Certificate three summary Module one
Flux balance analysis formulation usually starts with an objective function.
Objective function – Is a function that is maximized or minimized to identify optimal solutions.
Constraints – means to place limits on the allowable values the solutions can take.
Mathematical formulation of the objective functions
This illustrates the formation of the objective function using a simple mathematical model given as:
The coefficient on the ATP flux is positive since it is being maximized.
The growth function
This shows the requirements for making 1 g (one gram) of E. coli. It means that for the (E. coli) cell to grow, all the components must be provided in the right amount. Thus, a balance set of metabolic demands make up the growth objective function.
The shadow prices (S.P) – The shadow prices are the derivatives of the biomass objective function at the boundary with respect to an exchange flux. It is represented as:
The shadow prices can be used to determine whether a cell is limited by a particular constraint. S.P has proven to be useful in interpreting optimum solutions, and also in metabolic decision making.
Reduced costs - The reduced costs can be defined as the amount by which the objective function will be reduced if the corresponding enzyme is forced to carry a flux. The reduced costs can be used to analyze the presence of alternate equivalent flux distributions.
Robustness analysis – The sensitivity of the optimal properties of a network can be assessed by changing the network parameters over a given range of values, and repeatedly computing the optimal state. Both environmental and genetic parameters can be considered.
Phase plane analysis – In order for the objective function to remain constant, an increase in one exchange flux will be accompanied by a decrease in another. The parameter α is the slope of a line in the phenotypic phase plane along which the value of the objective function is constant. This line is called an isocline. This isocline defines four regions (also called phases) in the phenotypic phase plane. These regions are:
Single substrate limitation region (α = 0 or ∞)
Dual substrate limitation (α < 0)
Futile region (α >0)
Infeasible region (no flux balance)
Randomized sampling – The effect of imposing constraints can be studied using randomized sampling methods. This technique can be used to characterize flux solution space, concentration space, etc. It can also be used to understand network capabilities, disease states, etc.
Computational design of mutant strains – involves techniques such as optknock, optstrain, and optforce used for new strain development.
Optknock – Find gene deletions needed such that maximizing the biomass is coupled with maximizing the cellular (bioengineering objective). This can be applied for strain design such as that of lactate production, succinate, alanine, aspartate, etc.
Optstrain – This technique involves four steps for designing mutant strains. The steps are as follows:
1. Curate KEGG database to find balance reactions.
2. Identify reactions for databases with highest production yields for product.
3. Identify fewest number of reactions needed from database.
4. Run optknock to find reaction deletions which couple growth and production rates.
The linear basis for null space
The null space contains all steady state flux solution; that is, it contains all the balanced uses of the network. Thus, all the steady-state flux distributions, Vss are found in the null space. The null space has a dimension of n-r.
Convex basis for null space – extreme pathways involves convex analysis.
1. Involves the study of linear equations and inequalities
2. It is used to study metabolic networks
The linear equations are derived from the mass balances (S.V =0) and the inequalities are generated from thermodynamic information on the reversibility of reaction (V >0).
From linear algebra null space is defined which contains all of the solutions to the set of linear homogenous equations.
Biochemically, meaningful steady state flux solutions can be represented by a nonnegative linear combination of convex basis vectors as illustrated with the image below:
Extreme pathways can capture the phenotypic potential of metabolic reaction networks.
Types of extreme pathways
Type I – Primary metabolic pathway
Type II – Futile cycles
Type III – Internal loops
Elementary modes and extreme pathways
Elementary modes – A set of non-decomposable enzymes that characterize the convex solution core.
Extreme pathways - A minimal and unique set of pathways that define the edges of the convex solution.
Convex solution - A mathematically defined space of all steady-state flux distributions of a metabolic network.
Exchange flux - A reaction that crosses the system boundary.
The steps in the 13C MFA formulation include:
1. Feed cells with stable isotopes.
2. Incorporate isotope-labelled atoms into metabolite structure.
3. Quantify the relative abundance of the isotopes via GC/MS.
4. Predict flux distributions from network connectivity and atom mapping info.
5. Perform the steady-state mass balance of each isotopomer.
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