Daniel Gillespie

Daniel Thomas Gillespie (15 August 1938 – 19 April 2017) was a physicist who is best known for his derivation in 1976 of the stochastic simulation algorithm (SSA), also called the Gillespie algorithm. The SSA is a procedure for numerically simulating the time evolution of the molecular populations in a chemically reacting system in a way that takes account of the fact that molecules react in whole numbers and in a largely random way. Since the late 1990s, the SSA has been widely used to simulate chemical reactions inside living cells, where the small molecular populations of some reactant species often invalidate the differential equations of traditional deterministic chemical kinetics.

Gillespie's original derivation of the SSA began by considering how chemical reactions actually occur in a well-stirred dilute gas. Reasoning from physics (and not by heuristically extrapolating deterministic reaction rates to a stochastic context), he showed that the probability that a specific reaction will occur in the next very small time dt could be written as an explicit function of the current species populations multiplied by dt. From that result he deduced, using only the laws of probability, an exact formula for the joint probability density function p(τ,j) of the {time τ to the next reaction event} and the {index j of that reaction}. The SSA consists of first generating random values for τ and j according to p(τ,j), and then actualizing the next reaction accordingly. The generating step of the SSA can be accomplished using any of several different methods, and Gillespie's original paper presented two: the "direct method", which follows from a straightforward application of the well known Monte Carlo inversion method for generating random numbers; and the "first-reaction method", which is less straightforward but mathematically equivalent. Later workers derived additional methods for generating random numbers according to Gillespie's function p(τ,j) which offer computational advantages in various specific situations. Gillespie's original derivation of the SSA applied only to a well-stirred dilute gas. It was widely assumed/hoped that the SSA would also apply when the reactant molecules are solute molecules in a well-stirred dilute solution, a case more appropriate to cellular chemistry. In fact it does, but that was not definitively established until 2009. The SSA is one component of stochastic chemical kinetics, a field that Gillespie played a major role in developing and clarifying through his later publications.The SSA is physically accurate only for systems that are both dilute and well-mixed in the reactant (solute) molecules. An extension of the SSA which is aimed at circumventing the globally well-mixed requirement is the reaction-diffusion SSA (RD-SSA). It subdivides the system volume into cubic subvolumes or “voxels” which are small enough that each can be considered well-mixed. Chemical reactions are then considered to occur inside individual voxels and are modeled using the SSA. The diffusion of reactant molecules to adjacent voxels is modeled by special “voxel-hopping” reactions that accurately simulate the diffusion equation provided the voxels are, again, sufficiently small. But modeling a bimolecular reaction inside a voxel using the SSA’s reaction probability rate will be physically valid only if the reactant molecules are dilute inside the voxel, and that requires the voxels to be much larger than the reactant molecules. These opposing requirements (smaller vs. larger) on the voxel size for the RD-SSA often cannot be simultaneously met. In such cases, it will be necessary to adopt a much less restrictive simulation strategy that carefully tracks the location of every reactant molecule in the system. An algorithm of that kind was devised in 2014 by Gillespie and co-workers. Called the small-voxel tracking algorithm (SVTA), it subdivides the system volume into voxels that are smaller than the reactant molecules, and hence much smaller than the voxels used in the RD-SSA. Diffusion is therefore modeled much more accurately in the SVTA than in the RD-SSA. But inside such small voxels, the SSA’s bimolecular reaction probability rate will no longer be physically valid. So the SVTA instead models bimolecular reactions using a novel extension of the diffusional voxel-hopping rule. That extension rectifies the physical incorrectness of the standard diffusion equation on the small space-time scales where collision-induced reactions occur. The SVTA thus eliminates the requirements that the system be dilute and well-mixed, and it does so in a way that has theoretical support in molecular physics. The price for this major gain in robustness and accuracy is a simulation procedure that is more computationally intensive. Details of the SVTA and its justification in physical theory are given in the original paper; however, that paper does not develop a widely applicable, user-friendly software implementation of the SVTA.

Gillespie's broader research has produced articles on cloud physics, random variable theory, Brownian motion, Markov process theory, electrical noise, light scattering in aerosols, and quantum mechanics.