Gillespie-Driven kinetic Monte Carlo Algorithms to Model Events for Bulk or Solution (Bio)Chemical Systems Containing Elemental and Distributed Species

Abstract

Stochastic modeling techniques have emerged as a powerful tool to study the time evolution of processes in many research fields including (bio)­chemical engineering and biology. One of the most applied techniques is kinetic Monte Carlo (kMC) modeling according to the stochastic simulation algorithm (SSA) as pioneered by Gillespie, in which MC channels and time steps are discretely sampled from probability distributions. In the last decades, the SSA algorithm, as originally developed for systems with elemental species (e.g., A, B, C, etc.), has been further adapted (i) to also tackle systems with distributed species, therefore, populations and (ii) to enable faster algorithm execution. In the present contribution, we highlight the most important developments, taking bulk/solution polymerization as the reference distributed chemical process. We address SSA principles based on conventional array data structures, common acceleration methods (e.g., τ leaping and the scaling method), and the strength of tree- and matrix-based data structures for detailed storage of molecular information per distribution type and even individual population member. In addition, we report advancement regarding array programming and MC sampling methods complemented by the introduction of the use of higher-order trees and root-finding sampling tools. The contribution gives thus a detailed overview of the available main kMC algorithm steps to study kinetics, irrespective of the specific field of application due to their generic nature.

Translating Simulated Chain Length and Molar Mass Distributions in Chain‐Growth Polymerization for Experimental Comparison and Mechanistic Insight

“…The kMC simulations follow the stochastic simulation algorithm (SSA) [110,111] as developed by Gillespie, [123] in which reaction events are sampled consecutively using two random numbers between 0 and 1. The first random number determines the time elapsed with respect to the previous reaction event and the second one relates to the selection of the actual reaction event.…”

“…[97,129,130] Auxiliary tree data structures are here typically used to accelerate the retrieval of specific reactive moieties stored in the matrix. If only a chain length differentiation is needed one can rely solely on trees [86,110] and no matrices are required. For illustration purposes, we include in the present work simulation results that are both matrix-driven (Case study 1 and 4) and tree-driven (Case study 2 and 3).…”

“…The second guideline relates to the initialization phase of a kMC simulation. To properly reflect kMC probabilities if only nonmacromolecules are present at the start one needs a sufficiently large number of molecules, as recently summarized in Trigilio et al [110] A special case, as addressed in the present subsection, is the representation of polymeric reactants. If one deterministically models the number and mass CLD of a polymeric reactant through a continuous function (e.g., SF) one needs a transformation into a discrete distribution.…”

“…[101][102][103][104][105][106][107][108] Only toward the end of the previous century it became more realistic to simulate the actual distributions. [74,85,90,109] Specifically, in the recent decade, kinetic Monte Carlo (kMC) simulations [110][111][112][113][114][115][116][117] have gained significant interest as they do not suffer from Jacobian inversion induced computational stiffness issues and current computational capacities are much larger than before. In this way, a large ensemble of initial molecules can be easily used (e.g., above 10 6 ), guaranteeing a correct sampling of (stochastic) reaction probabilities and, hence, an accurate description of the polymerization kinetics.…”

“…In this way, a large ensemble of initial molecules can be easily used (e.g., above 10 6 ), guaranteeing a correct sampling of (stochastic) reaction probabilities and, hence, an accurate description of the polymerization kinetics. [98,110,111,[118][119][120] In the present contribution, the aim is to raise awareness in both the theoretical and experimental research community that several CLD and MMD representations exist and a correct translation from one representation to the other should be performed with care. Specific translation procedures are outlined and applied for polymerization recipes with basic and more complex reaction schemes.…”