Constraint and contingency in multifunctional gene regulatory circuits

Probabilistic Genotype-Phenotype Maps Reveal Mutational Robustness of RNA Folding, Spin Glasses, and Quantum Circuits

Recent studies of genotype-phenotype (GP) maps have reported universally enhanced phenotypic robustness to genotype mutations, a feature essential to evolution. Virtually all of these studies make a simplifying assumption that each genotype-represented as a sequence-maps deterministically to a single phenotype, such as a discrete structure. Here, we introduce probabilistic genotype-phenotype (PrGP) maps, where each genotype maps to a vector of phenotype probabilities, as a more realistic and universal language for investigating robustness in a variety of physical, biological, and computational systems. We study three model systems to show that PrGP maps offer a generalized framework which can handle uncertainty emerging from various physical sources: (1) thermal fluctuation in RNA folding, (2) external field disorder in spin glass ground state finding, and (3) superposition and entanglement in quantum circuits, which are realized experimentally on IBM quantum computers. In all three cases, we observe a novel biphasic robustness scaling which is enhanced relative to random expectation for more frequent phenotypes and approaches random expectation for less frequent phenotypes. We derive an analytical theory for the behavior of PrGP robustness, and we demonstrate that the theory is highly predictive of empirical robustness.

Frustration can Limit the Adaptation of Promiscuous Enzymes Through Gene Duplication and Specialisation

Virtually all enzymes catalyse more than one reaction, a phenomenon known as enzyme promiscuity. It is unclear whether promiscuous enzymes are more often generalists that catalyse multiple reactions at similar rates or specialists that catalyse one reaction much more efficiently than other reactions. In addition, the factors that shape whether an enzyme evolves to be a generalist or a specialist are poorly understood. To address these questions, we follow a three-pronged approach. First, we examine the distribution of promiscuity in empirical enzymes reported in the BRENDA database. We find that the promiscuity distribution of empirical enzymes is bimodal. In other words, a large fraction of promiscuous enzymes are either generalists or specialists, with few intermediates. Second, we demonstrate that enzyme biophysics is not sufficient to explain this bimodal distribution. Third, we devise a constraint-based model of promiscuous enzymes undergoing duplication and facing selection pressures favouring subfunctionalization. The model posits the existence of constraints between the catalytic efficiencies of an enzyme for different reactions and is inspired by empirical case studies. The promiscuity distribution predicted by our constraint-based model is consistent with the empirical bimodal distribution. Our results suggest that subfunctionalization is possible and beneficial only in certain enzymes. Furthermore, the model predicts that conflicting constraints and selection pressures can cause promiscuous enzymes to enter a 'frustrated' state, in which competing interactions limit the specialisation of enzymes. We find that frustration can be both a driver and an inhibitor of enzyme evolution by duplication and subfunctionalization. In addition, our model predicts that frustration becomes more likely as enzymes catalyse more reactions, implying that natural selection may prefer catalytically simple enzymes. In sum, our results suggest that frustration may play an important role in enzyme evolution.

Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve

Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer's graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype-phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.

Robustness and innovation in synthetic genotype networks

Genotype networks are sets of genotypes connected by small mutational changes that share the same phenotype. They facilitate evolutionary innovation by enabling the exploration of different neighborhoods in genotype space. Genotype networks, first suggested by theoretical models, have been empirically confirmed for proteins and RNAs. Comparative studies also support their existence for gene regulatory networks (GRNs), but direct experimental evidence is lacking. Here, we report the construction of three interconnected genotype networks of synthetic GRNs producing three distinct phenotypes in Escherichia coli. Our synthetic GRNs contain three nodes regulating each other by CRISPR interference and governing the expression of fluorescent reporters. The genotype networks, composed of over twenty different synthetic GRNs, provide robustness in face of mutations while enabling transitions to innovative phenotypes. Through realistic mathematical modeling, we quantify robustness and evolvability for the complete genotype-phenotype map and link these features mechanistically to GRN motifs. Our work thereby exemplifies how GRN evolution along genotype networks might be driving evolutionary innovation.

