Computational Systems Biology

Modern molecular biology has discovered and examined a vast number of cellular components. The mechanisms of regulation and functionality of the different molecules is still largely unknown. To understand the principles for cellular regulation mathematical models are now constructed and there is a need to simulate these models in silico.

The project group consists of one senior researcher Per Lötstedt, one researcher Stefan Engblom, as well as three PhD students Stefan Hellander, Lina Meinecke, and Pavol Bauer. This project is a collaboration with Docent Johan Elf at the Department of Cell and Molecular Biology at Uppsala University and the StochSS project at University of California, Santa Barbara, headed by professor Linda Petzold.

This project was also a part of a collaboration with the Department of Numerical Analysis and Computer Science at KTH, Stockholm.

Research

Macroscopic, mesoscopic, and microscopic models in molecular biology

The most accurate models for the chemical reactions in biological cells are based on differential equations. The reaction rate equations for the concentrations of the molecular species are a system of nonlinear ordinary differential equations and the macroscopic model equations. They can be solved using standard mathematical software but they are too inaccurate for certain problems e.g. involving a small number of molecules of each kind. This is often the situation in molecular biology. For such systems a more precise mesoscopic, stochastic modelling is necessary. The master equation of chemical kinetics is an equation for the probability density of the distribution of molecules. The master equation can be approximated by a Fokker-Planck equation which is a partial differential equation of parabolic type. A computational difficulty with the master and Fokker-Planck equations is the growth in the number of dimensions of the equations with the number of molecular species involved in the reactions. Standard numerical methods for solving these equations suffer from an exponential growth in computational work and memory.

A harder problem is the solution of the reaction-diffusion master equation where the species are not well-stirred but have a space dependence. There the master equation may depend on a state vector with thousands of dimensions making computational solution of the equation impossible. A code for simulation of such systems on unstructured meshes is found here. The method has been generalized to active transport problems and an adaptive switch between macroscopic and mesoscopic diffusion. A microscopic model where single molecules are tracked is necessary for some systems. Then the molecules follow Brownian dynamics and react with each other when they are close. An efficient method avoiding the small time steps in straightforward Brownian motion is GFRD [1]. The method is simplified and made more flexible by introducing operator splitting.

The Stochastic Simulation Algorithm (SSA) due to Gillespie [2] is widely used to simulate biochemical reaction networks modeled as a continuous-time discrete-space Markov process. CellMC is an XSLT-based, automated SBML (Systems Biology Markup Language) model compiler capable of producing very efficient SSA executables for the Cell/BE or multicore x86 PCs. CellMC was developed by Emmet Caulfield as a part of his master's thesis: "CellMC: An XSLT-based SBML model compiler for Cell/BE and IA32'.

1. J. S. van Zon and P. R. ten Wolde, Green's-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space, J. Chem. Phys., 123 (2005), art. 234910.
2. D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), pp. 403-434.

Circadian rythms

Living organisms have to adapt their behavior to different periodic changes in their environment e.g. the daily variation of light and the annual variation of temperature. Many of them have developed molecular clocks as their internal time-keeping mechanisms. An example of such an oscillatory process is the circadian rhythm with a period of about 24 h [3]. A model for this rhythm has been developed in [4] and is simplified in [5]. This model has two molecular species and the corresponding partial differential equation is two dimensional.

PDE approximation	Monte Carlo
The Fokker-Planck equation of the simplified model is solved with a numerical method for partial differential equations with space adaptivity. A description of this method is found here.	The chemical master equation is solved by the stochastic simulation algorithm (SSA) [1]
These two movies show the behavior of the probability density function in one period of the oscillation computed by two different numerical methods. A comparison between the methods is found here.

The original model has nine species and a reduction of the complexity is necessary with standard discretization methods. By combining the reaction rate equations on the macroscale with the Fokker-Planck equation on the mesoscale a two dimensional problem is obtained.

Dimension reduction
This movie shows one period of the projection of the probability function of the reduced problem on the same subspace as in the example above. This method is described here.

3. A. Golbeter, Computational approaches to cellular rhythms, Nature, 420 (2002), pp. 238-245.
4. N. Barkai, S. Leibler, Circadian clocks limited by noise, Nature, 403 (2000), pp. 267-268.
5. J. M. G. Vilar, H. Y. Kueh, N. Barkai, S. Leibler, Mechanisms of noise-resistance in genetic oscillators, Proc. Natl. Acad. Sci., 99 (2002), pp. 5988-5992.

Reaction-diffusion process
The cell with a nucleus is discretized by a tetrahedral mesh. A snapshot of the spatial distribution of one species in the cytosol (left) and another species in the nucleus (right). Red represents high copy number and blue low. The method for simulation of stochastic reaction-diffusion processes is described here.

