Width scaling of an interface constrained by a membrane

More recently, major progress has been made in showing that the long time distribution of the height follows that of the largest eigenvalue of a random matrix drawn from an ensemble that depends on the geometry of the interface Johansson ; PS ; SS . Remarkably, these different sub-universality classes have been observed experimentally TS ; TSSS . Very recently exact results for random initial conditions have been obtained CFS17 ; MS17 as well as large deviations for the tails of the height distribution MKV16 ; LDMRS16 .

In this work we consider how the shape of a growing interface is affected by an impenetrable flat membrane whose vertical position fluctuates (see Fig. 1). As we note below, different physical situations correspond to two distinct arrangements: one in which the membrane is ahead of the growing interface, and one in which it is behind. We refer to these as arrangements A and B respectively. Since the KPZ equation is not invariant under height-reversal ($h\to-h$), the two arrangements need not belong to the same universality class. On the basis of numerical results, an exact solution along a special line and crossover scaling analyses, we argue that this system has a rich phase diagram comprising two distinct phases where the interface is rough and bound to the membrane (in addition to an unbound and a smooth, bound phase). Specifically, rough interfaces in arrangement A are described by the usual KPZ scaling exponents, while those in arrangement B have distinct exponents $\alpha=\frac{1}{3}$ and $z=1$. We further argue that the latter behavior arises from dynamical arrest of the growing interface and therewith a generic mechanism for non-KPZ scaling.

Such deviations from KPZ behavior are of interest because the KPZ universality class is highly robust, due to the nonlinearity $(\nabla h)^{2}$ being the only relevant contribution to the dynamics of growing interfaces with local interactions at large length- and timescales for $d\leq 2$ BS ; LP . Known exceptions tend to occur in rather special cases KS88 ; MKHZ89 ; DLSSa ; DLSSb ; WT90 ; KM92 , for example, when parameters are tuned to generate a leading nonlinearity at higher-order or when the noise is colored.

The dynamics of the free interface is modified by the presence of the membrane, a flat horizontal wall that is positioned above the interface (see Fig. 1). Once per unit time, the membrane attempts to move by one lattice site: upwards with probability $u$, and downwards with probability $1-u$. Thus in the absence of the interface, the membrane moves upwards with velocity $v_{M}^{f}=2u-1$. The interaction between the interface and membrane is modeled as follows: Any move, of either the interface or the membrane, that would cause them to overlap is forbidden. This occurs when there are points of contact between the membrane and interface. Note that the ‘ahead’ (A) and ‘behind’ (B) arrangements are both included in the model definition: (A) when $p>\frac{1}{2}$, the interface’s growth is directed towards the membrane: i.e., any contacts with the membrane occur at interfacial peaks; (B) when $p<\frac{1}{2}$ the situation is reversed, and contacts lie at interfacial troughs. Since the height distribution functions of the free interface are asymmetric about the mean position Johansson ; PS ; SS , the effect of the membrane may be different in the two arrangements.

Arrangements A and B are also motivated by biophysical considerations. In cell growth, fluctuations of the cell membrane can allow a lamellipodium (a sheet of material packed with growing actin filaments) to expand into the space behind it Bray01 ; WBFEGM . At the same time, the membrane inhibits the lamellipodium, and the system can be modeled by arrangement A (see Fig. 1, lower-left panel). By contrast, some bacterial colonies advance through the division of living cells on the interface between the colony and its surroundings WSG17 . The consumption of nutrients by bacteria in the surface layer starve bacteria behind them: the boundary between the living and dead bacteria then forms a membrane behind the interface, as in arrangement B (Fig. 1, lower-right panel; BWpc ). More generally, the motion induced by growth behind a point-like fluctuating barrier has been modeled as a Brownian ratchet POO93 : the model we introduce here thus realizes a spatially-extended ratchet.

