Dynamic metabolic resource allocation based on the maximum entropy principle

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.

Combining Kinetic and Constraint-Based Modelling to Better Understand Metabolism Dynamics

To understand the phenotypic capabilities of organisms, it is useful to characterise cellular metabolism through the analysis of its pathways. Dynamic mathematical modelling of metabolic networks is of high interest as it provides the time evolution of the metabolic components. However, it also has limitations, such as the necessary mechanistic details and kinetic parameters are not always available. On the other hand, large metabolic networks exhibit a complex topological structure which can be studied rather efficiently in their stationary regime by constraint-based methods. These methods produce useful predictions on pathway operations. In this review, we present both modelling techniques and we show how they bring complementary views of metabolism. In particular, we show on a simple example how both approaches can be used in conjunction to shed some light on the dynamics of metabolic networks.

Microbial inefficient substrate use through the perspective of resource allocation models

Microorganisms extract energy from substrates following strategies that may seem suboptimal at first glance. Beyond the so-called yield-rate trade-off, resource allocation models, which focus on assigning different functional roles to the limited number of enzymes that a cell can support, offer a framework to interpret the inefficient substrate use by microorganisms. We review here relevant examples of substrate conversions where a significant part of the available energy is not utilised and how resource allocation models offer a mechanistic interpretation thereof, notably for open mixed cultures. Future developments are identified, in particular, the challenge of considering metabolic flexibility towards uncertain environmental changes instead of strict fixed optimality objectives, with the final goal of increasing the prediction capabilities of resource allocation models. Finally, we highlight the relevance of resource allocation to understand and enable a promising biorefinery platform revolving around lactate, which would increase the flexibility of waste-to-chemical biorefinery schemes.

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.

Persistence as an Optimal Hedging Strategy

Bacteria invest in a slow-growing subpopulation, called persisters, to ensure survival in the face of uncertainty. This hedging strategy is remarkably similar to financial hedging, where diversifying an investment portfolio protects against economic uncertainty. We provide a new, to our knowledge, theoretical foundation for understanding cellular hedging by unifying the study of biological population dynamics and the mathematics of financial risk management through optimal control theory. Motivated by the widely accepted role of volatility in the emergence of persistence, we consider several models of environmental volatility described by continuous-time stochastic processes. This allows us to study an emergent cellular hedging strategy that maximizes the expected per capita growth rate of the population. Analytical and simulation results probe the optimal persister strategy, revealing results that are consistent with experimental observations and suggest new opportunities for experimental investigation and design. Overall, we provide a new, to our knowledge, way of conceptualizing and modeling cellular decision making in volatile environments by explicitly unifying theory from mathematical biology and finance.