Introduction

Proteins and their interactions determine how cells behave. Genes are the blueprints for protein synthesis; their activation or suppression determines the absence or presence of a protein which in turn can give rise to further activation or suppression of other genes and proteins. This chemical chain reaction usually involves positive and negative feedback loops and is subjected to stochastic noise and the influence of environmental factors. Moreover, epistasis – the cancellation or modification of a gene's contribution to the phenotype by other genes – is generally the rule rather than the exception in genetics.

Despite all these factors, amazingly robust cell behavior is apparent in many biological systems although it is difficult to be modeled and/or predicted. Building the topology and quantifying the direct and indirect cause–effect (stimulus, expression, activation, behavior) relationships of the reactions leading to the phenotypes – in general, genetic regulatory networks (GRN) – is challenging in at least three ways.

Firstly, how are these relationships described? Traditionally, mathematical models are expressed in terms of transfer functions relating inputs to outputs expressed as a composition of differential equations with a time dimension. We argue though that the cellular signaling networks are probabilistic in nature and that diffusion-based models remain challenging due to lack of knowledge of essential system parameters, such as rate constants. Most importantly, treating intracellular protein and gene interactions as in-vitro chemical reactions might not be safe because the usual assumption of diffusion dynamics namely that of the free movement of a sufficiently large number of molecules is usually the exception rather than the rule due to the very small number of reactant molecules in highly confined and crowded space. Moreover, the concentration of a protein is highly dependent on sub cellular localization and thus the picture of the cell as a homogeneous mix container is simply wrong. Of even more profound impact, many signaling systems are centered on or around scaffolding proteins mimicking solid-state chemical environments and have little or no resemblance to diffusion limited systems.

The biochemical and molecular mechanisms underlying epistatic gene interactions observed in various living organisms are poorly understood. In this chapter, we introduce a mathematical framework linking epistasis to the redundancy of biological networks. The approach is based on network reliability, an engineering concept that allows for computing the probability of functional network operation under different network perturbations, such as the failure of specific components, which, in a genetic system, correspond to the knock-out or knock-down of specific genes. Using this framework, we provide a formal definition of epistasis in terms of network reliability and we show how this concept can be used to infer functional constraints in biological networks from observed genetic interactions. This formalism might help increase our understanding of the systemic properties of the cell that give rise to observed epistatic patterns.

Biological networks

A major goal of post genomic biomedical research consists in understanding how the genetic components interact with each other to form living cells and organisms. The systems-wide approach requires both novel experimental techniques for mapping out such interactions and new mathematical models to describe and to analyze them. Interacting biological systems are often represented as networks (or graphs), where vertices correspond to components (e.g., genes, proteins, or metabolites) and edges correspond to pair wise interactions (e.g., activation, molecular binding, or chemical reaction). This abstract representation provides the conceptual basis for network biology, which aims at understanding the cell's functional organization and the complex behavior of living systems through biological network analysis (Strogatz 2001, Barabási & Oltvai 2004).

Various experimental methods have been developed to measure physical interactions (molecular binding events) among proteins and several computational methods exist for predicting such interactions. These data give rise to protein–protein interaction (PPI) networks which are available from dedicated databases (Schwikowski et al. 2000, Xenarios et al. 2000, Jensen et al. 2009). Genetic interactions, or epistasis, refers to functional relationships between genes.

Interactions between genes can be experimentally determined by combining multiple mutations and identifying combinations where the resulting phenotype differs from the expected one. Such genetic interactions, for example measured in yeast for cell proliferation and growth phenotypes, provided intricate insights into the genetic architecture and interplay of pathways. Due to the lack of comprehensive deletion libraries, similar experiments in higher eukaryotic cells have been challenging. Recently, we and others described methods to perform systematic, comprehensive double-perturbation analyses in Drosophila and human cells using RNA interference. We also introduced methods to use multiple phenotypes to map genetic interactions across a broad spectrum of processes.

This chapter focuses on the systematic mapping of genetic interactions and the use of image-based phenotypes to improve genetic interaction calling. It also describes experimental approaches for the analysis of genetic interactions in human cells and discusses concepts to expand genetic interaction mapping towards a genomic scale.

A short history of genetic interaction analysis

Using quantitative traits to map genetic interactions has a long tradition in Drosophila. In the 1960s and 1970s, Dobzhansky, Rendel, and others used externally visible pheno-types or overall fitness to study non-mendelian inheritance and dissect the heritability of complex traits (Fig. 3.1a). One of the underlying assumptions was that genetic loci in the Drosophila genome interact to shape complex phenotypes or buffer detrimental alleles.

In 1965, Dobzhansky and colleagues analyzed epistatic interactions between the components of genetic variants in Drosophila. They crossed flies carrying mutant alleles into a wild-type background obtained from a natural habitat and found that the combination of particular chromosomes showed synthetic sick phenotypes, whereas both chromosomes alone did not. This for the first time demonstrated the presence of bi-chromosomal synthetic interactions in Drosophila populations. Similarly, Rendel and colleagues demonstrated the existence of epistatic modifiers in Drosophila by analysis of scute alleles, which reduce the number of scutellar bristles on the dorsal thorax from four to an average of one.

