SunlightCBM - introduction

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Constraint-based modelling of metabolic networks is well described by Bernhard Palsson in his book Systems Biology: Properties of Reconstructed Networks (Cambridge University Press, 2006). Briefly, a metabolic network is a system of chemical reactions between metabolites, representing the metabolic capabilities of an organism in terms of raw materials processing and energy (ATP) generation (see Wikipedia article on metabolism). The kinetic approach to modelling such a network is to write down and solve the corresponding set of chemical rate equations as a large list of ODEs. This requires an vast number of rate constants which are not necessarily well known. However, since the relaxation time of such a system of ODEs is often quite fast (perhaps minutes or less), it can often be assumed that the metabolic network is in a steady state. In this case the so-called flux-balance condition holds: for each metabolite the sum of the fluxes of the reactions producing that metabolite is equal to the sum of the fluxes of the reactions consuming that metabolite, weighted of course by the stoichiometric coefficients. This is the basis of flux-balance analysis (FBA).

In constraint-based modelling (CBM), the flux-balance condition is treated as a constraint in the space of reaction fluxes. Additionally one can constrain the magnitudes of the fluxes, typically those reactions which are expected to be thermodynamically irreversible are restricted to have non-negative fluxes. Other constraints can be imposed, and other demands on the network represented by additional reactions. The most common of these is a biomass demand reaction, which drains the end-points of metabolism at a rate proportional to the growth rate. The biomass demand usually also incorporates a growth-associated maintenance (GAM) reaction, burning ATP at a rate proportional to the growth rate. For accurate modelling a non-growth-associated maintenance (NGAM) reaction may also be included, setting a minimum burn rate of ATP for the organism to remain viable. In order to replenish the metabolic network with raw materials (eg carbon / energy sources), additional exchange reactions are usually added. Exchange and demand reactions are unbalanced chemical reactions of the form 'A ↔ ' which effectively release the corresponding metabolite from the flux balance condition. By convention, a negative flux through an exchange reaction represents uptake of the corresponding metabolite, and a positive flux represent discharge. Schematically, the typical set-up is as follows:

exchange reactions ↔ metabolic network → biomass demand reaction

In most applications of CBM, exchange reactions are provided only for extra-organism metabolites, and explicit transporter reactions are included in the model to allow metabolites to enter the organism.

Constraint-based modelling has been most often applied to micro-organisms such as bacteria and yeast. For these systems it makes sense that the network adjusts itself to maximise the growth rate, in other words the flux through the biomass demand reaction. In fact, in continuous culture (for example a chemostat) one can show that the fastest-growing organism always comes to dominate the system. Moreover, whilst in principle all the reaction fluxes in the network are 'capped' by maximum values, in practice what is done is to limit the uptake rate of key metabolites in the exchange reactions. This is typically done by specifying finite negative lower bounds for certain of the exchange reaction fluxes, as in Eq. (5) below (recall that a negative flux corresponds to uptake).

The mathematics of constraint-based modelling

Underpinning the flux balance condition is the so-called stoichiometry matrix S_iα which specifies the number of moles of the i-th metabolite which is consumed by the α-th reaction. If the flux through the α-th reaction is v_α, then the flux balance condition is

Σ_α S_iα v_α = 0.

(1)

The fluxes are constrained by

l_α ≤ v_α ≤ u_α

(2)

where l_α and u_α are the lower and upper bounds for the flux, for example

0 ≤ v_α ≤ ∞	(irreversible reaction),	(3)
−∞ ≤ v_α ≤ ∞	(reversible reaction),	(4)
−v_m ≤ v_α ≤ ∞	(limiting exchange reaction, with v_m > 0).	(5)

Also, typically, one of the reactions is a biomass demand reaction with a flux v_biomass. The stoichiometry coefficients for this reaction, S_i,biomass, specify the number of moles per unit biomass of the i-th metabolite which are required for growth (for example the units could be millimoles per gram-dry-weight or mmol / gDW). Let's suppose that the units of l_α and u_α are moles per unit biomass per unit time (eg mmol / (gDW-hr). Then the units of v_biomass are an inverse time (eg hr⁻¹), exactly equal to the specific growth rate, so that ln 2 / v_biomass is the doubling time. Eqs. (1)–(5) are all linear constraints in the space of reaction fluxes v_α. A typical problem is to maximise one of the fluxes, most usually v_biomass, subject to these constraints. This is a linear programming (LP) problem, and it is precisely this kind of problem that SunlightCBM is designed to solve.

The Dual Problem

Every linear programming (LP) problem has a so-called dual problem. In the present case, in the primal LP problem the unknowns are the fluxes through the various reactions. In the dual LP problem, the unknowns are so-called shadow prices, associated with metabolites. Palsson has a nice discussion on shadow prices, and their use in classifying solution spaces, in his book. Briefly, the shadow price is the rate at which the target function (for example the flux through the biomass demand reaction) changes per unit increase in the availability of the metabolite in question. Some more insights into shadow prices are contained in our publications :

Flux networks in metabolic graphs, P. B. Warren, S. M. Duarte Queiros and J. L. Jones, Phys. Biol. 6, 046006 (2009); see also arXiv:0808.0321.
Duality, thermodynamics, and the linear programming problem in constraint-based models of metabolism, P. B. Warren and J. L. Jones, Phys. Rev. Lett. 99, 108101 (2007); see also q-bio/0702021.

The reduced cost is a quantity similar to the shadow price, but associated with reactions rather than metabolites. For example, the exchange reaction with the non-zero reduced cost will be the one limiting growth. With experience, shadow prices and reduced costs can give as much insight as the actual pattern of reaction fluxes. One should always beware though that the set of constraints may not uniquely constrain the optimal solution, so one should always check for flux variability before drawing conclusions.