Genotype networks in metabolic reaction spaces

Open-endedness in synthetic biology: A route to continual innovation for biological design

Design in synthetic biology is typically goal oriented, aiming to repurpose or optimize existing biological functions, augmenting biology with new-to-nature capabilities, or creating life-like systems from scratch. While the field has seen many advances, bottlenecks in the complexity of the systems built are emerging and designs that function in the lab often fail when used in real-world contexts. Here, we propose an open-ended approach to biological design, with the novelty of designed biology being at least as important as how well it fulfils its goal. Rather than solely focusing on optimization toward a single best design, designing with novelty in mind may allow us to move beyond the diminishing returns we see in performance for most engineered biology. Research from the artificial life community has demonstrated that embracing novelty can automatically generate innovative and unexpected solutions to challenging problems beyond local optima. Synthetic biology offers the ideal playground to explore more creative approaches to biological design.

The Architecture of Metabolic Networks Constrains the Evolution of Microbial Resource Hierarchies

Microbial strategies for resource use are an essential determinant of their fitness in complex habitats. When facing environments with multiple nutrients, microbes often use them sequentially according to a preference hierarchy, resulting in well-known patterns of diauxic growth. In theory, the evolutionary diversification of metabolic hierarchies could represent a mechanism supporting coexistence and biodiversity by enabling temporal segregation of niches. Despite this ecologically critical role, the extent to which substrate preference hierarchies can evolve and diversify remains largely unexplored. Here, we used genome-scale metabolic modeling to systematically explore the evolution of metabolic hierarchies across a vast space of metabolic network genotypes. We find that only a limited number of metabolic hierarchies can readily evolve, corresponding to the most commonly observed hierarchies in genome-derived models. We further show how the evolution of novel hierarchies is constrained by the architecture of central metabolism, which determines both the propensity to change ranks between pairs of substrates and the effect of specific reactions on hierarchy evolution. Our analysis sheds light on the genetic and mechanistic determinants of microbial metabolic hierarchies, opening new research avenues to understand their evolution, evolvability, and ecology.

What makes a reaction network "chemical"?

BACKGROUND

Reaction networks (RNs) comprise a set X of species and a set [Formula: see text] of reactions [Formula: see text], each converting a multiset of educts [Formula: see text] into a multiset [Formula: see text] of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix [Formula: see text] have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for [Formula: see text] that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively.

RESULTS

Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix [Formula: see text]. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and-in a stricter sense-futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of "atoms". In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless [Formula: see text] implies that they are "obligatory isomers". In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN.

CONCLUSIONS

"Chemical" RNs are directed hypergraphs with a stoichiometric matrix [Formula: see text] whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain [Formula: see text] by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

Diversity begets diversity during community assembly until ecological limits impose a diversity ceiling

Microbial communities are hugely diverse, but we do not yet understand how species invasions and extinctions drive and limit their diversity. On the one hand, the ecological limits hypothesis posits that diversity is primarily limited by environmental resources. On the other hand, the diversity‐begets‐diversity hypothesis posits that such limits can be easily lifted when new ecological niches are created by biotic interactions. To find out which hypothesis better explains the assembly of microbial communities, we used metabolic modelling. We represent each microbial species by a metabolic network that harbours thousands of biochemical reactions. Together, these reactions determine which carbon and energy sources a species can use, and which metabolic by‐products—potential nutrients for other species—it can excrete in a given environment. We assemble communities by modelling thousands of species invasions in a chemostat‐like environment. We find that early during the assembly process, diversity begets diversity. By‐product excretion transforms a simple environment into one that can sustain dozens of species. During later assembly stages, the creation of new niches slows down, existing niches become filled, successful invasions become rare, and species diversity plateaus. Thus, ecological limitations dominate the late assembly process. We conclude that each hypothesis captures a different stage of the assembly process. Species interactions can raise a community's diversity ceiling dramatically, but only within limits imposed by the environment.

From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics

Understanding how genotypes map onto phenotypes, fitness, and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution. We refer to this generally as the problem of the genotype-phenotype map. Though we are still far from achieving a complete picture of these relationships, our current understanding of simpler questions, such as the structure induced in the space of genotypes by sequences mapped to molecular structures, has revealed important facts that deeply affect the dynamical description of evolutionary processes. Empirical evidence supporting the fundamental relevance of features such as phenotypic bias is mounting as well, while the synthesis of conceptual and experimental progress leads to questioning current assumptions on the nature of evolutionary dynamics-cancer progression models or synthetic biology approaches being notable examples. This work delves with a critical and constructive attitude into our current knowledge of how genotypes map onto molecular phenotypes and organismal functions, and discusses theoretical and empirical avenues to broaden and improve this comprehension. As a final goal, this community should aim at deriving an updated picture of evolutionary processes soundly relying on the structural properties of genotype spaces, as revealed by modern techniques of molecular and functional analysis.

Quantum aspects of evolution: a contribution towards evolutionary explorations of genotype networks via quantum walks

Quantum biology seeks to explain biological phenomena via quantum mechanisms, such as enzyme reaction rates via tunnelling and photosynthesis energy efficiency via coherent superposition of states. However, less effort has been devoted to study the role of quantum mechanisms in biological evolution. In this paper, we used transcription factor networks with two and four different phenotypes, and used classical random walks (CRW) and quantum walks (QW) to compare network search behaviour and efficiency at finding novel phenotypes between CRW and QW. In the network with two phenotypes, at temporal scales comparable to decoherence time TD, QW are as efficient as CRW at finding new phenotypes. In the case of the network with four phenotypes, the QW had a higher probability of mutating to a novel phenotype than the CRW, regardless of the number of mutational steps (i.e. 1, 2 or 3) away from the new phenotype. Before quantum decoherence, the QW probabilities become higher turning the QW effectively more efficient than CRW at finding novel phenotypes under different starting conditions. Thus, our results warrant further exploration of the QW under more realistic network scenarios (i.e. larger genotype networks) in both closed and open systems (e.g. by considering Lindblad terms).

