Statistical mechanics for metabolic networks during steady state growth

Information Processing in Biochemical Networks

Living systems are characterized by controlled flows of matter, energy, and information. While the biophysics community has productively engaged with the first two, addressing information flows has been more challenging, with some scattered success in evolutionary theory and a more coherent track record in neuroscience. Nevertheless, interdisciplinary work of the past two decades at the interface of biophysics, quantitative biology, and engineering has led to an emerging mathematical language for describing information flows at the molecular scale. This is where the central processes of life unfold: from detection and transduction of environmental signals to the readout or copying of genetic information and the triggering of adaptive cellular responses. Such processes are coordinated by complex biochemical reaction networks that operate at room temperature, are out of equilibrium, and use low copy numbers of diverse molecular species with limited interaction specificity. Here we review how flows of information through biochemical networks can be formalized using information-theoretic quantities, quantified from data, and computed within various modeling frameworks. Optimization of information flows is presented as a candidate design principle that navigates the relevant time, energy, crosstalk, and metabolic constraints to predict reliable cellular signaling and gene regulation architectures built of individually noisy components.

Predicting the Rate Structure of an Evolved Metabolic Network

Background: When glucose molecules are metabolized by a biological cell, the molecules are constrained to flow along distinct reaction trajectories, which are defined by the cell's underlying metabolic network. Methods: Using the computational technique of Elementary Mode Analysis, the entire set of all possible trajectories can be enumerated, effectively allowing metabolism to be viewed in a discretized space. Results: With the resulting set of Elementary Flux Modes (EMs), macroscopic fluxes, (of both mass and energy) that cross the cell envelope can be computed by a simple, linear combination of the individual EM trajectories. The challenge in this approach is that the usage probability of each EM is unknown. But, because the analytical framework we have adopted allows metabolism to be viewed in a discrete space, we can use the mathematics of statistical thermodynamics to derive the usage probabilities when the system entropy is maximized. The resulting probabilities, which obey a Boltzmann-type distribution, predict a rate structure for the metabolic network that is in remarkable agreement with experimentally measured rates of adaptively evolved E. coli strains. Conclusions: Thus, in principle, the intracellular dynamic properties of such bacteria can be predicted, using only the knowledge of the DNA sequence, to reconstruct the metabolic reaction network, and the measurement of the specific glucose uptake rate.

Metabolic Objectives and Trade-Offs: Inference and Applications

Background/Objectives: Determining appropriate cellular objectives is crucial for the system-scale modeling of biological networks for metabolic engineering, cellular reprogramming, and drug discovery applications. The mathematical representation of metabolic objectives can describe how cells manage limited resources to achieve biological goals within mechanistic and environmental constraints. While rapidly proliferating cells like tumors are often assumed to prioritize biomass production, mammalian cell types can exhibit objectives beyond growth, such as supporting tissue functions, developmental processes, and redox homeostasis. Methods: This review addresses the challenge of determining metabolic objectives and trade-offs from multiomics data. Results: Recent advances in single-cell omics, metabolic modeling, and machine/deep learning methods have enabled the inference of cellular objectives at both the transcriptomic and metabolic levels, bridging gene expression patterns with metabolic phenotypes. Conclusions: These in silico models provide insights into how cells adapt to changing environments, drug treatments, and genetic manipulations. We further explore the potential application of incorporating cellular objectives into personalized medicine, drug discovery, tissue engineering, and systems biology.

A multi-tissue metabolome atlas of primate pregnancy

Pregnancy induces dramatic metabolic changes in females; yet, the intricacies of this metabolic reprogramming remain poorly understood, especially in primates. Using cynomolgus monkeys, we constructed a comprehensive multi-tissue metabolome atlas, analyzing 273 samples from 23 maternal tissues during pregnancy. We discovered a decline in metabolic coupling between tissues as pregnancy progressed. Core metabolic pathways that were rewired during primate pregnancy included steroidogenesis, fatty acid metabolism, and arachidonic acid metabolism. Our atlas revealed 91 pregnancy-adaptive metabolites changing consistently across 23 tissues, whose roles we verified in human cell models and patient samples. Corticosterone and palmitoyl-carnitine regulated placental maturation and maternal tissue progenitors, respectively, with implications for maternal preeclampsia, diabetes, cardiac hypertrophy, and muscle and liver regeneration. Moreover, we found that corticosterone deficiency induced preeclampsia-like inflammation, indicating the atlas's potential clinical value. Overall, our multi-tissue metabolome atlas serves as a framework for elucidating the role of metabolic regulation in female health during pregnancy.