Robustness and stability of spin-glass ground states to perturbed interactions

Across many problems in science and engineering, it is important to consider how much the output of a given system changes due to perturbations of the input. Here, we investigate the glassy phase of ±J spin glasses at zero temperature by calculating the robustness of the ground states to flips in the sign of single interactions. For random graphs and the Sherrington-Kirkpatrick model, we find relatively large sets of bond configurations that generate the same ground state. These sets can themselves be analyzed as subgraphs of the interaction domain, and we compute many of their topological properties. In particular, we find that the robustness, equivalent to the average degree, of these subgraphs is much higher than one would expect from a random model. Most notably, it scales in the same logarithmic way with the size of the subgraph as has been found in genotype-phenotype maps for RNA secondary structure folding, protein quaternary structure, gene regulatory networks, as well as for models for genetic programming. The similarity between these disparate systems suggests that this scaling may have a more universal origin.

Emergence of co-expression in gene regulatory networks

Transcriptomes are known to organize themselves into gene co-expression clusters or modules where groups of genes display distinct patterns of coordinated or synchronous expression across independent biological samples. The functional significance of these co-expression clusters is suggested by the fact that highly coexpressed groups of genes tend to be enriched in genes involved in common functions and biological processes. While gene co-expression is widely assumed to reflect close regulatory proximity, the validity of this assumption remains unclear. Here we use a simple synthetic gene regulatory network (GRN) model and contrast the resulting co-expression structure produced by these networks with their known regulatory architecture and with the co-expression structure measured in available human expression data. Using randomization tests, we found that the levels of co-expression observed in simulated expression data were, just as with empirical data, significantly higher than expected by chance. When examining the source of correlated expression, we found that individual regulators, both in simulated and experimental data, fail, on average, to display correlated expression with their immediate targets. However, highly correlated gene pairs tend to share at least one common regulator, while most gene pairs sharing common regulators do not necessarily display correlated expression. Our results demonstrate that widespread co-expression naturally emerges in regulatory networks, and that it is a reliable and direct indicator of active co-regulation in a given cellular context.

On the evolution and development of morphological complexity: A view from gene regulatory networks

How does morphological complexity evolve? This study suggests that the likelihood of mutations increasing phenotypic complexity becomes smaller when the phenotype itself is complex. In addition, the complexity of the genotype-phenotype map (GPM) also increases with the phenotypic complexity. We show that complex GPMs and the above mutational asymmetry are inevitable consequences of how genes need to be wired in order to build complex and robust phenotypes during development. We randomly wired genes and cell behaviors into networks in EmbryoMaker. EmbryoMaker is a mathematical model of development that can simulate any gene network, all animal cell behaviors (division, adhesion, apoptosis, etc.), cell signaling, cell and tissues biophysics, and the regulation of those behaviors by gene products. Through EmbryoMaker we simulated how each random network regulates development and the resulting morphology (i.e. a specific distribution of cells and gene expression in 3D). This way we obtained a zoo of possible 3D morphologies. Real gene networks are not random, but a random search allows a relatively unbiased exploration of what is needed to develop complex robust morphologies. Compared to the networks leading to simple morphologies, the networks leading to complex morphologies have the following in common: 1) They are rarer; 2) They need to be finely tuned; 3) Mutations in them tend to decrease morphological complexity; 4) They are less robust to noise; and 5) They have more complex GPMs. These results imply that, when complexity evolves, it does so at a progressively decreasing rate over generations. This is because as morphological complexity increases, the likelihood of mutations increasing complexity decreases, morphologies become less robust to noise, and the GPM becomes more complex. We find some properties in common, but also some important differences, with non-developmental GPM models (e.g. RNA, protein and gene networks in single cells).

Control for multifunctionality: bioinspired control based on feeding in Aplysia californica

Animals exhibit remarkable feats of behavioral flexibility and multifunctional control that remain challenging for robotic systems. The neural and morphological basis of multifunctionality in animals can provide a source of bioinspiration for robotic controllers. However, many existing approaches to modeling biological neural networks rely on computationally expensive models and tend to focus solely on the nervous system, often neglecting the biomechanics of the periphery. As a consequence, while these models are excellent tools for neuroscience, they fail to predict functional behavior in real time, which is a critical capability for robotic control. To meet the need for real-time multifunctional control, we have developed a hybrid Boolean model framework capable of modeling neural bursting activity and simple biomechanics at speeds faster than real time. Using this approach, we present a multifunctional model of Aplysia californica feeding that qualitatively reproduces three key feeding behaviors (biting, swallowing, and rejection), demonstrates behavioral switching in response to external sensory cues, and incorporates both known neural connectivity and a simple bioinspired mechanical model of the feeding apparatus. We demonstrate that the model can be used for formulating testable hypotheses and discuss the implications of this approach for robotic control and neuroscience.

