Relationship between fitness and heterogeneity in exponentially growing microbial populations.

Despite major environmental and genetic differences, microbial metabolic networks are known to generate consistent physiological outcomes across vastly different organisms. This remarkable robustness suggests that, at least in bacteria, metabolic activity may be guided by universal principles. The constrained optimization of evolutionarily motivated objective functions, such as the growth rate, has emerged as the key theoretical assumption for the study of bacterial metabolism. While conceptually and practically useful in many situations, the idea that certain functions are optimized is hard to validate in data. Moreover, it is not always clear how optimality can be reconciled with the high degree of single-cell variability observed in experiments within microbial populations. To shed light on these issues, we develop an inverse modeling framework that connects the fitness of a population of cells (represented by the mean single-cell growth rate) to the underlying metabolic variability through the maximum entropy inference of the distribution of metabolic phenotypes from data. While no clear objective function emerges, we find that, as the medium gets richer, the fitness and inferred variability for Escherichia coli populations follow and slowly approach the theoretically optimal bound defined by minimal reduction of variability at given fitness. These results suggest that bacterial metabolism may be crucially shaped by a population-level trade-off between growth and heterogeneity.

A scalable algorithm to explore the Gibbs energy landscape of genome-scale metabolic networks.

The integration of various types of genomic data into predictive models of biological networks is one of the main challenges currently faced by computational biology. Constraint-based models in particular play a key role in the attempt to obtain a quantitative understanding of cellular metabolism at genome scale. In essence, their goal is to frame the metabolic capabilities of an organism based on minimal assumptions that describe the steady states of the underlying reaction network via suitable stoichiometric constraints, specifically mass balance and energy balance (i.e. thermodynamic feasibility). The implementation of these requirements to generate viable configurations of reaction fluxes and/or to test given flux profiles for thermodynamic feasibility can however prove to be computationally intensive. We propose here a fast and scalable stoichiometry-based method to explore the Gibbs energy landscape of a biochemical network at steady state. The method is applied to the problem of reconstructing the Gibbs energy landscape underlying metabolic activity in the human red blood cell, and to that of identifying and removing thermodynamically infeasible reaction cycles in the Escherichia coli metabolic network (iAF1260). In the former case, we produce consistent predictions for chemical potentials (or log-concentrations) of intracellular metabolites; in the latter, we identify a restricted set of loops (23 in total) in the periplasmic and cytoplasmic core as the origin of thermodynamic infeasibility in a large sample (10(6)) of flux configurations generated randomly and compatibly with the prior information available on reaction reversibility.

Collective catalysis under spatial constraints: Phase separation and size-scaling effects on mass action kinetics.

Chemical reactions are usually studied under the assumption that both substrates and catalysts are well-mixed (WM) throughout the system. Although this is often applicable to test-tube experimental conditions, it is not realistic in cellular environments, where biomolecules can undergo liquid-liquid phase separation (LLPS) and form condensates, leading to important functional outcomes, including the modulation of catalytic action. Similar processes may also play a role in protocellular systems, like primitive coacervates, or in membrane-assisted prebiotic pathways. Here we explore whether the demixing of catalysts could lead to the formation of microenvironments that influence the kinetics of a linear (multistep) reaction pathway, as compared to a WM system. We implemented a general lattice model to simulate LLPS of a collection of different catalysts and extended it to include diffusion and a sequence of reactions of small substrates. We carried out a quantitative analysis of how the phase separation of the catalysts affects reaction times depending on the affinity between substrates and catalysts, the length of the reaction pathway, the system size, and the degree of homogeneity of the condensate. A key aspect underlying the differences reported between the two scenarios is that the scale invariance observed in the WM system is broken by condensation processes. The main theoretical implications of our results for mean-field chemistry are drawn, extending the mass action kinetics scheme to include substrate initial "hitting times" to reach the catalysts condensate. We finally test this approach by considering open nonlinear conditions, where we successfully predict, through microscopic simulations, that phase separation inhibits chemical oscillatory behavior, providing a possible explanation for the marginal role that this complex dynamic behavior plays in real metabolisms.

