Guy Shinar

Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space

Managing Trade-Offs Most organisms experience selection on a host of traits to determine their likelihood to succeed evolutionarily. However, specific traits may experience trade-offs in determining an organisms optimal phenotype. Shoval et al. (p. 1157; see the Perspective by Noor and Milo) relate physical traits to the task that they are optimizing using a Pareto curve, a power law probability distribution, to show that a single set of trait values optimizes performance at a given task and that performance decreases as an organisms phenotype moves away from this set of trait values. The results suggest how selection makes the best trade-offs for an arbitrary number of tasks and traits and may explain examples of evolutionary variation. The fitness of an organism can be modeled graphically to determine how phenotypic trade-offs are maximized. Biological systems that perform multiple tasks face a fundamental trade-off: A given phenotype cannot be optimal at all tasks. Here we ask how trade-offs affect the range of phenotypes found in nature. Using the Pareto front concept from economics and engineering, we find that best–trade-off phenotypes are weighted averages of archetypes—phenotypes specialized for single tasks. For two tasks, phenotypes fall on the line connecting the two archetypes, which could explain linear trait correlations, allometric relationships, as well as bacterial gene-expression patterns. For three tasks, phenotypes fall within a triangle in phenotype space, whose vertices are the archetypes, as evident in morphological studies, including on Darwin’s finches. Tasks can be inferred from measured phenotypes based on the behavior of organisms nearest the archetypes.

Structural Sources of Robustness in Biochemical Reaction Networks

Steady As She Blows A fundamental characteristic of many biological control networks is the capacity to maintain the concentration of a particular component at steady state within a narrow range, in spite of variations in the amounts of other network components that might change as a result of environmental variables in the state of a cell. In a mathematical analysis, Shinar and Feinberg (p. 1389) reveal the essential requirements of a network robust to perturbation. Using this method, the sources of robustness in two bacterial systems—one that functions in osmoregulation and another that controls carbon flux in metabolism—were explained. Models of metabolic regulation show how the stability of specific components is maintained within a varying environment. In vivo variations in the concentrations of biomolecular species are inevitable. These variations in turn propagate along networks of chemical reactions and modify the concentrations of still other species, which influence biological activity. Because excessive variations in the amounts of certain active species might hamper cell function, regulation systems have evolved that act to maintain concentrations within tight bounds. We identify simple yet subtle structural attributes that impart concentration robustness to any mass-action network possessing them. We thereby describe a large class of robustness-inducing networks that already embraces two quite different biochemical modules for which concentration robustness has been observed experimentally: the Escherichia coli osmoregulation system EnvZ-OmpR and the glyoxylate bypass control system isocitrate dehydrogenase kinase-phosphatase–isocitrate dehydrogenase. The structural attributes identified here might confer robustness far more broadly.

Input–output robustness in simple bacterial signaling systems

Biological signaling systems produce an output, such as the level of a phosphorylated protein, in response to defined input signals. The output level as a function of the input level is called the systems input–output relation. One may ask whether this input–output relation is sensitive to changes in the concentrations of the systems components, such as proteins and ATP. Because component concentrations often vary from cell to cell, it might be expected that the input–output relation will likewise vary. If this is the case, different cells exposed to the same input signal will display different outputs. Such variability can be deleterious in systems where survival depends on accurate match of output to input. Here we suggest a mechanism that can provide input–output robustness, that is, an input–output relation that does not depend on variations in the concentrations of any of the systems components. The mechanism is based on certain bacterial signaling systems. It explains how specific molecular details can work together to provide robustness. Moreover, it suggests an approach that can help identify a wide family of nonequilibrium mechanisms that potentially have robust input–output relations.

Concordant chemical reaction networks.

We describe a large class of chemical reaction networks, those endowed with a subtle structural property called concordance. We show that the class of concordant networks coincides precisely with the class of networks which, when taken with any weakly monotonic kinetics, invariably give rise to kinetic systems that are injective - a quality that, among other things, precludes the possibility of switch-like transitions between distinct positive steady states. We also provide persistence characteristics of concordant networks, instability implications of discordance, and consequences of stronger variants of concordance. Some of our results are in the spirit of recent ones by Banaji and Craciun, but here we do not require that every species suffer a degradation reaction. This is especially important in studying biochemical networks, for which it is rare to have all species degrade.

Robustness in glyoxylate bypass regulation.

