How reaction kinetics with time-dependent rate coefficients differs from generalized mass action.

Reactions occurring in low-dimensional media have been the subject of intense study for over two decades. These media can be either homogeneous or heterogeneous. Reactions in intracellular environments are a typical example of biochemical reactions occurring in low-dimensional and heterogeneous media. Other examples are excitation trapping in molecular aggregates, charge recombination in colloids, reactions taking place at interfaces of different phases and surface catalysis. Reactions occurring in such conditions have been investigated by Monte Carlo simulations of point particles diffusing and reacting in a discrete space. 4] This space is characterized by a dimension d. Fractal spaces (those in which d is fractional), such as percolation clusters, provide a useful representation of a heterogeneous low-dimensional medium. Using Monte Carlo simulations, Kopelmann showed that the diffusion-limited reaction rate for the elementary reaction A+A!ø in a fractal medium is proportional to t!h[A]2 for batch conditions. Here the square brackets denote concentrations. The parameter h is a function of the dimensionality of the reaction media, such that 0<h<1. For example, in three dimensions h=0 (concordant with the law of mass action), on a percolation cluster h=1/3 and in a one-dimensional channel h=1/2. The increase in h with decreasing dimensionality reflects deviations from the classical law of mass action. These deviations are the result of dimensional or topological constraints in which convective or diffusive stirring is inefficient. The above results are also correct for a reaction A+B!ø with initial conditions [A0]= [B0] , [5] with the exception that h is different. Reaction kinetics characterized by a time-dependent rate coefficient k(t) is also known as fractal-like reaction kinetics. Such effective rate equations have been shown to describe the kinetics of more complex reactions such as batch reactions of the type A+B!ø with general initial conditions [A0]1⁄46 [B0] . Savageau introduced the power-law formalism in the context of biochemical systems theory. It is also known as generalized mass action kinetics. In this general framework the rate of reaction, v is of the form represented in Equation (1):