Per Lötstedt - Research

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Stochastic simulation of reaction-diffusion processes in molecular biology
Ice sheets and ice shelves
Computational finance

 

Stochastic simulation of reaction-diffusion processes 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 in a well stirred system are a system of nonlinear ordinary differential equations at a macroscopic level of modeling. They can be solved using standard mathematical software but they are too inaccurate for certain problems involving a small number of molecules of each kind. This is often the situation in molecular biology. For such processes a more precise stochastic modeling is necessary. The master equation of chemical kinetics is an equation at the mesoscopic level of modeling for the probability density of the distribution of molecules. Each species corresponds to one dimension. A computational difficulty with the master equation is the exponential growth of the work in the number of dimensions of the equations.

The problem of high dimension is even more severe if the species are space dependent. Then the cell domain in partitioned into voxels or subcompartments in a mesoscopic model and the state of the system is defined by the copy number of the species in each voxel. The molecules react with each other and diffuse to neighboring voxels. The only computationally feasible alternative is to simulate the system with a Monte Carlo method. A method to compute a trajectory has been developed for unstructured meshes, for active transport, and for one dimensional structures embedded in 3D space based on Gillespie's Stochastic Simulation Algorithm.

An even more detailed model is microscopic simulation of single molecules. The Smoluchowski model for spherical molecules to react with each other and diffuse in Brownian motion has been implemented efficiently using the the ideas in Green's Function Reaction Dynamics (GFRD) by van Zon and ten Wolde. The new position of a free molecule is sampled from a Gaussian distribution. For two molecules in the neighborhood of each other their new relative position is determined by a probability distribution satisfying the Smoluchowski equation.

The macrosopic model for the concentrations of the species is the reaction-diffusion partial differential equation. Methods have been developed for the coupling of the mesoscopic model with the macroscopic model and for the coupling of the mesoscopic model with the microscopic model in GFRD. In this way the most accurate and computationally expensive method is used only where it is necessary for the accuracy of the simulations.

A difficulty with a discretization of space by an unstructured mesh is that the computed jump probabilities by the finite element method are negative due to poor mesh quality. Two remedies have been devised. The first exit time also defines jump probabilities and they are always non-negative. By discretizing a problem with slightly different diffusion coefficients the jump probabilities are shown to be non-negative. The diffusion in a crowded environment is observed to behave anomalously in a transient time interval. Diffusion and reactions are modified in the mesoscopic model by introducing internal states mimicking this behavior.

Part of the project is a collaboration with the Elf group at the Department of Cell and Molecular biology at Uppsala University. Important subsystems in a cell have been analyzed mathematically and in simulations and biological conclusions have been drawn.

Financial support has been obtained from the Swedish Research Council, Swedish Foundation for Strategic Research, Göran Gustafsson Foundation, Centre for Interdisciplinary Mathematics, and the Graduate School in Mathematics and Computing. See also the homepage.

Recent papers
E Blanc, S Engblom, A Hellander, P Lötstedt, Mesoscopic modeling of stochastic reaction-diffusion kinetics in the subdiffusive regime, revision in Multiscale Model. Simul., 14 (2016), 668-707.

L Meinecke, S Engblom, A Hellander, P Lötstedt, Analysis and design of jump coefficients in discrete stochastic diffusion models, revision in SIAM J. Sci. Comput., 38 (2016), A55-A83.

L Meinecke, P Lötstedt, Stochastic Diffusion Processes on Cartesian Meshes, J. Comput. Appl. Math., 294 (2016), 1-11.

P Lötstedt, L Meinecke, Simulation of Stochastic Diffusion via First Exit Times, revised version in J. Comput. Phys., 300 (2015), 862-886.

S Wang, J Elf, S Hellander, P Lötstedt, Stochastic reaction-diffusion processes with embedded lower dimensional structures, revised version in Bull. Math. Biol., 76 (2014), 819-853.

A Grönlund, P Lötstedt, J Elf, Transcription factor binding kinetics constrain noise suppression by negative feedback, Nature Commun., 4 (2013), 1864.

M H Bani-Hashemian, S Hellander, P Lötstedt, Efficient sampling in event-driven algorithms for reaction-diffusion processes, revised version in Commun. Comput. Phys., 13 (2013), 958-984.

A Hellander, S Hellander, P Lötstedt, Coupled mesoscopic and microscopic simulation of stochastic reaction--diffusion processes in mixed dimensions, revised version in Multiscale Model. Simul., 10 (2012), 585-611.

A Grönlund, P Lötstedt, J Elf, Delayed induced anomalous fluctuations in intracellular regulation, Nature Commun., 2 (2011), 419.

S Hellander, P Lötstedt, Flexible Single Molecule Simulation of Reaction-Diffusion Processes, a revised version in J. Comput. Phys. 230 (2011), 3948-3965.

A Grönlund, P Lötstedt, J Elf, Costs and constraints from time-delayed feed-back in small gene regulatory motifs, Proc. Natl. Acad. Sci. USA, 107 (2010), 8171-8176

A Hellander, P Lötstedt, Incorporating Active Transport of Cellular Cargo in Stochastic Mesoscopic Models of Living Cells, revised version in Multiscale Model. Simul., 8 (2010), 1691-1714.

L Ferm, A Hellander, P Lötstedt, An adaptive algorithm for simulation of stochastic reaction-diffusion processes, revised version in J. Comput. Phys., 229 (2010), pp. 343-360.