Emergence of collective self-oscillations in minimal lattice models with feedback.

The emergence of collective oscillations and synchronization is a widespread phenomenon in complex systems. While widely studied in the setting of dynamical systems, this phenomenon is not well understood in the context of out-of-equilibrium phase transitions in many-body systems. Here we consider three classical lattice models, namely the Ising, the Blume-Capel, and the Potts models, provided with a feedback among the order and control parameters. With the help of the linear response theory we derive low-dimensional nonlinear dynamical systems for mean-field cases. These dynamical systems quantitatively reproduce many-body stochastic simulations. In general, we find that the usual equilibrium phase transitions are taken over by more complex bifurcations where nonlinear collective self-oscillations emerge, a behavior that we illustrate by the feedback Landau theory. For the case of the Ising model, we obtain that the bifurcation that takes over the critical point is nontrivial in finite dimensions. Namely, we provide numerical evidence that in two dimensions the most probable value of a cycle's amplitude follows the Onsager law for slow feedback. We illustrate multistability for the case of discontinuously emerging oscillations in the Blume-Capel model, whose tricritical point is substituted by the Bautin bifurcation. For the Potts model with q=3 colors we highlight the appearance of two mirror stable limit cycles at a bifurcation line and characterize the onset of chaotic oscillations that emerge at low temperature through either the Feigenbaum cascade of period doubling or the Afraimovich-Shilnikov scenario of a torus destruction. We also demonstrate that entropy production singularities as a function of the temperature correspond to qualitative change in the spectrum of Lyapunov exponents. Our results show that mean-field collective behavior can be described by the bifurcation theory of low-dimensional dynamical systems, which paves the way for the definition of universality classes of collective oscillations.

Statistical modeling of adaptive neural networks explains co-existence of avalanches and oscillations in resting human brain.

Neurons in the brain are wired into adaptive networks that exhibit collective dynamics as diverse as scale-specific oscillations and scale-free neuronal avalanches. Although existing models account for oscillations and avalanches separately, they typically do not explain both phenomena, are too complex to analyze analytically or intractable to infer from data rigorously. Here we propose a feedback-driven Ising-like class of neural networks that captures avalanches and oscillations simultaneously and quantitatively. In the simplest yet fully microscopic model version, we can analytically compute the phase diagram and make direct contact with human brain resting-state activity recordings via tractable inference of the model's two essential parameters. The inferred model quantitatively captures the dynamics over a broad range of scales, from single sensor oscillations to collective behaviors of extreme events and neuronal avalanches. Importantly, the inferred parameters indicate that the co-existence of scale-specific (oscillations) and scale-free (avalanches) dynamics occurs close to a non-equilibrium critical point at the onset of self-sustained oscillations.

Statistical mechanics of biomolecular condensates via cavity methods.

Physical mechanisms of phase separation in living systems play key physiological roles and have recently been the focus of intensive studies. The strongly heterogeneous nature of such phenomena poses difficult modeling challenges that require going beyond mean-field approaches based on postulating a free energy landscape. The pathway we take here is to calculate the partition function starting from microscopic interactions by means of cavity methods, based on a tree approximation for the interaction graph. We illustrate them on the binary case and then apply them successfully to ternary systems, in which simpler one-factor approximations are proved inadequate. We demonstrate the agreement with lattice simulations and contrast our theory with coacervation experiments of associative de-mixing of nucleotides and poly-lysine. Different types of evidence are provided to support cavity methods as ideal tools for modeling biomolecular condensation, giving an optimal balance between the consideration of spatial aspects and fast computational results.

Probing Single-Cell Fermentation Fluxes and Exchange Networks via pH-Sensing Hybrid Nanofibers.

