Interface growth driven by a single active particle

We study pattern formation, fluctuations, and scaling induced by a growth-promoting active walker on an otherwise static interface. Active particles on an interface define a simple model for energy-consuming proteins embedded in the plasma membrane, responsible for membrane deformation and cell movement. In our model, the active particle overturns local valleys of the interface into hills, simulating growth, while itself sliding and seeking new valleys. In one dimension, this “overturn-slide-search” dynamics of the active particle causes it to move superdiffusively in the transverse direction while pulling the immobile interface upward. Using Monte Carlo simulations, we find an emerging tentlike mean profile developing with time, despite large fluctuations. The roughness of the interface follows scaling with the growth, dynamic, and roughness exponents, derived using simple arguments as $\beta =2/3$, $z=3/2$, and $\alpha =1/2$, respectively, implying a breakdown of the usual scaling law $\beta =\alpha /z$, due to very local growth of the interface. The transverse displacement of the puller on the interface scales as $\sim t^{2/3}$ and the probability distribution of its displacement is bimodal, with an unusual linear cusp at the origin. Both the mean interface pattern and probability distribution display scaling. A puller on a static two-dimensional interface also displays aspects of scaling in the mean profile and probability distribution. We also show that a pusher on a fluctuating interface moves subdiffusively leading to a separation of timescale in pusher motion and interface response.

Figure 1

Cell membrane modeled as a flexible 1D interface with active proteins studded on it. The extensile and contractile forces exerted by the cytoskeleton are mediated by the transmembrane components, and they result, respectively, in a pushing (blue) or pulling (red) effect on the interface.

Figure 2

Dynamical moves of a puller performing an “overturn-slide-search” motion on an interface. (a) When in a local valley the puller overturns it to a hill, inducing interface growth. (b) Once it is on top of a local hill, the particle chooses at random a direction along which it will slide down, (c) searching, until it finds a local valley, which it then overturns.

Figure 3

(a) Puller on a static, initially “flat,” interface. The particle pulls the interface upward as it slides over it. The black solid line represents the profile averaged over many realization, at a given time. Profiles in individual evolutions show large deviations from the mean. (b) Puller on a fluctuating interface constructing a mean mountainlike profile. (c) A puller that started out from the bottom of a macroscopic valley, riding on a rough expanding front. (d) Pusher on a fluctuating interface giving rise to a mean valleylike profile.

Figure 4

Puller on a static, flat 2D interface gives rise to a circular tentlike structure analogous to the profile seen in one dimension.

Figure 5

(a) Probability rates when no particle is present. (b) Probability rates when a particle is present. If ${\beta }_o>0$, we have $p_{+}>p_{-}$.

Figure 6

The particle diffuses on either side when on top of a hill, drifts downward when on a slope, and faces a barrier to jump, when in a valley.

Figure 7

The parameter space for our model. The parameter $u$ represents the rate of intrinsic fluctuations of the interface while ${\beta }_o$ is the activity parameter. The left half: the ${\beta }_o<0$ (puller) regime was studied in [11]. The $u=1$ line corresponds to their $\omega =1$ regime. The focus of our paper is on the regimes ${\beta }_o<0,u=0$ (puller) and ${\beta }_o>0,u=1$ (pusher).

Figure 8

The colored lines are the height profiles at the end of a different realization at $t=5000$, for a particle starting from $x=0$ at $t=0$. The black tentlike structure is an ensemble-averaged profile at a given time with height $d$ and base length $b$ as depicted. Evidently fluctuations are large.

Figure 9

The scaled form of the mean interface profile constructed by the puller, obtained after averaging over an ensemble of histories. The inset shows the unscaled mean profile at different times.

Figure 10

The local roughness in scaled form. The black line shows the scaled mean profile; the local fluctuations are larger in magnitude than the mean height.