Control Theory and Systems Biology: Potential Applications in Neurodegeneration and Search for Therapeutic Targets

Control theory, a well-established discipline in engineering and mathematics, has found novel applications in systems biology. This interdisciplinary approach leverages the principles of feedback control and regulation to gain insights into the complex dynamics of cellular and molecular networks underlying chronic diseases, including neurodegeneration. By modeling and analyzing these intricate systems, control theory provides a framework to understand the pathophysiology and identify potential therapeutic targets. Therefore, this review examines the most widely used control methods in conjunction with genomic-scale metabolic models in the steady state of the multi-omics type. According to our research, this approach involves integrating experimental data, mathematical modeling, and computational analyses to simulate and control complex biological systems. In this review, we find that the most significant application of this methodology is associated with cancer, leaving a lack of knowledge in neurodegenerative models. However, this methodology, mainly associated with the Minimal Dominant Set (MDS), has provided a starting point for identifying therapeutic targets for drug development and personalized treatment strategies, paving the way for more effective therapies.

A Markov constraint to uniquely identify elementary flux mode weights in unimolecular metabolic networks

Elementary flux modes (EFMs) are minimal, steady state pathways characterizing a flux network. Fundamentally, all steady state fluxes in a network are decomposable into a linear combination of EFMs. While there is typically no unique set of EFM weights that reconstructs these fluxes, several optimization-based methods have been proposed to constrain the solution space by enforcing some notion of parsimony. However, it has long been recognized that optimization-based approaches may fail to uniquely identify EFM weights and return different feasible solutions across objective functions and solvers. Here we show that, for flux networks only involving single molecule transformations, these problems can be avoided by imposing a Markovian constraint on EFM weights. Our Markovian constraint guarantees a unique solution to the flux decomposition problem, and that solution is arguably more biophysically plausible than other solutions. We describe an algorithm for computing Markovian EFM weights via steady state analysis of a certain discrete-time Markov chain, based on the flux network, which we call the cycle-history Markov chain. We demonstrate our method with a differential analysis of EFM activity in a lipid metabolic network comparing healthy and Alzheimer's disease patients. Our method is the first to uniquely decompose steady state fluxes into EFM weights for any unimolecular metabolic network.

The interplay between metabolic stochasticity and cAMP-CRP regulation in single E. coli cells

The inherent stochasticity of metabolism raises a critical question for understanding homeostasis: are cellular processes regulated in response to internal fluctuations? Here, we show that, in E. coli cells under constant external conditions, catabolic enzyme expression continuously responds to metabolic fluctuations. The underlying regulatory feedback is enabled by the cyclic AMP (cAMP) and cAMP receptor protein (CRP) system, which controls catabolic enzyme expression based on metabolite concentrations. Using single-cell microscopy, genetic constructs in which this feedback is disabled, and mathematical modeling, we show how fluctuations circulate through the metabolic and genetic network at sub-cell-cycle timescales. Modeling identifies four noise propagation modes, including one specific to CRP regulation. Together, these modes correctly predict noise circulation at perturbed cAMP levels. The cAMP-CRP system may thus have evolved to control internal metabolic fluctuations in addition to external growth conditions. We conjecture that second messengers may more broadly function to achieve cellular homeostasis.

Optimal metabolic strategies for microbial growth in stationary random environments

In order to grow in any given environment, bacteria need to collect information about the medium composition and implement suitable growth strategies by adjusting their regulatory and metabolic degrees of freedom. In the standard sense, optimal strategy selection is achieved when bacteria grow at the fastest rate possible in that medium. While this view of optimality is well suited for cells that have perfect knowledge about their surroundings (e.g. nutrient levels), things are more involved in uncertain or fluctuating conditions, especially when changes occur over timescales comparable to (or faster than) those required to organize a response. Information theory however provides recipes for how cells can choose the optimal growth strategy under uncertainty about the stress levels they will face. Here we analyse the theoretically optimal scenarios for a coarse-grained, experiment-inspired model of bacterial metabolism for growth in a medium described by the (static) probability density of a single variable (the 'stress level'). We show that heterogeneity in growth rates consistently emerges as the optimal response when the environment is sufficiently complex and/or when perfect adjustment of metabolic degrees of freedom is not possible (e.g. due to limited resources). In addition, outcomes close to those achievable with unlimited resources are often attained effectively with a modest amount of fine tuning. In other terms, heterogeneous population structures in complex media may be rather robust with respect to the resources available to probe the environment and adjust reaction rates.

