Reaction and Diffusion on Fractal Sets


Systems biologists are interested in modelling chemical reactions in the intracellular environment, and to date much of what is done is based on the use of mass action kinetics to construct models of elementary reactions. Mass action kinetic models are based on a number assumptions which are not obviously valid in the intracellular environment. The cytoplasm is far from an ideal, isotropic wellmixed solution and often the concentrations of important chemical species are very small. Molecular crowding can have significant thermodynamic effects, but also must play an important dynamical role. An interesting approach that has been adopted to this has its roots in fractal geometry - a given molecule, depending upon its size and shape and the sizes and shapes of the molecules which surround it will find itself able to move in an environment of restricted dimension (see for example[1, 2]). Simple ideas have been suggested which give spatially homogeneous rate-like equations which attempt to account for this. It has been suggested, for example, that rate laws which depend on non-integer powers of the concentration of species might be used, and alternatively that the rate constants for elementary reactions which involve the encounter of different species (as opposed to spontaneous decomposition of individual molecules) should be time-dependent[1]. In this case the rates decay in time - the suggested form is the Zipf-Mandlebrot law which tends to a power law decay at long times, it is suggested that this power law characterises the dimension of the restricted environment of each chemical species[2]. Both of these approaches suffer from shortcomings. The use of non-integer powers of concentrations can only be justified in very limited circumstances, and has been shown to be inferior to the time-dependent rate parameter when describing certain lattice gas computer simulations of chemical reactions. However, the latter is clearly not invariant to time translation - the origin of time has a particular significance, and it is not clear as a general principle what the correct choice of time origin should be. Moreover, experimental techniques are being refined to the extent that spatio-temporal resolution of the species within a single cell is becoming possible. We might, therefore, aspire to constructing theories which describe the dynamics for spatially non-uniform distributions of active species. We have recently been working on a class of simple models of this type. These are spatio-temporal dynamical systems which model reaction and diffusion on a certain class of fractal sets. It has been known for some time now that it is possible to define random walks, and hence diffusion, on a certain class of fractals (indeed, it was this observation that motivated the work described above[1]). A simple example if this class is the Sierpinsky Gasket which has constrictions to the diffusion process in the sense that it can be disconnected by the removal of a finite set of points. The talk will focus mainly on this example, but we shall also suggest ways which could lead to more general models. Supported by the Manchester Institute for Mathematical Science (MIMS).
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