Simulating metabolism with statistical thermodynamics

Probabilistic and maximum entropy modeling of chemical reaction systems: Characteristics and comparisons to mass action kinetic models

We demonstrate and characterize a first-principles approach to modeling the mass action dynamics of metabolism. Starting from a basic definition of entropy expressed as a multinomial probability density using Boltzmann probabilities with standard chemical potentials, we derive and compare the free energy dissipation and the entropy production rates. We express the relation between entropy production and the chemical master equation for modeling metabolism, which unifies chemical kinetics and chemical thermodynamics. Because prediction uncertainty with respect to parameter variability is frequently a concern with mass action models utilizing rate constants, we compare and contrast the maximum entropy model, which has its own set of rate parameters, to a population of standard mass action models in which the rate constants are randomly chosen. We show that a maximum entropy model is characterized by a high probability of free energy dissipation rate and likewise entropy production rate, relative to other models. We then characterize the variability of the maximum entropy model predictions with respect to uncertainties in parameters (standard free energies of formation) and with respect to ionic strengths typically found in a cell.

Dynamic optimization reveals alveolar epithelial cells as key mediators of host defense in invasive aspergillosis

Aspergillus fumigatus is an important human fungal pathogen and its conidia are constantly inhaled by humans. In immunocompromised individuals, conidia can grow out as hyphae that damage lung epithelium. The resulting invasive aspergillosis is associated with devastating mortality rates. Since infection is a race between the innate immune system and the outgrowth of A. fumigatus conidia, we use dynamic optimization to obtain insight into the recruitment and depletion of alveolar macrophages and neutrophils. Using this model, we obtain key insights into major determinants of infection outcome on host and pathogen side. On the pathogen side, we predict in silico and confirm in vitro that germination speed is an important virulence trait of fungal pathogens due to the vulnerability of conidia against host defense. On the host side, we found that epithelial cells, which have been underappreciated, play a role in fungal clearance and are potent mediators of cytokine release. Both predictions were confirmed by in vitro experiments on established cell lines as well as primary lung cells. Further, our model affirms the importance of neutrophils in invasive aspergillosis and underlines that the role of macrophages remains elusive. We expect that our model will contribute to improvement of treatment protocols by focusing on the critical components of immune response to fungi but also fungal virulence traits.

Association of the malate dehydrogenase-citrate synthase metabolon is modulated by intermediates of the Krebs tricarboxylic acid cycle

Mitochondrial malate dehydrogenase (MDH)-citrate synthase (CS) multi-enzyme complex is a part of the Krebs tricarboxylic acid (TCA) cycle ‘metabolon’ which is enzyme machinery catalyzing sequential reactions without diffusion of reaction intermediates into a bulk matrix. This complex is assumed to be a dynamic structure involved in the regulation of the cycle by enhancing metabolic flux. Microscale Thermophoresis analysis of the porcine heart MDH-CS complex revealed that substrates of the MDH and CS reactions, NAD⁺ and acetyl-CoA, enhance complex association while products of the reactions, NADH and citrate, weaken the affinity of the complex. Oxaloacetate enhanced the interaction only when it was present together with acetyl-CoA. Structural modeling using published CS structures suggested that the binding of these substrates can stabilize the closed format of CS which favors the MDH-CS association. Two other TCA cycle intermediates, ATP, and low pH also enhanced the association of the complex. These results suggest that dynamic formation of the MDH-CS multi-enzyme complex is modulated by metabolic factors responding to respiratory metabolism, and it may function in the feedback regulation of the cycle and adjacent metabolic pathways.

