A multi-scaled approach for simulating chemical reaction systems

In this paper we give an overview of some very recent work, as well as presenting a new approach, on the stochastic simulation of multi-scaled 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 present a new approach which couples the three regimes mentioned above. We then apply this approach to a biologically inspired problem involving the expression and activity of LacZ and LacY proteins in E. coli, and conclude with a discussion on the significance of this work.     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.     

Hybrid simulation of cellular behavior

MOTIVATION: To be valuable to biological or biomedical research, in silico methods must be scaled to complex pathways and large numbers of interacting molecular species. The correct method for performing such simulations, discrete event simulation by Monte Carlo generation, is computationally costly for large complex systems. Approximation of molecular behavior by continuous models fails to capture stochastic behavior that is essential to many biological phenomena. RESULTS: We present a novel approach to building hybrid simulations in which some processes are simulated discretely, while other processes are handled in a continuous simulation by differential equations. This approach preserves the stochastic behavior of cellular pathways, yet enables scaling to large populations of molecules. We present an algorithm for synchronizing data in a hybrid simulation and discuss the trade-offs in such simulation. We have implemented the hybrid simulation algorithm and have validated it by simulating the statistical behavior of the well-known lambda phage switch. Hybrid simulation provides a new method for exploring the sources and nature of stochastic behavior in cells.     

Adaptive stochastic-deterministic chemical kinetic simulations

MOTIVATION: Biochemical signaling pathways and genetic circuits often involve very small numbers of key signaling molecules. Computationally expensive stochastic methods are necessary to simulate such chemical situations. Single-molecule chemical events often co-exist with much larger numbers of signaling molecules where mass-action kinetics is a reasonable approximation. Here, we describe an adaptive stochastic method that dynamically chooses between deterministic and stochastic calculations depending on molecular count and propensity of forward reactions. The method is fixed timestep and has first order accuracy. We compare the efficiency of this method with exact stochastic methods. RESULTS: We have implemented an adaptive stochastic-deterministic approximate simulation method for chemical kinetics. With an error margin of 5%, the method solves typical biologically constrained reaction schemes more rapidly than exact stochastic methods for reaction volumes >1-10 micro m(3). We have developed a test suite of reaction cases to test the accuracy of mixed simulation methods. AVAILABILITY: Simulation software used in the paper is freely available from http://www.ncbs.res.in/kinetikit/download.html     

Catabolite inhibition: a general phenomenon in the control of carbohydrate utilization

When Escherichia coli is grown in synthetic medium with radioactive galactose or lactose as the carbon source, the addition of glucose rapidly inhibited utilization of the radioactive substrate, whether the formation of (14)CO(2) or acid-insoluble products was measured. The inhibition was reversed after the removal of glucose. Experiments with mutants blocked in subsequent steps of galactose and lactose metabolism demonstrated that the inhibition occurs prior to the formation of the first metabolic product. The utilization of a variety of sugars, including maltose, lactose, mannose, galactose, l-arabinose, xylose, and glycerol was inhibited by glucose. Of a number of carbohydrates tested as potential inhibitors, only glucose and, to a lesser extent, glucose-6-phosphate (G-6-P) were capable of inhibiting the utilization of all of the substrates. Glucose did not inhibit G-6-P utilization but G-6-P inhibited glucose utilization. With all substrates, except glycerol, there was a delay before the onset of inhibition by G-6-P. We conclude that E. coli has a general regulatory mechanism, termed catabolite inhibition, which controls the activity of early reactions in carbohydrate metabolism, allowing certain substrates to be utilized preferentially.     

Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli

MOTIVATION: Many biochemical networks involve reactions localized on the cell membrane. This can give rise to spatial gradients of the concentration of cytosolic species. Moreover, the number of membrane molecules can be small and stochastic effects can become relevant. Pathways usually consist of a complex interaction network and are characterized by a large set of parameters. The inclusion of spatial and stochastic effects is a major challenge in developing quantitative and dynamic models of pathways. RESULTS: We have developed a particle-based spatial stochastic method (GMP) to simulate biochemical networks in space, including fluctuations from the diffusion of particles and reactions. Gradients emerging from membrane reactions can be resolved. As case studies for the GMP method we used a simple gene expression system and the phosphoenolpyruvate:glucose phosphotransferase system pathway. AVAILABILITY: The source code for the GMP method is available at http://www.science.uva.nl/research/scs/CellMath/GMP.     

Dynamic partitioning for hybrid simulation of the bistable HIV-1 transactivation network

MOTIVATION: The stochastic kinetics of a well-mixed chemical system, governed by the chemical Master equation, can be simulated using the exact methods of Gillespie. However, these methods do not scale well as systems become more complex and larger models are built to include reactions with widely varying rates, since the computational burden of simulation increases with the number of reaction events. Continuous models may provide an approximate solution and are computationally less costly, but they fail to capture the stochastic behavior of small populations of macromolecules. RESULTS: In this article we present a hybrid simulation algorithm that dynamically partitions the system into subsets of continuous and discrete reactions, approximates the continuous reactions deterministically as a system of ordinary differential equations (ODE) and uses a Monte Carlo method for generating discrete reaction events according to a time-dependent propensity. Our approach to partitioning is improved such that we dynamically partition the system of reactions, based on a threshold relative to the distribution of propensities in the discrete subset. We have implemented the hybrid algorithm in an extensible framework, utilizing two rigorous ODE solvers to approximate the continuous reactions, and use an example model to illustrate the accuracy and potential speedup of the algorithm when compared with exact stochastic simulation. AVAILABILITY: Software and benchmark models used for this publication can be made available upon request from the authors.