Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method

BackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations

Simulation methods with extended stability for stiff biochemical Kinetics

Background: With increasing computer power, simulating the dynamics of complex systems in chemistry and biology is becoming increasingly routine. The modelling of individual reactions in (bio)chemical systems involves a large number of random events that can be simulated by the stochastic simulation algorithm (SSA). The key quantity is the step size, or waiting time, τ, whose value inversely depends on the size of the propensities of the different channel reactions and which needs to be re-evaluated after every firing event. Such a discrete event simulation may be extremely expensive, in particular for stiff systems where τ can be very short due to the fast kinetics of some of the channel reactions. Several alternative methods have been put forward to increase the integration step size. The so-called τ-leap approach takes a larger step size by allowing all the reactions to fire, from a Poisson or Binomial distribution, within that step. Although the expected value for the different species in the reactive system is maintained with respect to more precise methods, the variance at steady state can suffer from large errors as τ grows.Results: In this paper we extend Poisson τ-leap methods to a general class of Runge-Kutta (RK) τ-leap methods. We show that with the proper selection of the coefficients, the variance of the extended τ-leap can be well-behaved, leading to significantly larger step sizes.Conclusions: The benefit of adapting the extended method to the use of RK frameworks is clear in terms of speed of calculation, as the number of evaluations of the Poisson distribution is still one set per time step, as in the original τ-leap method. The approach paves the way to explore new multiscale methods to simulate (bio)chemical systems

Effective simulation techniques for biological systems

In this paper we give an overview of some very recent work on the stochastic simulation of systems involving chemical reactions. In many biological systems (such as genetic regulation and cellular dynamics) there is a mix between small numbers of key regulatory proteins, and medium and large numbers of molecules. In addition, it is important to be able to follow the trajectories of individual molecules by taking proper account of the randomness inherent in such a system. We describe different types of simulation techniques (including the stochastic simulation algorithm, Poisson Runge-Kutta methods and the Balanced Euler method) for treating simulations in the three different reaction regimes: slow, medium and fast. We then review some recent techniques on the treatment of coupled slow and fast reactions for stochastic chemical kinetics and discuss how novel computing implementations can enhance the performance of these simulations

Comparison of Approximation Schemes in Stochastic Simulation Methods for Stiff Chemical Systems

Interest in stochastic simulations of chemical systems is growing. One of the aspects of simulation of chemical systems that has been the prime focus over the past few years is accelerated simulation methods applicable when there is a separation of time scale. With so many new methods being developed we have decided to look at four methods that we consider to be the main foundation for this research area. The four methods that will be the focus of this thesis are: the slow scale stochastic simulation algorithm, the quasi steady state assumption applied to the stochastic simulation algorithm, the nested stochastic simulation algorithm and the implicit tau leaping method. These four methods are designed to deal with stiff chemical systems so that the computational time is decreased from that of the "gold standard" Gillespie algorithm, the stochastic simulation algorithm. These approximation methods will be tested against a variety of sti examples such as: a fast reversible dimerization, a network of isomerizations, a fast species acting as a catalyst, an oscillatory system and a bistable system. Also, these methods will be tested against examples that are marginally stiff, where the time scale separation is not that distinct. From the results of testing stiff examples, the slow scale SSA was typically the best approximation method to use. The slow scale SSA was highly accurate and extremely fast in comparison with the other methods. We also found for certain cases, where the time scale separation was not as distinct, that the nested SSA was the best approximation method to use

Efficient simulation techniques for biochemical reaction networks

Discrete-state, continuous-time Markov models are becoming commonplace in the modelling of biochemical processes. The mathematical formulations that such models lead to are opaque, and, due to their complexity, are often considered analytically intractable. As such, a variety of Monte Carlo simulation algorithms have been developed to explore model dynamics empirically. Whilst well-known methods, such as the Gillespie Algorithm, can be implemented to investigate a given model, the computational demands of traditional simulation techniques remain a significant barrier to modern research. In order to further develop and explore biologically relevant stochastic models, new and efficient computational methods are required. In this thesis, high-performance simulation algorithms are developed to estimate summary statistics that characterise a chosen reaction network. The algorithms make use of variance reduction techniques, which exploit statistical properties of the model dynamics, to improve performance. The multi-level method is an example of a variance reduction technique. The method estimates summary statistics of well-mixed, spatially homogeneous models by using estimates from multiple ensembles of sample paths of different accuracies. In this thesis, the multi-level method is developed in three directions: firstly, a nuanced implementation framework is described; secondly, a reformulated method is applied to stiff reaction systems; and, finally, different approaches to variance reduction are implemented and compared. The variance reduction methods that underpin the multi-level method are then re-purposed to understand how the dynamics of a spatially-extended Markov model are affected by changes in its input parameters. By exploiting the inherent dynamics of spatially-extended models, an efficient finite difference scheme is used to estimate parametric sensitivities robustly.Comment: Doctor of Philosophy thesis submitted at the University of Oxford. This research was supervised by Prof Ruth E. Baker and Dr Christian A. Yate

Learning-Based Importance Sampling via Stochastic Optimal Control for Stochastic Reaction Networks

