Stochastic theories of the activated complex and the activated collision: The RNA example

Abstract

We propose a common rigorous foundation to the classical collision theory and that of the classical activated complex. Based on the notion of activated complex, this foundation relies on a stochastic approach showing up the different influence factors of a chemical reaction. The thermodynamic formulation is obtained here by assuming the exponential statistical distribution within each stable chemical state. The general model we obtain yields two stochastic formulations called the stochastic transition state theory (denoted STST) and the stochastic activated collision theory (denoted SACT) respectively, depending on whether the rate of the reaction is of the same scale as, either the rate of passing over the potential energy barrier, for the STST, or the rate of reaching the activated complex state (which is a generalization of the collision rate), for the SACT. The modeling is first done in the case of a closed small system undergoing stochastic changes of chemical state and then, by extension, in the frame of classical chemical kinetics. Some properties of these two theories are studied. Furthermore the formulation is applied to the stochastic example of RNA structure changes and to the classical simple example of a reaction between two atoms A and B to form the diatomic molecule AB.

RNA folding at elementary step resolution

“…Computing the transition matrix using only free energies of the involved conformations is less rigorous than a treatment based on a stochastic theory of the activated complex (Jacob et al+, 1997a(Jacob et al+, , 1997b, but makes it easy to take into account new regularities of RNA structure as they are discovered+ It is straightforward to extend the folding analysis to include tertiary interactions for which sufficient experimental data become available+ Examples are H-type pseudoknots, coaxial continuation of stacks, extension of double helices by nonWatson-Crick base pairs (commonly purine-purine pairings), U-U pairs in interior loops, and base triplets+ Despite the relative simplicity of the master equation (2), analytic solutions are available only for further drastic restrictions on allowed transitions and equal values of their probabilities (k j, jϩ1 ϭ k r , k j, jϪ1 ϭ k R )+ Here we rely instead on numerical simulations to study the stochastic process as defined above+ To this end we use a variant of a Monte Carlo algorithm developed in the 1970s by Gillespie (1976Gillespie ( , 1977 to study stochastic kinetics in chemical reaction networks+ Gillespie's method is based on the same assumption as the derivation of the master equation: individual elementary steps are uncorrelated and the occurrence of an event follows a Poisson process on a proper time scale+ Probability distributions, expectation values, variances, and other ensemble properties are obtained through sampling sufficiently many trajectories with identical initial conditions+ The computer program we implemented is freely available upon request from Christoph Flamm+…”

Modelling ‘evo‐devo’ with RNA

“…The free energy associated with each shape gives rise to an energy landscape over the configuration space and the energy differences between adjacent shapes determine (roughly) the transition probabilities. (19,28,31) The energy landscape of a sequence is the RNA analogue of Waddington's developmental or epigenetic landscape. (32) Sequences folding into the same mfe shape can differ profoundly in their energy landscapes.…”

Probability distribution of the chemical states of a closed system and thermodynamic law of mass action from kinetics: The RNA example

“…However, because of the assumption of stochastic continuity which is not satisfied on the set of the activated complexes, we show that this kind of process can only be used under the stochastic activated collision theory. 4 Next, modeling the rate constants under this theory and further assuming a symmetry condition on the rate constants, we show that the stationary probability distribution of the structures is the same as under hypothesis ͑EXS͒ and that moreover the process is reversible, i.e.,…”

“…where v r and v r Ϫ1 are the collision volumes for the reaction r and its inverse r Ϫ1 respectively, 4 and ⌬ 1 G is the Gibbs free energy difference between the final state and the initial one, when assuming the mutual proximity of the various reactant molecules and the same for the final molecules. But, as for the thermodynamic law of mass action, the assumption ͑EXS͒ obtained by maximizing the entropy of the system under the condition that the mean internal energy of the system is constant, is only valid for systems without chemical reactions.…”