one excludes the $L=1000$ curve, which does not reach a full ballistic regime, the scaling exponents agree with the proposed values $\chi =z_2=1$.

Figure 15

Width scaling at $\omega =100$. The oscillating widths collapse on a single curve for $\alpha =0.175$ and $z_1=1$.

Figure 16

Particle trajectory for $\omega =400$, with $L$ as in the key. The depicted behavior is emblematic of the whole $1\ll \omega \lesssim L^2$ regime. The dynamics goes qualitatively as follows. Starting from a flat interface at $t=0$, the particle fluctuates diffusively. It eventually picks a direction at random and starts running, but it does so only after some finite time.