The homeostatic control of their environment is an essential task of living cells. It has been hypothesized that, when microenvironmental pH inhomogeneities are induced by high cellular metabolic activity, diffusing protons act as signaling molecules, driving the establishment of exchange networks sustained by the cell-to-cell shuttling of overflow products such as lactate. Despite their fundamental role, the extent and dynamics of such networks is largely unknown due to the lack of methods in single-cell flux analysis. In this study, we provide direct experimental characterization of such exchange networks. We devise a method to quantify single-cell fermentation fluxes over time by integrating high-resolution pH microenvironment sensing via ratiometric nanofibers with constraint-based inverse modeling. We apply our method to cell cultures with mixed populations of cancer cells and fibroblasts. We find that the proton trafficking underlying bulk acidification is strongly heterogeneous, with maximal single-cell fluxes exceeding typical values by up to 3 orders of magnitude. In addition, a crossover in time from a networked phase sustained by densely connected "hubs" (corresponding to cells with high activity) to a sparse phase dominated by isolated dipolar motifs (i.e., by pairwise cell-to-cell exchanges) is uncovered, which parallels the time course of bulk acidification. Our method addresses issues ranging from the homeostatic function of proton exchange to the metabolic coupling of cells with different energetic demands, allowing for real-time noninvasive single-cell metabolic flux analysis.

Dynamic facilitation picture of a higher-order glass singularity.

We show that facilitated spin mixtures with a tunable facilitation reproduce, on a Bethe lattice, the simplest higher-order singularity scenario predicted by the mode-coupling theory (MCT) of liquid-glass transition. Depending on the facilitation strength, they yield either a discontinuous glass transition or a continuous one, with no underlying thermodynamic singularity. Similar results are obtained for facilitated spin models on a diluted Bethe lattice. The mechanism of dynamical arrest in these systems can be interpreted in terms of bootstrap and standard percolation and corresponds to a crossover from a compact to a fractal structure of the incipient spanning cluster of frozen spins. Theoretical and numerical simulation results are fully consistent with MCT predictions.

Congestion phenomena on complex networks.

We define a minimal model of traffic flows in complex networks in order to study the trade-off between topological-based and traffic-based routing strategies. The resulting collective behavior is obtained analytically for an ensemble of uncorrelated networks and summarized in a rich phase diagram presenting second-order as well as first-order phase transitions between a free-flow phase and a congested phase. We find that traffic control improves global performance, enlarging the free-flow region in parameter space only in heterogeneous networks. Traffic control introduces nonlinear effects and, beyond a critical strength, may trigger the appearance of a congested phase in a discontinuous manner. The model also reproduces the crossover in the scaling of traffic fluctuations empirically observed on the Internet.

Oscillations in feedback-driven systems: Thermodynamics and noise.

Oscillations in nonequilibrium noisy systems are important physical phenomena. These oscillations can happen in autonomous biochemical oscillators such as circadian clocks. They can also manifest as subharmonic oscillations in periodically driven systems such as time crystals. Oscillations in autonomous systems and, to a lesser degree, subharmonic oscillations in periodically driven systems have been both thoroughly investigated, including their relation with thermodynamic cost and noise. We perform a systematic study of oscillations in a third class of nonequilibrium systems: feedback-driven systems. In particular, we use the apparatus of stochastic thermodynamics to investigate the role of noise and thermodynamic cost in feedback-driven oscillations. For a simple two-state model that displays oscillations, we analyze the relation between precision and dissipation, revealing that oscillations can remain coherent for an indefinite time in a finite system with thermal fluctuations in a limit of diverging thermodynamic cost. We consider oscillations in a more complex system with several degrees of freedom, an Ising model driven by feedback between the magnetization and the external field. This feedback-driven system can display subharmonic oscillations similar to the ones observed in time crystals. We illustrate the second law for feedback-driven systems that display oscillations. For the Ising model, the oscillating dissipated heat can be negative. However, when we consider the total entropy that also includes an informational term related to measurements, the oscillating total entropy change is always positive. We also study the finite-size scaling of the dissipated heat, providing evidence for the existence of a first-order phase transition for certain parameter regimes.

An introduction to the maximum entropy approach and its application to inference problems in biology.