Enzyme activities predicted by metabolite concentrations and solvent capacity in the cell

Experimental measurements or computational model predictions of the post-translational regulation of enzymes needed in a metabolic pathway is a difficult problem. Consequently, regulation is mostly known only for well-studied reactions of central metabolism in various model organisms. In this study, we use two approaches to predict enzyme regulation policies and investigate the hypothesis that regulation is driven by the need to maintain the solvent capacity in the cell. The first predictive method uses a statistical thermodynamics and metabolic control theory framework while the second method is performed using a hybrid optimization-reinforcement learning approach. Efficient regulation schemes were learned from experimental data that either agree with theoretical calculations or result in a higher cell fitness using maximum useful work as a metric. As previously hypothesized, regulation is herein shown to control the concentrations of both immediate and downstream product concentrations at physiological levels. Model predictions provide the following two novel general principles: (1) the regulation itself causes the reactions to be much further from equilibrium instead of the common assumption that highly non-equilibrium reactions are the targets for regulation; and (2) the minimal regulation needed to maintain metabolite levels at physiological concentrations maximizes the free energy dissipation rate instead of preserving a specific energy charge. The resulting energy dissipation rate is an emergent property of regulation which may be represented by a high value of the adenylate energy charge. In addition, the predictions demonstrate that the amount of regulation needed can be minimized if it is applied at the beginning or branch point of a pathway, in agreement with common notions. The approach is demonstrated for three pathways in the central metabolism of E. coli (gluconeogenesis, glycolysis-tricarboxylic acid (TCA) and pentose phosphate-TCA) that each require different regulation schemes. It is shown quantitatively that hexokinase, glucose 6-phosphate dehydrogenase and glyceraldehyde phosphate dehydrogenase, all branch points of pathways, play the largest roles in regulating central metabolism.

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.

Interrogating metabolism as an electron flow system

Metabolism is generally considered as a neatly organised system of modular pathways, shaped by evolution under selection for optimal cellular growth. This view falls short of explaining and predicting a number of key observations about the structure and dynamics of metabolism. We highlight these limitations of a pathway-centric view on metabolism and summarise studies suggesting how these could be overcome by viewing metabolism as a thermodynamically and kinetically constrained, dynamical flow system. Such a systems-level, first-principles based view of metabolism can open up new avenues of metabolic engineering and cures for metabolic diseases and allow better insights to a myriad of physiological processes that are ultimately linked to metabolism. Towards further developing this view, we call for a closer interaction among physical and biological disciplines and an increased use of electrochemical and biophysical approaches to interrogate cellular metabolism together with the microenvironment in which it exists.

Prediction of Metabolite Concentrations, Rate Constants and Post-Translational Regulation Using Maximum Entropy-Based Simulations with Application to Central Metabolism of Neurospora crassa

We report the application of a recently proposed approach for modeling biological systems using a maximum entropy production rate principle in lieu of having in vivo rate constants. The method is applied in four steps: (1) a new ordinary differential equation (ODE) based optimization approach based on Marcelin’s 1910 mass action equation is used to obtain the maximum entropy distribution; (2) the predicted metabolite concentrations are compared to those generally expected from experiments using a loss function from which post-translational regulation of enzymes is inferred; (3) the system is re-optimized with the inferred regulation from which rate constants are determined from the metabolite concentrations and reaction fluxes; and finally (4) a full ODE-based, mass action simulation with rate parameters and allosteric regulation is obtained. From the last step, the power characteristics and resistance of each reaction can be determined. The method is applied to the central metabolism of Neurospora crassa and the flow of material through the three competing pathways of upper glycolysis, the non-oxidative pentose phosphate pathway, and the oxidative pentose phosphate pathway are evaluated as a function of the NADP/NADPH ratio. It is predicted that regulation of phosphofructokinase (PFK) and flow through the pentose phosphate pathway are essential for preventing an extreme level of fructose 1,6-bisphophate accumulation. Such an extreme level of fructose 1,6-bisphophate would otherwise result in a glassy cytoplasm with limited diffusion, dramatically decreasing the entropy and energy production rate and, consequently, biological competitiveness.

Non-steady state mass action dynamics without rate constants: dynamics of coupled reactions using chemical potentials

Comprehensive and predictive simulation of coupled reaction networks has long been a goal of biology and other fields. Currently, metabolic network models that utilize enzyme mass action kinetics have predictive power but are limited in scope and application by the fact that the determination of enzyme rate constants is laborious and low throughput. We present a statistical thermodynamic formulation of the law of mass action for coupled reactions at both steady states and non-stationary states. The formulation uses chemical potentials instead of rate constants. When used to model deterministic systems, the method corresponds to a rescaling of the time dependent reactions in such a way that steady states can be reached on the same time scale but with significantly fewer computational steps. The relationships between reaction affinities, free energy changes and generalized detailed balance are central to the discussion. The significance for applications in systems biology are discussed as is the concept and assumption of maximum entropy production rate as a biological principle that links thermodynamics to natural selection.