Comparing Technological Development and Biological Evolution from a Network Perspective

We compare the "patterns of mutation" in biological and technological networks. Negative selection at central nodes in biological networks has been widely reported; however, we show technological networks have an opposite trend. This suggests a potential contrast: biological evolution involves random tinkering, whereas man-made systems change according to rational planning.

Syntrophy emerges spontaneously in complex metabolic systems

Syntrophy allows a microbial community as a whole to survive in an environment, even though individual microbes cannot. The metabolic interdependence typical of syntrophy is thought to arise from the accumulation of degenerative mutations during the sustained co-evolution of initially self-sufficient organisms. An alternative and underexplored possibility is that syntrophy can emerge spontaneously in communities of organisms that did not co-evolve. Here, we study this de novo origin of syntrophy using experimentally validated computational techniques to predict an organism’s viability from its metabolic reactions. We show that pairs of metabolisms that are randomly sampled from a large space of possible metabolism and viable on specific primary carbon sources often become viable on new carbon sources by exchanging metabolites. The same biochemical reactions that are required for viability on primary carbon sources also confer viability on novel carbon sources. Our observations highlight a new and important avenue for the emergence of metabolic adaptations and novel ecological interactions.

By exchanging resources, the members of a microbial community can survive in environments where individual species cannot. Despite the abundance of such syntrophy, little is known about its evolutionary origin. The predominant hypothesis is that syntrophy arises when originally independent organisms in the same community become interdependent by accumulating mutations. In this view, syntrophy arises when organisms co-evolve. In sharp contrast we find that different metabolism can interact syntrophically without a shared evolutionary history. We show that syntrophy is an inherent and emergent property of the complex chemical reaction networks that constitute metabolism.

Seeing the wood for the trees: a forest of methods for optimization and omic-network integration in metabolic modelling

Metabolic modelling has entered a mature phase with dozens of methods and software implementations available to the practitioner and the theoretician. It is not easy for a modeller to be able to see the wood (or the forest) for the trees. Driven by this analogy, we here present a 'forest' of principal methods used for constraint-based modelling in systems biology. This provides a tree-based view of methods available to prospective modellers, also available in interactive version at http://modellingmetabolism.net, where it will be kept updated with new methods after the publication of the present manuscript. Our updated classification of existing methods and tools highlights the most promising in the different branches, with the aim to develop a vision of how existing methods could hybridize and become more complex. We then provide the first hands-on tutorial for multi-objective optimization of metabolic models in R. We finally discuss the implementation of multi-view machine learning approaches in poly-omic integration. Throughout this work, we demonstrate the optimization of trade-offs between multiple metabolic objectives, with a focus on omic data integration through machine learning. We anticipate that the combination of a survey, a perspective on multi-view machine learning and a step-by-step R tutorial should be of interest for both the beginner and the advanced user.

Metabolic Determinants of Enzyme Evolution in a Genome-Scale Bacterial Metabolic Network

Different genes and proteins evolve at very different rates. To identify the factors that explain these differences is an important aspect of research in molecular evolution. One such factor is the role a protein plays in a large molecular network. Here, we analyze the evolutionary rates of enzyme-coding genes in the genome-scale metabolic network of Escherichia coli to find the evolutionary constraints imposed by the structure and function of this complex metabolic system. Central and highly connected enzymes appear to evolve more slowly than less connected enzymes, but we find that they do so as a by-product of their high abundance, and not because of their position in the metabolic network. In contrast, enzymes catalyzing reactions with high metabolic flux-high substrate to product conversion rates-evolve slowly even after we account for their abundance. Moreover, enzymes catalyzing reactions that are difficult to by-pass through alternative pathways, such that they are essential in many different genetic backgrounds, also evolve more slowly. Our analyses show that an enzyme's role in the function of a metabolic network affects its evolution more than its place in the network's structure. They highlight the value of a system-level perspective for studies of molecular evolution.

An enormous potential for niche construction through bacterial cross-feeding in a homogeneous environment

Microorganisms modify their environment by excreting by-products of metabolism, which can create new ecological niches that can help microbial populations diversify. A striking example comes from experimental evolution of genetically identical Escherichia coli populations that are grown in a homogeneous environment with the single carbon source glucose. In such experiments, stable communities of genetically diverse cross-feeding E. coli cells readily emerge. Some cells that consume the primary carbon source glucose excrete a secondary carbon source, such as acetate, that sustains other community members. Few such cross-feeding polymorphisms are known experimentally, because they are difficult to screen for. We studied the potential of bacterial metabolism to create new ecological niches based on cross-feeding. To do so, we used genome scale models of the metabolism of E. coli and metabolisms of similar complexity, to identify unique pairs of primary and secondary carbon sources in these metabolisms. We then combined dynamic flux balance analysis with analytical calculations to identify which pair of carbon sources can sustain a polymorphic cross-feeding community. We identified almost 10,000 such pairs of carbon sources, each of them corresponding to a unique ecological niche. Bacterial metabolism shows an immense potential for the construction of new ecological niches through cross feeding.

Biodiversity can emerge in a completely homogeneous environment from populations with initially genetically identical individuals. This striking observation comes from experimental evolution of bacteria, which create new ecological niches when they excrete nutrient-rich waste products that can sustain the life of other bacteria. It is difficult to estimate the potential of any one organism for such metabolic niche construction experimentally, because it is challenging to screen for novel metabolic abilities on a large scale. We therefore used experimentally validated models of bacterial metabolism to predict how many novel niches organisms like Escherichia coli can construct, if a novel niche must be able to sustain a stable community of microbes that differ in the nutrients they consume. We identify thousands of such niches. They differ in their primary carbon source and a secondary carbon source that is excreted by some microbes and used by others. Because we restricted ourselves to chemically simple environments, we may even have underestimated the enormous potential of microbes for niche construction.

