In biochemistry and systems biology, flux balance analysis (FBA) is a mathematical method for simulating the metabolism of cells or entire unicellular organisms, such as E. coli or yeast, using genome-scale reconstructions of metabolic networks. Genome-scale reconstructions describe all known or hypothesized biochemical reactions in an organism based on its entire genome. These reconstructions model metabolism by focusing on the interconversions between metabolites, identifying which metabolites are involved in the various reactions taking place in a cell or organism, and determining the genes that encode the enzymes which catalyze these reactions (if any).
History
Earliest work in FBA dates back to the early 1980s. Papoutsakis demonstrated that it was possible to construct flux balance equations using a metabolic map. It was Watson,Watson MR (1984) Metabolic maps for the Apple II. 12, 1093-1094 however, who first introduced the idea of using linear programming and an objective function to solve for the fluxes in a pathway. The first significant study was subsequently published by Fell and Small in 1986, who used flux balance analysis together with more elaborate objective functions to study the constraints in fat synthesis.
Uses FBA finds applications in bioprocess engineering to systematically identify modifications to the metabolic networks of microbes used in fermentation processes that improve product yields of industrially important chemicals such as ethanol and succinic acid. It has also been used for the identification of putative drug targets in cancer and pathogens, rational design of culture media, and host–pathogen interactions. The results of FBA can be visualized for smaller networks using flux maps similar to the image on the right, which illustrates the steady-state fluxes carried by reactions in glycolysis. The thickness of the arrows is proportional to the flux through the reaction.
Simulations
In comparison to traditional methods of modeling, FBA is less intensive in terms of the input data required for constructing the model. Simulations performed using FBA are computationally inexpensive and can calculate steady-state metabolic fluxes for large models (over 10,000 reactions) in a few seconds on modern personal computers. This means that the effect of deleting reactions from the network and/or changing flux constraints can be sensibly modelled on a single computer.
Gene/reaction deletion and perturbation studies
Single reaction deletion A frequently used technique to search a metabolic network for reactions that are particularly critical to the production of biomass. By removing each reaction in a network in turn and measuring the predicted flux through the biomass function or any other objective such as ATP production, each reaction can be classified as either essential (if the flux through the biomass function is substantially reduced) or non-essential (if the flux through the biomass function is unchanged or only slightly reduced).
Pairwise reaction deletion Pairwise reaction deletion of all possible pairs of reactions is useful when looking for drug targets, as it allows the simulation of multi-target treatments, either by a single drug with multiple targets or by drug combinations. Double deletion studies can also quantify the synthetic lethal interactions between different pathways providing a measure of the contribution of the pathway to overall network robustness.
Single and multiple gene deletions Genes are connected to enzyme-catalyzed reactions by Boolean expressions known as Gene-Protein-Reaction expressions (GPR). Typically a GPR takes the form (Gene A AND Gene B) to indicate that the products of genes A and B are protein sub-units that assemble to form the complete protein and therefore the absence of either would result in deletion of the reaction. On the other hand, if the GPR is (Gene A OR Gene B) it implies that the products of genes A and B are isozymes, meaning that the expression of either one is sufficient to maintain an active reaction. A reaction can also be regulated by a single gene, or in the case of diffusion, it may not be associated with any gene at all. Therefore, it is possible to evaluate the effect of single or multiple gene deletions by evaluation of the GPR as a Boolean expression. If the GPR evaluates to false, the reaction is constrained to zero in the model prior to performing FBA. Thus gene knockouts can be simulated using FBA. Logically, reactions that are not associated with any genes cannot be deleted.
Interpretation of gene and reaction deletion results The utility of reaction inhibition and deletion analyses becomes most apparent if a gene-protein-reaction matrix has been assembled for the network being studied with FBA. The gene-protein-reaction matrix is a binary matrix connecting genes with the proteins made from them. Using this matrix, reaction essentiality can be converted into gene essentiality indicating the gene defects which may cause a certain disease phenotype or the proteins/enzymes which are essential (and thus what enzymes are the most promising drug targets in pathogens). However, the gene-protein-reaction matrix does not specify the Boolean relationship between genes with respect to the enzyme, instead it merely indicates an association between them. Therefore, it should be used only if the Boolean GPR expression is unavailable.
