Peter V. Gordon

Local kinetics of morphogen gradients

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction–diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production. We present analytical results that characterize the dynamics of this process and are in quantitative agreement with numerical solutions of the underlying nonlinear equations. The derived results provide an explicit connection between the parameters of the problem and the time needed to reach a steady state value at a given position. Our approach can be used for the quantitative analysis of tissue patterning by morphogen gradients, a subject of active research in biophysics and developmental biology.

Fast subsonic combustion as a free-interface problem

The paper is concerned with the recently identified fast, yet subsonic, combustion waves occurring in obstacle-laden (e.g. porous) systems and driven not by thermal diffusivity but rather by the draginduced diffusion of pressure. In the framework of a quasi-one-dimensional formulation where the impact of obstacles is accounted for through a frictional drag term, an asymptotic expression for the wave propagation velocity D is derived. The propagation velocity is controlled by the temperature (T+) at the entrance to the reaction zone rather than at its exit (Tb) as occurs in deflagrative combustion. The evaluatedD(T+) dependence allows description of the subsonic detonation in terms of a free-interface problem. The latter is found to be dynamically akin to the problem of gasless combustion known for its rich pattern-forming dynamics. Premixed gas combustion is the combustion of gaseous reactants which are perfectly mixed prior to ignition. The most distinctive feature of premixed combustion is its ability to assume the form of a self-sustained reaction wave propagating subsonically or supersonically at a well-defined speed. Apart from their technological relevance, combustion waves constitute a truly fascinating dynamical system, displaying an amazingly rich variety of phenomena such as non-uniqueness of possible propagation regimes, their birth (ignition) and destruction (extinction), chaotic self-motion and fractal-like growth, various hysteretic transitions, etc. One of the most effective practices in the theoretical exploration of combustion waves is their description in terms of a free-interface problem where the reaction zone is considered as infinitely thin compared to the other length-scales involved. The current study is concerned with the formulation and analysis of a free-interface problem associated with the recently identified new mode of subsonic combustion arising in hydraulically resisted flows (e.g. porous beds), and where the combustion wave is driven by the drag-induced diffusion of pressure, rather than thermal diffusivity as occurs in conventional unconfined flames. In the simplest case (the so-called small-heat release approximation) the emerging free-interface problem is described by a single filtration equation (Sec. 6), = r 2 ,

Local accumulation times for source, diffusion, and degradation models in two and three dimensions.

We analyze the transient dynamics leading to the establishment of a steady state in reaction-diffusion problems that model several important processes in cell and developmental biology and account for the diffusion and degradation of locally produced chemical species. We derive expressions for the local accumulation time, a convenient characterization of the time scale of the transient at a given location, in two- and three-dimensional systems with first-order degradation kinetics, and analyze their dependence on the model parameters. We also study the relevance of the local accumulation time as a single measure of timing for the transient and demonstrate that, while it may be sufficient for describing the local concentration dynamics far from the source, a more delicate multi-scale description of the transient is needed near a tightly localized source in two and three dimensions.

On disintegration of near-limit cellular flames

A strongly non-linear geometrically-invariant model for the dynamics of near-limit cellular flame is proposed, where the flame evolution is governed by a system of equations for the flame interface and its temperature. The model generalizes its earlier weakly non-linear version pertinent to a mildly perturbed planar flame. Numerical simulations of the new model show that at sufficiently high levels of heat losses the cellular flame resulting from the diffusive instability exhibits a tendency toward self-fragmentation, quite in line with direct numerical simulations of the associated reaction-diffusion system.

Neural networks with prescribed large time behaviour

The generalized Hebb rule (with a non-symmetrical synaptic matrix) allows us to create simple neural networks with complicated large time behaviour. These networks can simulate, in a sense, any dynamics and, in particular, can generate any hyperbolic attractors and invariant sets. The explicit mathematical algorithm allows us, by adjusting the network parameters (the neuron number, coupling matrix and thresholds) to obtain a network with given large time dynamics.

The KPP system in a periodic flow with a heat loss

We consider a reaction–diffusion–advection system of the KPP type in a periodic flow with heat loss through the boundary. We show that, as in the case of a shear flow, the propagation speed is determined by the linearization ahead of the front and is thus independent of the Lewis number. Moreover, we show that a flame may be blown off or be extinguished by the presence of a periodic flow. We present an explicit procedure for constructing a flow which leads to the blow-off or extinction of the flame. The period cell size has to be sufficiently small for the flow to extinguish a flame if the channel is wider than critical.

Self-similar dynamics of morphogen gradients

Morphogen gradients are concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues and play a fundamental role in various aspects of embryonic development. We discovered a family of self-similar solutions in a canonical class of nonlinear reaction-diffusion models describing the formation of morphogen gradients. These solutions are realized in the limit of infinitely high production rate at the tissue boundary and are given by the product of the steady state concentration profile and a function of the diffusion similarity variable. We solved the boundary value problem for the similarity profile numerically and analyzed the implications of the discovered self-similarity on the dynamics of morphogenetic patterning.

An upper bound of the bulk burning rate in porous media combustion

The long-time behaviour of the system of degenerate reaction–diffusion equations describing detonation in porous media is considered. An upper bound of the bulk burning rate is found.

An elementary model for autoignition of laminar jets.

In this paper, we formulate and analyse an elementary model for autoignition of cylindrical laminar jets of fuel injected into an oxidizing ambient at rest. This study is motivated by renewed interest in analysis of hydrothermal flames for which such configuration is common. As a result of our analysis, we obtain a sharp characterization of the autoignition position in terms of the principal physical and geometrical parameters of the problem.

Gelfand-type problem for two-phase porous media

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in contrast to the classical Gelfand problem which uses a single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively owing to the absence of stationary temperature distribution. We also show that the presence of interphase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two-phase porous media reduces to the classical Gelfand problem with renormalized constants.