Dynamic optimization of metabolic networks coupled with gene expression

The regulation of metabolic activity by tuning enzyme expression levels is crucial to sustain cellular growth in changing environments. Metabolic networks are often studied at steady state using constraint-based models and optimization techniques. However, metabolic adaptations driven by changes in gene expression cannot be analyzed by steady state models, as these do not account for temporal changes in biomass composition. Here we present a dynamic optimization framework that integrates the metabolic network with the dynamics of biomass production and composition, explicitly taking into account enzyme production costs and enzymatic capacity. In contrast to the established dynamic flux balance analysis, our approach allows predicting dynamic changes in both the metabolic fluxes and the biomass composition during metabolic adaptations. We applied our algorithm in two case studies: a minimal nutrient uptake network, and an abstraction of core metabolic processes in bacteria. In the minimal model, we show that the optimized uptake rates reproduce the empirical Monod growth for bacterial cultures. For the network of core metabolic processes, the dynamic optimization algorithm predicted commonly observed metabolic adaptations, such as a diauxic switch with a preference ranking for different nutrients, re-utilization of waste products after depletion of the original substrate, and metabolic adaptation to an impending nutrient depletion. These examples illustrate how dynamic adaptations of enzyme expression can be predicted solely from an optimization principle

Elucidating temporal resource allocation and diurnal dynamics in phototrophic metabolism using conditional FBA

The computational analysis of phototrophic growth using constraint-based optimization requires to go beyond current time-invariant implementations of flux-balance analysis (FBA). Phototrophic organisms, such as cyanobacteria, rely on harvesting the sun’s energy for the conversion of atmospheric CO2 into organic carbon, hence their metabolism follows a strongly diurnal lifestyle. We describe the growth of cyanobacteria in a periodic environment using a new method called conditional FBA. Our approach enables us to incorporate the temporal organization and conditional dependencies into a constraint-based description of phototrophic metabolism. Specifically, we take into account that cellular processes require resources that are themselves products of metabolism. Phototrophic growth can therefore be formulated as a time- dependent linear optimization problem, such that optimal growth requires a differential allocation of resources during different times of the day. Conditional FBA then allows us to simulate phototrophic growth of an average cell in an environment with varying light intensity, resulting in dynamic time-courses for all involved reaction fluxes, as well as changes in biomass composition over a diurnal cycle. Our results are in good agreement with several known facts about the temporal organization of phototrophic growth and have implications for further analysis of resource allocation problems in phototrophic metabolism

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

Constrained Allocation Flux Balance Analysis

New experimental results on bacterial growth inspire a novel top-down approach to study cell metabolism, combining mass balance and proteomic constraints to extend and complement Flux Balance Analysis. We introduce here Constrained Allocation Flux Balance Analysis, CAFBA, in which the biosynthetic costs associated to growth are accounted for in an effective way through a single additional genome-wide constraint. Its roots lie in the experimentally observed pattern of proteome allocation for metabolic functions, allowing to bridge regulation and metabolism in a transparent way under the principle of growth-rate maximization. We provide a simple method to solve CAFBA efficiently and propose an "ensemble averaging" procedure to account for unknown protein costs. Applying this approach to modeling E. coli metabolism, we find that, as the growth rate increases, CAFBA solutions cross over from respiratory, growth-yield maximizing states (preferred at slow growth) to fermentative states with carbon overflow (preferred at fast growth). In addition, CAFBA allows for quantitatively accurate predictions on the rate of acetate excretion and growth yield based on only 3 parameters determined by empirical growth laws.Comment: 21 pages, 6 figures (main) + 33 pages, various figures and tables (supporting); for the supplementary MatLab code, see http://tinyurl.com/h763es

Models of protein production along the cell cycle: an investigation of possible sources of noise