Phenotype-specific estimation of metabolic fluxes using gene expression data

A cell's genome influences its metabolism via the expression of enzyme-related genes, but transcriptome and fluxome are not perfectly correlated as post-transcriptional mechanisms also regulate reaction's kinetics. Here, we addressed the question: given a transcriptome, how unobserved mechanisms of reaction kinetics should be systematically accounted for when inferring the fluxome? To infer the most likely and least biased fluxome, we present Pheflux, a constraint-based model maximizing Shannon's entropy of fluxes per mRNA. Benchmarked against 13C fluxes of yeast and bacteria, Pheflux accurately estimates the carbon core metabolism. We applied Pheflux to thousands of normal and tumor cell transcriptomes obtained from The Cancer Genome Atlas. Pheflux showed statistically significantly higher glucose yields on lactate in breast, kidney, and bronchus-lung tumoral cells than their normal counterparts. Results are consistent with the Warburg effect, a hallmark of cancer metabolism, suggesting that Pheflux can be efficiently used to study the metabolism of eukaryotic cells.

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.

Metabolic models of human gut microbiota: Advances and challenges

The human gut is a complex ecosystem consisting of hundreds of microbial species interacting with each other and with the human host. Mathematical models of the gut microbiome integrate our knowledge of this system and help to formulate hypotheses to explain observations. The generalized Lotka-Volterra model has been widely used for this purpose, but it does not describe interaction mechanisms and thus does not account for metabolic flexibility. Recently, models that explicitly describe gut microbial metabolite production and consumption have become popular. These models have been used to investigate the factors that shape gut microbial composition and to link specific gut microorganisms to changes in metabolite concentrations found in diseases. Here, we review how such models are built and what we have learned so far from their application to human gut microbiome data. In addition, we discuss current challenges of these models and how these can be addressed in the future.

How to Tackle Underdeterminacy in Metabolic Flux Analysis? A Tutorial and Critical Review

Metabolic flux analysis is often (not to say almost always) faced with system underdeterminacy. Indeed, the linear algebraic system formed by the steady-state mass balance equations around the intracellular metabolites and the equality constraints related to the measurements of extracellular fluxes do not define a unique solution for the distribution of intracellular fluxes, but instead a set of solutions belonging to a convex polytope. Various methods have been proposed to tackle this underdeterminacy, including flux pathway analysis, flux balance analysis, flux variability analysis and sampling. These approaches are reviewed in this article and a toy example supports the discussion with illustrative numerical results.

"Minimal metabolism": A key concept to investigate the origins and nature of biological systems

The systems view on life and its emergence from complex chemistry has remarkably increased the scientific attention on metabolism in the last two decades. However, during this time there has not been much theoretical discussion on what constitutes a metabolism and what role it actually played in biogenesis. A critical and updated review on the topic is here offered, including some references to classical models from last century, but focusing more on current and future research. Metabolism is considered as intrinsically related to the living but not necessarily equivalent to it. More precisely, the idea of "minimal metabolism", in contrast to previous, top-down conceptions, is formulated as a heuristic construct, halfway between chemistry and biology. Thus, rather than providing a complete or final characterization of metabolism, our aim is to encourage further investigations on it, particularly in the context of life's origin, for which some concrete methodological suggestions are provided. Also see the video abstract here: https://youtu.be/DP7VMKk2qpA.

Learning the distribution of single-cell chromosome conformations in bacteria reveals emergent order across genomic scales

The order and variability of bacterial chromosome organization, contained within the distribution of chromosome conformations, are unclear. Here, we develop a fully data-driven maximum entropy approach to extract single-cell 3D chromosome conformations from Hi-C experiments on the model organism Caulobacter crescentus. The predictive power of our model is validated by independent experiments. We find that on large genomic scales, organizational features are predominantly present along the long cell axis: chromosomal loci exhibit striking long-ranged two-point axial correlations, indicating emergent order. This organization is associated with large genomic clusters we term Super Domains (SuDs), whose existence we support with super-resolution microscopy. On smaller genomic scales, our model reveals chromosome extensions that correlate with transcriptional and loop extrusion activity. Finally, we quantify the information contained in chromosome organization that may guide cellular processes. Our approach can be extended to other species, providing a general strategy to resolve variability in single-cell chromosomal organization.

Addressing uncertainty in genome-scale metabolic model reconstruction and analysis

The reconstruction and analysis of genome-scale metabolic models constitutes a powerful systems biology approach, with applications ranging from basic understanding of genotype-phenotype mapping to solving biomedical and environmental problems. However, the biological insight obtained from these models is limited by multiple heterogeneous sources of uncertainty, which are often difficult to quantify. Here we review the major sources of uncertainty and survey existing approaches developed for representing and addressing them. A unified formal characterization of these uncertainties through probabilistic approaches and ensemble modeling will facilitate convergence towards consistent reconstruction pipelines, improved data integration algorithms, and more accurate assessment of predictive capacity.