The glyoxylate bypass allows Escherichia coli to grow on carbon sources with only two carbons by bypassing the loss of carbons as CO2 in the tricarboxylic acid cycle. The flux toward this bypass is regulated by the phosphorylation of the enzyme isocitrate dehydrogenase (IDH) by a bifunctional kinase–phosphatase called IDHKP. In this system, IDH activity has been found to be remarkably robust with respect to wide variations in the total IDH protein concentration. Here, we examine possible mechanisms to explain this robustness. Explanations in which IDHKP works simultaneously as a first-order kinase and as a zero-order phosphatase with a single IDH binding site are found to be inconsistent with robustness. Instead, we suggest a robust mechanism where both substrates bind the bifunctional enzyme to form a ternary complex.

Rules for biological regulation based on error minimization

The control of gene expression involves complex mechanisms that show large variation in design. For example, genes can be turned on either by the binding of an activator (positive control) or the unbinding of a repressor (negative control). What determines the choice of mode of control for each gene? This study proposes rules for gene regulation based on the assumption that free regulatory sites are exposed to nonspecific binding errors, whereas sites bound to their cognate regulators are protected from errors. Hence, the selected mechanisms keep the sites bound to their designated regulators for most of the time, thus minimizing fitness-reducing errors. This offers an explanation of the empirically demonstrated Savageau demand rule: Genes that are needed often in the natural environment tend to be regulated by activators, and rarely needed genes tend to be regulated by repressors; in both cases, sites are bound for most of the time, and errors are minimized. The fitness advantage of error minimization appears to be readily selectable. The present approach can also generate rules for multi-regulator systems. The error-minimization framework raises several experimentally testable hypotheses. It may also apply to other biological regulation systems, such as those involving protein-protein interactions.

Sensitivity and Robustness in Chemical Reaction Networks

For a wide class of chemical reaction networks, including all those governed by detailed balanced mass-action kinetics, we examine the robustness of equilibrium species concentrations against fluctuations in the overall reactant supply. In particular, we present lower bounds on the individual species-concentration sensitivities that derive from reaction network structure alone, independent of kinetic parameters or even of the particular equilibrium state at which sensitivities are calculated. These bounds suggest that, in the class of reaction networks considered here, very high robustness (i.e., very low sensitivities) should be expected only when the various molecules are constructed from a large number of distinct elemental building blocks that appear in high multiplicity or that combine gregariously. This situation is often encountered in biology.

Design principles for robust biochemical reaction networks: What works, what cannot work, and what might almost work

We bring together recent results that connect the structure of a mass-action reaction network to its capacity for concentration robustness - that is, its capacity to keep the concentration of a critical bio-active species within narrow limits, even against large fluctuations in the overall supply of the networks constituents.

Concordant chemical reaction networks and the Species-Reaction Graph.

In a recent paper it was shown that, for chemical reaction networks possessing a subtle structural property called concordance, dynamical behavior of a very circumscribed (and largely stable) kind is enforced, so long as the kinetics lies within the very broad and natural weakly monotonic class. In particular, multiple equilibria are precluded, as are degenerate positive equilibria. Moreover, under certain circumstances, also related to concordance, all real eigenvalues associated with a positive equilibrium are negative. Although concordance of a reaction network can be decided by readily available computational means, we show here that, when a nondegenerate networks Species-Reaction Graph satisfies certain mild conditions, concordance and its dynamical consequences are ensured. These conditions are weaker than earlier ones invoked to establish kinetic system injectivity, which, in turn, is just one ramification of network concordance. Because the Species-Reaction Graph resembles pathway depictions often drawn by biochemists, results here expand the possibility of inferring significant dynamical information directly from standard biochemical reaction diagrams.

Evolutionary Balancing of Fitness-Limiting Factors

Debates concerning the roles of different factors that may limit an organism’s reproductive success pervade evolutionary ecology. We suggest that a broad class of limiting‐factors problems involving essential resources or essential components of reproductive effort can be analyzed with an evolutionary application of Liebig’s law of the minimum. We explore life‐history evolution using the metaphor of an organism that must harvest two essential resources (resources 1 and 2) from a stochastically varying environment. Our models make three predictions. First, organisms should overinvest, relative to the deterministic case, in harvesting the resource whose per‐offspring harvest cost is smaller. Second, at the optimum, organisms balance multiple fitness‐limiting factors rather than being consistently limited by one factor. Third, the optimal investment in harvesting a resource is directly linked to the probability that the organism’s fitness will be limited by that resource. Under temporal variation, the optimal proportional investment in harvesting resource 1 is equal to the probability that resource 1 will limit fitness. Our results help to explain why the responses of populations to environmental perturbations are hard to predict: as an organism transitions between different limiting factors, its responses to perturbations of those factors will likewise change.