### BIRS Workshop on Multi-scale Stochastic Modeling of Cell Dynamics

This week I’ve been at the Banff International Research Station (BIRS), a mathematical research institute located at the Banff Centre, on the outskirts of Banff, Alberta, Canada. I was participating in a 5-day workshop on Multi-scale Stochastic Modeling of Cell Dynamics.

There can be few more picturesque workshop venues than the Banff Centre, located as it is is at the foot of Tunnel Mountain in the middle of Banff National Park. This was my second trip to BIRS at Banff. My previous visit was in the middle of summer, so it was interesting to contrast the summer and winter landscapes. That said, the weather was relatively mild, and the region has had relatively little snow this year. The workshop schedule was arranged to provide a long break each afternoon for outdoor activities. Many went skiing, but I went for a couple of hikes instead. On Monday afternoon a few of us hiked up Tunnel Mountain, and on Thursday a few more hiked from Bow Falls around Tunnel Mountain to the local hoodoos along the Bow River. Both walks offered great views of the Canadian Rockies, including nearby Mount Rundle. Wednesday afternoon’s session was rescheduled to allow people to see the Olympic torch arrive in Banff that evening, on its way to Vancouver for the 2010 Winter Olympics. A few of my photos from the trip are on flickr.

Essentially, the workshop was concerned with interesting outstanding mathematical, theoretical and computational issues in systems biology modelling, focusing in particular on stochastic and multi-scale issues. It featured many excellent participants from a diverse range of backgrounds. The quality of presentations was very high, so in combination with the outstanding location, this was one of the best workshops that I’ve attended for several years. I don’t have the time or the energy to give extensive notes for each talk, but I’ll go quickly through each talk in turn trying to give something of the flavour of the presentation, together with a link to the web page of the presenter, which will hopefully provide links to further details.

## 18/1/10

### Des Higham

#### Discrete versus continuous

Des was looking at some simple first order reaction networks, and in particular at hitting times for the Markov jump process and the associated diffusion approximation (chemical Langevin equation). He showed a simple isomerisation network where the diffusion approximation is not guaranteed to stay non-negative. Be careful with diffusion approximations was the message.

### Jin Wang

#### Potential and flux landscape for stability

Jin-Wang was looking at how methods from statistical physics can be used to better understand the global dynamics of stochastic models. He looked at a couple of examples, including a circadian clock and a cell-cycle model.

### Ted Perkins

#### Trajectory inference

Ted was thinking about how methods from computing science can be used to construct the most likely trajectory of a stochastic kinetic model between a given start and end point. The talk was based on a recent NIPS paper.

### Ruth Williams

#### Enzymatic degradation

Observed correlations between protein expression levels could potentially be due to indirect coupling that arises due to a common enzyme being used for degredation. Ruth showed how the model could be re-written as a multiclass queueing system so that queueing theory results can be used to get at interesting properties of the stationary distribution such as the stationary correlation.

## 19/1/10

### Sam Kou

#### Parameter inference for SDEs

Sam was looking at parameter estimation for univariate stochastic differential equation (SDE) models from a Bayesian perspective. He explained the problems with naive MCMC samplers for the problem, and proposed a new sampler constructed by coupling together a collection of such samplers with different time discretisations. He also showed how more accurate estimates of the parameters can be obtained by extrapolating from the distributions corresponding to different degrees of discretisation.

### Matt Scott

#### Intrinsic noise in continuous (spatial) systems

Matt was thinking about how to do intrinsic network noise in space. He set up the model on a lattice, and then took a multi-scale limit (essentially a high concentration, linear noise approximation).

### Di Liu

#### Stochastic kinetics with multiple time scales

Di was looking at multi-scale (hybrid) simulation algorithms for stochastic networks with multiple time scales. In particular, he was looking at Nested SSA algorithms, and contrasting with slow-scale SSA, multiscale SSA and other implementation of nested SSA. He also talked about adaptation (re-classification of fast/slow), and looked at Tyson’s yeast cell cycle model as an example.

### Darren Wilkinson

#### Bayesian inference for stochastic network models

I gave an overview of statistical methods for parameter inference for stochastic kinetic models, with emphasis on Bayesian approaches and sequential likelihood free MCMC. I showed an example application to stochastic kinetic modelling of p53/Mdm2 oscillations.