Constraint and Contingency Pervade the Emergence of Novel Phenotypes in Complex Metabolic Systems

An evolutionary constraint is a bias or limitation in phenotypic variation that a biological system produces. We know examples of such constraints, but we have no systematic understanding about their extent and causes for any one biological system. We here study metabolisms, genomically encoded complex networks of enzyme-catalyzed biochemical reactions, and the constraints they experience in bringing forth novel phenotypes that allow survival on novel carbon sources. Our computational approach does not limit us to analyzing constrained variation in any one organism, but allows us to quantify constraints experienced by any metabolism. Specifically, we study metabolisms that are viable on one of 50 different carbon sources, and quantify how readily alterations of their chemical reactions create the ability to survive on a novel carbon source. We find that some metabolic phenotypes are much less likely to originate than others. For example, metabolisms viable on D-glucose are 1835 times more likely to give rise to metabolisms viable on D-fructose than on acetate. Likewise, we observe that some novel metabolic phenotypes are more contingent on parental phenotypes than others. Biochemical similarities among carbon sources can help explain the causes of these constraints. In addition, we study metabolisms that can be produced by recombination among 55 metabolisms of different bacterial strains or species, and show that their novel phenotypes are also contingent on and constrained by parental genotypes. To our knowledge, our analysis is the first to systematically quantify the incidence of constrained evolution in a broad class of biological system that is central to life and its evolution.

The genotype-phenotype map of an evolving digital organism

To understand how evolving systems bring forth novel and useful phenotypes, it is essential to understand the relationship between genotypic and phenotypic change. Artificial evolving systems can help us understand whether the genotype-phenotype maps of natural evolving systems are highly unusual, and it may help create evolvable artificial systems. Here we characterize the genotype-phenotype map of digital organisms in Avida, a platform for digital evolution. We consider digital organisms from a vast space of 10¹⁴¹ genotypes (instruction sequences), which can form 512 different phenotypes. These phenotypes are distinguished by different Boolean logic functions they can compute, as well as by the complexity of these functions. We observe several properties with parallels in natural systems, such as connected genotype networks and asymmetric phenotypic transitions. The likely common cause is robustness to genotypic change. We describe an intriguing tension between phenotypic complexity and evolvability that may have implications for biological evolution. On the one hand, genotypic change is more likely to yield novel phenotypes in more complex organisms. On the other hand, the total number of novel phenotypes reachable through genotypic change is highest for organisms with simple phenotypes. Artificial evolving systems can help us study aspects of biological evolvability that are not accessible in vastly more complex natural systems. They can also help identify properties, such as robustness, that are required for both human-designed artificial systems and synthetic biological systems to be evolvable.

The phenotype of an organism comprises the set of morphological and functional traits encoded by its genome. In natural evolving systems, phenotypes are organized into mutationally connected networks of genotypes, which increase the likelihood for an evolving population to encounter novel adaptive phenotypes (i.e., its evolvability). We do not know whether artificial systems, such as self-replicating and evolving computer programs—digital organisms—are more or less evolvable than natural systems. By studying how genotypes map onto phenotypes in digital organisms, we characterize many commonalities between natural and artificial evolving systems. In addition, we show that phenotypic complexity can both facilitate and constrain evolution, which harbors lessons not only for designing evolvable artificial systems, but also for synthetic biology.

The potential for non-adaptive origins of evolutionary innovations in central carbon metabolism

BACKGROUND

Biological systems are rife with examples of pre-adaptations or exaptations. They range from the molecular scale - lens crystallins, which originated from metabolic enzymes - to the macroscopic scale, such as feathers used in flying, which originally served thermal insulation or waterproofing. An important class of exaptations are novel and useful traits with non-adaptive origins. Whether such origins could be frequent cannot be answered with individual examples, because it is a question about a biological system's potential for exaptation. We here take a step towards answering this question by analyzing central carbon metabolism, and novel traits that allow an organism to survive on novel sources of carbon and energy. We have previously applied flux balance analysis to this system and predicted the viability of 10¹⁵ metabolic genotypes on each of ten different carbon sources.

RESULTS

We here use this exhaustive genotype-phenotype map to ask whether a central carbon metabolism that is viable on a given, focal carbon source C - the equivalent of an adaptation in our framework - is usually or rarely viable on one or more other carbon sources C _new - a potential exaptation. We show that most metabolic genotypes harbor potential exaptations, that is, they are viable on one or more carbon sources C _new . The nature and number of these carbon sources depends on the focal carbon source C itself, and on the biochemical similarity between C and C _new . Moreover, metabolisms that show a higher biomass yield on C, and that are more complex, i.e., they harbor more metabolic reactions, are viable on a greater number of carbon sources C _new .

CONCLUSIONS

A high potential for exaptation results from correlations between the phenotypes of different genotypes, and such correlations are frequent in central carbon metabolism. If they are similarly abundant in other metabolic or biological systems, innovations may frequently have non-adaptive ("exaptive") origins.

