Constraint-based analysis thermodynamic constraints

Analysis of biochemical systems, with their behaviors constrained by the known system stoichiometry, falls under the broad heading constraint-based analysis, a methodology that allows us to explore computationally metabolic fluxes and concentrations constrained by the physical chemical laws of mass conservation and thermodynamics. This chapter introduces the mathematical formulation of the constraints on reaction fluxes and reactant concentrations that arise from the stoichiometry of an integrated network and are the basis of constraint-based analysis. 

Despite its widely recognized limitations, flux balance analysis has resulted in a large number of successful applications [35, 67, 72 74], including several extensions and refinements. See Ref. [247] for a recent review. Of particular interest are recent efforts to augment the stoichiometric balance equations with thermodynamic constraints providing a link between concentration and flux in the constraint-based analysis of metabolic networks [74, 149, 150]. For a more comprehensive review, we refer to the very readable monograph of Palsson [50]. 

As we shall see, linear algebraic constraints arising from steady state mass balance form the basis of metabolic flux analysis (MFA) and flux balance analysis (FBA). Thermodynamic laws, while introducing inherent non-linearities into the mathematical description of the feasible flux space, allow determination of feasible reaction directions and facilitate the introduction of reactant concentrations to the constraint-based framework. 

Figure 9.6 Small network example to illustrate the application of thermodynamic constraints in constraint-based analysis.

Introducing the chemical potential (or free energy) and the thermodynamic constraint provides a solid physical chemistry foundation for the constraint-based analysis approach to metabolic systems analysis. Treatment of the network thermodynamics not only improves the accuracy of the predictions on the steady state fluxes, but can also be used to make predictions on the steady state concentrations of metabolites. To see this, we substitute the relation between reaction Gibbs free energy (ArG ) of the th reaction and the concentrations of biochemical reactants... 

One can view biochemical systems as represented at the most basic level as networks of given stoichiometry. Whether the steady state or the kinetic behavior is explored, the stoichiometry constrains the feasible behavior according to mass balance and the laws of thermodynamics. As we have seen in this chapter, some analysis is possible based solely on the stoichiometric structure of a given system. Mass balance provides linear constraints on reaction fluxes non-linear thermodynamic constraints provide information about feasible flux directions and reactant concentrations. 

Applying mass-balance and thermodynamic constraints typically leaves one without a precisely defined (unique) solution for reaction fluxes and reactant concentration, but instead with a mathematically constrained feasible space for these variables. Exploration of this feasible space is the purview of constraint-based analysis. It has so far been left unstated that any application in this area starts with the determination of the reactions in a system, from which the stoichiometric matrix arises. This first step, network reconstruction, integrates genomic and proteomic data to determine carefully the enzymes present in an organism, cell, or subcellular compartment. The network reconstruction process is described elsewhere [107]. 

Metabolic networks can be quantitatively and qualitatively studied without enzyme kinetic parameters by using a constraints-based approach. Metabolic networks must obey the fundamental physicochemical laws, such as mass, energy, redox balances, diffusion, and thermodynamics. Therefore, when kinetic constants are unavailable, cellular function can still be mathematically constrained based on the mass and energy balance. Flux balance analysis (FBA) is a mathematical modeling framework that can be used to study the steady-state metabolic capabilities of cell-based physicochemical constraints. ... 

Figure 26. The proposed workflow of structural kinetic modeling Rather than constructing a single kinetic model, an ensemble of possible models is evaluated, such that the ensemble is consistent with available biological information and additional constraints of interest. The analysis is based upon a (thermodynamically consistent) metabolic state, characterized by a vector S° and the associated flux v° v(S°). Since based only on the an evaluation of the eigenvalues of the Jacobian matrix are evaluated, the approach is (computationally) applicable to large scale system. Redrawn and adapted from Ref. 296.

The equality constraints composed of the mass and heat balances and the performance equations in each subsystem, thermodynamic properties of the flows, and specifications for design are represented by the functions h which are in the form of n equations with m+n variables. These equations are easily arranged in the order of precedence based on structural analysis. The number of independent variables (parameters), y, corresponds to the degrees of freedom in the system. When the value of the parameters is given, n equations are solved with respect to n variables, z. Thereupon, the inequality constraints, if any, are checked and the objective functions are calculated. Therefore, the problem is rewritten simply as follows ... 

Table 8.1 describes the steps of the methodology in more detail. The procedure starts with the Problem definition production rate, chemistry, product specifications, safety, health and environmental constraints, physical properties, available technologies. Then, a first evaluation of feasibility is performed by an equilibrium design. This is based on a thermodynamic analysis that includes simultaneous chemical and physical equilibrium (CPE). The investigation can be done directly by computer simulation, or in a more systematic way by building a residue curve map (RCM), as explained in the Appendix A. This step will identify additional thermodynamic experiments necessary to consolidate the design decisions, mainly phase-equilibrium measurements. Limitations set by chemical equilibrium or by thermodynamic boundaries should be analyzed here. 

Identifying constraints on reaction directions is essential for applications of metabolic flux analysis. However, in many applications the procedure used for determining reaction directions is not concretely defined. Typically, a subset of the reactions in a model is assigned as irreversible and the feasible directions are assigned based on information in pathway databases [59], In these applications, by treating certain reactions as implicitly unidirectional, biologically reasonable results can often be obtained without considering the system thermodynamics as outlined above. 

Thermodynamics plays an important role in the stability analysis of transport and rate processes, and the nonequilibrium thermodynamics approach in particular may enhance and broaden this role. This chapter reviews stability analysis based on the conventional Gibbs approach and tbe nonequilibrium thermodynamics theory. It considers the stability of equilibrium, near-equilibrium, and far-from-equilibrium states with some case studies. The entropy production approach for nonequilibrium systems appears to be more general for stability analysis. One major implication of the nonequilibrium thermodynamics theory is the introduction of distance from global equilibrium as a constraint for determining the stability of nonequilibrium systems. When a system is far from global equilibrium, the possibility of new organized structures of matter arise beyond an instability point. 

Second, we discuss front propagation in systems with multiple stationary states, again far from equilibrium. Consider a chemical system with two stable stationary states at given external constraints. Contact of two such systems, one in each of the two stable stationary states, leads to a front propagation of transition from the less to the more stable stationary state. We report on studies of this process by means of numerical solutions of reaction diffusion equations, experiments and a thermodynamic analysis of stability and relative stability based on the concept of excess work. 

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