# Daniel Gillespie

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Daniel Thomas Gillespie
Born 15 August 1938
Died 19 April 2017 (aged 78)
Residence California, USA
Nationality American
Fields Physics and Stochastic Processes
Institutions University of Maryland, College Park
NAWC China Lake
Alma mater Rice University
Johns Hopkins University
Doctoral advisor Aihud Pevsner
Known for Gillespie algorithm

Daniel Thomas Gillespie is a physicist who is best known for his derivation in 1976 of the stochastic simulation algorithm (SSA), also called the Gillespie algorithm.[1][2] 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[1] 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[1] 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[1][2][3] 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.[4] 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.[3][4][5][6][7][8][9][10][11][12][13][14]

The SSA is physically accurate only for systems that are both dilute and well-mixed in the reactant (solute) molecules.[14] 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.[14] 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.[15] 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;[15] 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,[16][17] random variable theory,[18] Brownian motion,[19][20] Markov process theory,[21][22] electrical noise,[23][24][25] light scattering in aerosols,[26][27] and quantum mechanics.[28][29]

## Education

Born in Missouri, Gillespie grew up in Oklahoma where he graduated from Shawnee High School in 1956. In 1960 he received his B.A. (Magna cum Laude) with a major in physics from Rice University.

Gillespie received his Ph.D. from Johns Hopkins University in 1968 with a dissertation in experimental elementary particle physics under Aihud Pevsner. Part of his dissertation derived procedures for stochastically simulating high-energy elementary particle reactions using digital computers, and Monte Carlo methodology would play a major role in his later work. During his graduate student years at JHU he was also a Jr. Instructor (1960–63) and an Instructor (1966-68) in the sophomore General Physics course.

## Career

From 1968 to 1971, Gillespie was a Faculty Research Associate at the University of Maryland College Park's Institute for Molecular Physics. He did research in classical transport theory with Jan Sengers. In 1971 he was also an Instructor in the University's Physics Department.

From 1971 to 2001, Gillespie was a civilian scientist at the Naval Weapons Center in China Lake, California. Initially he was a Research Physicist in the Earth and Planetary Sciences Division. There his research in cloud physics led to a procedure for simulating the growth of raindrops in clouds,[16] and that prompted his paper on the SSA.[1] In 1981 he became Head of the Research Department's Applied Mathematics Research Group, and in 1994 he was made a Senior Scientist in the Research Department. He retired from China Lake in 2001.

From 2001 to 2015, Gillespie was a private consultant in computational biochemistry, working under contract for various periods of time with the California Institute of Technology, the Molecular Sciences Institute (in Berkeley), the Beckman Institute at Caltech, and the University of California, Santa Barbara. Most of this was in collaboration with the Linda Petzold research group in the Computer Sciences Department of UCSB.

