Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

Equivalent genetic regulatory networks in different contexts recover contrasting spatial cell patterns that resemble those in Arabidopsis root and leaf epidermis: a dynamic model

In Arabidopsis thaliana, leaf and root epidermis hairs exhibit contrasting spatial arrangements even though the genetic networks regulating their respective cell-fate determination have very similar structures and components. We integrated available experimental data for leaf and root hair patterning in dynamic network models which may be reduced to activator-inhibitor models. This integration yielded expected results for these kinds of dynamic models, including striped and dotted cell patterns which are characteristic of root and leaf epidermis, respectively. However, these formal tools have led us to novel insights on current data and to put forward precise hypotheses which can be addressed experimentally. In particular, despite subtle differences in the root and leaf networks, these have equivalent dynamical behaviors. Our simulations also suggest that only when a biasing signal positively affects an activator in the network, the system recovers striped cellular patterns similar to those of root epidermis. We also postulate that cell shape may affect pattern stability in the root. Our results thus support the idea that in this and other cases, contrasting spatial cell patterns and other evolutionary morphogenetic novelties originate from conserved genetic network modules subject to divergent contextual traits.     

Spatial network structure and robustness of detritus-based communities in a patchy environment

The mechanisms that regulate the spatial distribution of species are an essential aid to understanding the effects of the environment on the persistence of populations and communities. The effects of spatial structure on the persistence and robustness of ecological communities can, in turn, prove useful in uncovering their functioning, e.g., in the decomposition of leaf detritus. We applied the framework of complex networks to evaluate the effects of spatial structure on the colonization process of leaf detritus in a patchy aquatic environment, with a spatial network of six pools at different salinity. We found three well-defined modules formed by groups of taxa sharing the same pools, observing an association between modularity and spatial proximity of pools. Modules maximize the number of links within modules, and minimize the number of links among modules, showing the presence of a strong site-specific association between taxa and pools. The topological characteristics of the network show robustness against random perturbations and a lower tolerance of targeted perturbations. These findings suggest that random events, such as flooding or heavy rains, slightly affect the robustness of the system, while localized perturbations on the most connected nodes could have a negative effect on the connectivity of the whole network. The consequences could lead to a structural and functional homogenization of the system, with potential effects for the entire trophic chain. Here we discuss the topological properties of the network in relation to the spatial distribution of pools, showing how network analysis can yield valuable insight for conservation and management.     

Evolution of networks for body plan patterning; interplay of modularity, robustness and evolvability

A major goal of evolutionary developmental biology (evo-devo) is to understand how multicellular body plans of increasing complexity have evolved, and how the corresponding developmental programs are genetically encoded. It has been repeatedly argued that key to the evolution of increased body plan complexity is the modularity of the underlying developmental gene regulatory networks (GRNs). This modularity is considered essential for network robustness and evolvability. In our opinion, these ideas, appealing as they may sound, have not been sufficiently tested. Here we use computer simulations to study the evolution of GRNs' underlying body plan patterning. We select for body plan segmentation and differentiation, as these are considered to be major innovations in metazoan evolution. To allow modular networks to evolve, we independently select for segmentation and differentiation. We study both the occurrence and relation of robustness, evolvability and modularity of evolved networks. Interestingly, we observed two distinct evolutionary strategies to evolve a segmented, differentiated body plan. In the first strategy, first segments and then differentiation domains evolve (SF strategy). In the second scenario segments and domains evolve simultaneously (SS strategy). We demonstrate that under indirect selection for robustness the SF strategy becomes dominant. In addition, as a byproduct of this larger robustness, the SF strategy is also more evolvable. Finally, using a combined functional and architectural approach, we determine network modularity. We find that while SS networks generate segments and domains in an integrated manner, SF networks use largely independent modules to produce segments and domains. Surprisingly, we find that widely used, purely architectural methods for determining network modularity completely fail to establish this higher modularity of SF networks. Finally, we observe that, as a free side effect of evolving segmentation and differentiation in combination, we obtained in-silico developmental mechanisms resembling mechanisms used in vertebrate development.     

