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Scaling methods for accelerating kinetic Monte Carlo simulations of chemical reaction networks

Yen Ting Lin, Song Feng, William S. Hlavacek

arXiv:1903.08615v2q-bio.QMphysics.chem-phq-bio.MN

TL;DR

Population scales can change dynamically, limiting static, homogeneous scaling, and standard scaling’s best λ depends non-obviously on parameter values. Partial scaling uses adaptive, reaction-specific factors; across three benchmark problems, it achieves greater acceleration at the same or better accuracy and better preserves second-moment information than standard scaling.

  • Problem

    Populations can span and dynamically change scales, limiting the applicability of static, homogeneous scaling.

    Source: D. Limitation by small but critical… · p. 7

  • Method

    Partial scaling assigns adaptive, reaction-specific scaling factors based on the smallest reactant or product population and a critical population size Nc.

    Source: §I · p. 2, §IV · pp. 7–8

  • Results

    Across three non-trivial benchmark problems, partial scaling yields greater acceleration for the same or better accuracy and better preserves second-moment information than standard scaling.

    Source: §VII · p. 13

  • Takeaways & Limitations

    Users can avoid overly aggressive scaling without numerical experiments when using partial scaling, provided Nc is not near or below 1.

    Source: §VII · p. 13

    Static, homogeneous scaling has limited applicability when populations span and dynamically change scales.

    Source: D. Limitation by small but critical… · p. 7

Abstract

from arXiv · show

Various kinetic Monte Carlo algorithms become inefficient when some of the population sizes in a system are large, which gives rise to a large number of reaction events per unit time. Here, we present a new acceleration algorithm based on adaptive and heterogeneous scaling of reaction rates and stoichiometric coefficients. The algorithm is conceptually related to the commonly used idea of accelerating a stochastic simulation by considering a sub-volume $\lambda \Omega$ ($0<\lambda<1$) within a system of interest, which reduces the number of reaction events per unit time occurring in a simulation by a factor $1/\lambda$ at the cost of greater error in unbiased estimates of first moments and biased overestimates of second moments. Our new approach offers two unique benefits. First, scaling is adaptive and heterogeneous, which eliminates the pitfall of overaggressive scaling. Second, there is no need for an \emph{a priori} classification of populations as discrete or continuous (as in a hybrid method), which is problematic when discreteness of a chemical species changes during a simulation. The method requires specification of only a single algorithmic parameter, $N_c$, a global critical population size above which populations are effectively scaled down to increase simulation efficiency. The method, which we term partial scaling, is implemented in the open-source BioNetGen software package. We demonstrate that partial scaling can significantly accelerate simulations without significant loss of accuracy for several published models of biological systems. These models characterize activation of the mitogen-activated protein kinase ERK, prion protein aggregation, and T-cell receptor signaling.

I. INTRODUCTION

Large populations and fast reactions can make exact kinetic Monte Carlo simulations impractical, while existing acceleration methods trade accuracy for efficiency. The paper introduces partial scaling, which adapts scaling by reaction and population to avoid overaggressive scaling without classifying populations in advance as discrete or continuous.

I. INTRODUCTION

  • Large populations or fast reactions can create an unmanageable number of events per unit time for exact KMC.Exact KMC updates the state one reaction event at a time.Source: §I · p. 1
  • Existing acceleration methods trade accuracy for efficiency, and their approximations may become intolerable as populations or parameters change during simulation.Hybrid methods divide variables into discrete and continuous sets; changing population scales can complicate that approach.Source: §I · p. 2
  • Standard scaling simulates a representative sub-volume λΩ, reducing events by 1/λ but risking biased higher-moment estimates and errors when critical populations become too small.A critical species should not be scaled near or below a population of 1.Source: §I · p. 2
  • Partial scaling adapts reaction-rate scaling to each reaction’s smallest reactant or product population, scaling stoichiometric coefficients inversely and leaving small populations unscaled.Its single parameter, Nc, determines when scaling is applied; the method aims to preserve unbiased first-moment and less-biased second-moment estimates.Source: §I · p. 2
  • Unlike hybrid methods, partial scaling requires no advance classification of populations as discrete or continuous, which can be problematic when discreteness changes during simulation.Scaling is heterogeneous across reactions and adaptive as populations change.Source: §I · p. 2
  • The paper evaluates partial scaling on three published biochemical network models and describes its implementation in BioNetGen.The models concern ERK activation, prion protein aggregation, and T-cell receptor signaling.Source: §I · pp. 2–3

II. NOTATION

The section establishes notation for a closed, well-mixed, isothermal reaction system and its stochastic and continuum descriptions.