Predicting Evolutionary Constraints by Identifying Conflicting Demands in Regulatory Networks

Gene regulation networks allow organisms to adapt to diverse environmental niches. However, the constraints underlying the evolution of gene regulation remain ill defined. Here, we show that partial order-a concept that ranks network output levels as a function of different input signals-identifies such constraints. We tested our predictions by experimentally evolving an engineered signal-integrating network in multiple environments. We find that populations: (1) expand in fitness space along the Pareto-optimal front associated with conflicts in regulatory demands, by fine-tuning binding affinities within the network, and (2) expand beyond the Pareto-optimal front through changes in the network structure. Our constraint predictions are based only on partial order and do not require information on the network architecture or underlying genetics. Overall, our findings show that limited knowledge of current regulatory phenotypes can provide predictions on future evolutionary constraints.

Using small samples to estimate neutral component size and robustness in the genotype-phenotype map of RNA secondary structure

In genotype-phenotype (GP) maps, the genotypes that map to the same phenotype are usually not randomly distributed across the space of genotypes, but instead are predominantly connected through one-point mutations, forming network components that are commonly referred to as neutral components (NCs). Because of their impact on evolutionary processes, the characteristics of these NCs, like their size or robustness, have been studied extensively. Here, we introduce a framework that allows the estimation of NC size and robustness in the GP map of RNA secondary structure. The advantage of this framework is that it only requires small samples of genotypes and their local environment, which also allows experimental realizations. We verify our framework by applying it to the exhaustively analysable GP map of RNA sequence length L = 15, and benchmark it against an existing method by applying it to longer, naturally occurring functional non-coding RNA sequences. Although it is specific to the RNA secondary structure GP map in the first place, our framework can probably be transferred and adapted to other sequence-to-structure GP maps.

Genotype Components as Predictors of Phenotype in Model Gene Regulatory Networks

Models of gene regulatory networks (GRN) have proven useful for understanding many aspects of the highly complex behavior of biological control networks. Randomly generated non-Boolean networks were used in experimental simulations to generate data on dynamic phenotypes as a function of several genotypic parameters. We found that predictive relationships between some phenotypes and quantitative genotypic parameters such as number of network genes, interaction density, and initial condition could be derived depending on the strength of the topological (positional) genotype on specific phenotypes. We quantitated the strength of the topological genotype effect (TGE) on a number of phenotypes in multi-gene networks. For phenotypes with a low influence of topological genotype, derived and empirical relationships using quantitative genotype parameters were accurate in phenotypic outcomes. We found a number of dynamic network properties, including oscillation behaviors, that were largely dependent on genotype topology, and for which no such general quantitative relationships were determinable. It remains to be determined if these results are applicable to biological gene regulatory networks.

The architecture of an empirical genotype-phenotype map

Recent advances in high-throughput technologies are bringing the study of empirical genotype-phenotype (GP) maps to the fore. Here, we use data from protein-binding microarrays to study an empirical GP map of transcription factor (TF) -binding preferences. In this map, each genotype is a DNA sequence. The phenotype of this DNA sequence is its ability to bind one or more TFs. We study this GP map using genotype networks, in which nodes represent genotypes with the same phenotype, and edges connect nodes if their genotypes differ by a single small mutation. We describe the structure and arrangement of genotype networks within the space of all possible binding sites for 525 TFs from three eukaryotic species encompassing three kingdoms of life (animal, plant, and fungi). We thus provide a high-resolution depiction of the architecture of an empirical GP map. Among a number of findings, we show that these genotype networks are "small-world" and assortative, and that they ubiquitously overlap and interface with one another. We also use polymorphism data from Arabidopsis thaliana to show how genotype network structure influences the evolution of TF-binding sites in vivo. We discuss our findings in the context of regulatory evolution.

