Diffusion-limited aggregation with surface tension

Root-gel interactions and the root waving behavior of Arabidopsis

Arabidopsis roots grown on inclined agarose gels exhibit a sinusoidal growth pattern known as root waving. While root waving has been attributed to both intrinsic factors (e.g. circumnutation) and growth responses to external signals such as gravity, the potential for physical interactions between the root and its substrate to influence the development of this complex phenotype has been generally ignored. Using a rotating stage microscope and time-lapse digital imaging, we show that (1) root tip mobility is impeded by the gel surface, (2) this impedance causes root tip deflections by amplifying curvature in the elongation zone in a way that is distinctly nontropic, and (3) root tip impedance is augmented by normal gravitropic pressure applied by the root tip against the gel surface. Thus, both lateral corrective bending near the root apex and root tip impedance could be due to different vector components of the same graviresponse. Furthermore, we speculate that coupling between root twisting and bending is a mechanical effect resulting from root tip impedance.

Mean-field diffusion-limited aggregation: a "density" model for viscous fingering phenomena

We explore a universal "density" formalism to describe nonequilibrium growth processes, specifically, the immiscible viscous fingering in Hele-Shaw cells (usually referred to as the Saffman-Taylor problem). For that we develop an alternative approach to the viscous fingering phenomena, whose basic concepts have been recently published in a Rapid Communication [Phys. Rev. E 63, 045305(R) (2001)]. This approach uses the diffusion-limited aggregation (DLA) paradigm as a core: we introduce a mean-field DLA generalization in stochastic and deterministic formulations. The stochastic model, a quasicontinuum DLA, simulates Monte Carlo patterns, which demonstrate a striking resemblance to natural Hele-Shaw fingers and, for steady-state growth regimes, follow precisely the Saffman-Taylor analytical solutions in channel and sector configurations. The relevant deterministic theory, a complete set of differential equations for a time development of density fields, is derived from that stochastic model. As a principal conclusion, we prove an asymptotic equivalency of both the stochastic and deterministic mean-field DLA formulations to the classic Saffman-Taylor hydrodynamics in terms of an interface evolution.