Robustness and evolution: concepts, insights and challenges from a developmental model system

Robustness, the persistence of an organismal trait under perturbations, is a ubiquitous property of complex living systems. We here discuss key concepts related to robustness with examples from vulva development in the nematode Caenorhabditis elegans. We emphasize the need to be clear about the perturbations a trait is (or is not) robust to. We discuss two prominent mechanistic causes of robustness, namely redundancy and distributed robustness. We also discuss possible evolutionary causes of robustness, one of which does not involve natural selection. To better understand robustness is of paramount importance for understanding organismal evolution. Part of the reason is that highly robust systems can accumulate cryptic variation that can serve as a source of new adaptations and evolutionary innovations. We point to some key challenges in improving our understanding of robustness.     

Coherent coupling of feedback loops: a design principle of cell signaling networks

MOTIVATION: It is widely accepted that cell signaling networks have been evolved to be robust against perturbations. To investigate the topological characteristics resulting in such robustness, we have examined large-scale signaling networks and found that a number of feedback loops are present mostly in coupled structures. In particular, the coupling was made in a coherent way implying that same types of feedback loops are interlinked together. RESULTS: We have investigated the role of such coherently coupled feedback loops through extensive Boolean network simulations and found that a high proportion of coherent couplings can enhance the robustness of a network against its state perturbations. Moreover, we found that the robustness achieved by coherently coupled feedback loops can be kept evolutionarily stable. All these results imply that the coherent coupling of feedback loops might be a design principle of cell signaling networks devised to achieve the robustness.     

Weighted gene coexpression network analysis strategies applied to mouse weight

Systems-oriented genetic approaches that incorporate gene expression and genotype data are valuable in the quest for genetic regulatory loci underlying complex traits. Gene coexpression network analysis lends itself to identification of entire groups of differentially regulated genes—a highly relevant endeavor in finding the underpinnings of complex traits that are, by definition, polygenic in nature. Here we describe one such approach based on liver gene expression and genotype data from an F₂ mouse intercross utilizing weighted gene coexpression network analysis (WGCNA) of gene expression data to identify physiologically relevant modules. We describe two strategies: single-network analysis and differential network analysis. Single-network analysis reveals the presence of a physiologically interesting module that can be found in two distinct mouse crosses. Module quantitative trait loci (mQTLs) that perturb this module were discovered. In addition, we report a list of genetic drivers for this module. Differential network analysis reveals differences in connectivity and module structure between two networks based on the liver expression data of lean and obese mice. Functional annotation of these genes suggests a biological pathway involving epidermal growth factor (EGF). Our results demonstrate the utility of WGCNA in identifying genetic drivers and in finding genetic pathways represented by gene modules. These examples provide evidence that integration of network properties may well help chart the path across the gene–trait chasm. Electronic supplementary material The online version of this article (doi: ) contains supplementary material, which is available to authorized users. Tova F. Fuller, Anatole Ghazalpour contributed equally to this work.     

Emergent buffering balances evolvability and robustness in the evolution of phenotypic flexibility

Evolution of adaptive phenotypic flexibility requires a system that can dynamically restore and update a functional phenotype in response to environmental change. The architecture of such a system evolves under the conflicting demands of versatility and robustness, and resolution of these demands should be particularly evident in organisms that require external inputs for reiterative trait production within a generation, such as in metabolic networks that underlie yearly acquisition of diet‐dependent coloration in birds. Here, we show that a key structural feature of carotenoid networks–redundancy of biochemical pathways–enables these networks to translate variable environmental inputs into consistent phenotypic outcomes. We closely followed life‐long changes in structure and utilization of metabolic networks in a large cohort of free‐living birds and found that greater individual experience with dietary change between molts leads to wider occupancy of the metabolic network and progressive accumulation of redundant pathways in a functionally active network. This generated a regime of emergent buffering whereby greater dietary experience was mechanistically linked to greater robustness of resulting traits and an increasing ability to retain and implement previous adaptive solutions. Thus, experience‐related buffering links evolvability and robustness in carotenoid‐metabolizing networks and we argue that this mechanistic principle facilitates the evolution of phenotypic flexibility.     