A cornerstone of statistical inference, the maximum entropy framework is being increasingly applied to construct descriptive and predictive models of biological systems, especially complex biological networks, from large experimental data sets. Both its broad applicability and the success it obtained in different contexts hinge upon its conceptual simplicity and mathematical soundness. Here we try to concisely review the basic elements of the maximum entropy principle, starting from the notion of 'entropy', and describe its usefulness for the analysis of biological systems. As examples, we focus specifically on the problem of reconstructing gene interaction networks from expression data and on recent work attempting to expand our system-level understanding of bacterial metabolism. Finally, we highlight some extensions and potential limitations of the maximum entropy approach, and point to more recent developments that are likely to play a key role in the upcoming challenges of extracting structures and information from increasingly rich, high-throughput biological data.

Statistical mechanics for metabolic networks during steady state growth.

Which properties of metabolic networks can be derived solely from stoichiometry? Predictive results have been obtained by flux balance analysis (FBA), by postulating that cells set metabolic fluxes to maximize growth rate. Here we consider a generalization of FBA to single-cell level using maximum entropy modeling, which we extend and test experimentally. Specifically, we define for Escherichia coli metabolism a flux distribution that yields the experimental growth rate: the model, containing FBA as a limit, provides a better match to measured fluxes and it makes a wide range of predictions: on flux variability, regulation, and correlations; on the relative importance of stoichiometry vs. optimization; on scaling relations for growth rate distributions. We validate the latter here with single-cell data at different sub-inhibitory antibiotic concentrations. The model quantifies growth optimization as emerging from the interplay of competitive dynamics in the population and regulation of metabolism at the level of single cells.

Maximum entropy modeling of metabolic networks by constraining growth-rate moments predicts coexistence of phenotypes.

In this work maximum entropy distributions in the space of steady states of metabolic networks are considered upon constraining the first and second moments of the growth rate. Coexistence of fast and slow phenotypes, with bimodal flux distributions, emerges upon considering control on the average growth (optimization) and its fluctuations (heterogeneity). This is applied to the carbon catabolic core of Escherichia coli where it quantifies the metabolic activity of slow growing phenotypes and it provides a quantitative map with metabolic fluxes, opening the possibility to detect coexistence from flux data. A preliminary analysis on data for E. coli cultures in standard conditions shows degeneracy for the inferred parameters that extend in the coexistence region.

Scales and multimodal flux distributions in stationary metabolic network models via thermodynamics.

In this work it is shown that scale-free tails in metabolic flux distributions inferred in stationary models are an artifact due to reactions involved in thermodynamically unfeasible cycles, unbounded by physical constraints and in principle able to perform work without expenditure of free energy. After implementing thermodynamic constraints by removing such loops, metabolic flux distributions scale meaningfully with the physical limiting factors, acquiring in turn a richer multimodal structure potentially leading to symmetry breaking while optimizing for objective functions.

Quantifying the entropic cost of cellular growth control.

Viewing the ways a living cell can organize its metabolism as the phase space of a physical system, regulation can be seen as the ability to reduce the entropy of that space by selecting specific cellular configurations that are, in some sense, optimal. Here we quantify the amount of regulation required to control a cell's growth rate by a maximum-entropy approach to the space of underlying metabolic phenotypes, where a configuration corresponds to a metabolic flux pattern as described by genome-scale models. We link the mean growth rate achieved by a population of cells to the minimal amount of metabolic regulation needed to achieve it through a phase diagram that highlights how growth suppression can be as costly (in regulatory terms) as growth enhancement. Moreover, we provide an interpretation of the inverse temperature ß controlling maximum-entropy distributions based on the underlying growth dynamics. Specifically, we show that the asymptotic value of ß for a cell population can be expected to depend on (i) the carrying capacity of the environment, (ii) the initial size of the colony, and (iii) the probability distribution from which the inoculum was sampled. Results obtained for E. coli and human cells are found to be remarkably consistent with empirical evidence.

Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding.

The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales. In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the Lovazs ellipsoid method (LEM). Our results show that a rounding procedure dramatically improves the performances of the HR in these inference problems and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.

Quantitative constraint-based computational model of tumor-to-stroma coupling via lactate shuttle.

