Interaction Posterior Probability Correlation

The interaction posterior probability is a measure that quanties the chance of ob- serving regulator to response interactions xn

i → yn (see Section 4.3.4). In the case of

MCMC simulations, such as with the Bayesian regression methods, the probability is marginalized over a specic number of samples, typically with2000iterations in size or larger. The convergence of a MCMC can be evaluated by comparing the probabilities of dierent MCMC chains. The interactions posteriors can be displayed in a scatter

60 Chapter 3 plot that compares two MCMC chains as shown in the left plot of Figure 5.6, or as a time-varying trajectory of the correlation value between the interaction probabilities of two chains as shown in the right plot of Figure 5.6. Convergence is indicated with higher correlation values, which means that the sampled interactions of two chains are likely to occur at the same rate, independent of the starting condition.

Chapter 4

Statistical Inference of Gene Regu-

latory Networks

In this chapter I assess the accuracy of various state-of-the-art statistics and machine learning methods described in Chapter 2 for reconstructing gene and protein regula- tory networks in the context of circadian regulation. The study draws on the increasing availability of gene expression and protein concentration time series for key circadian clock components in Arabidopsis thaliana. In addition, gene expression and protein concentration time series are simulated from a recently published regulatory network of the circadian clock in A.thaliana, in which protein and gene interactions are de- scribed by a Markov jump process based on Michaelis-Menten kinetics. I closely follow recent experimental protocols, including the entrainment of seedlings to dierent light- dark cycles and the knock-out of various key regulatory genes. The study provides relative network reconstruction accuracy scores for a critical comparative performance evaluation, and sheds light on a series of highly pertinent questions: it investigates the inuence of systematically missing values related to unknown protein concentrations and mRNA transcription rates, it quanties the dependence of the performance on the network topology and the degree of recurrency, it provides deeper insight into when and why non-linear methods fail to outperform linear ones, and it provides a comparison between inferred and published gene regulatory network structures.

This study was published in collaboration with Marco Grzegorczyk in [1]. Both of us contributed in the discussion of the experimental setup, in running simulations, and evaluating the results. Marco Grzegorzyk performed simulations with the methods MBN, SSM, BGe, and ARD-SBR and I performed simulations with the remaining

62 Chapter 4 methods. The work is based on preliminary studies that was published in [4] and [57].

4.1 Introduction

Plants have to carefully manage their resources. The process of photosynthesis allows them to utilize sunlight to produce essential carbohydrates during the day. How- ever, the earth's rotation predictably removes sunlight, and hence the opportunity for photosynthesis, for a signicant part of each day, and plants need to orchestrate the accumulation, utilization and storage of photosynthetic products in the form of starch over the daily cycle to avoid periods of starvation, and thus optimize growth rates.

In the last few years, substantial progress has been made to model the central pro- cesses of circadian regulation, i.e. the mechanism of internal time-keeping that allows the plant to anticipate each new day, at the molecular level [113, 62]. Moreover, sim- ple mechanistic models have been developed to describe the feedback between carbon metabolism and the circadian clock, by which the plant adjusts the rates of starch ac- cumulation and consumption in response to changes in the light-dark cycle [42]. What is needed is the elucidation of the detailed structure of the molecular regulatory net- works and signalling pathways of these processes, by utilization and integration of tran- scriptomic, proteomic and metabolic concentration proles that become increasingly available from international research collaborations like AGRON-OMICS1 _{and TiMet-}

Consortium [138]. The inference of molecular regulatory networks from post-genomic data has been a central topic in computational systems biology for over a decade. A variety of methods have been proposed and several procedures have been pursued to objectively assess the network reconstruction accuracy [80, 149, 148]. The objective of the present Chapter is to complement these studies in six important respects. Firstly, I have taken a particular focus on circadian regulation. To this end, I have taken the central circadian clock network in A.thaliana, as published by Guerriero et al. [62], as a ground truth for evaluation, and closely followed recent experimental protocols for data generation, including the entrainment of seedlings to dierent light-dark cycles, and the knock-out of various key regulatory genes. To make the data generated from this network as realistic as possible, I have modelled gene and protein interactions as a Markov jump process (MJP) based on Michaelis-Menten kinetics. This is to be preferred over mechanistic models based on ordinary dierential equations [used e.g. by 113], as MJPs capture the intrinsic stochasticity of molecular interactions. MJPs

4.2. REGRESSION MODEL 63