Phenotypic innovation through recombination in genome-scale metabolic networks

Recombination is an important source of metabolic innovation, especially in prokaryotes, which have evolved the ability to survive on many different sources of chemical elements and energy. Metabolic systems have a well-understood genotype-phenotype relationship, which permits a quantitative and biochemically principled understanding of how recombination creates novel phenotypes. Here, we investigate the power of recombination to create genome-scale metabolic reaction networks that enable an organism to survive in new chemical environments. To this end, we use flux balance analysis, an experimentally validated computational method that can predict metabolic phenotypes from metabolic genotypes. We show that recombination is much more likely to create novel metabolic abilities than random changes in chemical reactions of a metabolic network. We also find that phenotypic innovation is more likely when recombination occurs between parents that are genetically closely related, phenotypically highly diverse, and viable on few rather than many carbon sources. Survival on a new carbon source preferentially involves reactions that are superessential, that is, essential in many metabolic networks. We validate our observations with data from 61 reconstructed prokaryotic metabolic networks. Our systematic and quantitative analysis of metabolic systems helps understand how recombination creates innovation.

Adaptive Genetic Robustness of Escherichia coli Metabolic Fluxes

Genetic robustness refers to phenotypic invariance in the face of mutation and is a common characteristic of life, but its evolutionary origin is highly controversial. Genetic robustness could be an intrinsic property of biological systems, a result of direct natural selection, or a byproduct of selection for environmental robustness. To differentiate among these hypotheses, we analyze the metabolic network of Escherichia coli and comparable functional random networks. Treating the flux of each reaction as a trait and computationally predicting trait values upon mutations or environmental shifts, we discover that 1) genetic robustness is greater for the actual network than the random networks, 2) the genetic robustness of a trait increases with trait importance and this correlation is stronger in the actual network than in the random networks, and 3) the above result holds even after the control of environmental robustness. These findings demonstrate an adaptive origin of genetic robustness, consistent with the theoretical prediction that, under certain conditions, direct selection is sufficiently powerful to promote genetic robustness in cellular organisms.

Nonlinear Dynamics in Gene Regulation Promote Robustness and Evolvability of Gene Expression Levels

Cellular phenotypes underpinned by regulatory networks need to respond to evolutionary pressures to allow adaptation, but at the same time be robust to perturbations. This creates a conflict in which mutations affecting regulatory networks must both generate variance but also be tolerated at the phenotype level. Here, we perform mathematical analyses and simulations of regulatory networks to better understand the potential trade-off between robustness and evolvability. Examining the phenotypic effects of mutations, we find an inverse correlation between robustness and evolvability that breaks only with nonlinearity in the network dynamics, through the creation of regions presenting sudden changes in phenotype with small changes in genotype. For genotypes embedding low levels of nonlinearity, robustness and evolvability correlate negatively and almost perfectly. By contrast, genotypes embedding nonlinear dynamics allow expression levels to be robust to small perturbations, while generating high diversity (evolvability) under larger perturbations. Thus, nonlinearity breaks the robustness-evolvability trade-off in gene expression levels by allowing disparate responses to different mutations. Using analytical derivations of robustness and system sensitivity, we show that these findings extend to a large class of gene regulatory network architectures and also hold for experimentally observed parameter regimes. Further, the effect of nonlinearity on the robustness-evolvability trade-off is ensured as long as key parameters of the system display specific relations irrespective of their absolute values. We find that within this parameter regime genotypes display low and noisy expression levels. Examining the phenotypic effects of mutations, we find an inverse correlation between robustness and evolvability that breaks only with nonlinearity in the network dynamics. Our results provide a possible solution to the robustness-evolvability trade-off, suggest an explanation for the ubiquity of nonlinear dynamics in gene expression networks, and generate useful guidelines for the design of synthetic gene circuits.

Evolutionary escape on complex genotype-phenotype networks

We study the problem of evolutionary escape that is the process whereby a population under sudden changes in the selective pressures acting upon it try to evade extinction by evolving from previously well-adapted phenotypes to those that are favoured by the new selective pressure. We perform a comparative analysis between results obtained by modelling genotype space as a regular hypercube (H-graphs), which is the scenario considered in previous work on the subject, to those corresponding to a complex genotype-phenotype network (B-graphs). In order to analyse the properties of the escape process on both these graphs, we apply a general theory based on multi-type branching processes to compute the evolutionary dynamics and probability of escape. We show that the distribution of distances between phenotypes in B-graphs exhibits a much larger degree of heterogeneity than in H-graphs. This property, one of the main structural differences between both types of graphs, causes heterogeneous behaviour in all results associated to the escape problem. We further show that, due to the heterogeneity characterising escape on B-graphs, escape probability can be underestimated by assuming a regular hypercube genotype network, even if we compare phenotypes at the same distance in H-graphs. Similarly, it appears that the complex structure of B-graphs slows down the rate of escape.

Genetic Correlations Greatly Increase Mutational Robustness and Can Both Reduce and Enhance Evolvability

Mutational neighbourhoods in genotype-phenotype (GP) maps are widely believed to be more likely to share characteristics than expected from random chance. Such genetic correlations should strongly influence evolutionary dynamics. We explore and quantify these intuitions by comparing three GP maps-a model for RNA secondary structure, the HP model for protein tertiary structure, and the Polyomino model for protein quaternary structure-to a simple random null model that maintains the number of genotypes mapping to each phenotype, but assigns genotypes randomly. The mutational neighbourhood of a genotype in these GP maps is much more likely to contain genotypes mapping to the same phenotype than in the random null model. Such neutral correlations can be quantified by the robustness to mutations, which can be many orders of magnitude larger than that of the null model, and crucially, above the critical threshold for the formation of large neutral networks of mutationally connected genotypes which enhance the capacity for the exploration of phenotypic novelty. Thus neutral correlations increase evolvability. We also study non-neutral correlations: Compared to the null model, i) If a particular (non-neutral) phenotype is found once in the 1-mutation neighbourhood of a genotype, then the chance of finding that phenotype multiple times in this neighbourhood is larger than expected; ii) If two genotypes are connected by a single neutral mutation, then their respective non-neutral 1-mutation neighbourhoods are more likely to be similar; iii) If a genotype maps to a folding or self-assembling phenotype, then its non-neutral neighbours are less likely to be a potentially deleterious non-folding or non-assembling phenotype. Non-neutral correlations of type i) and ii) reduce the rate at which new phenotypes can be found by neutral exploration, and so may diminish evolvability, while non-neutral correlations of type iii) may instead facilitate evolutionary exploration and so increase evolvability.

