The economic basis of periodic enzyme dynamics

Periodic enzyme activities can improve the metabolic performance of cells. As an adaptation to periodic environments or by driving metabolic cycles that can shift fluxes and rearrange metabolic processes in time to increase their efficiency. To study what benefits can ensue from rhythmic gene expression or posttranslational modification of enzymes, I propose a theory of optimal enzyme rhythms in periodic or static environments. The theory is based on kinetic metabolic models with predefined metabolic objectives, scores the effects of harmonic enzyme oscillations, and determines amplitudes and phase shifts that maximise cell fitness. In an expansion around optimal steady states, the optimal enzyme profiles can be computed by solving a quadratic optimality problem. The formulae show how enzymes can increase their efficiency by oscillating in phase with their substrates and how cells can benefit from adapting to external rhythms and from spontaneous, intrinsic enzyme rhythms. Both types of behaviour may occur different parameter regions of the same model. Optimal enzyme profiles are not passively adapted to existing substrate rhythms, but shape them actively to create opportunities for further fitness advantage: in doing so, they reflect the dynamic effects that enzymes can exert in the network. The proposed theory combines the dynamics and economics of metabolic systems and shows how optimal enzyme profiles are shaped by network structure, dynamics, external rhythms, and metabolic objectives. It covers static enzyme adaptation as a special case, reveals the conditions for beneficial metabolic cycles, and predicts optimally combinations of gene expression and posttranslational modification for creating enzyme rhythms

How enzyme economy shapes metabolic fluxes

Metabolic fluxes are governed by physical and economic principles. Stationarity constrains them to a subspace in flux space and thermodynamics makes them lead from higher to lower chemical potentials. At the same time, fluxes in cells represent a compromise between metabolic performance and enzyme cost. To capture this, some flux prediction methods penalise larger fluxes by heuristic cost terms. Economic flux analysis, in contrast, postulates a balance between enzyme costs and metabolic benefits as a necessary condition for fluxes to be realised by kinetic models with optimal enzyme levels. The constraints are formulated using economic potentials, state variables that capture the enzyme labour embodied in metabolites. Generally, fluxes must lead from lower to higher economic potentials. This principle, which resembles thermodynamic constraints, can complement stationarity and thermodynamic constraints in flux analysis. Futile modes, which would be incompatible with economic potentials, are defined algebraically and can be systematically removed from flux distributions. Enzymes that participate in potential futile modes are likely targets of regulation. Economic flux analysis can predict high-yield and low-yield strategies, and captures preemptive expression, multi-objective optimisation, and flux distributions across several cells living in symbiosis. Inspired by labour value theories in economics, it justifies and extends the principle of minimal fluxes and provides an intuitive framework to model the complex interplay of fluxes, metabolic control, and enzyme costs in cells

Flux cost functions and the choice of metabolic fluxes

Metabolic fluxes in cells are governed by physical, biochemical, physiological, and economic principles. Cells may show "economical" behaviour, trading metabolic performance against the costly side-effects of high enzyme or metabolite concentrations. Some constraint-based flux prediction methods score fluxes by heuristic flux costs as proxies of enzyme investments. However, linear cost functions ignore enzyme kinetics and the tight coupling between fluxes, metabolite levels and enzyme levels. To derive more realistic cost functions, I define an apparent "enzymatic flux cost" as the minimal enzyme cost at which the fluxes can be realised in a given kinetic model, and a "kinetic flux cost", which includes metabolite cost. I discuss the mathematical properties of such flux cost functions, their usage for flux prediction, and their importance for cells' metabolic strategies. The enzymatic flux cost scales linearly with the fluxes and is a concave function on the flux polytope. The costs of two flows are usually not additive, due to an additional "compromise cost". Between flux polytopes, where fluxes change their directions, the enzymatic cost shows a jump. With strictly concave flux cost functions, cells can reduce their enzymatic cost by running different fluxes in different cell compartments or at different moments in time. The enzymactic flux cost can be translated into an approximated cell growth rate, a convex function on the flux polytope. Growth-maximising metabolic states can be predicted by Flux Cost Minimisation (FCM), a variant of FBA based on general flux cost functions. The solutions are flux distributions in corners of the flux polytope, i.e. typically elementary flux modes. Enzymatic flux costs can be linearly or nonlinearly approximated, providing model parameters for linear FBA based on kinetic parameters and extracellular concentrations, and justified by a kinetic model

