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Summary of the paper: Metabolic Flux Balance Analysis and the in Silico Analysis of Escherichia coli K-12 Gene Deletions by Jeremy S. Edwards1,2 and Bernhard O. Palsson1 1Departament of Bioengineering, University of California 2Harvard Medical School Department of Genetics Catalina Alupoaei OBJECTIVE • To analyze the integrated function of the metabolic pathways with the goal of the development of dynamic models for the complete simulations of cellular metabolism • To computationally examine the condition dependent optimal metabolic pathway utilization using E. coli in silico • To show that the flux balance analysis can be used to analyze and interpret the metabolic behavior of wild-type and mutant E.coli strains. INTRODUCTION •The integrated function of biological systems involves many complex interactions among the components within the cell •The properties of complex biological process cannot be analyzed or predicted on a description of the individual components, and integrated systems based approaches must be applied •The engineering approach to analyses and design of complex systems is to have a mathematical or computer model FLUX BALANCE ANALYSIS (FBA) MODEL • All biological processes are subject to physico-chemical constrains (mass balance, osmotic pressure, etc) •Flux balance analysis analyze the metabolic capabilities of a cellular system based on the metabolic (reaction) network and mass balance constraints •The mass balance constrains can be assigned on a genome scale for a number of organisms •The mass balance constraints in a metabolic network can be represented mathematically by a matrix equation: S v 0 S = m x n stoichiometric matrix m = number of metabolites n = number of reactions in the network v = vector of all fluxes in the metabolic network (internal, transport, growth) ADDITIONAL CONSTRAINTS Used to enforce the reversibility of each metabolic reaction and the maximal flux in the transport reactions. On the magnitude of individual metabolic fluxes: i i i The transport flux for some metabolites was unrestrained i ; i ; The transport flux for some metabolites was constrained: 0 i imax The transport flux was constrained to zero - no metabolite FEASIBLE SET – OBJECTIVE FUNCTION • The intersection of the nullspace and the region defined by the linear inequalities define a region in flux space – feasible set • The feasible set define the capabilities of the metabolic network subject to the imposed cellular constraints •The feasible point can be further reduced by imposing additional constraints (kinetic or gene regulatory constraints) • A particular metabolic flux distribution within the feasible set was found using the linear programming (LP) – identified a solution that minimize a metabolic objective function, Z: Minimize Z , Z ci vi c = the unit vector in the direction of the growth flux GROWTH FLUX • Was defined in terms of the biosynthetic requirements: v growth Biomass d m X m All _ m dm = biomass composition of metabolite Xm The growth flux was modeled as a single reaction that converts all the biosynthetic precursors into biomass. FLUX BALANCE ANALYSIS - EXAMPLE A flux balance was written for each metabolite (Xi) within the metabolic network to yield the dynamic mass balance equation for each metabolite in the network The rate of accumulation of Xi was equated to its net rate of production yielding the dynamic mass balance for Xi: dX i Vsyn Vdeg Vuse Vtrans dt Vsyn, Vdeg, Vtrans, Vuse = metabolic fluxes Vsyn, Vdeg = refer to the synthesis and degradation reactions of metabolite Xi Vtrans = correspond to exchange fluxes that bring metabolism into or out of the system boundary Vuse = refers to the growth and maintenance requirements FLUX BALANCE ANALYSIS EXAMPLE CONTINUE or dX i Vsyn Vdeg Vuse bi dt Xi = external metabolite bi = the net transport of Xi into the defined metabolic system For the E. coli metabolic network all the transient material balances were represented by a single matrix equation, dX S v b dt X = m dimensional vector defining the quantity of metabolites in a cell v = vector of n metabolic fluxes S = m x n stoichiometric matrix b = vector of metabolic exchange fluxes FLUX BALANCE ANALYSIS - EXAMPLE CONTINUE The time constants characterizing metabolic transients are very rapid compared to the time constants of cell growth and process dynamics, therefore, the transient mass balances were simplified to only consider the steady state behavior. S v I b 0 I = identity matrix U = matrix br = vector S v U br 0 vreactions Sreactions Suse U vuse 0 br S ' Sreactions Suse U vreactions v ' vuse br S ' v' 0 Sreaction = metabolic reactions within the system boundary Suse = biomasses and maintenance requirement fluxes U = allow certain metabolites to be transported into and out of the system PHENOTYPE PHASE PLANE (PhPP) ANALYSIS • Is a two-dimensional projection of the feasible set • Two parameters that describes the growth conditions were defined as the two axes of the two dimensional space • The optimal flux distribution was calculated by solving the LP problem while adjusting the exchange flux constraints • A finite number of qualitatively different patterns of metabolic pathway utilizations were identified and the regions were demarcated by lines • Line of optimality - one demarcation line – represents the optimal relation between exchange fluxes defined on the axes of the PhPP ALTERATION of the GENOTYPE • FBA and E. coli in silico were used to examine the systemic effects of in silico gene deletions • To simulate a gene deletion, all metabolic reactions catalyzed by a given gene product were simultaneously constrained to zero • The optimal metabolic flux distribution for the generation of biomass was calculated for each in silico deletion strain • The in silico gene deletion analysis was performed with the transport flux constraints defined by the wild-type PhPP RESULTS Gene deletions: • The growth characteristics of all in silico gene deletions strains were examined at each point from the PhPP • The gene were categorized as: essential, critical or non-essential • The effects of the in silico gene deletions were phase-dependent, optimal growth phenotypes for each growth condition were identified • The optimal utilization of the metabolic pathways was dependent on the specific transport flux constraints • Metabolic phenotypes based on the optimal biomass yield and biosynthetic production capabilities were computationally analyzed DISCUSSION • The study presented is an example of the rapidly growing field of in silico biology • An in silico representation of E. coli was utilized to study the condition dependent phenotype of E. coli and the central metabolism gene deletion strains • It was shown that a computational analysis on the metabolic behavior can provide valuable insight into cellular metabolism • FBA can be defined as a metabolic constraining approach • Metabolic functions based on most reliable information were constrained • The E. coli FBA results, with maximal growth rate as the objective function are consistent with experimental data CONCLUSIONS • The in silico representation of E. coli was utilized to study the condition dependent phenotype of E. coli and the central metabolism gene deletion strains • It was shown that a computational analysis on the metabolic behavior can provide valuable insight into cellular metabolism • The in silico study builds on the ability to define metabolic genotypes in bacteria and mathematical methods to analyze the possible and optimal phenotypes that they can express • This approach enables the analysis or study of mutant strains and their metabolic pathways