Systems genetics is an emerging field based on old approaches going back to the genetic studies performed by Gregor Mendel (Mendel 1866). Mendel's experiments primarily focused on explaining inheritance of single traits and their phenotypes – for example how specific genetic alleles influence colour or size of peas – but recently developed technologies can comprehensively dissect the genetic architecture of complex traits and quantify how genes interact to shape phenotypes by using natural variation or experimental perturbations as a basis to understand links from genotypes to phenotypes. This exciting new area has recently been termed ‘systems genetics’ (Civelek & Lusis 2014).

While the basic, underlying questions are not new, systems genetics builds upon major methodological advances that facilitate the measurement of genotypes and pheno-types in a previously unforeseen and comprehensive manner. With this arsenal at hand, one of the major aims of systems genetics is to understand “how genetic information is integrated, coordinated and ultimately transmitted through molecular, cellular and physiological networks to enable the higher-order functions and emergent properties of biological systems” (Nadeau & Dudley 2011).

Definition of systems genetics

Systems genetics is born out of a synthesis of multiple fields: it integrates approaches of genetics, genomics, systems biology and ‘phenomics’, that is, our increased ability to obtain quantitative and detailed measurements on a broad spectrum of phenotypes. One of the first papers using the term ‘systems genetics’ defines it as “the integration and anchoring of multi-dimensional data-types to underlying genetic variation” (Threadgill 2006). Since then, many studies have aimed at integrating genome-wide data across many different levels, and possibly different environments, in approaches that are closely related to quantitative genetics.

In our view, a systems genetic approach should bring together three dimensions: it should combine (i) a genome-wide analysis with (ii) many quantitative phenotypes, both at the molecular and organismal level, (iii) in many different conditions or environments (Fig. 1.1).

In recent years many efforts have been invested in comprehensively evaluating the behavior and relationships of all genes/proteins in a particular biological system and at a particular state. Here, we review how genome-wide RNAi screens together with mass spectrometry can be integrated to generate high-confidence functional interac- tome networks. Next we review the mathematical modeling methods available today that allow the computational reconstruction of such networks. Network modeling will play an important role in generating hypotheses, driving further experimentation and thus novel insights into network structure and behavior.

Introduction

Most biologists study a specific biological problem by investigating the activities of a limited number of genes or proteins involved in a particular biological process. This traditional approach is critical and has proven to be extremely successful to reveal the detailed molecular functions of individual genes and proteins. For example, genetic studies of embryonic patterning in Drosophila identified about 40 genes with striking segmentation defects that fell into distinct phenotypic classes: gap genes, pair rule genes, segment polarity genes, and homeotic genes (Nusslein-Volhard & Wieschaus 1980). Detailed analyses of the mutant phenotypes and functions of even this relatively small set of genes led to a comprehensive molecular framework of the process of embryonic patterning (St Johnston & Nusslein-Volhard 1992). Reductionist approaches, however, are not sufficient for generating the big picture of how a biological system, including multiple levels of many different gene products and the interactions among them, works at different physiological states or developmental stages (Friedman & Perrimon 2007). Thus, as our knowledge of individual genes and proteins accumulates, there is a need to comprehensively evaluate the behavior and relationships of all genes/proteins in a particular biological system and at a particular state. In recent years, progress has been made in multi cellular organisms towards this goal mostly in tissue culture, a platform that allows a sufficient amount of homogeneous material to be easily obtained.

In the analysis of any data using statistical modelling, it is imperative that the choice of model is informed by expert knowledge and that its adequacy is determined based on the extent to which it captures and describes the patterns observed in the data. This is especially true in systems where a subset of the constituent components may not be known or cannot be observed. In this chapter, we demonstrate how statistical inference can be used to inform model selection and, by identifying where existing models are unable to sufficiently capture observed behaviour, that statistical inference can help indicate which model refinements may be required.

In this chapter, we use Bayesian statistical methodology – specifically, Riemannian manifold population MCMC – to model interactions between molecular species in the JAK/STAT pathway in chronic myeloid leukaemia (CML) and compare two candidate models. We set out the biological context for this inference in Sections 9.1–9.1.4 and describe the two candidate models in Section 9.3. With the biology established, we describe our statistical methodology (Section 9.4) which we successfully apply in a simulation study to provide a proof of concept (Section 9.5), before we consider a subsequent, more biologically realistic dataset (Section 9.6) to assess which model best describes the behaviour observed in vitro. We relate the findings from this second synthetic study back to our model and dataset construction, thereby highlighting what further in vitro and in silico work is required (Section 9.7).

The oncology of chronic myeloid leukaemia

The condition that we now recognise as chronic myeloid leukaemia (CML) was first described in 1845, in quick succession, by two pathologists, Dr John Hughes Bennett (Bennett 1845) and Dr Rudolf Virchow (Virchow 1845).