II. NOTATION

  • The system has constant volume Ω and contains up to M chemical species participating in R mass-action reactions with rate constants κr.The discrete state N records the non-negative population Ni of each species Xi.Source: §II · p. 3
  • PN(t) denotes the probability that the discrete system is in state N at time t.The discrete state vector is N = (N1, …, NM).Source: §II · p. 3
  • In the continuum limit, concentrations ni = Ni/Ω define state n, whose probability density at time t is ρ(n,t).The limit takes Ω and Ni to infinity while keeping Ni/Ω constant for each species.Source: §II · p. 3

III. STATIC AND HOMOGENEOUS SCALING

The section explains why exact stochastic simulation becomes costly with increasing system size and reviews standard sub-volume scaling. It describes the method’s efficiency gain and its risks to accuracy when critical populations become too small.

III. STATIC AND HOMOGENEOUS SCALING

  • As system size Ω increases, propensity functions and reaction events per unit time increase proportionally, eventually making exact SSA computationally impractical.The SSA advances through one discrete reaction event at a time.Source: §III · p. 3
  • Standard scaling simulates a representative sub-volume λΩ, with expected cost reduction by 1/λ as λ decreases.The approach assumes a well-mixed system and takes sub-volume dynamics to represent the whole system.Source: §III · p. 3
  • Standard scaling can leave first-moment estimates unbiased but overestimate second moments, and can distort dynamics when a critical population is reduced to 0 or O(1).A small scaled population may introduce bursty behavior unlike that of the full system.Source: §III · p. 3
  • At Ω=3, the figure shows a right-skewed stationary distribution for N1/Ω and a first-moment estimate that departs noticeably from the continuum limit.The figure compares stochastic sample paths and averages with deterministic ODE trajectories across system sizes.Source: Figure 1 · p. 4

A. A simple reaction network

The section introduces a two-species reaction network with zeroth-, first-, and second-order reactions.

A. A simple reaction network

B. Deterministic and stochastic chemical kinetics

The section connects deterministic mass-action kinetics to a stochastic description in which reactions cause state transitions at rates determined by how many ways they can occur.

B. Deterministic and stochastic chemical kinetics

  • Mass-action ODEs describe concentration changes through reaction-rate terms that incorporate the reactions’ stoichiometric coefficients.The concentrations n1 and n2 correspond to population densities in the continuum limit.Source: B. Deterministic and stochastic… · p. 4
  • A stochastic description enumerates possible state transitions and assigns each a transition rate.For example, the number of ways two X1 molecules can react is N1 choose 2.Source: B. Deterministic and stochastic… · p. 5
  • In the continuum limit, the stochastic rate constants relate to mass-action constants as κ̄r = κrΩ^(1-k) for a kth-order reaction.The examples derive this relationship for injection, bimolecular conversion, and removal reactions.Source: B. Deterministic and stochastic… · p. 5

C. Efficiency, accuracy and precision of standard scaling

Standard scaling simulates a sub-volume to reduce reaction events, while interpreting its dynamics as representative of the full system. This efficiency gain trades off precision and can inflate higher moments.

C. Efficiency, accuracy and precision of standard scaling

  • 1/λ fewer reaction events per unit time result from reducing system size from Ω to λΩ.Transition rates are reduced by λ while stoichiometric coefficients are increased by 1/λ.Source: C. Efficiency, accuracy and precision… · p. 5
  • Standard scaling reduces all transition rates by λ and increases all stoichiometric coefficients by 1/λ to represent the full system from a sub-volume.An event’s population change in the scaled system is interpreted as a change of ΔNi/λ in the unscaled state space.Source: C. Efficiency, accuracy and precision… · p. 5
  • The Kramers–Moyal analysis applies to scaled systems only when λNi remains much greater than 1 for each species.Under this condition, the diffusion terms are amplified relative to the unscaled system.Source: C. Efficiency, accuracy and precision… · pp. 5, 7
  • 1/λ higher diffusivity in the scaled system inflates variance, while matching first-moment estimates remains plausible when both system sizes are large.The first-moment comparison relies on drift being identical and multiplicative-noise effects being negligible.Source: C. Efficiency, accuracy and precision… · p. 7
  • Matching unscaled first-moment precision requires 1/λ as many sample paths, so scaling accelerates only if some precision loss is allowed.With the same number of paths, scaled estimates have greater standard error.Source: C. Efficiency, accuracy and precision… · p. 7

D. Limitation by small but critical populations

Standard scaling can fail when it makes populations small enough that their discrete dynamics matter, particularly when those populations influence system behavior.

