# Workshop "Statistics for Biological Networks"

## 8-9 September 2011

## Outline

Networks have become a paradigm of science and society. Social developments, such as the rise of the internet, has brought the idea of networks to the public's attention. Many people take part in social networks and the idea of six-degrees-of-separation is almost taken for granted.

Within science, networks also start to play an important role, although in a more complex way. Attempts to model complex systems, such as in climate science, system biology and system chemistry, are very complex. Chaotic behaviour of the, mathematically-encoded, system always lurk around the corner. Often, this is undesirable. Many systems are well-known to be stable and artificially induced instability is both impractical and theoretically unproductive. Networks offer a way to describe complex relationships via a set pairwise dependence relationships.

## Registration

Places are limited and therefore registration is required. Please register for this workshop by emailing your name and affiliation to k.m.e.schelhaas@rug.nl.

## Programme:

### Thursday 8 September 2011 in the Bernoulliborg 5161.0162

- 14:00 - 14:45 Mr Ivan Vujacic
- 14:45 - 15:30 Dr Javier Gonzalez
- 15:45 - 16:30 Dr Veronica Vinciotti: Keynote Speech

### Friday 9 September 2011 in Nijenborgh 4 5114.0001

- 9:30 - 10:15 Dr Vilda Purutcuoglu: Keynote Speech
- 10:30 - 11:15 Mr Antonio Abbruzzo
- 11:15 - 12:00 Dr Fentaw Abegaz

## Abstracts

### Dr Veronica Vinciotti: Modelling gene regulation at the transcriptional and post-transcriptional level

Transcription factors are key elements in the regulation of genes as they increase (activation) or decrease (repression) the amount of mRNA produced from a gene. They often work together with other transcription factors, either in a cooperative or competitive way, creating a complex network of dynamic interactions. Whereas transcription factors work at the transcriptional level of regulation, microRNAs have a key role at the post-transcriptional level, as they have been found to induce mRNA degradation.

In this talk, a mathematical model is presented that quantifies the mRNA production by transcription factors at the transcription level, as well as the effect of microRNA on mRNA degradation at the post-transcriptional level. The parameters in the model are estimated by embedding the mathematical model in a statistical framework to account for the noisy nature of biological data.

### Mr Ivan Vujacic: Inference of Ordinary Differential Equations

Models that are defined by a system of ordinary differential equations have been widely used in various fields of science. They describe the state of some system ( biological, physical etc.) through the relationship between the derivatives of the state and the state itself. Usually these ODEs contain some parameters that are unknown from theoretical considerations and need to be estimated from some noisy data. In nonlinear systems, that are common in practical applications, an analytic solution usually does not exist. Standard statistical approaches for parameter estimation involve evaluating likelihood function, which requires the explicit numerical solution of the differential equations describing the model. This implies that these methods are computationally very expensive. In this presentation we describe two recently proposed methods, that avoid numerical solving of differential equation while still producing reasonably good estimates. The both methods use some sort of data smoothing in order to avoid solving the dierential equation. We also discuss consistency of one of the described estimators and consistency of the estimator that is described in other talk "Reproducing Kernel Hilbert Space approach for ODE inference".

### Dr Javier Gonzalez: Reproducing Kernel Hilbert Space approach for ODE inference

Non-linear differential systems of equation have attracted the interest of many researchers in System Biology, Ecology or Chemistry among others fields due to their flexibility and ability to describe dynamical systems. Despite the importance of such models in many branches of science they have not been the focus of statistical analysis until the last few years. In this work we propose a general approach to estimate the parameters of differential systems of equations measured with noise. Our approach is based on the maximization of a likelihood where the differential system of equations is used as a constrain. By introducing a reproducing kernel Hilbert space approach, we transform the constraint maximization problem into an unconstraint numeric maximization problem easy to solve in practice. The method is tested in several real and simulated examples showing its utility in a wide range of scenarios.

### Dr Vilda Purutcuoglu: Diffusion Bridge Modelling of Stochastic Networks

The complexity of the actual biological networks direct us to find mathematical approaches in order to understand the major biological processes beyond the single gene analyses. But since the activation of a pathway depends on many conditions from stochastic to spatial constraints and none of the pathway has an isolated structure, rather has densely interconnections with many other cell processes, the mechanism of the actual biological processes has not been yet fully captured. Moreover the large number of nodes and interactions between the nodes increase this complexity. Whereas under such challenges we can still predict the best fitted model for the available data based on the known constraints of the biological system.

The stochastic models which enable us to consider the random feature or stochasticity of the biological processes when the exact number of the molecules in the reactions are known. Since these models need very detailed knowledge about the systems, the final models represent more comprehensive view of the actual process with respect to the ordinary differential equation approaches that are commonly used.

In this study the stochasticity is modelled via the diffusion approximation and the estimation of the model parameters, which are the stochastic reaction rate constants, is implemented within the Bayesian framework. Hereby the inference is conducted by the Metropolis-within-Gibbs algorithm via the columnwise update of states and the modified diffusion bridge methods. Both suggested approaches are initially applied for a small system. Then their performances are evaluated in a complex biological network whose real-life time-course observations are assumed to come from a model with measurement error.

### Mr Antonio Abbruzzo: Penalized Gaussian Graphical Modelling of Dynamic Networks

The concept of a gene network is central in systems biology. It describes the idea of the stability and interconnectedness of molecular reactions. The challenge is to give this a precise statistical interpretation. In this talk we present promising work on structured approaches using penalized graphical models. Ordinary graphical models are computable for small systems – not the type that one would encounter in genomics. Sparse versions of these methods are beginning to make some of these methods more relevant for their systems biological applications.

### Dr Fentaw Abegaz: Copula approach for non-Gaussian Graphical Models

Multivariate data that include binary, ordinal and continuous types often arise
in quantitative studies. Gaussian graphical models have been successful to provide a
convenient setting for modeling multivariate Gaussian data. For non-Gaussian data,
a way to model multivariate dependence is via a copula model, in which the depen-
dence among the variables is parameterized separately from the univariate marginal
distributions. We present a semiparametric l_{1} penalized likelihood estimation with
Expectation-Maximization algorithm. Moreover, we discuss convex optimization
algorithm for learning sparse time-varying non-Gaussian graphical models