Resource constrained flux balance analysis predicts selective pressure on the global structure of metabolic networks

Background

A universal feature of metabolic networks is their hourglass or bow-tie structure on cellular level. This architecture reflects the conversion of multiple input nutrients into multiple biomass components via a small set of precursor metabolites. However, it is yet unclear to what extent this structural feature is the result of natural selection.

Results

We extend flux balance analysis to account for limited cellular resources. Using this model, optimal structure of metabolic networks can be calculated for different environmental conditions. We observe a significant structural reshaping of metabolic networks for a toy-network and E. coli core metabolism if we increase the share of invested resources for switching between different nutrient conditions. Here, hub nodes emerge and the optimal network structure becomes bow-tie-like as a consequence of limited cellular resource constraint. We confirm this theoretical finding by comparing the reconstructed metabolic networks of bacterial species with respect to their lifestyle.

Conclusions

We show that bow-tie structure can give a system-level fitness advantage to organisms that live in highly competitive and fluctuating environments. Here, limitation of cellular resources can lead to an efficiency-flexibility tradeoff where it pays off for the organism to shorten catabolic pathways if they are frequently activated and deactivated. As a consequence, generalists that shuttle between diverse environmental conditions should have a more predominant bow-tie structure than specialists that visit just a few isomorphic habitats during their life cycle.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0232-5) contains supplementary material, which is available to authorized users.

Mechanisms of mutational robustness in transcriptional regulation

Robustness is the invariance of a phenotype in the face of environmental or genetic change. The phenotypes produced by transcriptional regulatory circuits are gene expression patterns that are to some extent robust to mutations. Here we review several causes of this robustness. They include robustness of individual transcription factor binding sites, homotypic clusters of such sites, redundant enhancers, transcription factors, redundant transcription factors, and the wiring of transcriptional regulatory circuits. Such robustness can either be an adaptation by itself, a byproduct of other adaptations, or the result of biophysical principles and non-adaptive forces of genome evolution. The potential consequences of such robustness include complex regulatory network topologies that arise through neutral evolution, as well as cryptic variation, i.e., genotypic divergence without phenotypic divergence. On the longest evolutionary timescales, the robustness of transcriptional regulation has helped shape life as we know it, by facilitating evolutionary innovations that helped organisms such as flowering plants and vertebrates diversify.

Exhaustive Analysis of a Genotype Space Comprising 10(15 )Central Carbon Metabolisms Reveals an Organization Conducive to Metabolic Innovation

All biological evolution takes place in a space of possible genotypes and their phenotypes. The structure of this space defines the evolutionary potential and limitations of an evolving system. Metabolism is one of the most ancient and fundamental evolving systems, sustaining life by extracting energy from extracellular nutrients. Here we study metabolism’s potential for innovation by analyzing an exhaustive genotype-phenotype map for a space of 10¹⁵ metabolisms that encodes all possible subsets of 51 reactions in central carbon metabolism. Using flux balance analysis, we predict the viability of these metabolisms on 10 different carbon sources which give rise to 1024 potential metabolic phenotypes. Although viable metabolisms with any one phenotype comprise a tiny fraction of genotype space, their absolute numbers exceed 10⁹ for some phenotypes. Metabolisms with any one phenotype typically form a single network of genotypes that extends far or all the way through metabolic genotype space, where any two genotypes can be reached from each other through a series of single reaction changes. The minimal distance of genotype networks associated with different phenotypes is small, such that one can reach metabolisms with novel phenotypes – viable on new carbon sources – through one or few genotypic changes. Exceptions to these principles exist for those metabolisms whose complexity (number of reactions) is close to the minimum needed for viability. Increasing metabolic complexity enhances the potential for both evolutionary conservation and evolutionary innovation.

Genotype-phenotype mapping is one of the ultimate goals of computational systems biology, and can provide new insights into the function and evolution of biological systems. We present a comprehensive genotype-phenotype map for a space of metabolic genotypes that comprises more than 10¹⁵ central carbon metabolisms. Only one in a million of these metabolisms can sustain life on any one of 10 carbon sources we consider, but these viable metabolisms form connected genotype networks that extend far through genotype space. In addition, they render multiple novel metabolic phenotypes in their immediate neighborhood accessible through small evolutionary changes that require only the alteration of single metabolic reactions. The map we construct reveals an organization of core metabolism that simultaneously facilitates evolutionary conservation of existing metabolic phenotypes, and the origination of novel metabolic traits that allow viability on novel carbon sources. Such metabolic innovation is essential, particularly for organisms that experience unexpected environmental changes, and that explore or invade new habitats.

Phenotypic constraints promote latent versatility and carbon efficiency in metabolic networks

System-level properties of metabolic networks may be the direct product of natural selection or arise as a by-product of selection on other properties. Here we study the effect of direct selective pressure for growth or viability in particular environments on two properties of metabolic networks: latent versatility to function in additional environments and carbon usage efficiency. Using a Markov chain Monte Carlo (MCMC) sampling based on flux balance analysis (FBA), we sample from a known biochemical universe random viable metabolic networks that differ in the number of directly constrained environments. We find that the latent versatility of sampled metabolic networks increases with the number of directly constrained environments and with the size of the networks. We then show that the average carbon wastage of sampled metabolic networks across the constrained environments decreases with the number of directly constrained environments and with the size of the networks. Our work expands the growing body of evidence about nonadaptive origins of key functional properties of biological networks.