Thermodynamics of random reaction networks

Reaction networks are useful for analyzing reaction systems occurring in chemistry, systems biology, or Earth system science. Despite the importance of thermodynamic disequilibrium for many of those systems, the general thermodynamic properties of reaction networks are poorly understood. To circumvent the problem of sparse thermodynamic data, we generate artificial reaction networks and investigate their non-equilibrium steady state for various boundary fluxes. We generate linear and nonlinear networks using four different complex network models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz, Pan-Sinha) and compare their topological properties with real reaction networks. For similar boundary conditions the steady state flow through the linear networks is about one order of magnitude higher than the flow through comparable nonlinear networks. In all networks, the flow decreases with the distance between the inflow and outflow boundary species, with Watts-Strogatz networks showing a significantly smaller slope compared to the three other network types. The distribution of entropy production of the individual reactions inside the network follows a power law in the intermediate region with an exponent of circa -1.5 for linear and -1.66 for nonlinear networks. An elevated entropy production rate is found in reactions associated with weakly connected species. This effect is stronger in nonlinear networks than in the linear ones. Increasing the flow through the nonlinear networks also increases the number of cycles and leads to a narrower distribution of chemical potentials. We conclude that the relation between distribution of dissipation, network topology and strength of disequilibrium is nontrivial and can be studied systematically by artificial reaction networks.

Comparison of Optimal Thermodynamic Models of the Tricarboxylic Acid Cycle from Heterotrophs, Cyanobacteria, and Green Sulfur Bacteria

We have applied a new stochastic simulation approach to predict the metabolite levels, material flux, and thermodynamic profiles of the oxidative TCA cycles found in E. coli and Synechococcus sp. PCC 7002, and in the reductive TCA cycle typical of chemolithoautotrophs and phototrophic green sulfur bacteria such as Chlorobaculum tepidum. The simulation approach is based on modeling states using statistical thermodynamics and employs an assumption similar to that used in transition state theory. The ability to evaluate the thermodynamics of metabolic pathways allows one to understand the relationship between coupling of energy and material gradients in the environment and the self-organization of stable biological systems, and it is shown that each cycle operates in the direction expected due to its environmental niche. The simulations predict changes in metabolite levels and flux in response to changes in cofactor concentrations that would be hard to predict without an elaborate model based on the law of mass action. In fact, we show that a thermodynamically unfavorable reaction can still have flux in the forward direction when it is part of a reaction network. The ability to predict metabolite levels, energy flow, and material flux should be significant for understanding the dynamics of natural systems and for understanding principles for engineering organisms for production of specialty chemicals.

Concepts, challenges, and successes in modeling thermodynamics of metabolism

The modeling of the chemical reactions involved in metabolism is a daunting task. Ideally, the modeling of metabolism would use kinetic simulations, but these simulations require knowledge of the thousands of rate constants involved in the reactions. The measurement of rate constants is very labor intensive, and hence rate constants for most enzymatic reactions are not available. Consequently, constraint-based flux modeling has been the method of choice because it does not require the use of the rate constants of the law of mass action. However, this convenience also limits the predictive power of constraint-based approaches in that the law of mass action is used only as a constraint, making it difficult to predict metabolite levels or energy requirements of pathways. An alternative to both of these approaches is to model metabolism using simulations of states rather than simulations of reactions, in which the state is defined as the set of all metabolite counts or concentrations. While kinetic simulations model reactions based on the likelihood of the reaction derived from the law of mass action, states are modeled based on likelihood ratios of mass action. Both approaches provide information on the energy requirements of metabolic reactions and pathways. However, modeling states rather than reactions has the advantage that the parameters needed to model states (chemical potentials) are much easier to determine than the parameters needed to model reactions (rate constants). Herein, we discuss recent results, assumptions, and issues in using simulations of state to model metabolism.