Structural properties of genotype-phenotype maps

The map between genotype and phenotype is fundamental to biology. Biological information is stored and passed on in the form of genotypes, and expressed in the form of phenotypes. A growing body of literature has examined a wide range of genotype-phenotype (GP) maps and has established a number of properties that appear to be shared by many GP maps. These properties are 'structural' in the sense that they are properties of the distribution of phenotypes across the point-mutation network of genotypes. They include: a redundancy of genotypes, meaning that many genotypes map to the same phenotypes, a highly non-uniform distribution of the number of genotypes per phenotype, a high robustness of phenotypes and the ability to reach a large number of new phenotypes within a small number of mutational steps. A further important property is that the robustness and evolvability of phenotypes are positively correlated. In this review, I give an overview of the study of GP maps with particular emphasis on these structural properties, and discuss a model that attempts to explain why these properties arise, as well as some of the fundamental ways in which the structure of GP maps can affect evolutionary outcomes.

Approaches to Macroevolution: 1. General Concepts and Origin of Variation

Approaches to macroevolution require integration of its two fundamental components, i.e. the origin and the sorting of variation, in a hierarchical framework. Macroevolution occurs in multiple currencies that are only loosely correlated, notably taxonomic diversity, morphological disparity, and functional variety. The origin of variation within this conceptual framework is increasingly understood in developmental terms, with the semi-hierarchical structure of gene regulatory networks (GRNs, used here in a broad sense incorporating not just the genetic circuitry per se but the factors controlling the timing and location of gene expression and repression), the non-linear relation between magnitude of genetic change and the phenotypic results, the evolutionary potential of co-opting existing GRNs, and developmental responsiveness to nongenetic signals (i.e. epigenetics and plasticity), all requiring modification of standard microevolutionary models, and rendering difficult any simple definition of evolutionary novelty. The developmental factors underlying macroevolution create anisotropic probabilities-i.e., an uneven density distribution-of evolutionary change around any given phenotypic starting point, and the potential for coordinated changes among traits that can accommodate change via epigenetic mechanisms. From this standpoint, "punctuated equilibrium" and "phyletic gradualism" simply represent two cells in a matrix of evolutionary models of phenotypic change, and the origin of trends and evolutionary novelty are not simply functions of ecological opportunity. Over long timescales, contingency becomes especially important, and can be viewed in terms of macroevolutionary lags (the temporal separation between the origin of a trait or clade and subsequent diversification); such lags can arise by several mechanisms: as geological or phylogenetic artifacts, or when diversifications require synergistic interactions among traits, or between traits and external events. The temporal and spatial patterns of the origins of evolutionary novelties are a challenge to macroevolutionary theory; individual events can be described retrospectively, but a general model relating development, genetics, and ecology is needed. An accompanying paper (Jablonski in Evol Biol 2017) reviews diversity dynamics and the sorting of variation, with some general conclusions.

A spectrum of modularity in multi-functional gene circuits

A major challenge in systems biology is to understand the relationship between a circuit's structure and its function, but how is this relationship affected if the circuit must perform multiple distinct functions within the same organism? In particular, to what extent do multi‐functional circuits contain modules which reflect the different functions? Here, we computationally survey a range of bi‐functional circuits which show no simple structural modularity: They can switch between two qualitatively distinct functions, while both functions depend on all genes of the circuit. Our analysis reveals two distinct classes: hybrid circuits which overlay two simpler mono‐functional sub‐circuits within their circuitry, and emergent circuits, which do not. In this second class, the bi‐functionality emerges from more complex designs which are not fully decomposable into distinct modules and are consequently less intuitive to predict or understand. These non‐intuitive emergent circuits are just as robust as their hybrid counterparts, and we therefore suggest that the common bias toward studying modular systems may hinder our understanding of real biological circuits.

