Oleson, Björn E.;Möller, Mark;Prank, Klaus - 4DiCeS: Simulating Diffusion of Signalling Molecules within a Cell

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Title: 4DiCeS: Simulating Diffusion of Signalling Molecules within a Cell	P116
Oleson, Björn E.; Möller, Mark; Prank, Klaus boleson@techfak.uni-bielefeld.de, mmoeller@techfak.uni-bielefeld.de, kprank@techfak.uni-bielefeld.de NRW Graduate School in Bioinformatics and Genome Research / Centre of Biotechnology (CeBiTec) / Bielefeld University, Bielefeld, Germany

Introduction:
Cells must be able to sense and respond to their environment, in particular to communicate with other cells. The exchange of information, called signal transduction, between and inside cells is essential for their survival. Here we, use a computational approach to explore the processes of communication within cells, i.e. intracellular signal transduction. This is done by mathematical modelling and simulation of signalling, the interpretation of information transferred in as well as between cells, and the interpretation of signals they receive.

Signal Transduction:
Signal reception commences at the point where an extra cellular signal activates a target molecule in the cellular membrane or inside the cell. In many cases the target molecule is a receptor protein which is typically triggered by just one type of signal. It carries out the primary transduction step: receiving the external signal, and creating a new intracellular signal in response to the stimulus. This is only the start in a subsequent chain of intracellular signal transduction processes. The information is passed from one set of intracellular signalling molecules to another, each set causing the production of the next. The final outcome is a cellular response, as e.g. the activation of a metabolic enzyme, gene expression, or changes within the cytoskeleton. Signal transduction is based on dynamic networks which can not be understood anymore without the use of mathematical modelling and simulations.

Stochastic (Monte Carlo) Simulation:
The modelling of cellular processes, such as signal transduction often involve the representation of biochemical reactions which a very small number of molecules, where deterministic approaches to a continuous-variation in the concentrations of molecular species by ordinary differential equations (ODE) fail. In order to correctly model the dynamics of cellular signalling stochastic effects have to be taken into account. In these approaches, individual molecular encounters are explicitly simulated using computer generated random numbers drawn from an appropriate probability distribution (Press et al., 1992). Stochastic simulations try to imitate natural processes my using computational methods. There have been already several attempts to use Monte Carlo simulations for the study of biochemical kinetics, such as the Gillespie-algorithm (Gillespie, 1977) for exact stochastic simulations or the computationally more efficient Gibson-Bruck-algorithm (Gibson and Bruck, 2000). However, these approaches possess a major problem. Although these algorithms are elegant and strait forward, the understanding of a model and its transformation to the simulation both become rather complicated. The user here has to handle with abstract parameters not easily found in the lab.

Diffusion:
Mathematical modelling and simulation of particle-movements as diffusion is an important topic in physics and chemistry. The relevance of this approach in biology is steadily growing. A Random Walk simulation is in its description close to reality and due to that easy to under-stand by users (Gil et al., 2000). Especially for small numbers of particles, Random Walk simulations show their benefits. Due to the precise knowledge of the whereabouts of every single particle, the produced data is much more detailed than any differential equation or Monte Carlo based simulation. One has to accept though that speed drops and memory usage becomes prohibitive with increasing numbers of particles. Therefore it is best to use Random Walks for small and Monte Carlo approaches as well as differential equations for large numbers of particles involved.
To simulate signal transduction of cells, cellular and subcellular structures have to be considered (Weng and Bhalla, 1999). All cell boundaries are membranes - motional lipid bilayers with bound reactive and transductive enzymes. Therefore a two dimensional diffusion of molecules on the membranes and a three dimensional one in free solution exists. The implementation of Random Walk diffusion is pretty strait forward. The movement of each molecule is determined either by one angle in two dimensions or two angles in three dimensions, a distance length and coordinates in space. The angles and the distance length all are made by random numbers and during a defined time step they transform the old coordinates into new ones. With a number of M molecules this procedure needs O(M) time. So for large M a different Random Walk approach is used. Here the cell is divided into a three dimensional mesh of subvolumes, called volume elements (VEs). An equation then takes care of how many particles stay in their home VE or spread over all the neighbouring VEs, using random numbers on all results (Benichou et al., 2000). This takes O(V) time for V subvolumes. The problem with this procedure is the assumption of equally distributed particles in every VE and the loss of distinct coordinates for every molecule or ion. But if the VE size is wisely adjusted to the mean distance length of a particle species this problem gets insignificant. So if the quantity of molecules or ions exceed the number of VEs this algorithm is used. Else an ordinary Random Walk is performed on all molecules.

Future work:
There are different processes of membrane passage - either active or inactive. The active form needs an energy rich molecule to perform and this molecule again has to diffuse within the cell. Also binding of two or more molecules has to be considered. When molecules combine, their coordinate systems become one, and this will reduce computation time. And last but not least a free particle in solution might attach to the membrane or vice versa. This means the diffusion algorithm has to change to either situation.

[1] Benichou, O., M. Moreau, and G. Oshanin. 2000. Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 61:3388-406.
[2] Gibson, M.A., and J. Bruck. 2000. Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phy Chem. 104:1876-1889.
[3] Gil, A., J. Segura, J.A. Pertusa, and B. Soria. 2000. Monte carlo simulation of 3-D buffered Ca(2+) diffusion in neuroendocrine cells. Biophys J. 78:13-33.
[4] Gillespie, D.T. 1977. Exact stochastic simulation of coupled chemical reactions. J Phy Chem. 81:2340-2361.
[5] Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. 1992. 7. Random Numbers. In Numerical recipies in C: The art of scientific computing. Cambridge University Press, Cambridge. 274-328.
[6] Weng, G., and U.S. Bhalla. 1999. Complexity in biological signaling systems. Science. 284:92-96.