S Engblom, L Ferm, A Hellander, P Lötstedt, Simulation of stochastic reaction-diffusion processes on unstructured meshes, revised version in SIAM J. Sci. Comput. 31 (2009), 1774-1797.

L Ferm, P Lötstedt, Adaptive solution of the master equation in low dimensions, revised version in Appl. Numer. Math. 59 (2009), 187-204.

P Sjöberg, P Lötstedt, J Elf, Fokker-Planck approximation of the master equation in molecular biology, revised version in Comput. Visual. Sci. 12 (2009), 37-50.

L Ferm, P Lötstedt, A Hellander, A hierarchy of approximations of the master equation scaled by a size parameter, revised version in J. Sci. Comput., 34 (2008), 127-151.

M Hegland, A Hellander, P Lötstedt, Sparse grids and hybrid methods for the chemical master equation, BIT, 48 (2008), 265-284.

A Hellander, P Lötstedt, Hybrid method for the chemical master equation, revised version in J. Comput. Phys., 227 (2007), 100-122.

L Ferm, P Lötstedt, Numerical method for coupling the macro and meso scales in stochastic chemical kinetics, revised version in BIT, 47 (2007), 735-762.

P Lötstedt, L Ferm, Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions, revised version in Multiscale Model. Simul. 5 (2006), 593-614.

L Ferm, P Lötstedt, P Sjöberg, Conservative solution of the Fokker-Planck equation in molecular biology, revised version in BIT, 46 (2006), S61-S83.

J Elf, P Lötstedt, P Sjöberg, Problems of high dimension in molecular biology, Proceedings of the 19th GAMM-Seminar, Leipzig, January, 23th-25th, 2003

PhD and licentiate theses
Paul Sjöberg, Numerical Solution of the Fokker-Planck Approximation of the Chemical Master, Equation licentiate thesis, Uppsala University, 2005.

Stefan Engblom, Numerical Methods for the Chemical Master Equation, licentiate thesis, Uppsala University, 2006.

Paul Sjöberg, Numerical Methods for Stochastic Modeling of Genes and Proteins, PhD thesis, Uppsala University, 2007.

Andreas Hellander, Numerical Simulation of Well Stirred Biochemical Reaction Networks Governed by the Master Equation, licentiate thesis, Uppsala University, 2008.

Stefan Engblom, Numerical Solution Methods in Stochastic Chemical Kinetics, PhD thesis, Uppsala University, 2008.

Andreas Hellander, Multiscale stochastic simulation of reaction-transport processes. Applications in Molecular Systems Biology, PhD thesis, Uppsala University, 2011.

Stefan Hellander, Stochastic Simulation of Reaction-Diffusion Processes, PhD thesis, Uppsala University, 2013.

Lina Meinecke, Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times, PhD thesis, Uppsala University, 2016.

 

Ice sheet and ice shelf modeling
Ice sheets are important components of the global climate system. It is of interest to predict the future of the ice sheets of Antarctica and Greenland and to reconstruct the ice in the past in paleoglaciology. The ice is modeled by a system of partial differential equations with a non-linear constitutive stress-strain relation. This is a collaboration with Nina Kirchner at the Bert Bolin Centre for Climate Research at Stockholm University with support from FORMAS. See also the homepage.

Recent papers
G Cheng, P Lötstedt, L von Sydow, Accurate and stable time stepping in ice sheet modeling, J. Comput. Phys., 329 (2017), 29-47.

N Kirchner, J Ahlkrona, E J Gowan, P Lötstedt, J M Lea, R Noormets, L von Sydow, J A Dowdeswell, T Benham, Shallow Ice Approximation, Second Order Shallow Ice Approximation and Full Stokes models: a discussion of their roles in palaeo-ice sheet modelling and development, Quat. Sci. Rev., 135 (2016), 103-114.

J Ahlkrona, P Lötstedt, N Kirchner, T Zwinger, Dynamically coupling the non-linear Stokes equations with the Shallow Ice Approximation in glaciology: Description and first applications of the ISCAL method, revised version in J. Comput. Phys., 308 (2016), 1-19.

J Ahlkrona, N Kirchner, P Lötstedt, A numerical study of scaling relations for non-Newtonian thin film flows with applications in ice sheet modeling, Quart. J. Mech. Appl. Math., 66 (2013), 417-435.

J Ahlkrona, N Kirchner, P Lötstedt, Accuracy of the zeroth and second order shallow ice approximation - Numerical and theoretical results, Geosci. Model Dev., 6 (2013), 2135-2152.

PhD thesis
Josefin Ahlkrona, Computational Ice Sheet Dynamics - Error control and efficiency, PhD thesis, Uppsala University, 2016.

 

Computational finance
Finite difference methods are used to solve partial differential equations to price financial derivatives. See the homepage.

Recent papers
P Lötstedt, L von Sydow, Numerical option pricing without oscillations using flux limiters, Comp. Math. Appl., 70 (2015), 1-10.

E Ekström, P Lötstedt, L von Sydow, J Tysk, Numerical option pricing in the presence of bubbles, Quant. Finance, 11 (2011), 1125-1128.

E Ekström, P Lötstedt, J Tysk, Boundary values and finite difference methods for the term structure equation, Appl. Math. Finance, 16 (2009), 253-259.

P Lötstedt, J Persson, L von Sydow, J Tysk, Space-time adaptive finite difference method for European multi-asset options, Comp. Math. Appl., 53 (2007), 1159-1180.

 

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© Britta Lötstedt och Johanna Wedin 2006