# On Efficient Algorithms for Stochastic Simulation of Biochemical Reaction Systems

### Abstract

#### Computational techniques provide invaluable tools for developing a quantitative understanding the complexity of biological systems. The knowledge of the biological system under study is formalized in a precise form by a model. A simulation algorithm will realize the dynamic interactions encoded in the model. The simulation can uncover biological implications and derive further predictive experiments. Several successful approaches with different levels of detail have been introduced to deal with various biological pathways including regulatory networks, metabolic pathways and signaling pathways. The Stochastic simulation algorithm (SSA), in particular, is an exact method to realize the time evolution of a well-mixed biochemical reaction network. It takes the inherent randomness in biological reactions and the discrete nature of involved molecular species as the main source in sampling a reaction event. SSA is useful for reaction networks with low populations of molecular species, especially key species. The macroscopic response can be significantly affected when these species involved in the reactions both quantitatively and qualitatively. Even though the underlying assumptions of SSA are obviously simplified for real biological networks, it has been proved having the capability of reproducing the stochastic effects in biological behaviour. Essentially, SSA uses a Monte Carlo simulation technique to realize temporal behaviour of biochemical network. A reaction is randomly selected to fire at a time according to its propensity by conducting a search procedure. The fired reaction leads the system to a new configuration. At this new configuration, reactions have to update their propensities to reflect the changes. In this thesis we investigate new algorithms for improving performance of SSA. First, we study the application of tree-based search for improving the search of a reaction firing, and devise a solution to optimize the average search length. We prove that by a tree-based search the performance of SSA can be sensibly improved, moving the search from linear time complexity to logarithmic complexity. We combine this idea with others from the literature, and compare the performance of our algorithm with previous ones. Our experiments show that our algorithm is faster, especially on large models. Second, we focus on reducing the cost of propensity updates. Although the computational cost for evaluating one reaction propensity is small, the cumulative cost for a large number of reactions contributes a significant portion to the simulation performance. Typical experiments show that the propensity updates contribute 65% to 85%, and in some special cases up to 99%, of the total simulation time even though a dependency graph was applied. Moreover, sometimes one models the kinetics using a complex propensity formula, further increasing the cost of propensity updates. We study and propose a new exact simulation algorithm, called RSSA named after Rejection-based SSA, to reduce the cost of propensity updates. The principle of RSSA is using an over-approximation of propensities to select a reaction firing. The exact propensity value is evaluated only as needed. Thus, the propensity updates are postponed and collapsed as much as possible. We show through experiments that the propensity updates by our algorithm is significantly reduced, and hence substantially improving the simulation time. Third, we extend our study for reaction-diffusion processes. The simulation should explicitly account the diffusion of species in space. The compartment-based reaction-diffusion simulation is based on dividing the space into subvolumes so that the subvolumes are well-mixed. The diffusion of a species between subvolumes is modelled as an additional unimolecular reaction. We propose a new algorithm, called Rejection-based Reaction Diffusion (RRD), to efficiently simulate such reaction-diffusion systems. RRD combines the tree-based search and the idea of RSSA to select the next reaction firing in a subvolume. The highlight of RRD comparing with previous algorithms is the selection of both the subvolume and the reaction uses only the over-approximation of propensities. We prove the correctness and experimentally show performance improvement of RRD over other compartment-based approaches in literature. Finally, we focus on performing a statistical analysis of the targeted event by stochastic simulation. A direct application of SSA is generating trajectories and then counting the number of the successful ones. Rare events, which occur only with a very small probability, however, make this approach infeasible since a prohibitively large number of trajectories would need to be generated before the estimation becomes reasonably accurate. We propose a new method, called splitting SSA (sSSA), to improve the accuracy and efficiency of stochastic simulation while applying to this problem. Essentially, sSSA is a kind of biased simulation in which it encourages the evolution of the system making the target event more likely, yet in such a way that allows one to recover an unbiased estimated probability. We compare both performance and accuracy for sSSA and SSA by experimenting in some concrete scenarios. Experimental results prevail that sSSA is more efficient than the naive SSA approach.

### Paper Details

## Authors | ## T. Vo |

## Download | http://eprints-phd.biblio.unitn.it/1070/ |

## Language | ## English |