Gillespie's stochastic framework valid for particles in aqueous solution?

Gillespie proposed a stochastic framework for simulating chemical reactions which is predicated on non-reactive elastic collisions serving to 'uniformize' particle position so that the assumption of well-mixedness is always satisfied (see page 409 in the linked version). This is formulated from kinetic theory.

A corollary to this is that a non-reactive collision between two molecules that are able to react does not induce local correlation, i.e., two particles able to react with each other that just collided, but didn't react are no more likely to react with each other in the next dt than any other particle pair in the volume.

Gillespie's algorithm is commonly used in biology where biochemical species are modeled in the aqueous environment of cells. Is this valid, and if so why? It seems the validity may depend on an assumption of Boltzmann-distributed velocities which may or may not be valid in aqueous phase. I recently asked about this (question here), however, the question was deemed a duplicate even though there was some disagreement among the answers.

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I think that it is wrong, even the Boltzmann Stosszahlansatz only assumes no correlation before collision. Now as to the magnitude of this error, it is probably somewhat irrelevant in biochemistry, and is likely captured well by the effective reaction constants. I believe the lack of correlation is a much smaller sin than the lack of spatial heterogeneity (addressed by many methods, perhaps most notably by Green's Function Reaction Dynamics, gfrd.org), although one might argue that these are connected. –  alarge May 22 '14 at 6:57
@amlrg I think they're definitely connected concepts since spatial heterogeneity begets spatial correlation. Can you weigh in on the validity of Boltzmann velocity distributions in liquid? That, to me, seems as or more important to the validity of Gillespie in modeling aqueous systems. –  vector07 May 23 '14 at 18:05
From Liouville equation you get that the equilibrium phase space distribution Poisson bracketed with the Hamiltonian should vanish. Through repeated integration, then, you should be able to get the singlet reduced phase space distribution. I think this might in general be Maxwell-Boltzmann, but I'll get back to you if I get around to verifying this. –  alarge May 25 '14 at 9:50
$\{\mathcal{H}, f^{[N]}\} = 0 \Rightarrow f^{[N]} = \frac{\mathcal{F}(\mathcal{H})}{\mathcal{Z}}$, where $\mathcal{Z} = \int \mathrm{d}x^N \mathcal{F}(\mathcal{H}(x^N))$ is the most general statement. Now $\mathcal{F}$ is going to depend on the ensemble, in the canonical one you have $\mathcal{F}(\mathcal{H}) = \frac{1}{N!h^{3N}}\frac{\exp(-\beta \mathcal{H})}{\mathcal{Z}}$, and so for the singlet $f^{(1)}(x) = N\int \mathrm{d}x^{(N-1)} f^{[N]}(x^N)$, with one more integration over the position coordinate yielding Maxwell--Boltzmann for basically anything. I think. –  alarge May 26 '14 at 3:40

1 Answer

As I understand it, your question which was marked as a duplicate not only was not a duplicate of any previous question, but also seems to contain erroneous and/or misleading answers.

Let's start with the velocity distribution. Wikipedia says the following: "The Maxwell–Boltzmann distribution applies to the classical ideal gas, which is an idealization of real gases. In real gases, there are various effects (e.g., van der Waals interactions, relativistic speed limits, and quantum exchange interactions) that make their speed distribution sometimes very different from the Maxwell–Boltzmann form."

Contrast this with (my previous comment about a derivation and) the passage found in Landau Lifshitz Statistical Physics (pp. 79-80): "Let us consider the probability distribution for the momenta, and once again emphasize the very important fact that in classical statistics this distribution does not depend on the nature of the interaction of particles within the system or on the nature of the external field, so can be expressed in a form applicable to all bodies. (In quantum statistics this statement is not quite true in general.)" After a derivation of the distribution, the book then goes on explaining that it works also in molecular systems and in systems with Brownian motion. They also show the difference between quantum statistics and the classical description of a harmonic oscillator and how one gets different velocity distributions of the particle (in the latter case accordingly recovers the Maxwell-Boltzmann distribution).

Supposing that we can characterize everything by classical mechanics (a rather fair assumption given the application, certainly one made routinely in molecular dynamics simulations of biological matter), the velocity distributions ought to follow Maxwell-Boltzmann. Now is this to say that Gillespie's formulation is without any fault? No. You are entirely correct in being suspicious of the assumption of no velocity correlation between collisions. However, due to the abundant aqueous solution (and relatively dilute reactants), the non-reactive collisions between the chemical compound and water quickly thermalize and destroy any previous velocity correlations between the two reagants (you might want to see if hydrodynamical interactions exist; I doubt that they are of any relevance in 3D, but for diffusion on top of the cell membrane they might play a minor role. An in-depth analysis would probably even make a decent publication.). The spatial correlation, however, probably takes a longer time to vanish, and it is here that Gillespie's formulation is at its weakest (the Gillespie method can be supplanted by a true continuum correction, i.e. Green's Function Reaction Dynamics, or by dividing space into small boxes and having chemical constants for diffusing from one box to another). For a recent application where spatial heterogeneity is important see for example PNAS 110, 5927 (2013) and references contained therein.

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