### Moises Santillan

#### Distribution evolution for gene regulation models

Moises was looking at analytic multiscale approximations in the context of some relatively simple stochastic kinetic models, including the lac operon model. He was separating time-scales and deriving analytic approximations to the steady-state distribution.

### Paul Tupper

#### State-dependent diffusion

Paul was looking at spatial diffusion, and in particular, thinking about proteins diffusing at different rates inside and outside the nucleus. He gave a nice simple example of diffusion in a box with different diffusion coefficients in each half, and showed that the steady state distribution depends rather subtly on how exactly diffusion is interpreted (mathematically, on the precise formalism of stochastic calculus adopted). He showed by modelling the diffusion as a Lorenz gas that the apparent paradox can be understood.

### Sayan Mukherjee

#### Multi-scale factor models

Sayan was using sparse multiscale factor models to understand large and complex genomic data sets. He exploited linearity and sparsity for dimension reduction. The talk included discussion of network inference and links with graphical models.

## 20/1/10

### Rachel Kuske

#### Coherence resonance

Rachel was discussing mixed mode oscillations, and in particular, noise stabalised transients, such as found with the FitzHugh Nagumo model with added noise – occasional kick from near equilibrium onto an unstable limit cycle. The classic biological example of this (mentioned only in passing in the talk), is the transition of *B. Subtilis* between competent and non competent states (the competent state is thought to be an excitable unstable state).

### Dave Anderson

#### Simulation methods for population processes

Dave was concerned with understanding the asymptotic behaviour of approximate simulation algorithms (principally, tau-leaping and a mid-point version). He was using Kurtz’s random time change representation of stochastic kinetic models and the associated approximations for analysis. Classical asymptotics show the mid-point method to be no better than regular tau-leaping, but practical applications show it to be superior. By rescaling the problem appropriately, an appropriate asymptotic limit can be constructed which shows the mid-point version to be better.

### Eldon Emberly

#### Polar localisation of proteins in bacteria

Eldon was thinking about protein localisation as a mechanism for cellular differentiation for bacteria such as Caulobacter Crescentus (which has stalks and swarmers). PopZ localises at a pole, and this protein can be expressed in E. Coli (which does not have a PopZ homologue) to better understand the mechanisms of localisation. A passive model was shown to be adequate to explain the observed behaviour.

### John Fricks

#### Neck linker extension in kinesin molecular motors

John has been working for some time on detailed molecular models of kinesin molecular motors. Here he was focusing on the importance of the neck linker (the bit between the two heads and the tail), and developing a model which is predictive in the context of artificial extension of the neck linker. The model was a combination of diffusion and discrete chemical kinetics, and he simplified the model as a renewal reward process to obtain analytic approximations.

### Hye-Won Kang

#### Space discretisation in reaction-diffusion models

Hye-Won is interested in stochastic models of pattern formation – especially pattering along an axis. Here reaction-limited kinetics is more relevant than diffusion limited kinetics – lots of well-mixed compartments. By exploiting upper and lower bounds on compartment sizes, an appropriate scaling limit can be constructed which leads to a system of ODEs for the first two moments which can be solved for the steady-state distribution. The talk was mainly 1-d, but apparently there are 3-d extensions.

### Peter Pfaffelhuber

#### Spatial multiscale chemical reaction networks

Peter described some joint work with Lea Popovic on multiscale reaction-diffusion across multiple compartments when the diffusion is fast, but compartment-dependent. The basic method was extended to multiscale reaction networks within each compartment, but then we clearly have to make assumptions about the relative time scale of the diffusion and the fast reactions.

### Mads Kaern

#### Stochastic simulation in evolving heterogeneous cell populations

Mads is interested in comparing stochastic models to experimental data on (heterogeneous) cell populations – especially flow cytometry data. For this one needs a framework for simulating exponentially growing cell populations, and Mads was using “constant number Monte Carlo”, which essentially randomly throws stuff out to maintain a fixed population size that is representative of the bigger population. He had some nice examples of how noise can give robustness to uncertain stress.

## 21/1/10

### Hans Othmer

#### Multi-scale analysis of reacting systems

Hans is interested in deterministic and stochastic models of patterning in drosophila. Think Gillespie across multiple boxes with one box per nucleus. He was then doing multi-scale analysis of the system to obtain analytic approximations.