Cancer cells utilize large amounts of ATP to sustain growth, relying primarily on non-oxidative, fermentative pathways for its production. In many types of cancers this leads, even in the presence of oxygen, to the secretion of carbon equivalents (usually in the form of lactate) in the cell's surroundings, a feature known as the Warburg effect. While the molecular basis of this phenomenon are still to be elucidated, it is clear that the spilling of energy resources contributes to creating a peculiar microenvironment for tumors, possibly characterized by a degree of toxicity. This suggests that mechanisms for recycling the fermentation products (e.g. a lactate shuttle) may be active, effectively inducing a mutually beneficial metabolic coupling between aberrant and non-aberrant cells. Here we analyze this scenario through a large-scale in silico metabolic model of interacting human cells. By going beyond the cell-autonomous description, we show that elementary physico-chemical constraints indeed favor the establishment of such a coupling under very broad conditions. The characterization we obtained by tuning the aberrant cell's demand for ATP, amino-acids and fatty acids and/or the imbalance in nutrient partitioning provides quantitative support to the idea that synergistic multi-cell effects play a central role in cancer sustainment.

Identifying all moiety conservation laws in genome-scale metabolic networks.

The stoichiometry of a metabolic network gives rise to a set of conservation laws for the aggregate level of specific pools of metabolites, which, on one hand, pose dynamical constraints that cross-link the variations of metabolite concentrations and, on the other, provide key insight into a cell's metabolic production capabilities. When the conserved quantity identifies with a chemical moiety, extracting all such conservation laws from the stoichiometry amounts to finding all non-negative integer solutions of a linear system, a programming problem known to be NP-hard. We present an efficient strategy to compute the complete set of integer conservation laws of a genome-scale stoichiometric matrix, also providing a certificate for correctness and maximality of the solution. Our method is deployed for the analysis of moiety conservation relationships in two large-scale reconstructions of the metabolism of the bacterium E. coli, in six tissue-specific human metabolic networks, and, finally, in the human reactome as a whole, revealing that bacterial metabolism could be evolutionarily designed to cover broader production spectra than human metabolism. Convergence to the full set of moiety conservation laws in each case is achieved in extremely reduced computing times. In addition, we uncover a scaling relation that links the size of the independent pool basis to the number of metabolites, for which we present an analytical explanation.

Thermodynamics of biochemical networks and duality theorems.

One interesting yet difficult computational issue has recently been posed in biophysics in regard to the implementation of thermodynamic constraints to complex networks. Biochemical networks of enzymes inside cells are among the most efficient, robust, differentiated, and flexible free-energy transducers in nature. How is the second law of thermodynamics encoded for these complex networks? In this article it is demonstrated that for chemical reaction networks in the steady state the exclusion (presence) of closed reaction cycles makes possible (impossible) the definition of a chemical potential vector. Interestingly, this statement is encoded in one of the key results in combinatorial optimization, i.e., the Gordan theorem of the alternatives. From a computational viewpoint, the theorem reveals that calculating a reaction's free energy and identifying infeasible loops in flux states are dual problems whose solutions are mutually exclusive, and this opens the way for efficient and scalable methods to perform the energy balance analysis of large-scale biochemical networks.

Counting and correcting thermodynamically infeasible flux cycles in genome-scale metabolic networks.

Thermodynamics constrains the flow of matter in a reaction network to occur through routes along which the Gibbs energy decreases, implying that viable steady-state flux patterns should be void of closed reaction cycles. Identifying and removing cycles in large reaction networks can unfortunately be a highly challenging task from a computational viewpoint. We propose here a method that accomplishes it by combining a relaxation algorithm and a Monte Carlo procedure to detect loops, with ad hoc rules (discussed in detail) to eliminate them. As test cases, we tackle (a) the problem of identifying infeasible cycles in the E. coli metabolic network and (b) the problem of correcting thermodynamic infeasibilities in the Flux-Balance-Analysis solutions for 15 human cell-type-specific metabolic networks. Results for (a) are compared with previous analyses of the same issue, while results for (b) are weighed against alternative methods to retrieve thermodynamically viable flux patterns based on minimizing specific global quantities. Our method, on the one hand, outperforms previous techniques and, on the other, corrects loopy solutions to Flux Balance Analysis. As a byproduct, it also turns out to be able to reveal possible inconsistencies in model reconstructions.