Stochastic Simulation for Systems Biology

 

Systems Biology is a buzzword that describes a range of modern and interdisciplinary activities. The field of Systems Biology has exposed the life sciences to ideas from computer science, engineering, physics and most notably, mathematics and statistics in order to shed light on the complex interactions that are observed in biological systems. At the heart of this study is the recognition that we need to understand not only the individual components of an organism but also how they interact to form complex dynamic systems at the level of cells, tissues, organs, organisms and populations.

 

 

As with any quantitative science, questions in systems biology must be posed, and tackled, using the language of mathematics.

 

Mathematical modelling is used to derive equations that describe

physical processes and statistical techniques are used to flesh out the models with realistic parameters and calibrate them against biological observations. Computational methods can then be used to simulate the resulting models.

 

You may have heard of two categories of experiment:

 

1. In vivo (Latin: within the living) studies are performed on living

organisms, or at least on living tissue

 

2. In vitro (Latin: within the glass) studies are performed in a controlled environment outside of a living organism; traditionally in a test tube.

 

Mathematical modelling throws up a third possibility that has been coined in silico (Latin: within silicon), where computer simulation replaces the wet-lab experiments. It is not hard to think of advantages: faster results, greater flexibility, elimination of ethical headaches and the ability to look at “what-if” scenarios beyond the scope of experimental biology.

 

Computational Biology

 

Ultimately computational biology, when successful, provides a predictive tool for answering questions such as:

 

1. Would an increase in the level of protein X inhibit or enhance the activity of protein Y?

 

2. How quickly can the immune system respond an attack by disease Z?

 

3. What happens if we increase the dose of drug A in a patient with disease B?

Focussing now on the level of the cell, computational biology has successfully exploited the framework of chemical kinetics to describe the interactions between genes, mRNAs, proteins and other objects, such as microRNAs. In this setting, a key feature is that one or more of the “species” may be present in small quantities. If this species is critical to the system, such as the transcription factor in gene networks, any slight change in molecular numbers may have a significant influence on the system dynamics. It does not make sense to think about the concentration of the species as a real number that changes continuously over time. Hence there is a trend in systems biology to move away from standard deterministic differential equations and to use stochastic simulation techniques to describe the dynamics in the cell.

 

Stochastic Simulation Framework

 

In the stochastic simulation framework, we think in terms of the probability of a reaction taking place, based on the current state of the system. This leads to the Chemical Master Equation (CME), which assigns a probability to each possible state of the system. The stochastic simulation algorithm (SSA) turns the CME into a practical tool, allowing statistically correct computational experiments to be performed. These ideas date back to the work of Gillespie in the 1970s, but have undergone a new lease of life because of their relevance in systems biology; see, for example, the text Stochastic Modelling for Systems Biology, by D. J. Wilkinson, Chapman & Hall/CRC, 2006 or the research article Modelling and Simulating Chemical Reactions by D. J. Higham, Society for Industrial and Applied Mathematics (SIAM) Review, April 2008.

 

Although straightforward to implement, the SSA can be impractically slow when at least one type of reaction occurs frequently. Pushing the approximation further leads to the Chemical Langevin Equation (CLE), a set of stochastic differential equations (SDEs) with one component for each chemical species, which is relatively cheap to simulate. As a further simplification, we can completely ignore fluctuations in the CLE and return to the purely deterministic Reaction Rate Equations (RREs).

 

A key challenge in computational cell biology is that stochastic and discrete models are needed for at least some of the interactions taking place, and yet a fully discrete and stochastic model will require a prohibitive amount of computational time. Hence, multi-scale modelling and algorithmics, intelligently combining features of the CME/CLE/RRE regimes are needed.

Deriving such “multi-scale” algorithms is an extremely active interdisciplinary research area that requires new insights from mathematics, statistics and the life sciences. Further breakthroughs must rely on the most complex multi-scale system that we know of - the human brain.

 

Article By:

Professor Desmond Higham
Department of Mathematics
University of Strathclyde

Science / Engineering courses / colleges