Noncommutative Biology: Sequential Regulation of Complex Networks

Single-cell variability in gene expression is important for generating distinct cell types, but it is unclear how cells use the same set of regulatory molecules to specifically control similarly regulated genes. While combinatorial binding of transcription factors at promoters has been proposed as a solution for cell-type specific gene expression, we found that such models resulted in substantial information bottlenecks. We sought to understand the consequences of adopting sequential logic wherein the time-ordering of factors informs the final outcome. We showed that with noncommutative control, it is possible to independently control targets that would otherwise be activated simultaneously using combinatorial logic. Consequently, sequential logic overcomes the information bottleneck inherent in complex networks. We derived scaling laws for two noncommutative models of regulation, motivated by phosphorylation/neural networks and chromosome folding, respectively, and showed that they scale super-exponentially in the number of regulators. We also showed that specificity in control is robust to the loss of a regulator. Lastly, we connected these theoretical results to real biological networks that demonstrate specificity in the context of promiscuity. These results show that achieving a desired outcome often necessitates roundabout steps.

DNA is the blueprint of life. Yet the order in which a cell follows these instructions makes it capable of generating thousands of different fates. How this information is extracted from underlying gene regulatory networks is unclear, especially given that biological networks are highly interconnected, and that the number of signaling pathways is relatively small (approximately 5–10). The conventional approach for increasing the information capacity of a limited set of regulators is to use them in combination. Surprisingly, combinatorial logic does not increase the diversity of target configurations or cell fates, but instead causes information bottlenecks. A different approach, called sequential logic, uses noncommutative sequences of a small set of regulators to drive networks to a large number of novel configurations. If certain targets are first protected, then even promiscuous regulators can activate specific subsets of lineage-specific targets. In this paper we show how sequential logic outperforms combinatorial logic, and argue that noncommutative sequences underlie a number of cases of biological regulation, e.g. how a small number of signaling pathways generates a large diversity of cell types in development. In addition to explaining biological networks, sequential logic may be a general experimental design strategy in synthetic and single-cell biology.

Genonets server-a web server for the construction, analysis and visualization of genotype networks

A genotype network is a graph in which vertices represent genotypes that have the same phenotype. Edges connect vertices if their corresponding genotypes differ in a single small mutation. Genotype networks are used to study the organization of genotype spaces. They have shed light on the relationship between robustness and evolvability in biological systems as different as RNA macromolecules and transcriptional regulatory circuits. Despite the importance of genotype networks, no tool exists for their automatic construction, analysis and visualization. Here we fill this gap by presenting the Genonets Server, a tool that provides the following features: (i) the construction of genotype networks for categorical and univariate phenotypes from DNA, RNA, amino acid or binary sequences; (ii) analyses of genotype network topology and how it relates to robustness and evolvability, as well as analyses of genotype network topography and how it relates to the navigability of a genotype network via mutation and natural selection; (iii) multiple interactive visualizations that facilitate exploratory research and education. The Genonets Server is freely available at http://ieu-genonets.uzh.ch.

Boolean Models of Biological Processes Explain Cascade-Like Behavior

Biological networks play a key role in determining biological function and therefore, an understanding of their structure and dynamics is of central interest in systems biology. In Boolean models of such networks, the status of each molecule is either “on” or “off” and along with the molecules interact with each other, their individual status changes from “on” to “off” or vice-versa and the system of molecules in the network collectively go through a sequence of changes in state. This sequence of changes is termed a biological process. In this paper, we examine the common perception that events in biomolecular networks occur sequentially, in a cascade-like manner, and ask whether this is likely to be an inherent property. In further investigations of the budding and fission yeast cell-cycle, we identify two generic dynamical rules. A Boolean system that complies with these rules will automatically have a certain robustness. By considering the biological requirements in robustness and designability, we show that those Boolean dynamical systems, compared to an arbitrary dynamical system, statistically present the characteristics of cascadeness and sequentiality, as observed in the budding and fission yeast cell- cycle. These results suggest that cascade-like behavior might be an intrinsic property of biological processes.