### Greg Rempala

#### Statistical and algebraic methods for mass-action kinetics

Greg is interested in using algebraic methods for evidence synthesis in systems biology as an alternative to more conventional hierarchical statistical models. He was using the approach to take rate constant estimates from different models and data sets and to carry out network inference by deciding which rate constants are significantly different from zero.

### David McMillen

#### Bacterial gene expression

Dave gave a nice talk on synthetic and systems biology, and on problems and issues associated with using, measuring and modelling fluorescent proteins. He focused particularly on issues of folding, maturation and inclusion body formation, and on the population dynamics of cells growing in batch culture. He also gave a nice example of developing a synthetic circuit to give *E. Coli* resistance to infection from Bacteriophage lambda.

### Peter Swain

#### Quantifying and modelling stochastic biochemical networks

Peter discussed the study of intrinsic and extrinsic noise in bacteria. He showed how this could be studied experimentally via inclusion of two copies of a simple circuit with different coloured fluorescent reporters. Intrinsic noise can be modelled in the usual way, and extrinsic noise can be injected by allowing rate constants to vary according to (say) a Gaussian OU process. He argued that it will often make sense to correlated the extrinsic noise, and showed some nice examples of how feed-forward networks can attenuate noise fluctuations.

### Konstantin Mischaikow

#### Databases for global dynamics of multi-parameter systems

Konstantin is interested in categorising the global dynamic properties of deterministic iterated maps (and ultimately extending to ODE models and potentially also stochastic models). He argued that conventional descriptions are too complex and of limited practical value, and provided more robust descriptions that can be stored in a reasonable amount of space in a database.

### Hong Qian

#### Non-equilibrium phase transition

Hong is interested in understanding stochastic kinetic reaction network models from a statistical physics viewpoint. He showed how quasi-stationary analysis can give insight into the origins of irreversibility, and illustrated his ideas with some example phosphorylation/dephosphorylation networks. He also explained how the methods give insight into the fitness landscapes and the forces driving cellular evolution.

### Lev Tsimring

#### Synthetic gene oscillators

Lev described work on the development of synthetic gene oscillators *in vivo* and the associated detailed stochastic modelling. The first oscillator developed was single cell based, so oscillations at the population level decayed away due to gradual loss of synchrony. A new oscillator was developed using a protein which diffuses in and out of the cell, allowing synchronisation of the cells via quorum sensing. It worked, but diffusion effect led to spatial effects and wave propagation. This too can be modelled nicely. The work is described in a (very!) recent Nature paper.

## 22/1/10

### Tom Kurtz

#### Diffusion approximation for multiscale reaction networks

Tom is interested in the mathematical analysis of multiscale reaction networks. He gave a nice overview of diffusion approximations, and then argued that Martingale representations are a powerful method for gaining deeper insight into multiscale approximations, due to the Martingale central limit theorem. He showed how the technique could be used on the classic Michaelis-Menten system.

### Katharina Best/Surovcik

#### Spatial scaling in quorum sensing

Katharinia is interested in modelling quorum sensing in the symbiotic bacterium *Sinorhizobuim meliloti* found in plant roots. She modelled a reaction network for each cell, and then the diffusion of certain species in and out of cells. She did a multi-scale approximation assuming that the environment in which the cells live is a much larger volume than that occupied by the bacteria.

### Tomas Gedeon

#### Somitogenesis clock-wave initiation

Tomas is interested in complex spatio-temporal pattern formation in embryos – especially spine formation in vertebrates – zebrafish. He has been thinking about modelling, stochasticity, model selection and parameter estimation. He built deterministic models, and did a simple, intuitive “Bayes theorem via the rejection method” type algorithm, to infer parameters and select models. He concluded that a model of the delta-notch signalling pathway that included two binding sites and monomor-only decay was most compatible with the data.

### David Cottrell

#### Putting diffusion into stochastic networks (analytic)

David described some joint work with Peter Swain and Paul Tupper on using the theory of branching processes (analysis of the dual process) to obtain some powerful approximations to models of RNA and protein synthesis that takes spatial diffusion into account (assuming RNA degredation much faster than protein degredation). The technique is very flexible, but relies heavily on linearity of the model (no 2nd order reactions).

All in all, a useful and interesting week.

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