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Register a new userAdaptive Hybrid Simulations for Multiscale Stochastic Reaction Networks
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The probability distribution describing the state of a Stochastic Reaction Network evolves according to the Chemical Master Equation (CME). It is common to estimated its solution using Monte Carlo methods such as the Stochastic Simulation Algorithm (SSA). In many cases these simulations can take an impractical amount of computational time. Therefore many methods have been developed that approximate the Stochastic Process underlying the Chemical Master Equation. Prominent strategies are Hybrid Models that regard the firing of some reaction channels as being continuous and applying the quasi-stationary assumption to approximate the dynamics of fast subnetworks. However as the dynamics of a Stochastic Reaction Network changes with time these approximations might have to be adapted during the simulation. We develop a method that approximates the solution of a CME by automatically partitioning the reaction dynamics into discrete/continuous components and applying the quasi-stationary assumption on identifiable fast subnetworks. Our method does not require user intervention and it adapts to exploit the changing timescale separation between reactions and/or changing magnitudes of copy numbers of constituent species. We demonstrate the efficiency of the proposed method by considering examples from Systems Biology and showing that very good approximations to the exact probability distributions can be achieved in significantly less computational time.
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In the past few decades, the development of fluorescent technologies and microscopic techniques has greatly improved scientists ability to observe real-time single-cell activities. In this paper, we consider the filtering problem associate with these advanced technologies, i.e., how to estimate latent dynamic states of an intracellular multiscale stochastic reaction network from time-course measurements of fluorescent reporters. A good solution to this problem can further improve scientists ability to extract information about intracellular systems from time-course experiments. A straightforward approach to this filtering problem is to use a particle filter where particles are generated by simulation of the full model and weighted according to observations. However, the exact simulation of the full dynamic model usually takes an impractical amount of computational time and prevents this type of particle filters from being used for real-time applications, such as transcription regulation networks. Inspired by the recent development of hybrid approximations to multiscale chemical reaction networks, we approach the filtering problem in an alternative way. We first prove that accurate solutions to the filtering problem can be constructed by solving the filtering problem for a reduced model that represents the dynamics as a hybrid process. The model reduction is based on exploiting the time-scale separations in the original network and, therefore, can greatly reduce the computational effort required to simulate the dynamics. As a result, we are able to develop efficient particle filters to solve the filtering problem for the original model by applying particle filters to the reduced model. We illustrate the accuracy and the computational efficiency of our approach using several numerical examples.
Biochemical reaction networks frequently consist of species evolving on multiple timescales. Stochastic simulations of such networks are often computationally challenging and therefore various methods have been developed to obtain sensible stochastic approximations on the timescale of interest. One of the rigorous and popular approaches is the multiscale approximation method for continuous time Markov processes. In this approach, by scaling species abundances and reaction rates, a family of processes parameterized by a scaling parameter is defined. The limiting process of this family is then used to approximate the original process. However, we find that such approximations become inaccurate when combinations of species with disparate abundances either constitute conservation laws or form virtual slow auxiliary species. To obtain more accurate approximation in such cases, we propose here an appropriate modification of the original method.
We consider the problem of estimating the dynamic latent states of an intracellular multiscale stochastic reaction network from time-course measurements of fluorescent reporters. We first prove that accurate solutions to the filtering problem can be constructed by solving the filtering problem for a reduced model that represents the dynamics as a hybrid process. The model reduction is based on exploiting the time-scale separations in the original network, and it can greatly reduce the computational effort required to simulate the dynamics. This enables us to develop efficient particle filters to solve the filtering problem for the original model by applying particle filters to the reduced model. We illustrate the accuracy and the computational efficiency of our approach using a numerical example.
In most natural sciences there is currently the insight that it is necessary to bridge gaps between different processes which can be observed on different scales. This is especially true in the field of chemical reactions where the abilities to form bonds between different types of atoms and molecules create much of the properties we experience in our everyday life, especially in all biological activity. There are essentially two types of processes related to biochemical reaction networks, the interactions among molecules and interactions involving their conformational changes, so in a sense, their internal state. The first type of processes can be conveniently approximated by the so-called mass-action kinetics, but this is not necessarily so for the second kind where molecular states do not define any kind of density or concentration. In this paper we demonstrate the necessity to study reaction networks in a stochastic formulation for which we can construct a coherent approximation in terms of specific space-time scales and the number of particles. The continuum limit procedure naturally creates equations of Fokker-Planck type where the evolution of the concentration occurs on a slower time scale when compared to the evolution of the conformational changes, for example triggered by binding or unbinding events with other (typically smaller) molecules. We apply the asymptotic theory to derive the effective, i.e. macroscopic dynamics of the biochemical reaction system. The theory can also be applied to other processes where entities can be described by finitely many internal states, with changes of states occuring by arrival of other entities described by a birth-death process.
Motivation: Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortu-nately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user. Results: We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks. Availability: The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu.
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