Genome-Wide Detection and Analysis of Multifunctional Genes

Many genes can play a role in multiple biological processes or molecular functions. Identifying multifunctional genes at the genome-wide level and studying their properties can shed light upon the complexity of molecular events that underpin cellular functioning, thereby leading to a better understanding of the functional landscape of the cell. However, to date, genome-wide analysis of multifunctional genes (and the proteins they encode) has been limited. Here we introduce a computational approach that uses known functional annotations to extract genes playing a role in at least two distinct biological processes. We leverage functional genomics data sets for three organisms—H. sapiens, D. melanogaster, and S. cerevisiae—and show that, as compared to other annotated genes, genes involved in multiple biological processes possess distinct physicochemical properties, are more broadly expressed, tend to be more central in protein interaction networks, tend to be more evolutionarily conserved, and are more likely to be essential. We also find that multifunctional genes are significantly more likely to be involved in human disorders. These same features also hold when multifunctionality is defined with respect to molecular functions instead of biological processes. Our analysis uncovers key features about multifunctional genes, and is a step towards a better genome-wide understanding of gene multifunctionality.

Almost every aspect of cellular function depends on protein activity. In spite of being fine-tuned to carry out highly specific functions, proteins can also multitask. Experimental studies have identified genes and proteins endowed with more than one molecular function, or participating in very different biological processes. These studies suggest that the degree of functional plasticity exhibited by proteins might go well beyond a simple “one protein—one function” relationship. However, systematic studies of the properties of multifunctional genes (and their encoded proteins) have been limited. Here we present a computational framework to identify putative multifunctional genes, and compare their properties with those of other genes. We find that multifunctional genes are significantly different from other genes with respect to their physicochemical properties, expression profiles, and interaction properties. We also observe that multifunctional genes tend to be more conserved, and that a greater fraction of them are associated with human disorders. Taken together, these results represent a step towards a more complete understanding of the role multifunctional genes play in the functional organization of the cell.

Function does not follow form in gene regulatory circuits

Gene regulatory circuits are to the cell what arithmetic logic units are to the chip: fundamental components of information processing that map an input onto an output. Gene regulatory circuits come in many different forms, distinct structural configurations that determine who regulates whom. Studies that have focused on the gene expression patterns (functions) of circuits with a given structure (form) have examined just a few structures or gene expression patterns. Here, we use a computational model to exhaustively characterize the gene expression patterns of nearly 17 million three-gene circuits in order to systematically explore the relationship between circuit form and function. Three main conclusions emerge. First, function does not follow form. A circuit of any one structure can have between twelve and nearly thirty thousand distinct gene expression patterns. Second, and conversely, form does not follow function. Most gene expression patterns can be realized by more than one circuit structure. And third, multifunctionality severely constrains circuit form. The number of circuit structures able to drive multiple gene expression patterns decreases rapidly with the number of these patterns. These results indicate that it is generally not possible to infer circuit function from circuit form, or vice versa.

Intersecting transcription networks constrain gene regulatory evolution

Epistasis—the non-additive interactions between different genetic loci—constrains evolutionary pathways, blocking some and permitting others^1–8. For biological networks such as transcription circuits, the nature of these constraints and their consequences are largely unknown. Here we describe the evolutionary pathways of a transcription network that controls the response to mating pheromone in yeasts⁹. A component of this network, the transcription regulator Ste12, has evolved two different modes of binding to a set of its target genes. In one group of species, Ste12 binds to specific DNA binding sites, while in another lineage it occupies DNA indirectly, relying on a second transcription regulator to recognize DNA. We show, through the construction of various possible evolutionary intermediates, that evolution of the direct mode of DNA binding was not directly accessible to the ancestor. Instead, it was contingent on a lineage-specific change to an overlapping transcription network with a different function, the specification of cell type. These results show that analyzing and predicting the evolution of cis-regulatory regions requires an understanding of their positions in overlapping networks, as this placement constrains the available evolutionary pathways.

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

A Boolean gene regulatory model of heterosis and speciation

BACKGROUND

Modelling genetic phenomena affecting biological traits is important for the development of agriculture as it allows breeders to predict the potential of breeding for certain traits. One such phenomenon is heterosis or hybrid vigor: crossing individuals from genetically distinct populations often results in improvements in quantitative traits, such as growth rate, biomass production and stress resistance. Heterosis has become a very useful tool in global agriculture, but its genetic basis remains controversial and its effects hard to predict. We have taken a computational approach to studying heterosis, developing a simulation of evolution, independent reassortment of alleles and hybridization of Gene Regulatory Networks (GRNs) in a Boolean framework. These artificial regulatory networks exhibit topological properties that reflect those observed in biology, and fitness is measured as the ability of a network to respond to external inputs in a pre-defined way.

