Constraints-based Models

All of the formalisms that have been discussed so far are for the development of kinetic models. In order for any one of them to be applicable, knowledge of the intricate regulatory features of the system in question is required. Unfortunately, such information is normally difficult to obtain. Indeed, it is often the case that enzyme kinetic parameters are available

for only a limited number of model organisms. This, not even considering the fact that

KM values are typically derived from in vitro assays, a condition that casts suspicion

on their validity in vivo (Teusink et al., 2000; Theobald et al., 1997; Rizzi et al., 1997). Accordingly, kinetic models of metabolism have mostly been limited to relatively small networks.

Figure 4.1: Principles of constraint-based modeling. As the name suggests, the idea of constraints-based modeling is to begin with a large, unconstrained solution space, and then proceed by narrowing it down through the addition of constraints. A three-dimensional flux space for a given metabolic network is depicted here. Without any constraints the fluxes can take on any real value (left). After application of stoichiometric, thermodynamic and enzyme capacity constraints, the possible solutions are confined to a region in the total flux space (center), termed the allowable solution space. Any point outside of this space violates one or more of the applied constraints. Given that it is typically the case that even with all constraints in place the solution space is still large, particular solutions can be identified using linear programming by introducing cellular objective assumptions, such as optimal ATP or biomass production. The figure was taken from Reed et al., 2003.

In part due to the difficulties mentioned above, the constraints-based framework has emerged as a successful alternative to kinetic models. Rather than requiring detailed information that can be difficult to obtain, constraints-based models need only generally available physicochemical information such as stoichiometry, reversibility, energy balance, and, when available, reaction velocities (Edwards et al., 2000; Edwards and Palsson, 1999; Ramakrishna et al., 2001). As the name suggests, the principle of the framework is to begin with a large solution space, and then proceed by narrowing it down through the addition of constraints (Figure 4.1). To illustrate, consider a metabolic network with three reactions. Without any further information, the flux through each reaction can be any real number. However, if it is known that all three reactions are irreversible, then we can add the constraint that the flux through each must be greater than or equal to zero. Similarly,

if the exact flux through one of the reactions is known, for example through experimental measurement, then an equality constraint can be enforced for that reaction. Finally, given that it is typically the case with real networks that even with all the constraints in place the solution space is still large, particular solutions can be identified using linear programming by introducing cellular objective assumptions, such as optimality with respect to ATP or biomass production. This procedure is referred to as flux balance analysis.

Figure 4.2: A small illustrative network. It is composed of the four metabolites: A, B, C and D; and the six reactions r1, r2, r3, r4, r5 and r6. The box is used to differentiate between inter- and intracellular space.

Since the metabolic network for Halobacterium salinarum has 695 reactions (Section (3.4), it is not very convenient for illustrating the development of a stoichiometric matrix, which is central to constraints-based models. For this purpose, we use the small network depicted in Figure 4.2. The network is essentially a branching pathway consisting of four metabolites (A, B, C and D) and six reactions (r1 through r6). It can be conveniently represented as the stoichiometric matrix

S = r1 r2 r3 r4 r5 r6 A B C D ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 0 −1 1 0 0 −1 0 1 0 0 −1 0 1 0 0 −1 0 0 0 0 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.9)

where each row corresponds to a metabolite and each column to a reaction. The entries of S are the stoichiometric coefficients that define the relationships between the reactions and compounds. A positive value for_sij indicates that compound _i is produced in the left to right direction of reaction _j, while a negative value indicates that it is consumed. For example, column two of S is defined as [−1 1 0 0]T _{indicating that A is consumed and B is}

produced by reaction r2.

As discussed in the the previous chapter, metabolism is responsible for converting the available nutrients into usable energy and biomass (growth). In the case ofHalobacterium

salinarum, defined media used for the archaeon typically consist of amino acids. Therefore, in order for growth to be possible, its metabolism should be able to convert the amino acids into the unsupplied components of its biomass, such as nucleotides, vitamins, and other cofactors. The processes involved in these conversions can easily be translated into con- straints by writing material balance around the metabolic network, using its stoichiometric matrix S as in

S·_v=_b (4.10)

where _v is a vector of reaction rates, and _b is a vector containing the net metabolite concentration changes. Note that unlike in kinetic models where reaction velocities are directly computed from rate equations, metabolic fluxes are the unknown quantities that need to be determined in flux balance models.

In order to simulate growth, a pseudoreaction that is often called the “growth function” is added to the network. Formally, it is defined as a reaction where the reactants are cellular constituents, and the product is a unit of biomass. The stoichiometric coefficients used for the reactants are determined with respect to the unit of biomass used. For example, if the product is defined to be one gram of biomass, and it is known that this quantity on average contains_a moles of alanine, then_a is used as the stoichiometric coefficient of alanine. The growth function is added to the stoichiometric matrix S, just like any other reaction in the network. Although terms relating to energy maintenance are often integrated into the growth reaction, this was not done in this study because we investigate systems where growth rates vary with time.

As mentioned earlier, it is often the case with real networks that the available con- straints are not enough to determine a unique solution; i.e, the systems are frequently underdetermined. Nevertheless, particular solutions can be obtained using linear program- ming, using the assumption that cells optimize their metabolism with respect to certain objectives, such as growth and energy production. For example, the linear programming problem can be posed as

Maximize f =_vobj (4.11) Subject To

S·_v=_b (4.12)

−∞ ≤vj ≤+∞ where rj ∈Rrev (4.13)

vj =cj where rj ∈Rknown (4.15)

where _obj is the index of the objective reaction (e.g., the growth reaction), _Rrev is the set of reversible reactions, _Rirrev is the set of irreversible reactions, _Rknown is the set of reactions for which fluxes are already known, and_cj is the known flux through each reaction

in _Rknown. Note that metabolic flux models are based on the separation of time scales

between cellular growth rates and metabolic transients. That is, metabolism typically has transients that are shorter than a few minutes, while growth is typically measured in hours. Accordingly, metabolic fluxes are assumed to be in a quasi-steady state relative to growth.

The mechanisms by which metabolic flux distributions are chosen is a complex interplay of regulatory events at different levels. More often than not, only a limited subset of these are known in detail. Therefore, one may be surprised at the attempt of determining a unique solution for considerably underdetermined systems by just using an objective function. Nevertheless, the argument is that although the regulatory mechanisms for the most part are unknown, metabolic networks, because of evolutionary pressures, are adapted to work in concert to achieve “optimal metabolism”, so that the survivability of cells are enhanced. It is expected that wild-type microorganism strains have metabolic phenotypes that are defined by a tendency to optimize their growth rates, at least in the environments where they are found.

To investigate the extent to which optimality principles can describe the operation of metabolic networks, Schuetz and coworkers systematically evaluated flux balance predic- tions using 11 objective functions against actual in vivo fluxes. This was done in Es- cherichia coli using data derived from13_{C-based flux analysis (Schuetz et al., 2007; Sauer,}

2006). They reported that although no single objective adequately described the flux states under all conditions tested, they were able to identify two sets that allowed for biologi- cally meaningful predictions without the need for further, potentially artificial constraints. Specifically, they found that growth on glucose in oxygen or nitrate respiring batch cultures is best described by nonlinear maximization of the ATP yield per flux unit, and that linear maximization of the overall ATP or biomass yield achieves the highest predictive accuracy in continuous cultures under nutrient scarcity. It was concluded that the identified opti- mality principles reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states.

4.2

Methods

This study includes both computational and experimental work; a considerable fraction of the data that were used for computational analysis were generated here. The computational and experimental methods that were employed are described in this section.