Surviving evolutionary escape on complex genotype-phenotype networks

We study the problem of evolutionary escape and survival of cell populations with a genotype-phenotype structure. We refer to evolutionary escape as the process where a cell of a given ill-adapted population to reach a well-adapted phenotype. Similarly, survival refers to the dynamics of the population once the escape phenotype has been reached. The aim of this paper is to analyse the influence of topological properties associated to robustness and evolvability on the probability of escape and on the probability of survival. In order to explore these issues, we formulate a population dynamics model, consisting of a multi-type time-continuous branching process, where types are associated to genotypes and their birth and death probabilities depend on the associated phenotype (non-escape or escape). We exploit the separation of time scales introduced by the the difference in reproductive ratios between the ill-adapted phenotypes and the escape phenotype. Two dynamical regimes emerge: a fast-decaying regime associated to the escape process itself, and a slow regime which corresponds to the survival dynamics of the population once the escape phenotype has been reached. We exploit this separation of time scales to analyse the topological factors which determine escape and survival probabilities. We show that, while the escape probability depends on the degree of escape phenotype, the probability of survival is essentially determined by its robustness, measured in terms of a weighted clustering coefficient.

Human genome variation and the concept of genotype networks

Genotype networks are a concept used in systems biology to study sets of genotypes having the same phenotype, and the ability of these to bring forth novel phenotypes. In the past they have been applied to determine the genetic heterogeneity, and stability to mutations, of systems such as metabolic networks and RNA folds. Recently, they have been the base for reconciling the neutralist and selectionist views on evolution. Here, we adapted this concept to the study of population genetics data. Specifically, we applied genotype networks to the human 1000 genomes dataset, and analyzed networks composed of short haplotypes of Single Nucleotide Variants (SNV). The result is a scan of how properties related to genetic heterogeneity and stability to mutations are distributed along the human genome. We found that genes involved in acquired immunity, such as some HLA and MHC genes, tend to have the most heterogeneous and connected networks, and that coding regions tend to be more heterogeneous and stable to mutations than non-coding regions. We also found, using coalescent simulations, that regions under selection have more extended and connected networks. The application of the concept of genotype networks can provide a new opportunity to understand the evolutionary processes that shaped our genome. Learning how the genotype space of each region of our genome has been explored during the evolutionary history of the human species can lead to a better understanding on how selective pressures and neutral factors have shaped genetic diversity within populations and among individuals. Combined with the availability of larger datasets of sequencing data, genotype networks represent a new approach to the study of human genetic diversity that looks to the whole genome, and goes beyond the classical division between selection and neutrality methods.

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.

The topology of robustness and evolvability in evolutionary systems with genotype-phenotype map

In this paper we formulate a topological definition of the concepts of robustness and evolvability. We start our investigation by formulating a multiscale model of the evolutionary dynamics of a population of cells. Our cells are characterised by a genotype-phenotype map: their chances of survival under selective pressure are determined by their phenotypes, whereas the latter are determined their genotypes. According to our multiscale dynamics, the population dynamics generates the evolution of a genotype-phenotype network. Our representation of the genotype-phenotype network is similar to previously described ones, but has a novel element, namely, our network contains two types of nodes: genotype and phenotype nodes. This network representation allows us to characterise robustness and evolvability in terms of its topological properties: phenotypic robustness by means of the clustering coefficient of the phenotype nodes, and evolvability as the emergence of giant connected component which allows navigation between phenotypes. This topological definition of evolvability allows us to characterise the so-called robustness of evolvability, which is defined in terms of the robustness against attack (i.e. edge removal) of the giant connected component. An investigation of the factors that affect the robustness of evolvability shows that phenotypic robustness and the cryptic genetic variation are key to the integrity of the ability to innovate. These results fit within the framework of a number of models which point out that robustness favours rather than hindering evolvability. We further show that the corresponding phenotype network, defined as the one-component projection of the whole genotype-phenotype network, exhibits the small-world phenomenon, which implies that in this type of evolutionary system the rate of adaptability is enhanced.

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.

A latent capacity for evolutionary innovation through exaptation in metabolic systems

Some evolutionary innovations may originate non-adaptively as exaptations, or pre-adaptations, which are by-products of other adaptive traits. Examples include feathers, which originated before they were used in flight, and lens crystallins, which are light-refracting proteins that originated as enzymes. The question of how often adaptive traits have non-adaptive origins has profound implications for evolutionary biology, but is difficult to address systematically. Here we consider this issue in metabolism, one of the most ancient biological systems that is central to all life. We analyse a metabolic trait of great adaptive importance: the ability of a metabolic reaction network to synthesize all biomass from a single source of carbon and energy. We use novel computational methods to sample randomly many metabolic networks that can sustain life on any given carbon source but contain an otherwise random set of known biochemical reactions. We show that when we require such networks to be viable on one particular carbon source, they are typically also viable on multiple other carbon sources that were not targets of selection. For example, viability on glucose may entail viability on up to 44 other sole carbon sources. Any one adaptation in these metabolic systems typically entails multiple potential exaptations. Metabolic systems thus contain a latent potential for evolutionary innovations with non-adaptive origins. Our observations suggest that many more metabolic traits may have non-adaptive origins than is appreciated at present. They also challenge our ability to distinguish adaptive from non-adaptive traits.