Cooperative metabolic resource allocation in spatially-structured systems

Natural selection has shaped the evolution of cells and multi-cellular organisms such that social cooperation can often be preferred over an individualistic approach to metabolic regulation. This paper extends a framework for dynamic metabolic resource allocation based on the maximum entropy principle to spatiotemporal models of metabolism with cooperation. Much like the maximum entropy principle encapsulates 'bet-hedging' behaviour displayed by organisms dealing with future uncertainty in a fluctuating environment, its cooperative extension describes how individuals adapt their metabolic resource allocation strategy to further accommodate limited knowledge about the welfare of others within a community. The resulting theory explains why local regulation of metabolic cross-feeding can fulfil a community-wide metabolic objective if individuals take into consideration an ensemble measure of total population performance as the only form of global information. The latter is likely supplied by quorum sensing in microbial systems or signalling molecules such as hormones in multi-cellular eukaryotic organisms.

Metabolic flux configuration determination using information entropy

Constraint-based models use steady-state mass balances to define a solution space of flux configurations, which can be narrowed down by measuring as many fluxes as possible. Due to loops and redundant pathways, this process typically yields multiple alternative solutions. To address this ambiguity, flux sampling can estimate the probability distribution of each flux, or a flux configuration can be singled out by further minimizing the sum of fluxes according to the assumption that cellular metabolism favors states where enzyme-related costs are economized. However, flux sampling is susceptible to artifacts introduced by thermodynamically infeasible cycles and is it not clear if the economy of fluxes assumption (EFA) is universally valid. Here, we formulated a constraint-based approach, MaxEnt, based on the principle of maximum entropy, which in this context states that if more than one flux configuration is consistent with a set of experimentally measured fluxes, then the one with the minimum amount of unwarranted assumptions corresponds to the best estimation of the non-observed fluxes. We compared MaxEnt predictions to Escherichia coli and Saccharomyces cerevisiae publicly available flux data. We found that the mean square error (MSE) between experimental and predicted fluxes by MaxEnt and EFA-based methods are three orders of magnitude lower than the median of 1,350,000 MSE values obtained using flux sampling. However, only MaxEnt and flux sampling correctly predicted flux through E. coli’s glyoxylate cycle, whereas EFA-based methods, in general, predict no flux cycles. We also tested MaxEnt predictions at increasing levels of overflow metabolism. We found that MaxEnt accuracy is not affected by overflow metabolism levels, whereas the EFA-based methods show a decreasing performance. These results suggest that MaxEnt is less sensitive than flux sampling to artifacts introduced by thermodynamically infeasible cycles and that its predictions are less susceptible to overfitting than EFA-based methods.

Information Theory in Computational Biology: Where We Stand Today

"A Mathematical Theory of Communication" was published in 1948 by Claude Shannon to address the problems in the field of data compression and communication over (noisy) communication channels. Since then, the concepts and ideas developed in Shannon's work have formed the basis of information theory, a cornerstone of statistical learning and inference, and has been playing a key role in disciplines such as physics and thermodynamics, probability and statistics, computational sciences and biological sciences. In this article we review the basic information theory based concepts and describe their key applications in multiple major areas of research in computational biology-gene expression and transcriptomics, alignment-free sequence comparison, sequencing and error correction, genome-wide disease-gene association mapping, metabolic networks and metabolomics, and protein sequence, structure and interaction analysis.

Dynamic metabolic resource allocation based on the maximum entropy principle

Organisms have evolved a variety of mechanisms to cope with the unpredictability of environmental conditions, and yet mainstream models of metabolic regulation are typically based on strict optimality principles that do not account for uncertainty. This paper introduces a dynamic metabolic modelling framework that is a synthesis of recent ideas on resource allocation and the powerful optimal control formulation of Ramkrishna and colleagues. In particular, their work is extended based on the hypothesis that cellular resources are allocated among elementary flux modes according to the principle of maximum entropy. These concepts both generalise and unify prior approaches to dynamic metabolic modelling by establishing a smooth interpolation between dynamic flux balance analysis and dynamic metabolic models without regulation. The resulting theory is successful in describing 'bet-hedging' strategies employed by cell populations dealing with uncertainty in a fluctuating environment, including heterogenous resource investment, accumulation of reserves in growth-limiting conditions, and the observed behaviour of yeast growing in batch and continuous cultures. The maximum entropy principle is also shown to yield an optimal control law consistent with partitioning resources between elementary flux mode families, which has important practical implications for model reduction, selection, and simulation.