Reaction inhibition The effect of inhibiting a reaction, rather than removing it entirely, can be simulated in FBA by restricting the allowed flux through it. The effect of an inhibition can be classified as lethal or non-lethal by applying the same criteria as in the case of a deletion where a suitable threshold is used to distinguish "substantially reduced" from "slightly reduced". Generally the choice of threshold is arbitrary but a reasonable estimate can be obtained from growth experiments where the simulated inhibitions/deletions are actually performed and growth rate is measured.
Growth media optimization To design optimal growth media with respect to enhanced growth rates or useful by-product secretion, it is possible to use a method known as Phenotypic Phase Plane analysis. PhPP involves applying FBA repeatedly on the model while co-varying the nutrient uptake constraints and observing the value of the objective function (or by-product fluxes). PhPP makes it possible to find the optimal combination of nutrients that favor a particular phenotype or a mode of metabolism resulting in higher growth rates or secretion of industrially useful by-products. The predicted growth rates of bacteria in varying media have been shown to correlate well with experimental results, as well as to define precise minimal media for the culture of Salmonella typhimurium.
Host-pathogen interactions
The human microbiota is a complex system with as many as 400 trillion microbes and bacteria interacting with each other and the host. To understand key factors in this system; a multi-scale, dynamic flux-balance analysis is proposed as FBA is classified as less computationally intensive.
Mathematical description
FBA formalizes the system of equations describing the concentration changes in a metabolic network as the dot product of a matrix of the stoichiometric coefficients (the stoichiometric matrix S) and the vector v of the unsolved fluxes. The right-hand side of the dot product is a vector of zeros representing the system at steady state. At steady state, metabolite concentrations remain constant as the rates of production and consumption are balanced, resulting in no net change over time. Since the system of equations is often underdetermined, there can be multiple possible solutions. To obtain a single solution, the flux that maximizes a reaction of interest, such as biomass or ATP production, is selected. Linear programming is then used to calculate one of the possible solutions of fluxes corresponding to the steady state.
In contrast to the traditionally followed approach of metabolic modeling using coupled ordinary differential equations, flux balance analysis requires very little information in terms of the enzyme kinetic parameters and concentration of metabolites in the system. It achieves this by making two assumptions, steady state and optimality. The first assumption is that the modeled system has entered a steady state, where the metabolite concentrations no longer change, i.e. in each metabolite node the producing and consuming fluxes cancel each other out. The second assumption is that the organism has been optimized through evolution for some biological goal, such as optimal growth or conservation of resources. The steady-state assumption reduces the system to a set of linear equations, which is then solved to find a flux distribution that satisfies the steady-state condition subject to the stoichiometry constraints while maximizing the value of a pseudo-reaction (the objective function) representing the conversion of biomass precursors into biomass.
The steady-state assumption dates to the ideas of material balance developed to model the growth of microbial cells in fermenters in bioprocess engineering. During microbial growth, a substrate consisting of a complex mixture of carbon, hydrogen, oxygen and nitrogen sources along with trace elements are consumed to generate biomass. The material balance model for this process becomes:
If we consider the system of microbial cells to be at steady state then we may set the accumulation term to zero and reduce the material balance equations to simple algebraic equations. In such a system, substrate becomes the input to the system which is consumed and biomass is produced becoming the output from the system. The material balance may then be represented as:
Mathematically, the algebraic equations can be represented as a dot product of a matrix of coefficients and a vector of the unknowns. Since the steady-state assumption puts the accumulation term to zero. The system can be written as:
Extending this idea to metabolic networks, it is possible to represent a metabolic network as a stoichiometry balanced set of equations. Moving to the matrix formalism, we can represent the equations as the dot product of a matrix of stoichiometry coefficients (stoichiometric matrix ) and the vector of fluxes as the unknowns and set the right hand side to 0 implying the steady state.
Metabolic networks typically have more reactions than metabolites and this gives an under-determined system of linear equations containing more variables than equations. The standard approach to solve such under-determined systems is to apply linear programming.
Linear programs are problems that can be expressed in canonical form:
where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients, A is a (known) matrix of coefficients, and is the matrix transpose. The expression to be maximized or minimized is called the objective function (cTx in this case). The inequalities Ax ≤ b are the constraints which specify a convex polytope over which the objective function is to be optimized.
Linear Programming requires the definition of an objective function. The optimal solution to the LP problem is considered to be the solution which maximizes or minimizes the value of the objective function depending on the case in point. In the case of flux balance analysis, the objective function Z for the LP is often defined as biomass production. Biomass production is simulated by an equation representing a lumped reaction that converts various biomass precursors into one unit of biomass.
Therefore, the canonical form of a Flux Balance Analysis problem would be: