Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interactions: a positive (resp. negative) arrow from a gene to another represents an activation (resp. inhibition) of the expression of the latter gene by some product of the former. Once such an interaction graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. There mathematical tools, among others, can be of some help. In this paper we discuss a rule proposed by Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential [C. Soulé, Graphic requirements for multistationarity, ComPlexUs 1 (2003) 123-133] or Boolean [E. Rémy, P. Ruet, D. Thieffry, Graphic requirements for multistability and attractive cycles in a boolean dynamical framework, 2005, Preprint] type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models. We then discuss open problems.     

A Factor Graph Nested Effects Model To Identify Networks from Genetic Perturbations

Complex phenotypes such as the transformation of a normal population of cells into cancerous tissue result from a series of molecular triggers gone awry. We describe a method that searches for a genetic network consistent with expression changes observed under the knock-down of a set of genes that share a common role in the cell, such as a disease phenotype. The method extends the Nested Effects Model of Markowetz et al. (2005) by using a probabilistic factor graph to search for a network representing interactions among these silenced genes. The method also expands the network by attaching new genes at specific downstream points, providing candidates for subsequent perturbations to further characterize the pathway. We investigated an extension provided by the factor graph approach in which the model distinguishes between inhibitory and stimulatory interactions. We found that the extension yielded significant improvements in recovering the structure of simulated and Saccharomyces cerevisae networks. We applied the approach to discover a signaling network among genes involved in a human colon cancer cell invasiveness pathway. The method predicts several genes with new roles in the invasiveness process. We knocked down two genes identified by our approach and found that both knock-downs produce loss of invasive potential in a colon cancer cell line. Nested effects models may be a powerful tool for inferring regulatory connections and genes that operate in normal and disease-related processes.     

From minimal signed circuits to the dynamics of Boolean regulatory networks

It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the two minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the three stable states corresponding to the Th0, Th1 and Th2 cells. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Quantitative analysis of fitness and genetic interactions in yeast on a genome scale

Global quantitative analysis of genetic interactions is a powerful approach for deciphering the roles of genes and mapping functional relationships among pathways. Using colony size as a proxy for fitness, we developed a method for measuring fitness-based genetic interactions from high-density arrays of yeast double mutants generated by synthetic genetic array (SGA) analysis. We identified several experimental sources of systematic variation and developed normalization strategies to obtain accurate single- and double-mutant fitness measurements, which rival the accuracy of other high-resolution studies. We applied the SGA score to examine the relationship between physical and genetic interaction networks, and we found that positive genetic interactions connect across functionally distinct protein complexes revealing a network of genetic suppression among loss-of-function alleles.     

Exploring a Local Genetic Interaction Network Using Evolutionary Replay Experiments

Understanding how genes interact is a central challenge in biology. Experimental evolution provides a useful, but underutilized, tool for identifying genetic interactions, particularly those that involve non-loss-of-function mutations or mutations in essential genes. We previously identified a strong positive genetic interaction between specific mutations in KEL1 (P344T) and HSL7 (A695fs) that arose in an experimentally evolved Saccharomyces cerevisiae population. Because this genetic interaction is not phenocopied by gene deletion, it was previously unknown. Using “evolutionary replay” experiments, we identified additional mutations that have positive genetic interactions with the kel1-P344T mutation. We replayed the evolution of this population 672 times from six timepoints. We identified 30 populations where the kel1-P344T mutation reached high frequency. We performed whole-genome sequencing on these populations to identify genes in which mutations arose specifically in the kel1-P344T background. We reconstructed mutations in the ancestral and kel1-P344T backgrounds to validate positive genetic interactions. We identify several genetic interactors with KEL1, we validate these interactions by reconstruction experiments, and we show these interactions are not recapitulated by loss-of-function mutations. Our results demonstrate the power of experimental evolution to identify genetic interactions that are positive, allele specific, and not readily detected by other methods, shedding light on an underexplored region of the yeast genetic interaction network.     

Dosage suppression genetic interaction networks enhance functional wiring diagrams of the cell

Dosage suppression is a genetic interaction in which overproduction of one gene rescues a mutant phenotype of another gene. Although dosage suppression is known to map functional connections among genes, the extent to which it might illuminate global cellular functions is unclear. Here we analyze a network of interactions linking dosage suppressors to 437 essential genes in yeast. For 424 genes, we curated interactions from the literature. Analyses revealed that many dosage suppression interactions occur between functionally related genes and that the majority do not overlap with other types of genetic or physical interactions. To confirm the generality of these network properties, we experimentally identified dosage suppressors for 29 genes from pooled populations of temperature-sensitive mutant cells transformed with a high-copy molecular-barcoded open reading frame library, MoBY-ORF 2.0. We classified 87% of the 1,640 total interactions into four general types of suppression mechanisms, which provided insight into their relative frequencies. This work suggests that integrating the results of dosage suppression studies with other interaction networks could generate insights into the functional wiring diagram of a cell.     