Statistical mechanics of interacting metabolic networks

We cast the metabolism of interacting cells within a statistical mechanics framework considering both the actual phenotypic capacities of each cell and its interaction with its neighbors. Reaction fluxes will be the components of high-dimensional spin vectors, whose values will be constrained by the stochiometry and the energy requirements of the metabolism. Within this picture, finding the phenotypic states of the population turns out to be equivalent to searching for the equilibrium states of a disordered spin model. We provide a general solution of this problem for arbitrary metabolic networks and interactions. We apply this solution to a simplified model of metabolism and to a complex metabolic network, the central core of Escherichia coli, and demonstrate that the combination of selective pressure and interactions defines a complex phenotypic space. We also present numerical results for cells fixed in a grid. These results reproduce the qualitative picture discussed for the mean-field model. Cells may specialize in producing or consuming metabolites complementing each other, and this is described by an equilibrium phase space with multiple minima, like in a spin-glass model.

Boost carbon availability and value in algal cell for economic deployment of biomass

This study elucidated storage carbon metabolism in a dynamic manner through kinetic model, metabolomics and stable metabolic flux analysis. Results revealed nutrient uptake rate, carbon availability and synthetic path rate accounted for the integration of process-compatible products. The uptake rate could be enhanced by promoting carbohydrate accumulation, inducing high performance of tricarboxylic acid cycle and anaplerotic routes. Values of specific rate for lipid from kinetic model and synthetic path rate from metabolic flux analysis revealed that conversion of carbon sinks occupied a key position in increasing productivities of lipid and astaxanthin to 302.34 and 1.83 mg g^-1 d^-1, respectively. Additionally, economic estimation was applied to link cultivation factors with market scenario and demonstrated that regulating such carbon metabolism raised 30% increase of biomass value. This study therefore provided a new orientation to boost carbon efficiency that helped to engineer carbon flux from carbon source to targeted products precisely and rapidly.

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.

Predicting the decision making chemicals used for bacterial growth

Predicting the contribution of media components to bacterial growth was first initiated by introducing machine learning to high-throughput growth assays. A total of 1336 temporal growth records corresponding to 225 different media, which were composed of 13 chemical components, were generated. The growth rate and saturated density of each growth curve were automatically calculated with the newly developed data processing program. To identify the decision making factors related to growth among the 13 chemicals, big datasets linking the growth parameters to the chemical combinations were subjected to decision tree learning. The results showed that the only carbon source, glucose, determined bacterial growth, but it was not the first priority. Instead, the top decision making chemicals in relation to the growth rate and saturated density were ammonium and ferric ions, respectively. Three chemical components (NH₄⁺, Mg²⁺ and glucose) commonly appeared in the decision trees of the growth rate and saturated density, but they exhibited different mechanisms. The concentration ranges for fast growth and high density were overlapped for glucose but distinguished for NH₄⁺ and Mg²⁺. The results suggested that these chemicals were crucial in determining the growth speed and growth maximum in either a universal use or a trade-off manner. This differentiation might reflect the diversity in the resource allocation mechanisms for growth priority depending on the environmental restrictions. This study provides a representative example for clarifying the contribution of the environment to population dynamics through an innovative viewpoint of employing modern data science within traditional microbiology to obtain novel findings.

Maximum entropy and population heterogeneity in continuous cell cultures

Continuous cultures of mammalian cells are complex systems displaying hallmark phenomena of nonlinear dynamics, such as multi-stability, hysteresis, as well as sharp transitions between different metabolic states. In this context mathematical models may suggest control strategies to steer the system towards desired states. Although even clonal populations are known to exhibit cell-to-cell variability, most of the currently studied models assume that the population is homogeneous. To overcome this limitation, we use the maximum entropy principle to model the phenotypic distribution of cells in a chemostat as a function of the dilution rate. We consider the coupling between cell metabolism and extracellular variables describing the state of the bioreactor and take into account the impact of toxic byproduct accumulation on cell viability. We present a formal solution for the stationary state of the chemostat and show how to apply it in two examples. First, a simplified model of cell metabolism where the exact solution is tractable, and then a genome-scale metabolic network of the Chinese hamster ovary (CHO) cell line. Along the way we discuss several consequences of heterogeneity, such as: qualitative changes in the dynamical landscape of the system, increasing concentrations of byproducts that vanish in the homogeneous case, and larger population sizes.