Modelling a Living Cell - Mathematics to Model Metabolic Pathways
by Joke Blom and Annette Kik
Can biologists describe a living cell in a computer? Mathematicians of the Centrum voor Wiskunde en Informatica (CWI) cooperate within the Silicon Cell project, where biologists and mathematicians try to reach this goal. Their aim is to relate theory and models closely with data from biological experiments. In future this research could give an indication whether new medication will be effective and how we can keep food good and tasteful as long as possible. This shortens research time and saves expensive biological experiments.
The twentieth century saw many inventions in the frontier areas of life sciences, physics, chemistry, and mathematics. Biological experiments improved: It even became possible to look inside a living cell. Biologists would gladly model its functioning. This is precisely the aim of the Silicon Cell Consortium Amsterdam, a cooperation of the Institute for Molecular Biological Sciences (IMBS) of the Vrije Universiteit in Amsterdam, the Swammerdam Institute for Life Sciences (SILS) and the Section Computational Science (SCS) of the University of Amsterdam, and CWI.
In the living cell, processes of metabolism - metabolic pathways or networks - can be described as partial differential equations (PDEs), where concentrations can vary in space and time: a reaction-diffusion model. There are two lines of research, the qualitative and quantitative one. Qualitative research could describe a process as: 'If cell-size increases it will affect the functioning in such a way' or: 'If the concentration of protein A becomes infinite, then that effect will occur'. In practice the situation is often more balanced and the parameters are limited, because the cell will not survive infinite values.
With quantitative research, realistic parameters for living cells are used. The values for constants and parameters are determined in real biological experiments, for example on metabolic pathways in yeast cells and the E. coli bacterium. One of the recent analytic results a contribution to the theory of Metabolic Control Analysis - is a new type of control coefficient summation theorem, which relates the control by the membrane transport, the diffusion control, and the size of a cell. Quantitative numerical experiments indicate that diffusion through a membrane in the E. coli bacterium is no limitative factor for the uptake of glucose. However, if the bacterium were ten times larger, the sugar admission from its living surroundings (human intestines) would falter.
Besides reaction-diffusion models for cellular pathways, developmental regulatory networks are studied. CWI develops models for simulating networks that are capable of quantitatively reproducing expression patterns in developmental processes. One example is the embryonic development of fruit flies, where biologists supply the experimental data. Mathematically speaking this research adds to a continuum-discrete hybrid model where discrete, moving and deformable objects, in which biochemical reactions take place, exchange species with the surrounding environment modelled as a continuum.
For the mathematicians, one challenge is to relate models that work on the different scales that can be measured in biological experiments by now or in the near future: From nanoseconds to weeks and from nanometers to millimeters. Current research at CWI is built around the concepts of simplification and integration. Simplification is essential: a straightforward simulation of a model comprising all the biochemical knowledge is computationally too demanding. Depending on the cellular phenomenon considered, models and methods of appropriate temporal and spatial scales will be developed: ordinary differential equations for simple cells that can be described as homogeneous objects, partial differential equations for moderate spatial and temporal variations, and particle methods for even smaller structures.
Furthermore, techniques for modularisation, model reduction, flexible gridding and flexible time discretization are developed. An effective use of these techniques requires that they are adaptive. Variations in spatial, temporal, and chemical complexity are handled most efficiently if the simplification techniques can be adjusted accordingly. Therefore it is necessary to integrate different model descriptions and numerical approximations into an aggregate simulation without sacrificing reliability. The dynamic heterogeneity of the living cell and the flexibility of the simplification techniques imply that the degree and type of simplification will vary in space and time, thus placing further constraints on the integration.
Another challenge for the mathematicians is to perfect the models to a level that shows in advance whether it is useful to perform a certain experiment. This way, expensive and elaborate biological experiments could perhaps decrease by half.
Joke Blom, CWI, The Netherlands
Tel: +31 20 592 4263