We summarize our findings from the simulations in the form of a phase diagram comprising four phases, Fig. 2. When $v_{M}^{f}=2u-1>v_{I}^{f}=p-\frac{1}{2}$, a free membrane and interface recede from each other. Thus for $p<p_{1}=(4u-1)/2$ we expect an unbound phase where $\overline{y}$ increases indefinitely, $C\sim 0$ and the interface has KPZ scaling. Meanwhile, the membrane cannot move faster than $v_{M}=u$, even when pushed by the interface. Therefore, if $v_{I}^{f}=p-\frac{1}{2}>u$, we expect for $p>p_{2}=(2u+1)/2$ a smooth phase where $C\sim O(L)$ and $W\sim O(1)$. Simulation data (see Supplementary Figures) are consistent with these expectations and suggest that the boundary between the rough and smooth phase lies slightly below $p_{2}$, as indicated in Fig. 2.

Of greatest interest are the two rough phases that lie between $p_{1}$ and $p_{2}$, with a transition between them near $p=\frac{1}{2}$. From the upper panel in Fig. 3 we find that for $p<\frac{1}{2}$ (arrangement B), the interface width scales as $L^{\alpha}$ with an exponent $\alpha=\frac{1}{3}$, smaller than the usual KPZ exponent of $\alpha=\frac{1}{2}$. By contrast, for $p>\frac{1}{2}$ (arrangement A), $\alpha=\frac{1}{2}$ is obtained (see Supplementary Figures). In both rough phases the contact count is $C\sim O(1)$ i.e. the interface touches the membrane only a few times, with long excursions in between. In the following, we exclude the possibility that the non-KPZ exponent is a finite-size effect (as is sometimes the case BE2001 ) on the basis that an exact calculation, a crossover scaling analysis and a dynamical scaling collapse, all consistently point to $\alpha=\frac{1}{3}$. These analyses also suggest that in the limit $L\to\infty$ the transition between the two rough phases occurs exactly at $p=\frac{1}{2}$, where the switch between arrangements A and B takes place. We thus designate these phases as RA and RB, respectively.

To determine the stationary height distribution $P(y)$, we introduce basis vectors $\{|0\rangle,|1\rangle,\ldots\}$ that span the space of heights, and transfer matrix defined by $T|0\rangle=|1\rangle$ and $T|y\rangle=q^{y}(|y-1\rangle+|y+1\rangle)$ for $y>0$. The expression

builds in the constraint that the height changes by exactly one unit along each bond on the lattice. Here we have used the fact that, for large $L$, $T^{L}\simeq|\phi\rangle\mu^{L}\langle\psi|$ if $\mu$ is the largest eigenvalue of $T$ and $|\phi\rangle$ and $\langle\psi|$ are the corresponding right and left eigenvectors.

The eigenvalue equation $T|\phi\rangle=\mu|\phi\rangle$ reads

We now make a continuum approximation $\langle y|\phi\rangle=\phi(y)$, put $q=1-\frac{\epsilon}{2}$ where $\epsilon=\frac{4c(u)}{L}$, and expand to leading non-trivial order for the case $y>0$. We find

which, under a change of variable $z=(\mu-2)\epsilon^{-\frac{2}{3}}+\epsilon^{\frac{1}{3}}y$ and $\phi(y)\to f(z)$ transforms to Airy’s equation

The physical solution, which vanishes as $z\to\infty$, is $f(z)=\mathrm{Ai}(z)$. The appearance of the combination $\epsilon^{\frac{1}{3}}y$ is almost sufficient to establish that characteristic lengthscale of the interface scales as $\epsilon^{-\frac{1}{3}}\sim L^{\frac{1}{3}}$, but to confirm this requires knowledge of how the eigenvalue $\mu$ behaves as $\epsilon\to 0$. This we deduce from the boundary case $y=0$ in Eq. (6), which becomes