RESULTS

Our model reproduced common experimental observations on heterosis using only biologically justified parameters, such as mutation rates. Hybrid vigor was observed and its extent was seen to increase as parental populations diverged, up until a point of sudden collapse of hybrid fitness. Thus, the model also describes a process akin to speciation due to genetic incompatibility of the separated populations. We also reproduce, for the first time in a model, the fact that hybrid vigor cannot easily be fixed by within a breeding line, currently an important limitation of the use of hybrid crops. The simulation allowed us to study the effects of three standard models for the genetic basis of heterosis: dominance, over-dominance, and epistasis.

CONCLUSION

This study describes the most detailed simulation of heterosis using gene regulatory networks to date and reproduces several phenomena associated with heterosis for the first time in a model. The level of detail in our model allows us to suggest possible warning signs of the impending collapse of hybrid vigor in breeding. In addition, the simulation provides a framework that can be extended to study other aspects of heterosis and alternative evolutionary scenarios.

Phenotypic robustness and the assortativity signature of human transcription factor networks

Many developmental, physiological, and behavioral processes depend on the precise expression of genes in space and time. Such spatiotemporal gene expression phenotypes arise from the binding of sequence-specific transcription factors (TFs) to DNA, and from the regulation of nearby genes that such binding causes. These nearby genes may themselves encode TFs, giving rise to a transcription factor network (TFN), wherein nodes represent TFs and directed edges denote regulatory interactions between TFs. Computational studies have linked several topological properties of TFNs — such as their degree distribution — with the robustness of a TFN's gene expression phenotype to genetic and environmental perturbation. Another important topological property is assortativity, which measures the tendency of nodes with similar numbers of edges to connect. In directed networks, assortativity comprises four distinct components that collectively form an assortativity signature. We know very little about how a TFN's assortativity signature affects the robustness of its gene expression phenotype to perturbation. While recent theoretical results suggest that increasing one specific component of a TFN's assortativity signature leads to increased phenotypic robustness, the biological context of this finding is currently limited because the assortativity signatures of real-world TFNs have not been characterized. It is therefore unclear whether these earlier theoretical findings are biologically relevant. Moreover, it is not known how the other three components of the assortativity signature contribute to the phenotypic robustness of TFNs. Here, we use publicly available DNaseI-seq data to measure the assortativity signatures of genome-wide TFNs in 41 distinct human cell and tissue types. We find that all TFNs share a common assortativity signature and that this signature confers phenotypic robustness to model TFNs. Lastly, we determine the extent to which each of the four components of the assortativity signature contributes to this robustness.

The cells of living organisms do not concurrently express their entire complement of genes. Instead, they regulate their gene expression, and one consequence of this is the potential for different cells to adopt different stable gene expression patterns. For example, the development of an embryo necessitates that cells alter their gene expression patterns in order to differentiate. These gene expression phenotypes are largely robust to genetic mutation, and one source of this robustness may reside in the network structure of interacting molecules that underlie genetic regulation. Theoretical studies of regulatory networks have linked network structure to robustness; however, it is also necessary to more extensively characterize real-world regulatory networks in order to understand which structural properties may be biologically meaningful. We recently used theoretical models to show that a particular structural property, degree assortativity, is linked to robustness. Here, we measure the assortativity of human regulatory networks in 41 distinct cell and tissue types. We then develop a theoretical framework to explore how this structural property affects robustness, and we find that the gene expression phenotypes of human regulatory networks are more robust than expected by chance alone.