Figure 11

The roughness plot for a pulled static interface (lower) and KPZ interface (upper). The plot in (a) shows the system size dependence of the roughness at early times for the pulled interface, causing the deviation from the relation in Eq. (8). The scaled plots in (b) demonstrate that $\alpha =1/2$ and $z=3/2$, while $\beta =2/3$ for the pulled interface and $\beta =1/3$ for the KPZ interface. The roughness in the figures is plotted after subtracting its value at $t=0$, i.e., $w^2(t=0,L)=0.25$.

Figure 12

The interface speed in steady state as $-{\beta }_o$ is varied. The speed increases linearly for small values of ${\beta }_o$ until it finally saturates, scaling with interface size as $1/L$.

Figure 13

Mean-square displacement of the active particle with reference to a random walker. The motion is faster than diffusive.

Figure 14

Scaled $P(x,t)$ for the puller on a static interface. Inset: unscaled plots show how $P(x,t)$ spreads with time.

Figure 15

Probability distribution of return time intervals $\tau$ recorded until some finite time (here 50 000 MC).

Figure 16

Showing how the probability $P(x,t)$ of being at a given site $x$ evolves with time $t$. In view of Eq. (16), the area under the probability decay curve for a given site until time $t$ is a measure of the mean height for that site at $t$. The area under $P(x,t)$ until $t$ being the largest for $x=0$ suggests that $H(x,t)$ at $x=0$ is largest at $t$, as shown in the inset.

Figure 17

Scaled mean-square displacement. At large times $t\gg L^{z_p}$, the walker is free and uncorrelated. The MSD takes the form $r^2\left(t\right)\approx D_{L}t$ in steady state. For early times $t\ll L^{z_p}$, the motion of the particle is superdiffusive, $r^2\left(t\right)\sim t^{2\lambda }$, and is independent of system size.

Figure 18

Scaled mean profile of a fluctuating interface coupled to a puller at different times. Unlike the tentlike pattern for the pulled static interface, here the peak at the origin is smooth.

Figure 19

Scaled $P(x,t)$ for the puller on a fluctuating interface. The inset shows how $P(x,t)$ spreads with time. Note the slight dip at the origin.

Figure 20

Starting from an interface in the form of a macroscopic triangular valley with a puller at the bottom, snapshots of the interface profiles are shown at different times. Colored curves represent the rough advancing front led by the puller, while the black curve shows the mean profile of the interface at that epoch.

Figure 21

Growth of the average height $d$ and base $b$ with time.

Figure 22

Mean-square roughness $w^2\left(t\right)$ of the front constructed by the puller in a macroscopic valley, growing with time. The inset shows that $w^2\left(t\right)$ is proportional to $b\left(t\right)$.

Figure 23

Typical profiles of a fluctuating interface coupled to a pusher (color), laid over the mean profile (black) averaged over many realization at $t=1000$.

Figure 24

(a) Scaled mean profile in the pre-steady-state regime. (b) $P(x,t)$ of the particle as defined in the text. One can see that the scaling exponents are quite different.

Figure 25

Mean-square displacement for a pusher on a fluctuating interface. Scaling with different sizes was seen with the dynamic exponent $z^{\prime }=2$ as the best fit. The dashed lines delineate the early subdiffusive $r^2\left(t\right)\sim t^{2{\lambda }^{\prime }}$ and later diffusive $r^2\left(t\right)\sim t$ regime.

Figure 26

The ensemble average height profile (a) on a 2D interface, (b) projection in the $H\text{−}x$ plane, and (c) top view of the profile.

Figure 27

The scaled differential area $A_s$ at each $x$ as seen from the $y=0$ plane. The area under the curve represents the volume under the mean profile.

Figure 28

The particle probability distribution $P(x,y,t)$ (a) on a 2D interface, (b) projection in the $P\text{−}x$ plane, and (c) top view of the distribution.

Figure 29

A cross section of scaled $P(x,y,t)$ along the $y=0$ plane. Scaling holds for the cross section of $P(x,y,t)$ along any plane passing through $x=0,y=0$.

Figure 30

In two dimensions, the rms displacement for the active puller is close to diffusive.