# CME2011

Workshop on the Numerical Solution of the Chemical Master Equation in Molecular Biology

**TUM Institute for Advanced Study**

**Focus Group HPC**

**12. September 2011**

**09:30 AM - 18:00 PM**

Conference Room 1-2 (1rst floor)

TUM Institute for Advanced Study

Lichtenbergstraße 2 a

85748 Garching

Organisors: TUM-IAS Hans Fischer Senior Fellow Markus Hegland (Australian National University) and Per Lötstedt (Uppsala University)

## Contents

# Description

Chemical reactions in living organisms exhibit stochastic behavior and are described by continuous time discrete state Markov models. The governing equation for the temporal evolution of the probability distribution of such systems is the chemical master equation. Traditional numerical techniques for the solution of the chemical systems frequently fail because of the curse of dimensionality which affects high-dimensional problems (with dimension larger than 5). One cause of the high dimensionality are the large numbers of different kinds of molecules involved. Often stochastic simulation is the only tool which is feasible to computationally explore such systems. In this workshop we will discuss alternatives based on novel numerical techniques which have the potential to overcome the curse.

# Participation

If you would like to attend, please register in advance via email to Dirk Pflüger, |

# How to get there

Directions can be found here

# Program

09:30-10:00 | Coffee and registration |

10:00-10:45 | Verena Wolf (abstract) |

10:45-11:30 | Wilhelm Huisinga (abstract) |

11:30-11:45 | break |

11:45-12:30 | Jochen Garcke (abstract) |

12:30-14:00 | Lunch |

14:00-14:45 | Tobias Jahnke (abstract) |

14:45-15:30 | Susanna Röblitz (abstract) |

15:30-16:00 | Coffee |

16:00-16:45 | Per Lötstedt (abstract) |

16:45-17:30 | Markus Hegland (abstract) |

# Abstracts

## Verena Wolf

**Stochastic Hybrid Analysis of Markov Population Models**

In this talk we will consider discrete-state Markov processes that exhibit a population structure. They find various applications in different domains and are particularly well-suited to describe the interactions between different molecular species in living cells. The direct analysis of such Markov population models is very challenging due to the "curse of dimension" but their regular structure allows fluid approximations on different levels. Such approximations result in ordinary or partial differential equations that describe the evolution of stochastic (hybrid) models in purely continuous or continuous-discrete state spaces. I will discuss numerical techniques to analyze such models over time and in equilibrium. Moreover, the appropriateness of fluid approximations and their relation to model reduction techniques will be discussed based on several examples from systems biology.

## Wilhelm Huisinga

**Hybrid stochastic-deterministic approach to directly solve the chemical master equation based on multiscale expansion**

*Joint work with Stephan Menz, Juan Latorre and Christof Schütte (FU Berlin)*

The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. In most applications it is impossible to solve the CME directly due to its high dimensionality. Instead indirect approaches based on realizations of the underlying Markov jump process are used such as the stochastic simulation algorithm (SSA). In the SSA, however, every reaction event has to be resolved explicitly such that it becomes numerically inefficient when the system’s dynamics include fast reaction processes or species with high population levels. In many hybrid approaches, such fast reactions are approximated as continuous processes or replaced by quasi-stationary distributions either in a stochastic or deterministic context. Current hybrid approaches, however, almost exclusively rely on the computation of ensembles of stochastic realizations. We present a novel hybrid stochastic–deterministic approach to solve the CME directly. Starting point is a partitioning of the molecular species into discrete and continuous species that induces a partitioning of the reactions into discrete–stochastic and continuous–deterministic. The approach is based on a WKB approximation of a conditional probability distribution function (PDF) of the continuous species (given a discrete state) combined with a multiscale expansion of the CME. The resulting hybrid stochastic–deterministic evolution equations comprise a CME with averaged propensities for the PDF of the discrete species that is coupled to an evolution equation of the partial expectation of the continuous species for each discrete state. In contrast to indirect hybrid methods, the impact of the evolution of discrete species on the dynamics of the continuous species has to be taken into account explicitly. The proposed approach is efficient whenever the number of discrete molecular species is small. We illustrate the performance of the new hybrid stochastic–deterministic approach in application to model systems of biological interest.

## Jochen Garcke

**On signaling cascades and tensor trains**

We study the chemical master equations for chemical signaling cascades, one of the main components of biological switches. The have been shown to simultaneously perform thresholding and signal amplification. This despite the intrinsic noise which is introduced by the chemical reactions which constitute the cascade stages.

Using the chemical master equations for a signaling cascase one obtains a tool for the determination of the marginal probability distributions over the domain of copy numbers of the species involved. This leads to a natural representation of intrinsic noise.

The solution of the master equations and even the representation of the resulting multidimensional probability distribution suffers under the curse of dimensionality. We propose the use of the recently introduced tensor-train framework for the representation of the arising high-dimensional objects. Theoretical considerations show a relation of the structure of a cascade to the structure of a tensor train. Numerical studies give evidence that the chemical master equation for a signaling cascade and its stationary solution have the necessary low rank structure allowing its efficient computation in 10 dimensions.

## Tobias Jahnke

**Efficient simulation of stochastic reaction systems**

*Joint work with Derya Altintan*

Abstract: In this talk an efficient simulation method for stochastic reaction systems is presented. The method is based on a partitioning of the reaction channels into subsystems which are propagated by the Strang splitting, and on the fact that many subsystems can be efficiently simulated because an explicit formula for the exact solution of these subsystems is available. We prove error bounds for the splitting error and present numerical examples which demonstrate the advantages of the new approach.

## Susanna Röblitz

**Meshfree methods for the chemical master equation**

For the solution of the chemical master equation (CME), error oriented adaptive solution methods based on h-p finite elements [Wulkow 1996, Deuflhard et al. 2008] or wavelets [Jahnke 2010] have been successfully developed over the last few years. These methods, originally motivated by one- or two-dimensional problems, are based on meshes and tensor product structures which complicates their applicability to high dimensional problems. This motivated our approach to use meshfree discretization techniques which are only based on a set of independent points. The price to pay for this flexibility, however, is the loss of concise error estimates. Nevertheless we believe that meshfree methods, maybe in combination with other techniques, are a valuable tool for the solution of the CME and we want to discuss their applicability and limitations during the workshop.

## Per Loetstedt

**The Master Equation and Stochastic Simulation with Time Delays**

In some models of biochemical reactions, it is important to include a time delay because there is a delay between the initiation of a reaction and its completion. A master equation for a delayed system is proposed and simplifying assumptions are made. Approximate equations for the mean values and the covariances are derived and applied to two regulatory motifs. A more accurate and expensive model is to introduce a multistep reaction instead of the fixed time delay. The relations between these two approaches are analyzed and compared for a simple feedback system.

## Markus Hegland

**The FSP algorithm and an optimal variant**

*Joint work with Vikram Sunkara*

The FSP method introduced by Munsky and Khammash is a popular method to solve the chemical master equation for moderate dimensions. In this talk I will review this method and discuss a variant which is provably optimal for some classes of problems.