We have analyzed and compared dynamic and constraint-based formulations of the same model for the central carbon metabolism of E. coli
(Chassagnole et al., 2002
). The constraint-based version does not account for metabolite concentrations, and it does not express transient behavior. Therefore, the formulations can only be compared in their common domain, which is the steady-state flux distribution.
The constraint-based model defines a solution space for the steady-state flux distribution (called the flux cone). This space is difficult to visualize due to its high dimensionality. We addressed this problem by developing sampling and projection approaches that facilitate the visualization of the shape of the solution space.
The steady state of the dynamic model contains the same constraints as the constraint-based model (stoichiometry, thermodynamic reversibility, and maximum uptake rates) and also any additional constraints imposed by the kinetic rate laws, kinetic parameters, and initial metabolite concentrations. Therefore, its solution space is a subset of the constraint-based solution space.
For a predefined set of initial conditions and parameter values, the dynamic model usually determines one steady-state solution. In fact, the initial metabolite concentrations of dynamic models determine their transient behavior, but, for the steady-state flux determination, they serve only to determine which steady state is chosen in the case of multistability. In this case, sampling the metabolite concentration space revealed a second steady-state characterized by a flux distribution with lower values of the fluxes and an accumulation of external glucose.
Instead, we also verified, as expected, that the location of the steady-state solution(s) inside the solution space is determined by the kinetic parameters, because by varying the kinetic parameters, the solution moves inside the solution space. The sampling of the kinetic parameter space revealed that, with unconstrained parameter values, the solutions of the dynamic model cover the whole steady-state solution space identified by the constraint-based model. This overlapping may seem unintuitive, as one would expect the rate laws to impose one additional layer of constraint into the steady-state solution space. However, besides having observed this with our sampling approaches, we also observe that, given any valid steady-state flux distribution, one can find kinetic parameter values that make the rate laws produce those steady-state flux values by solving each equation separately. This separation is only possible because the parameters are specific for each rate law, which defines a partition over the parameter set. The running example contains an average of 4 parameters per rate law, yielding many degrees of freedom for each equation. Thus, it is not surprising that, generally, parameter values can be found that satisfy the equations.
Interestingly, we found that by varying only one class of rate constants (Vmax
), the dynamical model formulation was able to achieve all of the same steady states as the constraint-based model (data not shown). This is an important observation, because it suggests that by changing only the expression levels of proteins ([E
’s), which can be achieved through regulation, a cell can adapt to reach essentially any possible steady state, without the need to introduce mutations that change rate constants. This observation reflects the adaptability of cell under different conditions and is in agreement with observations that microorganisms can undergo adaptive evolution to attain their optimal theoretical yields when placed under conditions where they originally performed sub-optimally (Ibarra et al., 2002
The observations stated above show that, in theory, a dynamic model can be fitted to any steady-state flux distribution inside the constraint-based solution space. However, there are physical limitations to the values of the kinetic parameters. Also, by querying parameter databases such as BRENDA (Schomburg et al., 2002
) and SABIO-RK (Rojas et al., 2007
), it is possible to observe that for each kinetic parameter there is a range of possible values determined by experimental conditions (such as temperature and pH) in which the cells are able to grow. Therefore, we evaluated how the imposition of parameter ranges map into flux ranges within the steady-state solution space. Although the rate laws do not constrain the solution space by themselves, they influence the probability distribution of the steady-state solutions. This is evidenced by the imposition of the kinetic parameter constraints. As the constraints become tighter, the solutions of lower probability disappear and the reachable solution space becomes smaller. Our results show that the impact of these constraints depends on the size of the solution space of the genome-scale model, which is mainly determined by the uptake rate of the limiting substrates, and on the allowable ranges of the kinetic parameters in the dynamic model.
The subset of the solution spaced obtained by constrained variation of kinetic parameters reveals that it is possible to map parameter ranges into flux ranges. This can be performed by sampling the parameter space and determining the respective steady-states. The generated flux ranges can be directly added into the FBA formulation as flux bounds. A similar sampling procedure, although with a different goal, is performed in the ensemble modeling approach (Tan et al., 2010