Analysis of Discrete Bioregulatory Networks Using Symbolic Steady States

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.     

Genetic regulation networks: circuits,regulons and attractors

We deal in this paper with the concept of genetic regulation network. The genes expression observed through the bio-array imaging allows the geneticist to obtain the intergenic interaction matrix W of the network. The interaction graph G associated to W presents in general interesting features like connected components, gardens of Eden, positive and negative circuits (or loops), and minimal components having 1 positive and 1 negative loop called regulons. Depending on parameters values like the connectivity coefficient K(W) and the mean inhibition weight I(W), the genetic regulation network can present several dynamical behaviours (fixed configuration, limit cycle of configurations) called attractors, when the observation time increases. We give some examples of such genetic regulation networks and analyse their dynamical properties and their biological consequences.     

Toward an integrated map of genetic interactions in cancer cells

Cancer genomes often harbor hundreds of molecular aberrations. Such genetic variants can be drivers or passengers of tumorigenesis and create vulnerabilities for potential therapeutic exploitation. To identify genotype‐dependent vulnerabilities, forward genetic screens in different genetic backgrounds have been conducted. We devised MINGLE, a computational framework to integrate CRISPR/Cas9 screens originating from different libraries building on approaches pioneered for genetic network discovery in model organisms. We applied this method to integrate and analyze data from 85 CRISPR/Cas9 screens in human cancer cells combining functional data with information on genetic variants to explore more than 2.1 million gene‐background relationships. In addition to known dependencies, we identified new genotype‐specific vulnerabilities of cancer cells. Experimental validation of predicted vulnerabilities identified GANAB and PRKCSH as new positive regulators of Wnt/β‐catenin signaling. By clustering genes with similar genetic interaction profiles, we drew the largest genetic network in cancer cells to date. Our scalable approach highlights how diverse genetic screens can be integrated to systematically build informative maps of genetic interactions in cancer, which can grow dynamically as more data are included.     

How to reconstruct a large genetic network from n gene perturbations in fewer than n(2) easy steps.

MOTIVATION: The reconstruction of genetic networks is the holy grail of functional genomics. Its core task is to identify the causal structure of a gene network, that is, to distinguish direct from indirect regulatory interactions among gene products. In other words, to reconstruct a genetic network is to identify, for each network gene, which other genes and their activity the gene influences directly. Crucial to this task are perturbations of gene activity. Genomic technology permits large-scale experiments perturbing the activity of many genes and assessing the effect of each perturbation on all other genes in a genome. However, such experiments cannot distinguish between direct and indirect effects of a genetic perturbation. RESULTS: I present an algorithm to reconstruct direct regulatory interactions in gene networks from the results of gene perturbation experiments. The algorithm is based on a graph representation of genetic networks and applies to networks of arbitrary size and complexity. Algorithmic complexity in both storage and time is low, less than O(n(2)). In practice, the algorithm can reconstruct networks of several thousand genes in mere CPU seconds on a desktop workstation. AVAILABILITY: A perl implementation of the algorithm is given in the Appendix. CONTACT: wagnera@unm.edu     

Breast Cancer Genes PSMC3IP and EPSTI1 Play a Role in Apoptosis Regulation

A key element to delineate the biology of individual tumors is the regulation of apoptosis. In this work, we functionally characterize two breast cancer associated genes, the proteasome 26S subunit ATPase 3 interacting protein (PSMC3IP) and the epithelial-stromal interaction 1 (EPSTI1), to explore their potential apoptotic role in breast cancer. We first explore the existence of direct physical interactions with annotated BC-apoptotic genes. Based on the generated interaction network, we examine several apoptotic markers to determine the effect of PSMC3IP and EPSTI1 gene expression modulation in two different human breast cancer cell lines to suggest potential molecular mechanisms to unveil their role in the disease. Our results show that PSMC3IP and EPSTI1 are able to modulate the extrinsic apoptotic pathway in estrogen receptor positive and triple negative breast cancer cell lines, highlighting them as potential therapeutic targets.     

Evolutionary conservation and network structure characterize genes of phenotypic relevance for mitosis in human

The impact of gene silencing on cellular phenotypes is difficult to establish due to the complexity of interactions in the associated biological processes and pathways. A recent genome-wide RNA knock-down study both identified and phenotypically characterized a set of important genes for the cell cycle in HeLa cells. Here, we combine a molecular interaction network analysis, based on physical and functional protein interactions, in conjunction with evolutionary information, to elucidate the common biological and topological properties of these key genes. Our results show that these genes tend to be conserved with their corresponding protein interactions across several species and are key constituents of the evolutionary conserved molecular interaction network. Moreover, a group of bistable network motifs is found to be conserved within this network, which are likely to influence the network stability and therefore the robustness of cellular functioning. They form a cluster, which displays functional homogeneity and is significantly enriched in genes phenotypically relevant for mitosis. Additional results reveal a relationship between specific cellular processes and the phenotypic outcomes induced by gene silencing. This study introduces new ideas regarding the relationship between genotype and phenotype in the context of the cell cycle. We show that the analysis of molecular interaction networks can result in the identification of genes relevant to cellular processes, which is a promising avenue for future research.     