Geometric robustness theory and biological networks

We provide a geometric framework for investigating the robustness of information flows over biological networks. We use information measures to quantify the impact of knockout perturbations on simple networks. Robustness has two components, a measure of the causal contribution of a node or nodes, and a measure of the change or exclusion dependence, of the network following node removal. Causality is measured as statistical contribution of a node to network function, wheras exclusion dependence measures a distance between unperturbed network and reconfigured network function. We explore the role that redundancy plays in increasing robustness, and how redundacy can be exploited through error-correcting codes implemented by networks. We provide examples of the robustness measure when applied to familiar boolean functions such as the AND, OR and XOR functions. We discuss the relationship between robustness measures and related measures of complexity and how robustness always implies a minimal level of complexity.     

Parasites lead to evolution of robustness against gene loss in host signaling networks

Many biological networks can maintain their function against single gene loss. However, the evolutionary mechanisms responsible for such robustness remain unclear. Here, we demonstrate that antagonistic host–parasite interactions can act as a selective pressure driving the emergence of robustness against gene loss. Using a model of host signaling networks and simulating their coevolution with parasites that interfere with network function, we find that networks evolve both redundancy and specific architectures that allow them to maintain their response despite removal of proteins. We show that when the parasite pressure is removed, subsequent evolution can lead to loss of redundancy while architecture‐based robustness is retained. Contrary to intuition, increased parasite virulence hampers evolution of robustness by limiting the generation of population level diversity in the host. However, when robustness emerges under high virulence, it tends to be stronger. These findings predict an increased presence of robustness mechanisms in biological networks operating under parasite interference. Conversely, the presence of such mechanisms could indicate current or past parasite interference.     

Integrating genetic and network analysis to characterize genes related to mouse weight

Systems biology approaches that are based on the genetics of gene expression have been fruitful in identifying genetic regulatory loci related to complex traits. We use microarray and genetic marker data from an F2 mouse intercross to examine the large-scale organization of the gene co-expression network in liver, and annotate several gene modules in terms of 22 physiological traits. We identify chromosomal loci (referred to as module quantitative trait loci, mQTL) that perturb the modules and describe a novel approach that integrates network properties with genetic marker information to model gene/trait relationships. Specifically, using the mQTL and the intramodular connectivity of a body weight–related module, we describe which factors determine the relationship between gene expression profiles and weight. Our approach results in the identification of genetic targets that influence gene modules (pathways) that are related to the clinical phenotypes of interest.     

A new measure of the robustness of biochemical networks

MOTIVATION: The robustness of a biochemical network is defined as the tolerance of variations in kinetic parameters with respect to the maintenance of steady state. Robustness also plays an important role in the fail-safe mechanism in the evolutionary process of biochemical networks. The purposes of this paper are to use the synergism and saturation system (S-system) representation to describe a biochemical network and to develop a robustness measure of a biochemical network subject to variations in kinetic parameters. Since most biochemical networks in nature operate close to the steady state, we consider only the robustness measurement of a biochemical network at the steady state. RESULTS: We show that the upper bound of the tolerated parameter variations is related to the system matrix of a biochemical network at the steady state. Using this upper bound, we can calculate the tolerance (robustness) of a biochemical network without testing many parametric perturbations. We find that a biochemical network with a large tolerance can also better attenuate the effects of variations in rate parameters and environments. Compensatory parameter variations and network redundancy are found to be important mechanisms for the robustness of biochemical networks. Finally, four biochemical networks, such as a cascaded biochemical network, the glycolytic-glycogenolytic pathway in a perfused rat liver, the tricarboxylic acid cycle in Dictyostelium discoideum and the cAMP oscillation network in bacterial chemotaxis, are used to illustrate the usefulness of the proposed robustness measure.     