Design constraints on a synthetic metabolism

A metabolism is a complex network of chemical reactions that converts sources of energy and chemical elements into biomass and other molecules. To design a metabolism from scratch and to implement it in a synthetic genome is almost within technological reach. Ideally, a synthetic metabolism should be able to synthesize a desired spectrum of molecules at a high rate, from multiple different nutrients, while using few chemical reactions, and producing little or no waste. Not all of these properties are achievable simultaneously. We here use a recently developed technique to create random metabolic networks with pre-specified properties to quantify trade-offs between these and other properties. We find that for every additional molecule to be synthesized a network needs on average three additional reactions. For every additional carbon source to be utilized, it needs on average two additional reactions. Networks able to synthesize 20 biomass molecules from each of 20 alternative sole carbon sources need to have at least 260 reactions. This number increases to 518 reactions for networks that can synthesize more than 60 molecules from each of 80 carbon sources. The maximally achievable rate of biosynthesis decreases by approximately 5 percent for every additional molecule to be synthesized. Biochemically related molecules can be synthesized at higher rates, because their synthesis produces less waste. Overall, the variables we study can explain 87 percent of variation in network size and 84 percent of the variation in synthesis rate. The constraints we identify prescribe broad boundary conditions that can help to guide synthetic metabolism design.

Superessential reactions in metabolic networks

The metabolic genotype of an organism can change through loss and acquisition of enzyme-coding genes, while preserving its ability to survive and synthesize biomass in specific environments. This evolutionary plasticity allows pathogens to evolve resistance to antimetabolic drugs by acquiring new metabolic pathways that bypass an enzyme blocked by a drug. We here study quantitatively the extent to which individual metabolic reactions and enzymes can be bypassed. To this end, we use a recently developed computational approach to create large metabolic network ensembles that can synthesize all biomass components in a given environment but contain an otherwise random set of known biochemical reactions. Using this approach, we identify a small connected core of 124 reactions that are absolutely superessential (that is, required in all metabolic networks). Many of these reactions have been experimentally confirmed as essential in different organisms. We also report a superessentiality index for thousands of reactions. This index indicates how easily a reaction can be bypassed. We find that it correlates with the number of sequenced genomes that encode an enzyme for the reaction. Superessentiality can help choose an enzyme as a potential drug target, especially because the index is not highly sensitive to the chemical environment that a pathogen requires. Our work also shows how analyses of large network ensembles can help understand the evolution of complex and robust metabolic networks.

Towards understanding how molecular networks evolve in plants

Residing beneath the phenotypic landscape of a plant are intricate and dynamic networks of genes and proteins. As evolution operates on phenotypes, we expect its forces to shape somehow these underlying molecular networks. In this review, we discuss progress being made to elucidate the nature of these forces and their impact on the composition and structure of molecular networks. We also outline current limitations and open questions facing the broader field of plant network analysis.

The role of robustness in phenotypic adaptation and innovation

Phenotypes that vary in response to DNA mutations are essential for evolutionary adaptation and innovation. Therefore, it seems that robustness, a lack of phenotypic variability, must hinder adaptation. The main purpose of this review is to show why this is not necessarily correct. There are two reasons. The first is that robustness causes the existence of genotype networks--large connected sets of genotypes with the same phenotype. I discuss why genotype networks facilitate phenotypic variability. The second reason emerges from the evolutionary dynamics of evolving populations on genotype networks. I discuss how these dynamics can render highly robust phenotypes more variable, using examples from protein and RNA macromolecules. In addition, robustness can help avoid an important evolutionary conflict between the interests of individuals and populations-a conflict that can impede evolutionary adaptation.

Metabolic networks and their evolution

Since the last decade of the twentieth century, systems biology has gained the ability to study the structure and function of genome-scale metabolic networks. These are systems of hundreds to thousands of chemical reactions that sustain life. Most of these reactions are catalyzed by enzymes which are encoded by genes. A metabolic network extracts chemical elements and energy from the environment, and converts them into forms that the organism can use. The function of a whole metabolic network constrains evolutionary changes in its parts. I will discuss here three classes of such changes, and how they are constrained by the function of the whole. These are the accumulation of amino acid changes in enzyme-coding genes, duplication of enzyme-coding genes, and changes in the regulation of enzymes. Conversely, evolutionary change in network parts can alter the function of the whole network. I will discuss here two such changes, namely the elimination of reactions from a metabolic network through loss of function mutations in enzyme-coding genes, and the addition of metabolic reactions, for example through mechanisms such as horizontal gene transfer. Reaction addition also provides a window into the evolution of metabolic innovations, the ability of a metabolism to sustain life on new sources of energy and of chemical elements.

The molecular origins of evolutionary innovations

The history of life is a history of evolutionary innovations, qualitatively new phenotypic traits that endow their bearers with new, often game-changing abilities. We know many individual examples of innovations and their natural history, but we know little about the fundamental principles of phenotypic variability that permit new phenotypes to arise. Most phenotypic innovations result from changes in three classes of systems: metabolic networks, regulatory circuits, and macromolecules. I here highlight two important features that these classes of systems share. The first is the ubiquity of vast genotype networks - connected sets of genotypes with the same phenotype. The second is the great phenotypic diversity of small neighborhoods around different genotypes in genotype space. I here explain that both features are essential for the phenotypic variability that can bring forth qualitatively new phenotypes. Both features emerge from a common cause, the robustness of phenotypes to perturbations, whose origins are linked to life in changing environments.

Environmental versatility promotes modularity in genome-scale metabolic networks

BACKGROUND

The ubiquity of modules in biological networks may result from an evolutionary benefit of a modular organization. For instance, modularity may increase the rate of adaptive evolution, because modules can be easily combined into new arrangements that may benefit their carrier. Conversely, modularity may emerge as a by-product of some trait. We here ask whether this last scenario may play a role in genome-scale metabolic networks that need to sustain life in one or more chemical environments. For such networks, we define a network module as a maximal set of reactions that are fully coupled, i.e., whose fluxes can only vary in fixed proportions. This definition overcomes limitations of purely graph based analyses of metabolism by exploiting the functional links between reactions. We call a metabolic network viable in a given chemical environment if it can synthesize all of an organism's biomass compounds from nutrients in this environment. An organism's metabolism is highly versatile if it can sustain life in many different chemical environments. We here ask whether versatility affects the modularity of metabolic networks.