D. Limitation by small but critical populations

  • The scaled-system approximation requires λNi >> 1 for every species; when scaling is too aggressive, estimates can err because the scaled system no longer represents the original.The Kramers–Moyal expansion ceases to apply when populations become discrete.Source: D. Limitation by small but critical… · p. 7
  • Scaling a population to zero or order 1 may be acceptable if it is behaviorally unimportant, but can produce seriously misleading results for a critical species.The suitability of this scaling depends on the population’s role in system behavior.Source: D. Limitation by small but critical… · p. 7
  • Static, homogeneous scaling is limited when populations span multiple scales and change scale dynamically.The authors contrast this behavior with specialized methods that require discrete populations to remain discrete and well-separated.Source: D. Limitation by small but critical… · p. 7

IV. ADAPTIVE AND HETEROGENEOUS SCALING

Partial scaling assigns reaction-specific scaling factors from population sizes as the simulation runs. It adapts to heterogeneous, changing populations while protecting small populations from scaling.

IV. ADAPTIVE AND HETEROGENEOUS SCALING

  • Partial scaling addresses populations spread across broad ranges and changing scale by tying heterogeneous, adaptive scaling to population sizes on the fly.The authors call the outlined implementation the partial scaling algorithm (PSA).Source: §IV · p. 7
  • Each reaction has its own time-varying scaling factor rather than a single global factor set in advance.The factors are assigned according to a dynamic update schedule.Source: §IV · pp. 7–8
  • The PSA takes a reaction network, propensities, stoichiometry, initial populations, reporting times, and Nc as inputs to generate a sample path.Its simulation updates rates, samples the next reaction and waiting time, and reports the state at specified times.Source: §IV · p. 8
  • Nc is the method’s sole parameter, and each reaction’s smallest reactant or product population determines its scaling factor relative to Nc.No scaling is applied when any reactant or product population is below Nc.Source: §IV · p. 8
  • A reaction’s rate is scaled down by λr, and its stoichiometric coefficients are scaled up by 1/λr when the reaction fires.The amplified coefficients update the system state.Source: §IV · pp. 8–9
  • When λr < 1, participating populations are effectively O(Nc) or larger, and any species below Nc is never scaled out.Choosing Nc >> 1 prevents inappropriately bursty stochastic dynamics.Source: §IV · p. 9

A. Accuracy and precision of partial scaling

Partial scaling preserves first moments but can overestimate second moments; its variance bias is smaller than standard scaling because unaffected processes retain unscaled statistics.

A. Accuracy and precision of partial scaling

  • For Nc ≫ 1, partial scaling preserves means but not variances, as with standard scaling.The analysis considers changes over a short interval relative to the expected waiting time to the next reaction.Source: A. Accuracy and precision of partial… · p. 9
  • Partial scaling preserves first moments but overestimates second moments, with less overall second-moment bias than standard scaling.Processes unaffected by scaling have the same statistics as in an unscaled simulation.Source: A. Accuracy and precision of partial… · p. 9
  • A deterministic 1/λr multiplier on reaction stoichiometry is essential for unbiased first-moment estimates after rate scaling.A random multiplier with the same expected value would make variance estimates more biased.Source: A. Accuracy and precision of partial… · p. 9
  • Long-run bias is model dependent: noise-induced phenomena involving reactions with scaled rates may produce systematic errors through demographic fluctuations.Such behavior depends on second or higher moments.Source: A. Accuracy and precision of partial… · p. 9
  • Even at Nc = 300, partial scaling gives reasonable estimates of both first and second moments for ERK activation.The threshold is far below the largest population, which is on the order of 10^6.Source: Figure 3 · p. 10

V. COMPARISON OF SCALING METHODS

Across three biological benchmarks, partial scaling achieves greater acceleration than standard scaling for the same or better accuracy and better preserves second-moment information. It also remains close to exact results in reported ERK, prion, and TCR comparisons.

V. COMPARISON OF SCALING METHODS

  • The benchmarks cover ERK activation, prion aggregation, and TCR signaling, with changing population scales, discrete prion seeding, and stochastic bistable switching represented.The models were selected because each challenges standard scaling.Source: §V · p. 10
  • For the reported calculations, even with aggressive partial scaling at Nc = 300, first-moment estimates are barely distinguishable from exact-simulation estimates, unlike standard scaling, which deviates markedly.Source: §V · p. 11
  • For TCR signaling, partial scaling produces marginal distributions close to exact results, while standard-scaled distributions differ markedly.The distributions use 10^4 sample paths and focus on the initial transient.Source: §V · p. 11
  • Across three benchmark problems, partial scaling gives greater acceleration for the same or better accuracy and better preserves second-moment information than standard scaling.The comparisons include timing and accuracy results for all three models.Source: §V · p. 11, Figure 4 · p. 11
  • Partial scaling is dynamic and multiscale: reaction-specific scaling factors change over time in all three models.The report illustrates this for one setting of Nc.Source: §V · p. 11

VI. IMPLEMENTATION

Partial scaling is available in BioNetGen’s SSA through a parameter that sets the critical population size. The implementation is generate-first and requires enumerating species and their reactions.