Enzyme economy in metabolic networks

Metabolic systems are governed by a compromise between metabolic benefit and enzyme cost. This hypothesis and its consequences can be studied by kinetic models in which enzyme profiles are chosen by optimality principles. In enzyme-optimal states, active enzymes must provide benefits: a higher enzyme level must provide a metabolic benefit to justify the additional enzyme cost. This entails general relations between metabolic fluxes, reaction elasticities, and enzyme costs, the laws of metabolic economics. The laws can be formulated using economic potentials and loads, state variables that quantify how metabolites, reactions, and enzymes affect the metabolic performance in a steady state. Economic balance equations link them to fluxes, reaction elasticities, and enzyme levels locally in the network. Economically feasible fluxes must be free of futile cycles and must lead from lower to higher economic potentials, just like thermodynamics makes them lead from higher to lower chemical potentials. Metabolic economics provides algebraic conditions for economical fluxes, which are independent of the underlying kinetic models. It justifies and extends the principle of minimal fluxes and shows how to construct kinetic models in enzyme-optimal states, where all enzymes have a positive influence on the metabolic performance

Elasticity sampling links thermodynamics to metabolic control

Metabolic networks can be turned into kinetic models in a predefined steady state by sampling the reaction elasticities in this state. Elasticities for many reversible rate laws can be computed from the reaction Gibbs free energies, which are determined by the state, and from physically unconstrained saturation values. Starting from a network structure with allosteric regulation and consistent metabolic fluxes and concentrations, one can sample the elasticities, compute the control coefficients, and reconstruct a kinetic model with consistent reversible rate laws. Some of the model variables are manually chosen, fitted to data, or optimised, while the others are computed from them. The resulting model ensemble allows for probabilistic predictions, for instance, about possible dynamic behaviour. By adding more data or tighter constraints, the predictions can be made more precise. Model variants differing in network structure, flux distributions, thermodynamic forces, regulation, or rate laws can be realised by different model ensembles and compared by significance tests. The thermodynamic forces have specific effects on flux control, on the synergisms between enzymes, and on the emergence and propagation of metabolite fluctuations. Large kinetic models could help to simulate global metabolic dynamics and to predict the effects of enzyme inhibition, differential expression, genetic modifications, and their combinations on metabolic fluxes. MATLAB code for elasticity sampling is freely available

The protein cost of metabolic fluxes: prediction from enzymatic rate laws and cost minimization

Bacterial growth depends crucially on metabolic fluxes, which are limited by the cell's capacity to maintain metabolic enzymes. The necessary enzyme amount per unit flux is a major determinant of metabolic strategies both in evolution and bioengineering. It depends on enzyme parameters (such as kcat and KM constants), but also on metabolite concentrations. Moreover, similar amounts of different enzymes might incur different costs for the cell, depending on enzyme-specific properties such as protein size and half-life. Here, we developed enzyme cost minimization (ECM), a scalable method for computing enzyme amounts that support a given metabolic flux at a minimal protein cost. The complex interplay of enzyme and metabolite concentrations, e.g. through thermodynamic driving forces and enzyme saturation, would make it hard to solve this optimization problem directly. By treating enzyme cost as a function of metabolite levels, we formulated ECM as a numerically tractable, convex optimization problem. Its tiered approach allows for building models at different levels of detail, depending on the amount of available data. Validating our method with measured metabolite and protein levels in E. coli central metabolism, we found typical prediction fold errors of 3.8 and 2.7, respectively, for the two kinds of data. ECM can be used to predict enzyme levels and protein cost in natural and engineered pathways, establishes a direct connection between protein cost and thermodynamics, and provides a physically plausible and computationally tractable way to include enzyme kinetics into constraint-based metabolic models, where kinetics have usually been ignored or oversimplified

A Quantitative Study of the Hog1 MAPK Response to Fluctuating Osmotic Stress in Saccharomyces cerevisiae

Background Yeast cells live in a highly fluctuating environment with respect to temperature, nutrients, and especially osmolarity. The Hog1 mitogen-activated protein kinase (MAPK) pathway is crucial for the adaption of yeast cells to external osmotic changes. Methodology/Principal Findings To better understand the osmo-adaption mechanism in the budding yeast Saccharomyces cerevisiae, we have developed a mathematical model and quantitatively investigated the Hog1 response to osmotic stress. The model agrees well with various experimental data for the Hog1 response to different types of osmotic changes. Kinetic analyses of the model indicate that budding yeast cells have evolved to protect themselves economically: while they show almost no response to fast pulse-like changes of osmolarity, they respond periodically and are well-adapted to osmotic changes with a certain frequency. To quantify the signal transduction efficiency of the osmo-adaption network, we introduced a measure of the signal response gain, which is defined as the ratio of output change integral to input (signal) change integral. Model simulations indicate that the Hog1 response gain shows bell-shaped response curves with respect to the duration of a single osmotic pulse and to the frequency of periodic square osmotic pulses, while for up-staircase (ramp) osmotic changes, the gain depends on the slope. Conclusions/Significance The model analyses suggest that budding yeast cells have selectively evolved to be optimized to some specific types of osmotic changes. In addition, our work implies that the signaling output can be dynamically controlled by fine-tuning the signal input profiles