1 Departament of Bioengineering, University of California 2 Harvard Medical School Department of Genetics Metabolic Flux Balance Analysis and the in Silico Analysis of Escherichia coli K-12 Gene Deletions by Jeremy S. Edwards 1,2 and Bernhard O. Palsson 1 Catalina Alupoaei Summary of the paper:

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 = 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:  The transport flux for some metabolites was unrestrained  The transport flux for some metabolites was constrained:  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: c = the unit vector in the direction of the growth flux

GROWTH FLUX d m = biomass composition of metabolite X m The growth flux was modeled as a single reaction that converts all the biosynthetic precursors into biomass. Was defined in terms of the biosynthetic requirements:

FLUX BALANCE ANALYSIS - EXAMPLE A flux balance was written for each metabolite (X i ) within the metabolic network to yield the dynamic mass balance equation for each metabolite in the network The rate of accumulation of X i was equated to its net rate of production yielding the dynamic mass balance for X i : V syn, V deg, V trans, V use = metabolic fluxes V syn, V deg = refer to the synthesis and degradation reactions of metabolite X i V trans = correspond to exchange fluxes that bring metabolism into or out of the system boundary V use = refers to the growth and maintenance requirements

FLUX BALANCE ANALYSIS EXAMPLE CONTINUE or X i = external metabolite b i = the net transport of X i into the defined metabolic system For the E. coli metabolic network all the transient material balances were represented by a single matrix equation, 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 I = identity matrix U = matrix b r = vector 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 reaction = metabolic reactions within the system boundary S use = 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