The group of Modeling and Simulation is focussed around stochastic modeling and analysis techniques. We are particularly interested in discrete-state Markov processes and stochastic hybrid systems that describe gene regulatory networks.

Current and recent work includes

- Approximations of the Chemical Master Equation
- Parameter Estimation
- Computation of Rare Event Probabilities
- Multistable decision switches
- DNA methylation models

#### Approximations of the Chemical Master Equation

The Chemical Master Equation describes the evolution of a system of coupled chemical reactions in terms of a continuous-time Markov chain. This modeling approach has received increasing attention because it takes into account the randomness of microscopic events, which has significant influence on many cellular processes.

In this project, we are working on numerical solution methods for the Chemical Master Equation. The most promising approach is based on a hybrid description where for certain molecular types the conditional moments are integrated instead of the full (conditional) probability distribution (article is in press; see also this PDF). This hybrid approach has the advantage that it can be applied to systems where certain variables change deterministically and continuously in time but other variables show discrete and stochastic behavior. Further contributions focus on the approximation of steady-state probabilities (PDF).

#### Parameter Estimation

Often models come with unknown parameters that need to be fitted according to observations of the real system. This project focuses on the inference of parameters of Markov models of chemical reaction networks based on noisy time-series data. We compute approximations of the likelihood of observed data and determine reaction rate constants that maximize the likelihood. See THIS technical report for further details.

#### Multistable decision switches

Cellular systems use switch-like behavior to decide between different strategies, e.g., as a response to an external signal. Often the underlying mechanism is the multistability of the corresponding chemical reaction network. If a Markov process representation is chosen, multistability is reflected by the shape of the probability distribution (see image). The stability analysis of the model turns out to be challenging because computable stability criteria for such models have rarely been studied. We propose the use of drift functions to analyze the stability of the underlying Markov process and compute the stable regions and their probabilities. With our current approach we are able to derive geometric bounds for the steady-state distribution and detect multistable behavior. In ongoing work, we consider necessary and sufficient criteria for multistability which are of particular interest in the area of synthetic biology.

#### Computation of rare event probabilities in large structured Markov models

Rare events such as gene mutations and overcoming of gene repression play an important role for the development of certain diseases. Stochastic models of such rare event systems are difficult to simulate. The reason is that, even if a large number of trajectories of the model is generated, the rare event may not occur during the simulation since its probability is extremely small. Importance sampling techniques are helpful in such situations because they change the underlying probability measure in such a way that the rare event becomes more likely. Currently, we work on methods that combine importance sampling with previous numerical solution techniques in order to approximate the probabilities of rare events. The main idea is to determine sequences of events that lead to the rare event of interest and direct the numerical solution accordingly. Our approach has many advantages compared to traditional importance sampling techniques. During the numerical solution an approximation of the probability distribution is calculated. It can be used to decide in an on-the-fly fashion which change of measure is appropriate. Moreover, we can avoid using the correction factor for the importance sampling estimates by solving the original system and the system with the change of measure simultaneously. Our preliminary results indicate that our approach works surprisingly fast and accurate compared to classical rare event simulation techniques.

#### DNA methylation models

Together with the group of Jörn Walter we develop stochastic models that describe epigenetic modifications of the DNA. We focus on methyl groups that are attached to CpG dyads with the help of DNMTs. Epigenetic modifications such as DNA methylation play an important role in the understanding of embryogenesis. Moreover, changes in epigenetic factors can play a critical role in certain diseases (see this paper). We use Markov models to describe the methylation activities of the DNMTs and the loss of methylation due to DNA replication. Our most recent models were able to make accurate predictions of the methylation frequencies (paper link).