Reversion of developmental mode in insects: evolution from long germband to short germband in the polyembrionic wasp Macrocentrus cingulum Brischke

Germband size in insects has played a central role in our understanding of insect patterning mechanisms and their evolution. The polarity of evolutionary change in insect patterning has been viewed so far as the unidirectional shift from the ancestral short germband patterning of basal hemimetabolous insects to the long germband patterning observed in most modern Holometabola. However, some orders of holometabolic insects display both short and long germband development, though the absence of a clear phylogenetic context does not permit definite conclusions on the polarity of change. Derived hymenoptera, that is, bees and wasps, represent a classical textbook example of long germband development. Yet, in some wasps putative short germband development has been described correlating with lifestyle changes, namely with evolution of endoparasitism and polyembryony. To address the potential reversion from long to short germband, we focused on the family Braconidae, which displays ancestral long germband development, and examined the derived polyembryonic braconid Macrocentrus cingulum. Using SEM analysis of M. cingulum embryogenesis coupled with analyses of embryonic patterning markers, we show that this wasp evolved short germband embryogenesis secondarily, in a way that is reminiscent of embryogenesis in the beetle Tribolium castaneum. This work shows that the evolution of germband size in insects is a reversible process that may correlate with other life-history traits and suggests broader implications on the mechanisms and evolvability of insect development.

Latent phenotypes pervade gene regulatory circuits

BACKGROUND

Latent phenotypes are non-adaptive byproducts of adaptive phenotypes. They exist in biological systems as different as promiscuous enzymes and genome-scale metabolic reaction networks, and can give rise to evolutionary adaptations and innovations. We know little about their prevalence in the gene expression phenotypes of regulatory circuits, important sources of evolutionary innovations.

RESULTS

Here, we study a space of more than sixteen million three-gene model regulatory circuits, where each circuit is represented by a genotype, and has one or more functions embodied in one or more gene expression phenotypes. We find that the majority of circuits with single functions have latent expression phenotypes. Moreover, the set of circuits with a given spectrum of functions has a repertoire of latent phenotypes that is much larger than that of any one circuit. Most of this latent repertoire can be easily accessed through a series of small genetic changes that preserve a circuit's main functions. Both circuits and gene expression phenotypes that are robust to genetic change are associated with a greater number of latent phenotypes.

CONCLUSIONS

Our observations suggest that latent phenotypes are pervasive in regulatory circuits, and may thus be an important source of evolutionary adaptations and innovations involving gene regulation.

Modeling the evolution of molecular systems from a mechanistic perspective

Systems biology-inspired genotype-phenotype mapping models are increasingly being used to study the evolutionary properties of molecular biological systems, in particular the general emergent properties of evolving systems, such as modularity, robustness, and evolvability. However, the level of abstraction at which many of these models operate might not be sufficient to capture all relevant intricacies of biological evolution in sufficient detail. Here, we argue that in particular gene and genome duplications, both evolutionary mechanisms of potentially major importance for the evolution of molecular systems and of special relevance to plant evolution, are not adequately accounted for in most GPM modeling frameworks, and that more fine-grained mechanistic models may significantly advance understanding of how gen(om)e duplication impacts molecular systems evolution.

Historical contingency and the gradual evolution of metabolic properties in central carbon and genome-scale metabolisms

BACKGROUND

A metabolism can evolve through changes in its biochemical reactions that are caused by processes such as horizontal gene transfer and gene deletion. While such changes need to preserve an organism's viability in its environment, they can modify other important properties, such as a metabolism's maximal biomass synthesis rate and its robustness to genetic and environmental change. Whether such properties can be modulated in evolution depends on whether all or most viable metabolisms - those that can synthesize all essential biomass precursors - are connected in a space of all possible metabolisms. Connectedness means that any two viable metabolisms can be converted into one another through a sequence of single reaction changes that leave viability intact. If the set of viable metabolisms is disconnected and highly fragmented, then historical contingency becomes important and restricts the alteration of metabolic properties, as well as the number of novel metabolic phenotypes accessible in evolution.

RESULTS

We here computationally explore two vast spaces of possible metabolisms to ask whether viable metabolisms are connected. We find that for all but the simplest metabolisms, most viable metabolisms can be transformed into one another by single viability-preserving reaction changes. Where this is not the case, alternative essential metabolic pathways consisting of multiple reactions are responsible, but such pathways are not common.

CONCLUSIONS

Metabolism is thus highly evolvable, in the sense that its properties could be fine-tuned by successively altering individual reactions. Historical contingency does not strongly restrict the origin of novel metabolic phenotypes.