VI. IMPLEMENTATION

  • Partial scaling was added to BioNetGen’s SSA, an efficient variation of Gillespie’s direct method.BioNetGen is an open-source, general-purpose simulation package for biological modelers.Source: §VI · p. 11
  • The poplevel argument sets Nc and scaling aggressiveness; for accuracy, its value should be much larger than 1.The argument is used only with partial scaling, invoked through the simulate command.Source: §VI · p. 12
  • BioNetGen’s implementation is generate-first, requiring enumeration of chemical species and the reactions in which they participate.Network-free methods do not require that explicit enumeration.Source: §VI · p. 12

VII. DISCUSSION AND CONCLUSION

Across three benchmark problems, partial scaling outperforms standard scaling in acceleration and accuracy, including better preservation of second-moment information. The discussion also identifies settings where its adaptive scaling is useful and a network-free simulation challenge that remains unresolved.

VII. DISCUSSION AND CONCLUSION

  • Overaggressive partial scaling is possible only when Nc is near or below 1, whereas standard scaling requires testing λ and may need retuning as parameters change.The best standard-scaling setting depends non-obviously on parameter values, including initial conditions and rate constants.Source: §VII · p. 13
  • Across three benchmarks, partial scaling yields greater acceleration for the same or better accuracy and better preserves second-moment information.The comparison covers ERK activation, prion protein aggregation, and TCR signaling.Source: §VII · p. 13, Table I · p. 14
  • Partial scaling is similar to PW-DMC but uses one critical population parameter instead of multiple rate-range rules, which require trial-and-error setting.PW-DMC can scale rates without regard to reactant populations, potentially making a discrete population negative in a bimolecular reaction.Source: §VII · p. 14
  • Partial scaling needs no a priori classification of populations as discrete or continuous, unlike hybrid approaches, but the methods were not directly compared.This avoids a classification requirement that may become problematic when population discreteness changes during a simulation.Source: §VII · p. 14
  • The authors expect partial scaling's approximations to be tolerable in parameter identification, where simulation efficiency matters and inexactness may be less concerning for noisy data.A moving window of trial parameter sets may compensate for noisy but unbiased first-moment estimates when determining goodness of fit.Source: §VII · pp. 14–15
  • With Nc = 300, reaction-specific scaling factors vary over time in all three models, showing that partial scaling is dynamic and heterogeneous.The plotted λr(t) values are from single stochastic simulations.Source: Figure 7 · p. 15
  • A general-purpose network-free implementation remains future work because partial scaling needs population-tracked state, while current general-purpose algorithms track individual biomolecular sites.The discussed population-tracking approach can track only chemical species with non-zero populations without exhaustively enumerating potentially populated species.Source: §VII · p. 15

APPENDIX

The appendix defines measures for first-moment errors, scaling aggressiveness, and second-moment variation, then describes the benchmark sampling and reporting settings.

APPENDIX

  • First-moment errors compare scaled-simulation means with corresponding means from an equal number of exact simulations, for each species and report time.The error is calculated per species at evenly spaced report times and then temporally averaged.Source: APPENDIX · pp. 15–16
  • The effective scaling factor λeff normalizes the time-averaged κ from a scaled simulation by its value in the corresponding unscaled simulation.For standard scaling, λeff equals λ; the appendix uses λeff to compare scaling aggressiveness fairly.Source: APPENDIX · p. 16
  • Second-moment variation is characterized with the Van Valen multivariate coefficient of variation, averaged over report times and compared between scaled and unscaled simulations.The reported difference in ⟨CVVV⟩ reflects second-moment error attributable to scaling.Source: APPENDIX · p. 16
  • The three-model experiments used 500 sample paths for ERK and prion aggregation, and 10^4 for TCR signaling, with model-specific report spacing and durations.ERK used Δt = 8.64 sec and T = 8,640 sec; prion aggregation used Δt = 0.01 day and T = 300 days; TCR signaling used Δt = 10 sec and T = 10^4 sec.Source: APPENDIX · p. 16
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