## References

1. Gillespie, D. T. (1976). "A general method for numerically simulating the stochastic time evolution of coupled chemical reactions". Journal of Computational Physics. 22: 403–434. Bibcode:1976JCoPh..22..403G. doi:10.1016/0021-9991(76)90041-3.
2. ^ a b Gillespie, D. T. (1977). "Exact stochastic simulation of coupled chemical reactions". Journal of Physical Chemistry. 81: 2340–2361. doi:10.1021/j100540a008.
3. ^ a b Gillespie, D. T. (1992). "A rigorous derivation of the chemical master equation". Physica A. 188: 404–425. Bibcode:1992PhyA..188..404G. doi:10.1016/0378-4371(92)90283-V.
4. ^ a b Gillespie, D. T. (2009). "A diffusional bimolecular propensity function". Journal of Chemical Physics. 131: 164109. Bibcode:2009JChPh.131p4109G. doi:10.1063/1.3253798.
5. ^ Gillespie, D. T. (2000). "The chemical Langevin equation". Journal of Chemical Physics. 113: 297. Bibcode:2000JChPh.113..297G. doi:10.1063/1.481811.
6. ^ Cao, Y.; Gillespie, D. T.; Petzold, L. R. (2005). "The slow-scale stochastic simulation algorithm". Journal of Chemical Physics. 122: 014116. Bibcode:2005JChPh.122a4116C. doi:10.1063/1.1824902.
7. ^ Cao, Y.; Gillespie, D. T.; Petzold, L. R. (2006). "Efficient stepsize selection for the tau-leaping simulation method". Journal of Chemical Physics. 124 (4): 044109. Bibcode:2006JChPh.124d4109C. doi:10.1063/1.2159468. PMID 16460151.
8. ^ Gillespie, D. T. (2007). "Stochastic simulation of chemical kinetics". Annual Review of Physical Chemistry. 58: 35–55. Bibcode:2007ARPC...58...35G. doi:10.1146/annurev.physchem.58.032806.104637.
9. ^ Gillespie, D. T. (2008), Bernardo, M.; Degano, P.; Zavattaro, G., eds., Simulation methods in systems biology, Formal Methods for Computational Systems Biology, Springer, pp. 125–167, ISBN 978-3-540-68892-1
10. ^ Gillespie, D. T. (2009). "The deterministic limit of stochastic chemical kinetics". Journal of Physical Chemistry B. 113: 1640–1644. doi:10.1021/jp806431b.
11. ^ Gillespie, D. T.; Cao, Y.; Sanft, K. R.; Petzold, L. R. (2009). "The subtle business of model reduction for stochastic chemical kinetics". Journal of Chemical Physics. 130: 064103. Bibcode:2009JChPh.130f4103G. doi:10.1063/1.3072704.
12. ^ Roh, M. K.; Daigle Jr, B. J.; Gillespie, D. T.; Petzold, L. R. (2011). "State-dependent doubly weighted stochastic simulation algorithm for automatic characterization of stochastic biochemical rare events". Journal of Chemical Physics. 135: 234108. Bibcode:2011JChPh.135w4108R. doi:10.1063/1.3668100.
13. ^ Gillespie, D. T.; Hellander, A.; Petzold, L. R. (2013). "Perspective: Stochastic algorithms for chemical kinetics". Journal of Chemical Physics. 138: 170901. Bibcode:2013JChPh.138p0901G. doi:10.1063/1.4801941.
14. ^ a b c Gillespie, D. T.; Petzold, L. R.; Seitaridou, E. (2014). "Validity conditions for stochastic chemical kinetics in diffusion-limited systems". Journal of Chemical Physics. 140: 054111. Bibcode:2014JChPh.140e4111G. doi:10.1063/1.4863990.
15. ^ a b Gillespie, D. T.; Seitaridou, E.; Gillespie, C. A. (2014). "The small-voxel tracking algorithm for simulating chemical reactions among diffusing molecules". Journal of Chemical Physics. 141: 234115. Bibcode:2014JChPh.141w4115G. doi:10.1063/1.4903962.
16. ^ a b Gillespie, D. T. (1975). "An exact method for numerically simulating the stochastic coalescence process in a cloud". Journal of the Atmospheric Sciences. 32: 1977–1989. Bibcode:1975JAtS...32.1977G. doi:10.1175/1520-0469(1975)032<1977:AEMFNS>2.0.CO;2.
17. ^ Gillespie, D. T. (1981). "A stochastic analysis of the homogeneous nucleation of vapor condensation". Journal of Chemical Physics. 74: 661. Bibcode:1981JChPh..74..661G. doi:10.1063/1.440825.
18. ^ Gillespie, D. T. (1983). "A theorem for physicists in the theory of random variables". American Journal of Physics. 51: 520. Bibcode:1983AmJPh..51..520G. doi:10.1119/1.13221.
19. ^ Gillespie, D. T. (1993). "Fluctuation and dissipation in Brownian motion". American Journal of Physics. 61: 1077. Bibcode:1993AmJPh..61.1077G. doi:10.1119/1.17354.
20. ^ Gillespie, D. T. (1996). "The mathematics of Brownian motion and Johnson noise". American Journal of Physics. 64: 225. Bibcode:1996AmJPh..64..225G. doi:10.1119/1.18210.
21. ^ Gillespie, D. T. (1996). "Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral". Physical Review E. 54 (2): 2084–2091. Bibcode:1996PhRvE..54.2084G. doi:10.1103/PhysRevE.54.2084. PMID 9965289.
22. ^ Gillespie, D. T. (1996). "The multivariate Langevin and Fokker-Planck equations". American Journal of Physics. 64: 1246. Bibcode:1996AmJPh..64.1246G. doi:10.1119/1.18387.
23. ^ Gillespie, D. T. (1997). "Markovian modeling of classical thermal noise in two inductively coupled wire loops". Physical Review E. 55: 2588–2605. Bibcode:1997PhRvE..55.2588G. doi:10.1103/PhysRevE.55.2588.
24. ^ Gillespie, D. T. (1998). "Theory of electrical noise induced in a wire loop by the thermal motions of ions in solution". Journal of Applied Physics. 83: 3118. Bibcode:1998JAP....83.3118G. doi:10.1063/1.367068.
25. ^ Gillespie, D. T. (2000). "A mathematical comparison of simple models of Johnson noise and shot noise". Journal of Physics: Condensed Matter. 12: 4195. Bibcode:2000JPCM...12.4195G. doi:10.1088/0953-8984/12/18/305.
26. ^ Gillespie, D. T. (1985). "Stochastic-analytic approach to the calculation of multiply scattered lidar returns". Journal of the Optical Society of America A. 2: 1307. Bibcode:1985JOSAA...2.1307G. doi:10.1364/JOSAA.2.001307.
27. ^ Gillespie, D. T. (1990). "Calculation of single scattering effects in an idealized bi-static lidar". Journal of Modern Optics. 37: 1603–1616. Bibcode:1990JMOp...37.1603G. doi:10.1080/09500349014551771.
28. ^ Gillespie, D. T. (1986). "Untenability of simple ensemble interpretations of quantum mechanics". American Journal of Physics. 54: 889. Bibcode:1986AmJPh..54..889G. doi:10.1119/1.14784.
29. ^ Gillespie, D. T. (1989). "Is quantum mechanics crazy?". American Journal of Physics. 57: 1065. Bibcode:1989AmJPh..57.1065G. doi:10.1119/1.15790.