RESULTS

Using recently developed techniques to randomly sample large numbers of viable metabolic networks from a vast space of metabolic networks, we use flux balance analysis to study in silico metabolic networks that differ in their versatility. We find that highly versatile networks are also highly modular. They contain more modules and more reactions that are organized into modules. Most or all reactions in a module are associated with the same biochemical pathways. Modules that arise in highly versatile networks generally involve reactions that process nutrients or closely related chemicals. We also observe that the metabolism of E. coli is significantly more modular than even our most versatile networks.

CONCLUSIONS

Our work shows that modularity in metabolic networks can be a by-product of functional constraints, e.g., the need to sustain life in multiple environments. This organizational principle is insensitive to the environments we consider and to the number of reactions in a metabolic network. Because we observe this principle not just in one or few biological networks, but in large random samples of networks, we propose that it may be a generic principle of metabolic network organization.

Randomizing genome-scale metabolic networks

Networks coming from protein-protein interactions, transcriptional regulation, signaling, or metabolism may appear to have "unusual" properties. To quantify this, it is appropriate to randomize the network and test the hypothesis that the network is not statistically different from expected in a motivated ensemble. However, when dealing with metabolic networks, the randomization of the network using edge exchange generates fictitious reactions that are biochemically meaningless. Here we provide several natural ensembles of randomized metabolic networks. A first constraint is to use valid biochemical reactions. Further constraints correspond to imposing appropriate functional constraints. We explain how to perform these randomizations with the help of Markov Chain Monte Carlo (MCMC) and show that they allow one to approach the properties of biological metabolic networks. The implication of the present work is that the observed global structural properties of real metabolic networks are likely to be the consequence of simple biochemical and functional constraints.

The low cost of recombination in creating novel phenotypes: Recombination can create new phenotypes while disrupting well-adapted phenotypes much less than mutation

Recombination is often considered a disruptive force for well-adapted phenotypes, but recent evidence suggests that this cost of recombination can be small. A key benefit of recombination is that it can help create proteins and regulatory circuits with novel and useful phenotypes more efficiently than point mutation. Its effectiveness stems from the large-scale reorganization of genotypes that it causes, which can help explore far-flung regions in genotype space. Recent work on complex phenotypes in model gene regulatory circuits and proteins shows that the disruptive effects of recombination can be very mild compared to the effects of mutation. Recombination thus can have great benefits at a modest cost, but we do not understand the reasons well. A better understanding might shed light on the evolution of recombination and help improve evolutionary strategies in biochemical engineering.

Genotype networks, innovation, and robustness in sulfur metabolism

Background

A metabolism is a complex network of chemical reactions. This network synthesizes multiple small precursor molecules of biomass from chemicals that occur in the environment. The metabolic network of any one organism is encoded by a metabolic genotype, defined as the set of enzyme-coding genes whose products catalyze the network's reactions. Each metabolic genotype has a metabolic phenotype. We define this metabolic phenotype as the spectrum of different sources of a chemical element that a metabolism can use to synthesize biomass. We here focus on the element sulfur. We study properties of the space of all possible metabolic genotypes in sulfur metabolism by analyzing random metabolic genotypes that are viable on different numbers of sulfur sources.

Results

We show that metabolic genotypes with the same phenotype form large connected genotype networks - networks of metabolic networks - that extend far through metabolic genotype space. How far they reach through this space depends linearly on the number of super-essential reactions. A super-essential reaction is an essential reaction that occurs in all networks viable in a given environment. Metabolic networks can differ in how robust their phenotype is to the removal of individual reactions. We find that this robustness depends on metabolic network size, and on other variables, such as the size of minimal metabolic networks whose reactions are all essential in a specific environment. We show that different neighborhoods of any genotype network harbor very different novel phenotypes, metabolic innovations that can sustain life on novel sulfur sources. We also analyze the ability of evolving populations of metabolic networks to explore novel metabolic phenotypes. This ability is facilitated by the existence of genotype networks, because different neighborhoods of these networks contain very different novel phenotypes.

Conclusions

We show that the space of metabolic genotypes involved in sulfur metabolism is organized similarly to that of carbon metabolism. We demonstrate that the maximum genotype distance and robustness of metabolic networks can be explained by the number of superessential reactions and by the sizes of minimal metabolic networks viable in an environment. In contrast to the genotype space of macromolecules, where phenotypic robustness may facilitate phenotypic innovation, we show that here the ability to access novel phenotypes does not monotonically increase with robustness.

In silico evolution of early metabolism

We developed a simulation tool for investigating the evolution of early metabolism, allowing us to speculate on the formation of metabolic pathways from catalyzed chemical reactions and on the development of their characteristic properties. Our model consists of a protocellular entity with a simple RNA-based genetic system and an evolving metabolism of catalytically active ribozymes that manipulate a rich underlying chemistry. Ensuring an almost open-ended and fairly realistic simulation is crucial for understanding the first steps in metabolic evolution. We show here how our simulation tool can be helpful in arguing for or against hypotheses on the evolution of metabolic pathways. We demonstrate that seemingly mutually exclusive hypotheses may well be compatible when we take into account that different processes dominate different phases in the evolution of a metabolic system. Our results suggest that forward evolution shapes metabolic network in the very early steps of evolution. In later and more complex stages, enzyme recruitment supersedes forward evolution, keeping a core set of pathways from the early phase.

Challenges in experimental data integration within genome-scale metabolic models

A report of the meeting "Challenges in experimental data integration within genome-scale metabolic models", Institut Henri Poincaré, Paris, October 10-11 2009, organized by the CNRS-MPG joint program in Systems Biology.