Effective bet-hedging through growth rate dependent stability.

Microbes in the wild face highly variable and unpredictable environments and are naturally selected for their average growth rate across environments. Apart from using sensory regulatory systems to adapt in a targeted manner to changing environments, microbes employ bet-hedging strategies where cells in an isogenic population switch stochastically between alternative phenotypes. Yet, bet-hedging suffers from a fundamental trade-off: Increasing the phenotype-switching rate increases the rate at which maladapted cells explore alternative phenotypes but also increases the rate at which cells switch out of a well-adapted state. Consequently, it is currently believed that bet-hedging strategies are effective only when the number of possible phenotypes is limited and when environments last for sufficiently many generations. However, recent experimental results show that gene expression noise generally decreases with growth rate, suggesting that phenotype-switching rates may systematically decrease with growth rate. Such growth rate dependent stability (GRDS) causes cells to be more explorative when maladapted and more phenotypically stable when well-adapted, and we show that GRDS can almost completely overcome the trade-off that limits bet-hedging, allowing for effective adaptation even when environments are diverse and change rapidly. We further show that even a small decrease in switching rates of faster-growing phenotypes can substantially increase long-term fitness of bet-hedging strategies. Together, our results suggest that stochastic strategies may play an even bigger role for microbial adaptation than hitherto appreciated.

ecmtool: fast and memory-efficient enumeration of elementary conversion modes.

MOTIVATION: Characterizing all steady-state flux distributions in metabolic models remains limited to small models due to the explosion of possibilities. Often it is sufficient to look only at all possible overall conversions a cell can catalyze ignoring the details of intracellular metabolism. Such a characterization is achieved by elementary conversion modes (ECMs), which can be conveniently computed with ecmtool. However, currently, ecmtool is memory intensive, and it cannot be aided appreciably by parallelization. RESULTS: We integrate mplrs-a scalable parallel vertex enumeration method-into ecmtool. This speeds up computation, drastically reduces memory requirements and enables ecmtool's use in standard and high-performance computing environments. We show the new capabilities by enumerating all feasible ECMs of the near-complete metabolic model of the minimal cell JCVI-syn3.0. Despite the cell's minimal character, the model gives rise to 4.2×109 ECMs and still contains several redundant sub-networks. AVAILABILITY AND IMPLEMENTATION: ecmtool is available at https://github.com/SystemsBioinformatics/ecmtool. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Metabolic control of nitrogen fixation in rhizobium-legume symbioses.

Rhizobia induce nodule formation on legume roots and differentiate into bacteroids, which catabolize plant-derived dicarboxylates to reduce atmospheric N2 into ammonia. Despite the agricultural importance of this symbiosis, the mechanisms that govern carbon and nitrogen allocation in bacteroids and promote ammonia secretion to the plant are largely unknown. Using a metabolic model derived from genome-scale datasets, we show that carbon polymer synthesis and alanine secretion by bacteroids facilitate redox balance in microaerobic nodules. Catabolism of dicarboxylates induces not only a higher oxygen demand but also a higher NADH/NAD+ ratio than sugars. Modeling and 13C metabolic flux analysis indicate that oxygen limitation restricts the decarboxylating arm of the tricarboxylic acid cycle, which limits ammonia assimilation into glutamate. By tightly controlling oxygen supply and providing dicarboxylates as the energy and electron source donors for N2 fixation, legumes promote ammonia secretion by bacteroids. This is a defining feature of rhizobium-legume symbioses.

Population dynamics of microbial cross-feeding are determined by co-localization probabilities and cooperation-independent cheater growth.

As natural selection acts on individual organisms the evolution of costly cooperation between microorganisms is an intriguing phenomenon. Introduction of spatial structure to privatize exchanged molecules can explain the evolution of cooperation. However, in many natural systems cells can also grow to low cell concentrations in the absence of these exchanged molecules, thus showing "cooperation-independent background growth". We here serially propagated a synthetic cross-feeding consortium of lactococci in the droplets of a water-in-oil emulsion, essentially mimicking group selection with varying founder population sizes. The results show that when the growth of cheaters completely depends on cooperators, cooperators outcompete cheaters. However, cheaters outcompete cooperators when they can independently grow to only ten percent of the consortium carrying capacity. This result is the consequence of a probabilistic effect, as low founder population sizes in droplets decrease the frequency of cooperator co-localization. Cooperator-enrichment can be recovered by increasing the founder population size in droplets to intermediate values. Together with mathematical modelling our results suggest that co-localization probabilities in a spatially structured environment leave a small window of opportunity for the evolution of cooperation between organisms that do not benefit from their cooperative trait when in isolation or form multispecies aggregates.

Understanding FBA Solutions under Multiple Nutrient Limitations.