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators

On Improving Stochastic Simulation for Systems Biology

Mathematical modeling and computer simulation are powerful approaches for understanding the complexity of biological systems. In particular, computer simulation represents a strong validation and fast hypothesis verification tool. In the course of the years, several successful attempts have been made to simulate complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. These processes are stochastic in nature, and furthermore they are characterized by multiple time scale evolutions and great variability in the population size of molecules. The most known method to capture random time evolutions of well-stirred chemical reacting systems is the Gillespie's Stochastic Simulation Algorithm. This Monte carlo method generates exact realizations of the state of the system by stochastically determining when a reaction will occurs and what reaction it will be. Most of the assumptions and hypothesis are clearly simplifications but in many cases this method have been proved useful to capture the randomness typical of realistic biological systems. Unfortunately, often the Gillespie's stochastic simulation method results slow in practice. This posed a great challenge and a motivation toward the development of new efficient methods able to simulate stochastic and multiscale biological systems. In this thesis we address the problems of simulating metabolic experiments and develop efficient simulation methods for well-stirred chemically reacting systems. We showed as a Systems Biology approach can provide a cheap, fast and powerful method for validating models proposed in literature. In the present case, we specified the model of SRI photocycle proposed by Hoff et al. in a suitable developed simulator. This simulator was specifically designed to reproduce in silico wet-lab experiments performed on metabolic networks with several possible controls exerted on them by the operator. Thanks to this, we proved that the screened model is able to explain correctly many light responses but unfortunately it was unable to explain some critical experiments, due to some unresolvable time scale problems. This confirm that our simulator is useful to simulate metabolic experiments. Furthermore, it can be downloaded at the URL http://sourceforge.net/projects/gillespie-qdc. In order to accelerate the simulation of SSA we first proposed a data parallel implementation on General Purpose Graphics Processing Units of a revised version of the Gillespie's First Reaction Method. The simulations performed on a GeForce 8600M GS Graphic Card with 16 stream processors showed that the parallel computations halves the execution time, and this performance scales with the number of steps of the simulation. We also highlighted some specific problem of the programming environment to execute non trivial general purpose applications. Concluding we proved the extreme computational power of these low cost and widespread technologies, but the limitations emerged demonstrate that we are far from a general purpose application for GPU. In our investigation we also attempted to achieve higher simulation speed focusing on tau-leaping methods. We revealed that these methods implement a common basic algorithmic convention. This convention is the pre-computation of information necessary to estimate the size of the leap and the number of reactions that will fire on it. Often these pre-processing operations are used to avoid negative populations. The computational cost to perform these operations is often proportional to the size of the model (i.e. number of reactions). This means that larger models involve larger computational cost. The pre-processing operations result in very efficient simulation when the leap are long and many reactions can be fired. But at the contrary they represent a burden when leap are short and few reactions occur. So to efficiently deal with the latter cases we proposed a method that works differently respect to the trend. The SSALeaping method, SSAL for short, is a new method which lays in the middle between the direct method (DM) and a tau-leaping. The SSALeaping method adaptively builds leaps and stepwise updates the system state. Differently from methods like the Modified tau-leaping (MTL), SSAL neither shifts from tau-leaping to DM nor pre-selects the largest leap time consistent with the leap condition. Additionally whereas MTL prevents negative populations taking apart critical and non critical reactions, SSAL generates sequentially the reactions to fire verifying the leap condition after each reaction selection. We proved that a reaction overdraws one of its reactants if and only if the leap condition is violated. Therefore, this makes it impossible for the population to become negatives, because SSAL stops the leap generation in advance. To test the accuracy and the performance of our method we performed a large number of simulations upon realistic biological models. The tests aimed to span the number of reactions fired in a leap and the number of reactions of the system as much as possible. Sometimes orders of magnitude. Results showed that our method performs better than MTL for many of the tested cases, but not in all. Then to augment the number of models eligible to be simulated efficiently we exploiting the complementarity emerged between SSAL and MTL, and we proposed a new adaptive method, called Adaptive Modified SSALeaping (AMS). During the simulation, our method switches between SSALeaping (SSAL) and Modified tau-leaping, according to conditions on the number of reactions of the model and the predicted number of reactions firing in a leap. We were able to find both theoretically and experimentally how to estimate the number of reactions that will fire in a leap and the threshold that determines the switch from one method to the other and viceversa. Results obtained from realistic biological models showed that in practice AMS performs better than SSAL and MTL by augmenting the number of models eligible ro be simulated efficiently. In fact, the method selects correctly the best algorithm between SSAL and MTL according to the cases. In this thesis we also investigated other new parallelization techniques. The parallelization of biological systems stimulated the interest of many researchers because the nature of these systems is parallel and sometimes distributed. However, the nature of the Gillespie's SSA is strictly sequential. We presented a novel exact formulation of SSA based on the idea of partitioning the volume. We proved the equivalence between our method and DM, and we have given a simple test to show its accuracy in practice. Then we proposed a variant of SSALeaping based on the partitioning of the volume, called Partitioned SSALeaping. The main feature we pointed out is that the dynamics of a system in a leap can be obtained by the composition of the dynamics processed by each sub-volume of the partition. This form of independency gives a different view with respect to existing methods. We only tested the method on a simple model, and we showed that the method accurately matched the results of DM, independently of the number of sub-volumes in the partition. This confirmed that the method works and that independency is effective. We have not already given parallel implementation of this method because this work is still in progress and much work has to be done. Nevertheless, the Partitioned SSAleaping is a promising approach for a future parallelization on multi core (e.g. GPU's) or in many core (e.g. cluster) technologies