Width Scaling of an Interface Constrained by a Membrane

We investigate the shape of a growing interface in the presence of an impenetrable moving membrane. The two distinct geometrical arrangements of the interface and membrane, obtained by placing the membrane behind or ahead of the interface, are not symmetrically related. On the basis of numerical results and an exact calculation, we argue that these two arrangements represent two distinct universality classes for interfacial growth: while the well-established Kardar-Parisi-Zhang (KPZ) growth is obtained in the “ahead” arrangement, we find an arrested KPZ growth with a smaller roughness exponent in the “behind” arrangement. This suggests that the surface properties of growing cell membranes and expanding bacterial colonies, for example, are fundamentally distinct.

Figure 1

Upper panel: Membrane-interface model and mapping to an exclusion process. Lower left: Arrangement $A$, actin network with leading interface interacting with a cell membrane. Lower right: Arrangement $B$, bacterial colony growth of live cells at an interface with a dead layer following behind.

Figure 2

Semiquantitative phase diagram in $u-p$ plane. RA and RB denote two rough phases, found in arrangements $A$ and $B$, respectively, and in which the roughness exponents are $\alpha =\frac{1}{2}$ and $\alpha =\frac{1}{3}$, respectively. The dotted line indicates the finite-size offset of the transition line between RA and RB; the dashed line indicates the offset of the rough-smooth transition from the upper bound $p_2=u+\frac{1}{2}$.

Figure 3

Dynamical scaling collapse. Upper panel: RB phase ($p=0.15$, $u=0.15$), $\alpha =\frac{1}{3}$, $\beta =\frac{1}{3}$, $z=1$ (arrested KPZ). Lower panel: Transition line ($p=0.5$, $u=0.35$), $\alpha =\frac{1}{3}$, $\beta =\frac{1}{4}$, $z=\frac{4}{3}$ (arrested EW).

Figure 4

Top: Fit of $P(y)$ to Airy function solution (11) within RB phase. Top left: $L=1024$, $p=0.5$, $u=0.4$ (near the exactly solvable line). Top right: $L=2048$, $p=0.25$, $u=0.25$ (far from the exactly solvable line). Bottom: Crossover scaling functions. Bottom left: Scaling function (12) for width near $p=0$ at $u=0$ with $\alpha =\frac{1}{2}$ and crossover exponent $\gamma =0.6$. Bottom right: Scaling function (13) for width for $p\ge \frac{1}{2}$ at $u=0$ with $\alpha =\frac{1}{3}$ and crossover exponent $\delta =1$. In both lower panels, the dashed line corresponds to $x^{-1/3}$.