Genome-scale stoichiometric modeling methods, in particular Flux Balance Analysis (FBA) and variations thereof, are widely used to investigate cell metabolism and to optimize biotechnological processes. Given (1) a metabolic network, which can be reconstructed from an organism's genome sequence, and (2) constraints on reaction rates, which may be based on measured nutrient uptake rates, FBA predicts which reactions maximize an objective flux, usually the production of cell components. Although FBA solutions may accurately predict the metabolic behavior of a cell, the actual flux predictions are often hard to interpret. This is especially the case for conditions with many constraints, such as for organisms growing in rich nutrient environments: it remains unclear why a certain solution was optimal. Here, we rationalize FBA solutions by explaining for which properties the optimal combination of metabolic strategies is selected. We provide a graphical formalism in which the selection of solutions can be visualized; we illustrate how this perspective provides a glimpse of the logic that underlies genome-scale modeling by applying our formalism to models of various sizes.

Unlocking Elementary Conversion Modes: ecmtool Unveils All Capabilities of Metabolic Networks.

The metabolic capabilities of cells determine their biotechnological potential, fitness in ecosystems, pathogenic threat levels, and function in multicellular organisms. Their comprehensive experimental characterization is generally not feasible, particularly for unculturable organisms. In principle, the full range of metabolic capabilities can be computed from an organism's annotated genome using metabolic network reconstruction. However, current computational methods cannot deal with genome-scale metabolic networks. Part of the problem is that these methods aim to enumerate all metabolic pathways, while computation of all (elementally balanced) conversions between nutrients and products would suffice. Indeed, the elementary conversion modes (ECMs, defined by Urbanczik and Wagner) capture the full metabolic capabilities of a network, but the use of ECMs has not been accessible until now. We explain and extend the theory of ECMs, implement their enumeration in ecmtool, and illustrate their applicability. This work contributes to the elucidation of the full metabolic footprint of any cell.

Elementary Growth Modes provide a molecular description of cellular self-fabrication.

In this paper we try to describe all possible molecular states (phenotypes) for a cell that fabricates itself at a constant rate, given its enzyme kinetics and the stoichiometry of all reactions. For this, we must understand the process of cellular growth: steady-state self-fabrication requires a cell to synthesize all of its components, including metabolites, enzymes and ribosomes, in proportions that match its own composition. Simultaneously, the concentrations of these components affect the rates of metabolism and biosynthesis, and hence the growth rate. We here derive a theory that describes all phenotypes that solve this circular problem. All phenotypes can be described as a combination of minimal building blocks, which we call Elementary Growth Modes (EGMs). EGMs can be used as the theoretical basis for all models that explicitly model self-fabrication, such as the currently popular Metabolism and Expression models. We then use our theory to make concrete biological predictions. We find that natural selection for maximal growth rate drives microorganisms to states of minimal phenotypic complexity: only one EGM will be active when growth rate is maximised. The phenotype of a cell is only extended with one more EGM whenever growth becomes limited by an additional biophysical constraint, such as a limited solvent capacity of a cellular compartment. The theory presented here extends recent results on Elementary Flux Modes: the minimal building blocks of cellular growth models that lack the self-fabrication aspect. Our theory starts from basic biochemical and evolutionary considerations, and describes unicellular life, both in growth-promoting and in stress-inducing environments, in terms of EGMs.

The common message of constraint-based optimization approaches: overflow metabolism is caused by two growth-limiting constraints.

Living cells can express different metabolic pathways that support growth. The criteria that determine which pathways are selected in which environment remain unclear. One recurrent selection is overflow metabolism: the simultaneous usage of an ATP-efficient and -inefficient pathway, shown for example in Escherichia coli, Saccharomyces cerevisiae and cancer cells. Many models, based on different assumptions, can reproduce this observation. Therefore, they provide no conclusive evidence which mechanism is causing overflow metabolism. We compare the mathematical structure of these models. Although ranging from flux balance analyses to self-fabricating metabolism and expression models, we can rewrite all models into one standard form. We conclude that all models predict overflow metabolism when two, model-specific, growth-limiting constraints are hit. This is consistent with recent theory. Thus, identifying these two constraints is essential for understanding overflow metabolism. We list all imposed constraints by these models, so that they can hopefully be tested in future experiments.

The number of active metabolic pathways is bounded by the number of cellular constraints at maximal metabolic rates.

Growth rate is a near-universal selective pressure across microbial species. High growth rates require hundreds of metabolic enzymes, each with different nonlinear kinetics, to be precisely tuned within the bounds set by physicochemical constraints. Yet, the metabolic behaviour of many species is characterized by simple relations between growth rate, enzyme expression levels and metabolic rates. We asked if this simplicity could be the outcome of optimisation by evolution. Indeed, when the growth rate is maximized-in a static environment under mass-conservation and enzyme expression constraints-we prove mathematically that the resulting optimal metabolic flux distribution is described by a limited number of subnetworks, known as Elementary Flux Modes (EFMs). We show that, because EFMs are the minimal subnetworks leading to growth, a small active number automatically leads to the simple relations that are measured. We find that the maximal number of flux-carrying EFMs is determined only by the number of imposed constraints on enzyme expression, not by the size, kinetics or topology of the network. This minimal-EFM extremum principle is illustrated in a graphical framework, which explains qualitative changes in microbial behaviours, such as overflow metabolism and co-consumption, and provides a method for identification of the enzyme expression constraints that limit growth under the prevalent conditions. The extremum principle applies to all microorganisms that are selected for maximal growth rates under protein concentration constraints, for example the solvent capacities of cytosol, membrane or periplasmic space.