细胞信号网络功能鲁棒的简并性拓扑特征

细胞信号网络对于外界环境的干扰表现出优良的鲁棒性,但是其维持功能鲁棒的内在机制尚未明确,本文研究了细胞信号网络功能鲁棒性的拓扑特征。选择布尔网络模型模拟细胞网络的动态行为,利用网络节点状态的扰动模拟外界环境干扰。基于演化策略探寻不同网络拓扑的功能并分析其在干扰环境下的鲁棒性,采用埃德尔曼提出的基于信息论的计算方法评估网络拓扑的简并度、冗余度和复杂度等拓扑属性,对比分析它们与功能鲁棒度的相关性及作用机理。结果显示,在网络模型的演化过程中,其拓扑简并度与功能鲁棒度显著正相关,相关性水平高于拓扑冗余度与鲁棒度的相关性。并且,随着鲁棒度的提升,网络的节点数和复杂度也随之升高,同样简并度与网络的节点数和复杂度的相关性高于拓扑冗余度与网络的节点数和复杂度的相关性。这说明增加的网络节点以简并的方式同时提高了网络拓扑的鲁棒度和复杂度。因此,细胞网络功能鲁棒性的拓扑特征是简并而不是冗余,简并为解决生物系统的复杂问题提供了有效手段,为人工系统的可靠性设计提供有益的借鉴。     

Network robustness and structure depend on the phenological characteristics of plants and pollinators

Many structural patterns have been found to be important for the stability and robustness of mutualistic plant–pollinator networks. These structural patterns are impacted by a suite of variables, including species traits, species abundances, their spatial configuration, and their phylogenetic history. Here, we consider a specific trait: phenology, or the timing of life history events. We expect that timing and duration of activity of pollinators, or of flowering in plants, could greatly affect the species'' roles within networks in which they are embedded. Using plant–pollinator networks from 33 sites in southern British Columbia, Canada, we asked (a) how phenological species traits, specifically timing of first appearance in the network and duration of activity in a network, were related to species'' roles within a network, and (b) how those traits affected network robustness to phenologically biased species loss. We found that long duration of activity increased connection within modules for both pollinators and plants and among modules for plants. We also found that date of first appearance was positively related to interaction strength asymmetry in plants but negatively related to pollinators. Networks were generally more robust to the loss of pollinators than plants, and robustness increased if the models allow new interactions to form when old ones are lost, constrained by overlapping phenology of plants and pollinators. Robustness declined with the loss of late‐flowering plants, which tended to have higher interaction strength asymmetry. In addition, robustness declined with loss of early‐flying or long‐duration pollinators. These pollinators tended to be among‐module connectors. Our results point to networks being limited by early‐flying pollinators. If plants flower earlier due to climate change, plant fitness may decline as they will depend on early emerging pollinators, unless pollinators also emerge earlier.     

Selection for distinct gene expression properties favours the evolution of mutational robustness in gene regulatory networks

Mutational robustness is a genotype's tendency to keep a phenotypic trait with little and few changes in the face of mutations. Mutational robustness is both ubiquitous and evolutionarily important as it affects in different ways the probability that new phenotypic variation arises. Understanding the origins of robustness is specially relevant for systems of development that are phylogenetically widespread and that construct phenotypic traits with a strong impact on fitness. Gene regulatory networks are examples of this class of systems. They comprise sets of genes that, through cross‐regulation, build the gene activity patterns that define cellular responses, different tissues or distinct cell types. Several empirical observations, such as a greater robustness of wild‐type phenotypes, suggest that stabilizing selection underlies the evolution of mutational robustness. However, the role of selection in the evolution of robustness is still under debate. Computer simulations of the dynamics and evolution of gene regulatory networks have shown that selection for any gene activity pattern that is steady and self‐sustaining is sufficient to promote the evolution of mutational robustness. Here, I generalize this scenario using a computational model to show that selection for different aspects of a gene activity phenotype increases mutational robustness. Mutational robustness evolves even when selection favours properties that conflict with the stationarity of a gene activity pattern. The results that I present support an important role for stabilizing selection in the evolution of robustness in gene regulatory networks.