# Multi-scale Stochastic Modeling of Cell Dynamics (10w5058)

Arriving in Banff, Alberta Sunday, January 17 and departing Friday January 22, 2010

## Organizers

Brian Ingalls (University of Waterloo)

Jonathan Mattingly (Duke University)

Lea Popovic (Concordia University)

Peter S. Swain (SynthSys, University of Edinburgh)

## Objectives

The goal of this particular workshop is to explore new research directions and the ways in which researchers in stochastic dynamics, statistics and systems biology can join efforts in advancing our understanding of complex networks and stochastic dynamics in cells. In order to ensure that the models proposed and questions raised are of direct relevance to the biological community a synergistic workshop involving experts from all three areas is necessary. Significant effort will be made to ensure the workshop participants are gathered from both mathematical sciences and systems biology, and that their exchange of ideas during the workshop will be disseminated to their respective research communities.The goal of this particular workshop is to explore new research directions and the ways in which researchers in stochastic dynamics, statistics and systems biology can join efforts in advancing our understanding of complex networks and stochastic dynamics in cells. In order to ensure that the models proposed and questions raised are of direct relevance to the biological community a synergistic workshop involving experts from all three areas is necessary. Significant effort will be made to ensure the workshop participants are gathered from both mathematical sciences and systems biology, and that their exchange of ideas during the workshop will be disseminated to their respective research communities.

The main objectives of the workshop are:

(i) bringing together mathematical scientists working on stochastic analysis on multiple scales, stochastic dynamics and dynamical systems who have the expertise to contribute to analytical solutions of problems arising in cell and systems biology; and

(ii) facilitating the exchange of knowledge and ideas between biological scientists and mathematical scientists in order to identify biological problems that are in need of new mathematical results.

The workshop will focus on the following mathematical questions:

1. Controlled dynamical systems in regulation of chemical networks:

What types of positive, negative and mixed feedback interactions appear repeatedly in regulatory networks? How are they affected by stochastic inputs and a noisy environment? What are the beneficial reasons from the control point of view for the repeated appearance of cascades and feedback loops? Which stochastic control mechanisms can be employed in such systems? How do the sources of time delays affect the dynamical properties of reaction systems? What are the key properties of the counterpart stochastic delay differential equations? What are the possible stochastic effects in the networks whose behavior is oscillatory?

2. Stoichiometric structure of reaction networks and their relationship to steady state analysis: When does a reaction network admit a unique non-trivial stable solution? What can we deduce about stochastic properties of systems with unique stable solutions? What types of stochastic behavior do systems that admit multiple stable solutions exhibit? What relevant statistics can we use to characterize the different behavior within such systems with multiple equilibria? How can we use large deviation techniques to describe the transitions between different stable solutions for such systems? What are the appropriate techniques for doing a sensitivity analysis on multi-scale reaction systems? Which are appropriate for systems with a unique steady state and which for systems with multiple equilibria?

3. Stochastic models of reaction networks and appropriate model reduction techniques:

How can we identify in some systematic fashion the scaling parameters for a given reaction network? How can we identify from the scaling parameters which techniques for model reduction will capture the relevant properties of the system? What other reduction techniques, in addition to stochastic averaging and diffusion approximation, can be utilized for such complex network models? When can we exploit the existence of stationary distribution for a subsystem of the network to calculate relevant stochastic quantities of the full network?

4. Effective simulation of multi-scale reaction networks: How can we improve the existing methods for implementing simulations schemes for discrete and continuous valued stochastic processes? How can we rigorously estimate the errors of such integrated simulation schemes? How can we use the existence of multiple stable states to effectively simulate the stochastic behavior near and the transitions between the stable states? What are the efficient ways of simulating rare events in such complex systems?

5. Parameter estimation and model validation for multi-scale stochastic networks: Which are the quantities that can be measured, and which quantities are always hidden in the data sets? How can we effectively use the observed data to infer the inherent stochastic levels in the system? What estimation techniques can we use based on partial information of temporal data from such reaction systems? Can we guarantee consistency of estimates (e.g. maximum likelihood estimates) for reaction parameters, and what can we say about their distributional (e.g. asymptotic normality)? How can we exploit the known dynamics of a given network system to perform model validation of the stochastic model?

The main objectives of the workshop are:

(i) bringing together mathematical scientists working on stochastic analysis on multiple scales, stochastic dynamics and dynamical systems who have the expertise to contribute to analytical solutions of problems arising in cell and systems biology; and

(ii) facilitating the exchange of knowledge and ideas between biological scientists and mathematical scientists in order to identify biological problems that are in need of new mathematical results.

The workshop will focus on the following mathematical questions:

1. Controlled dynamical systems in regulation of chemical networks:

What types of positive, negative and mixed feedback interactions appear repeatedly in regulatory networks? How are they affected by stochastic inputs and a noisy environment? What are the beneficial reasons from the control point of view for the repeated appearance of cascades and feedback loops? Which stochastic control mechanisms can be employed in such systems? How do the sources of time delays affect the dynamical properties of reaction systems? What are the key properties of the counterpart stochastic delay differential equations? What are the possible stochastic effects in the networks whose behavior is oscillatory?

2. Stoichiometric structure of reaction networks and their relationship to steady state analysis: When does a reaction network admit a unique non-trivial stable solution? What can we deduce about stochastic properties of systems with unique stable solutions? What types of stochastic behavior do systems that admit multiple stable solutions exhibit? What relevant statistics can we use to characterize the different behavior within such systems with multiple equilibria? How can we use large deviation techniques to describe the transitions between different stable solutions for such systems? What are the appropriate techniques for doing a sensitivity analysis on multi-scale reaction systems? Which are appropriate for systems with a unique steady state and which for systems with multiple equilibria?

3. Stochastic models of reaction networks and appropriate model reduction techniques:

How can we identify in some systematic fashion the scaling parameters for a given reaction network? How can we identify from the scaling parameters which techniques for model reduction will capture the relevant properties of the system? What other reduction techniques, in addition to stochastic averaging and diffusion approximation, can be utilized for such complex network models? When can we exploit the existence of stationary distribution for a subsystem of the network to calculate relevant stochastic quantities of the full network?

4. Effective simulation of multi-scale reaction networks: How can we improve the existing methods for implementing simulations schemes for discrete and continuous valued stochastic processes? How can we rigorously estimate the errors of such integrated simulation schemes? How can we use the existence of multiple stable states to effectively simulate the stochastic behavior near and the transitions between the stable states? What are the efficient ways of simulating rare events in such complex systems?

5. Parameter estimation and model validation for multi-scale stochastic networks: Which are the quantities that can be measured, and which quantities are always hidden in the data sets? How can we effectively use the observed data to infer the inherent stochastic levels in the system? What estimation techniques can we use based on partial information of temporal data from such reaction systems? Can we guarantee consistency of estimates (e.g. maximum likelihood estimates) for reaction parameters, and what can we say about their distributional (e.g. asymptotic normality)? How can we exploit the known dynamics of a given network system to perform model validation of the stochastic model?