A novel method to compute the time dependence of state distributions in the stochastic kinetic description of an autocatalytic system

A Comparison of the Stochastic and Deterministic Approaches in a Nucleation–Growth Type Model of Nanoparticle Formation

“…In response to the suggestion of James Martin, 12 which called for a consideration of single nucleation events, and then the suggestion of Finke, 13 which called for considering the stochastic nature of those single nucleation events, in nanoparticle formation models, a comparison of the stochastic (continuous time discrete state) and the more common deterministic approaches was presented here in two nucleation−growth type nanoparticle formation models. Detectable fluctuations in reaction time were found, which are very similar to those demonstrated in simple autocatalytic reactions 50 and are by no means specific to nanoparticle formation. The results showed that even at initial monomer molecule numbers as low as 10 7 , the stochastic and deterministic approaches predict almost identical particle size distributions.…”

“…As expected, the individual Gillespie simulations are noticeably different, and the deviations are very much in line with the known time dependence of simple autocatalytic processes. 50 It is also clear that some of the simulation runs are faster, whereas others are slower than the deterministic expectation. Two more figures showing how the simulations depend on the values of n and k n /k g are found in the Supporting Information as Figures S1 and S2.…”

“…Also, it is an important observation in stochastic kinetics that comparisons of distributions should be done using the cumulative distribution function rather than the probability density function. 16,25,28,29,50 In nonexpert use, the probability density function is often preferred as it is the direct equivalent of a histogram. However, histograms involve highly arbitrary categorization of events, which is understood to fundamentally influence the final picture drawn.…”

Bayesian Objective Functions for Estimating Parameters in Nonlinear Stochastic Differential Equation Models with Limited Data

“…To account for the effects of unmeasured disturbances and model imperfections, stochastic terms are sometimes added to the right-hand sides of ordinary differential equations (ODEs). The resulting stochastic differential equations (SDEs) can provide improved model predictions. , SDEs have been used for simulation, design, and optimization of chemical processes and for model predictive control. − Parameter estimation in SDE models is complicated by the two types of uncertainties that are encountered: (i) measurement uncertainty and (ii) model uncertainty related to the stochastic terms in the SDEs. In some situations, modelers may assume that measurement uncertainties (i.e., measurement noise variances) and the magnitudes of stochastic terms (i.e., disturbance intensities, also called the power spectral density) are known.…”

“…In some situations, modelers may assume that measurement uncertainties (i.e., measurement noise variances) and the magnitudes of stochastic terms (i.e., disturbance intensities, also called the power spectral density) are known. In many cases, however, the magnitudes of one or both types of uncertainty are estimated along with the model parameters. − A short review of the parameter estimation techniques in ODE and SDE models is presented below.…”