SunlightCBM
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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 Siα
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 vα, then
the flux balance condition is
The fluxes are constrained by
where lα
and
uα
are the lower and upper bounds for the flux, for example
0
≤ vα
≤ ∞ |
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(irreversible reaction),
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(3)
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−∞
≤ vα
≤ ∞ |
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(reversible reaction),
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(4)
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−vm
≤ vα
≤ ∞ |
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(limiting exchange reaction,
with vm
> 0). |
(5)
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Also, typically, one of the reactions is a biomass demand reaction with
a flux vbiomass.
The
stoichiometry
coefficients for this reaction, Si,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 lα
and
uα
are moles per unit biomass per unit time (eg mmol / (gDW-hr).
Then
the units of vbiomass
are an inverse time (eg hr−1), exactly equal
to the specific growth rate, so that ln
2 / vbiomass
is the doubling time. Eqs. (1)–(5) are all linear constraints in
the space of reaction fluxes vα.
A
typical
problem is to maximise one of the fluxes, most usually vbiomass,
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 :
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.
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