or equivalently $(\mu-1)f(z^{*})=\epsilon^{\frac{1}{3}}f^{\prime}(z^{*})$ where $z^{*}=(\mu-2)\epsilon^{-\frac{2}{3}}$. This equation is satisfied by $z^{*}=z_{0}+\beta\epsilon^{\frac{1}{3}}$ where $z_{0}=-2.3381\ldots$ is the largest real root of the Airy function. Substituting the resulting expression for $\mu$ back into the boundary condition reveals that $\beta=1+O(\epsilon^{\frac{1}{3}})$. Putting this together we find

and a similar expression for the left eigenvector $\psi(y)$. Then, from (5) we finally obtain

which readily implies that $\langle\overline{y^{n}}\rangle\sim\epsilon^{-\frac{n}{3}}\sim L^{\frac{n}{3}}$, as $\epsilon\sim\frac{1}{L}$. In particular, the width $W\sim L^{\frac{1}{3}}$, confirming the roughness exponent $\alpha$ in the vicinity of the line $p=\frac{1}{2}$. In addition, $P(0)\sim\epsilon$ which is consistent with a mean number of contacts that is independent of $L$: $C=LP(0)=O(1)$.

Second, crossovers to known behavior are also consistent with $\alpha=\frac{1}{3}$. At $p=0$, the interface only grows downwards and cannot be affected by the membrane above it. Thus at $p=0$ the interface will exhibit the usual KPZ scaling with a roughness exponent $\alpha=\frac{1}{2}$. However, for $p>0$ in the RB phase, we expect $\alpha=\frac{1}{3}$ and at finite $L$ there must be a crossover

where $\gamma$ is a crossover exponent and $g(x)$ is a scaling function which tends to a constant for small $x$ and has asymptotic behaviour $g(x)\sim x^{\phi}$ for $x\gg 1$. For $\alpha$ to go from $\frac{1}{2}$ along $p=0$ to $\frac{1}{3}$ within the RB phase we require $\phi\gamma=-\frac{1}{6}$. The data collapse in Figure 4 supports the scaling (12) and is consistent with $\phi=-\frac{1}{3}$ and $\gamma=\frac{1}{2}$.

The system should also cross over to smooth phase behaviour for $p>\frac{1}{2}$ at $u=0$. To study this we introduce a second crossover scaling function

and find that for $p>\frac{1}{2}$ there is a scaling collapse with $\delta=1$. If the RB roughness exponent is $1/3$, as claimed, we expect $k(x)\sim x^{-\frac{1}{3}}$ in this regime, which is confirmed by Fig. 4. The scaling form (13) further implies a divergence of the width $W\sim|p-\frac{1}{2}|^{-\frac{1}{3}}$ in the smooth phase as $p\searrow\frac{1}{2}$, similar to a nonequilibrium wetting transition Hinrichsen1997 ; Hinrichsen2003 ; Barato2010 .

It would be of great interest to observe the dynamical arrest experimentally, for example, in the context of growing bacterial colonies that motivated arrangement B. Such experimentally-relevant interfaces would usually be two-dimensional (2D), however we expect similar phenomena to occur in 2D as for the one-dimensional case: preliminary simulations Francesco in 2D for $p=1$ indicate a transition from a smooth to a rough phase RA with the same scaling (2) for the width but with exponent values corresponding to 2D KPZ $\alpha\simeq 0.4$, $\beta\simeq 0.2$. Of further interest would be to investigate the 2D analogue of phase RB and associated exponents, however simulation times for the 2D single-step model with $p<1$ are known to become prohibitive KK89 .

Many other open questions remain. A more complete theory that explains why the dynamical arrest occurs only in arrangement B, where the membrane follows the interface from behind, is desirable. One possibility is that the asymmetric form of the universal height distribution of the growing KPZ interface Johansson ; PS ; SS implies that the interaction between interfacial peaks and troughs is fundamentally different. Finally, it should be noted that experimental interfaces such as those we invoke will naturally exhibit hydrodynamic couplings mediated by any fluid medium. Such couplings could generate effective long range interactions on the interface which, it is known, can change scaling properties BS .

Supplementary Figures