Functional topology in a network of protein interactions

MOTIVATION: The building blocks of biological networks are individual protein-protein interactions (PPIs). The cumulative PPI data set in Saccharomyces cerevisiae now exceeds 78 000. Studying the network of these interactions will provide valuable insight into the inner workings of cells. RESULTS: We performed a systematic graph theory-based analysis of this PPI network to construct computational models for describing and predicting the properties of lethal mutations and proteins participating in genetic interactions, functional groups, protein complexes and signaling pathways. Our analysis suggests that lethal mutations are not only highly connected within the network, but they also satisfy an additional property: their removal causes a disruption in network structure. We also provide evidence for the existence of alternate paths that bypass viable proteins in PPI networks, while such paths do not exist for lethal mutations. In addition, we show that distinct functional classes of proteins have differing network properties. We also demonstrate a way to extract and iteratively predict protein complexes and signaling pathways. We evaluate the power of predictions by comparing them with a random model, and assess accuracy of predictions by analyzing their overlap with MIPS database. CONCLUSIONS: Our models provide a means for understanding the complex wiring underlying cellular function, and enable us to predict essentiality, genetic interaction, function, protein complexes and cellular pathways. This analysis uncovers structure-function relationships observable in a large PPI network.     

A Robust Structural PGN Model for Control of Cell-Cycle Progression Stabilized by Negative Feedbacks

The cell division cycle comprises a sequence of phenomena controlled by a stable and robust genetic network. We applied a probabilistic genetic network (PGN) to construct a hypothetical model with a dynamical behavior displaying the degree of robustness typical of the biological cell cycle. The structure of our PGN model was inspired in well-established biological facts such as the existence of integrator subsystems, negative and positive feedback loops, and redundant signaling pathways. Our model represents genes interactions as stochastic processes and presents strong robustness in the presence of moderate noise and parameters fluctuations. A recently published deterministic yeast cell-cycle model does not perform as well as our PGN model, even upon moderate noise conditions. In addition, self stimulatory mechanisms can give our PGN model the possibility of having a pacemaker activity similar to the observed in the oscillatory embryonic cell cycle.     

Identification of Networks of Co-Occurring,Tumor-Related DNA Copy Number Changes Using a Genome-Wide Scoring Approach

Tumorigenesis is a multi-step process in which normal cells transform into malignant tumors following the accumulation of genetic mutations that enable them to evade the growth control checkpoints that would normally suppress their growth or result in apoptosis. It is therefore important to identify those combinations of mutations that collaborate in cancer development and progression. DNA copy number alterations (CNAs) are one of the ways in which cancer genes are deregulated in tumor cells. We hypothesized that synergistic interactions between cancer genes might be identified by looking for regions of co-occurring gain and/or loss. To this end we developed a scoring framework to separate truly co-occurring aberrations from passenger mutations and dominant single signals present in the data. The resulting regions of high co-occurrence can be investigated for between-region functional interactions. Analysis of high-resolution DNA copy number data from a panel of 95 hematological tumor cell lines correctly identified co-occurring recombinations at the T-cell receptor and immunoglobulin loci in T- and B-cell malignancies, respectively, showing that we can recover truly co-occurring genomic alterations. In addition, our analysis revealed networks of co-occurring genomic losses and gains that are enriched for cancer genes. These networks are also highly enriched for functional relationships between genes. We further examine sub-networks of these networks, core networks, which contain many known cancer genes. The core network for co-occurring DNA losses we find seems to be independent of the canonical cancer genes within the network. Our findings suggest that large-scale, low-intensity copy number alterations may be an important feature of cancer development or maintenance by affecting gene dosage of a large interconnected network of functionally related genes.     

Identifying candidate genes for wood formation in poplar based on microarray network analysis and graph theory

In poplar, genetic research on wood properties is very important for the improvement of wood quality. Studies of wood formation genes at each developmental stage using modern biotechnology have often been limited to several genes or gene families. Because of the complex regulatory network involved in the co-expression and interactions of thousands of genes, however, the genetic mechanisms of wood formation must be surveyed on a genome-wide scale. In this study, we identified wood formation-related genes using a differentially co-expressed (DCE) gene subset approach based on biological networks inferred from microarray data. Gene co-expression networks in leaf, root, and wood tissues were first constructed and topologically analyzed using microarray data collected from the Gene Expression Omnibus. The DCE gene modules in wood-forming tissue were then detected based on graph theory, which was followed by gene ontology (GO) enrichment analysis and GO annotation of probe sets. Finally, 72 probe sets were identified in the largest cohesive subgroup of the DCE gene network in wood tissue, with most of the probe sets associated with wood formation-related biological processes and GO cellular component categories. The approach described in this paper provides an effective strategy to identify wood formation genes in poplar and should contribute to the better understanding of the genetic and molecular mechanisms underlying wood properties in trees.