Publications

Recent papers

• M. H. Bani-Hashemian, S. Hellander, P. Lötstedt, Efficient sampling in event-driven algorithms for reaction-diffusion processes, Comm. Comput. Phys., 13 (2013), pp. 958-984.
• B. Drawert, S. Engblom, A. Hellander, URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries, BMC Syst. Biol., 6 (2012), 76.
• A. Hellander, S. Hellander, P. Lötstedt, Coupled mesoscopic and microscopic simulation of stochastic reaction-diffusion processes in mixed dimensions, Multiscale Model. Simul., 10 (2012), pp. 585-611.
• S. Hellander, A. Hellander, L. Petzold, Reaction-diffusion master equation in the microscopic limit, Phys. Rev. E 85 (2012), 042901.
• A. Grönlund, P. Lötstedt, J. Elf, Delay-induced anomalous fluctuations in intracellular regulation, Nat. Commun., 2 (2011), pp. 419:1-7.
• S. Hellander, P. Lötstedt, Flexible single molecule simulation of reaction-diffusion processes, J. Comput. Phys., 230 (2011), pp. 3948-3965.
• A. Hellander, P. Lötstedt, Incorporating active transport of cellular cargo in stochastic mesoscopic models of living cells, Multiscale Model. Simul., 8 (2010), pp. 1691-1714.
• A. Grönlund, P. Lötstedt, J. Elf, Costs and constraints from time-delayed feedback in small gene regulatory motifs, Proc. Natl. Acad. Sci. USA, 107 (2010), pp. 8171-8176.
• E. Caulfield, A. Hellander, CellMC - a multiplatform model compiler for the Cell Broadband Engine and x86, Bioinformatics, 26 (2010), pp. 426-428.
• L. Ferm, A. Hellander, P. Lötstedt, An adaptive algorithm for simulation of stochastic reaction-diffusion processes, J. Comput. Phys., 229 (2010), pp. 343-360.
• S. Engblom, Parallel in time simulation of multiscale stochastic chemical kinetics, Multiscale Model. Simul., 8 (2009), pp. 46-68.
• S. Engblom, Spectral approximation of solutions to the chemical master equation, J. Comput. Appl. Math., 229 (2009), pp. 208-221.
• S. Engblom, Galerkin spectral method applied to the chemical master equation, Comm. Comput. Phys., 5 (2009), pp. 871-896.
• M. Hegland, A. Hellander, P. Lötstedt, Sparse grids and hybrid methods for the chemical master equation, BIT, 48 (2008), pp. 265-283.
• S. Engblom, L. Ferm, A. Hellander, P. Lötstedt, Simulation of stochastic reaction-diffusion processes on unstructured meshes, SIAM J. Sci. Comput., 31 (2009), pp. 1774-1797.
• P. Sjöberg, PDE and Monte Carlo approaches to solving the master equation applied to gene regulation
• A. Hellander, Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo, J. Chem. Phys., 128 (2008), pp. 154109:1-7.
• L. Ferm, P. Lötstedt, Adaptive solution of the master equation in low dimensions, Appl. Numer. Math., 59 (2009), pp. 187-204.
• L. Ferm, P. Lötstedt, A. Hellander, A hierarchy of approximations of the master equation scaled by a size parameter, J. Sci. Comput., 34 (2008), pp. 127-151.
• A. Hellander, P. Lötstedt, Hybrid method for the chemical master equation, J. Comput. Phys., 227 (2007), pp. 100-122.
• P. Sjöberg, Partial approximation of the master equation by the Fokker-Planck equation, in Applied Parallel Computing: State of the Art in Scientific Computing, Vol. 4699 of LNCS, 2007, pp. 637-646.
• S. Engblom, Gaussian quadratures with respect to discrete measures
• L. Ferm, P. Lötstedt, Numerical method for coupling the macro and meso scales in stochastic chemical kinetics, BIT, 47 (2007), pp. 735-762.
• P. Sjöberg, P. Lötstedt, J. Elf, Fokker-Planck approximation of the master equation in molecular biology, Comput. Vis. Sci., 12 (2009), pp. 37-50.
• P. Lötstedt, L. Ferm, Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions, Multiscale Model. Simul., 5 (2006), pp. 593-614.
• S. Engblom, Computing the moments of high dimensional solutions of the master equation, Appl. Math. Comput., 180 (2006), pp. 498-515.
• L. Ferm, P. Lötstedt, P. Sjöberg, Conservative solution of the Fokker-Planck equation for stochastic chemical reactions, BIT, 46 (2006), pp. S61-S83.
• J. Elf, P. Lötstedt, P. Sjöberg, Problems of high dimension in molecular biology, in Proc. 19th GAMM Seminar Leipzig, 2003, pp. 21-30.

Doctoral theses

• Paul Sjöberg, Numerical Methods for Stochastic Modeling of Genes and Proteins, Uppsala University, 2007.
• Stefan Engblom, Numerical Solution Methods in Stochastic Chemical Kinetics, Uppsala University, 2008.
• Andreas Hellander, Multiscale stochastic simulation of reaction-transport processes. Applications in Molecular Systems Biology, Uppsala University, 2011.

Licentiate theses

• Paul Sjöberg, Numerical solution of the Fokker-Planck approximation of the chemical master equation, Uppsala University, 2005.
• Stefan Engblom, Numerical methods for the chemical master equation, Uppsala University, 2006.
• Andreas Hellander, Numerical Simulation of Well Stirred Biochemical Reaction Networks Governed by the Master Equation, Uppsala University, 2008.

History

This project was started in 2002 and has been supported by the National Graduate School in Scientific Computing, the Graduate School in Mathematics and Computing, the Swedish Foundation for Strategic Research, the Swedish Research Council, and NIH under contract 1R01EB014877 - 01.

Contact Information

Per Lötstedt