In this article, we quantitatively study, through stochastic models, the efects of several intracellular phenomena, such as cell volume growth, cell division, gene replication as well as fuctuations of available RNA polymerases and ribosomes. These phenomena are indeed rarely considered in classic models of protein production and no relative quantitative comparison among them has been performed. The parameters for a large and representative class of proteins are determined using experimental measures. The main important and surprising conclusion of our study is to show that despite the signifcant fuctuations of free RNA polymerases and free ribosomes, they bring little variability to protein production contrary to what has been previously proposed in the literature. After verifying the robustness of this quite counter-intuitive result, we discuss its possible origin from a theoretical view, and interpret it as the result of a mean-feld efect

Constraint-based modeling of metabolism - interpreting predictions of growth and ATP synthesis in human and yeast

Growth is the primary objective of the cell. Diseases arise when cells diverge from a healthy growth-pattern. An increased understanding of cellular growth may thus be translated into improved human health. The cell requires materials and free energy (in the form of ATP) in order to grow, metabolism supplies the cell with this. The rate of metabolism is ultimately constrained by the biophysical properties of the metabolic enzymes. Interactions between the constraints and the growth-objective gives rise to metabolic trade-offs, e.g. between ATP synthesis from respiration and fermentation. We can gain quantitative insight into these processes by simulating metabolism using mathematical models. In this thesis I simulated the metabolism of four biological systems: the infant, cancer, yeast and muscle. The simulations demonstrated how a shift in metabolic strategy may increase the rates of ATP synthesis and growth. These increased metabolic rates come at the expense of decreased resource efficiency, i.e. ATP produced per carbon consumed. The effect was primarily caused by the low catalytic efficiency of the respiratory enzyme complexes I and V. By shifting from respiratory to fermentative ATP synthesis, the cell was able to bypass these constraints. An intermediate strategy involved bypassing only complex I. The phenomenon was experimentally corroborated in the working muscle, and it is the native state of the yeast Saccharomyces cerevisiae (which lacks complex I). The differences in efficiency between the different metabolic pathways also explained why cells grow faster on some carbon sources, e.g. the specific growth rate for yeast is higher on glucose than on ethanol. These models were extended to predict the world-record running-speeds at different distances, by taking the sizes of the body’s nutrient-deposits into account. A metabolic strategy employed by cancer cells involved excretion of the amino acid glutamate. The simulations showed a mechanistic relation to catabolism of branched-chain amino acids and the localization of amino acid metabolism to different cellular compartments. By experimentally inhibiting glutamate excretion using an off-the-shelf drug (sulfasalazine), the growth rate of a cancer cell line was reduced. The metabolic modeling involved integration of various types of data and thus demonstrated the potential to unify knowledge from different studies and domains. This exposed contradictory claims in literature and highlighted knowledge-gaps that need to be filled to further improve human health

Elementary approaches to microbial growth rate maximisation

This thesis, called Elementary approaches to microbial growth rate maximisation, reports on a theoretical search for principles underlying single cell growth, in particular for microbial species that are selected for fast growth rates. First, the optimally growing cell is characterised in terms of its elementary modes. We prove an extremum principle: a cell that maximises a metabolic rate uses few Elementary Flux Modes (EFMs, the minimal pathways that support steady-state metabolism). The number of active EFMs is bounded by the number of growth-limiting constraints. Later, this extremum principle is extended in a theory that explicitly accounts for self-fabrication. For this, we had to define the elementary modes that underlie balanced self-fabrication: minimal self-supporting sets of expressed enzymes that we call Elementary Growth Modes (EGMs). It turns out that many of the results for EFMs can be extended to their more general self-fabrication analogue. Where the above extremum principles tell us that few elementary modes are used by a rate-maximising cell, it does not tell us how the cell can find them. Therefore, we also search for an elementary adaptation method. It turns out that stochastic phenotype switching with growth rate dependent switching rates provides an adaptation mechanism that is often competitive with more conventional regulatory-circuitry based mechanisms. The derived theory is applied in two ways. First, the extremum principles are used to review the mathematical fundaments of all optimisation-based explanations of overflow metabolism. Second, a computational tool is presented that enumerates Elementary Conversion Modes. These elementary modes can be computed for larger networks than EFMs and EGMs, and still provide an overview of the metabolic capabilities of an organism