Adding Thermodynamic Constraints to Metabolic Models

Learning Objectives

• Explain how metabolites stabilize at a set concentration during steady state flux balance solutions

• Explain what it means to constrain a model

• Understand how thermodynamics govern the directionality of a reaction

Lesson

Metabolic modeling relies heavily on flux balance analysis (FBA), a technique which describes the flow of metabolites through a network of reactions using mathematical representations[1]. In evaluating the flow of metabolites in a system, FBA identifies a solution in which all metabolites reach a stable concentration where input reaction fluxes and output reaction fluxes are balanced. This particular point of balance is known as steady state, and when a system is in steady state, the concentration of metabolites in the system is unchanged.

The ability of FBA to converge on a steady solution is only possible because of the primary constraint imposed on the model: mass cannot be created or destroyed within the system, and therefore a mass balance must be maintained for the sum of all of the network reactions[1]. This can be represented mathematically by:

Where S is the stoichiometry matrix, which defines the amount of all metabolites consumed or generated in all reactions in the network, and where v is the vector of flux through every reaction in the network. Thus, the vector v is the solution from FBA that satisfies Equation 1.

Where S is the stoichiometry matrix, which defines the amount of all metabolites consumed or generated in all reactions in the network, and where v is the vector of flux through every reaction in the network. Thus, the vector v is the solution from FBA that satisfies Equation 1.

While the mass balance constraint is required for FBA to function, it is not the only constraint that can be imposed. In fact, adding additional constraints can narrow the design space of the model, limiting the chance of the model landing in a local or trivial solution and letting the model reflect better the reality of the metabolism it represents. Many models include upper and lower bound constraints of the flux values for each reaction, and these may be set based on experimental evidence. These kinds of flux constraints can be related to the transport of metabolites in and out of the system[2]. 

Generating a mathematical model does not guarantee that a model represents a physical reality. Adding constraints to an FBA model can ensure that the model does not converge on unrealistic solutions. One such constraint comes from adding thermodynamics to the model. The thermodynamic favorability of a reaction can be defined by the Gibbs free energy of the reaction (ΔG0rxn). When ΔG0rxn< 0 kJ/mol, we call the reaction spontaneous. When ΔG0rxn> 0 kJ/mol, we say that this reaction is not spontaneous and even that the reverse reaction might take place. The value of ΔG0rxn can vary depending on the concentration of the reactants and products in a particular system. A build-up of reactants can lead to a more negative ΔG0rxn, while high concentrations of products can make the ΔG0rxn more positive (not spontaneous). Thus there is a range of ΔG0rxn where a reaction can be “reversible”; that is, depending on the concentrations of reactants and products, the reaction can proceed either forwards or backwards.

When a metabolic model is built, reactions are typically defined rigidly as irreversibly in the forward or reverse direction or as freely reversible. Because this definition can be variable according to metabolite concentration, incorporating ΔG0rxn as a constraint on reactions can be very powerful in ensuring that the model is depicting realistic behavior of the network of reactions[3]. The free energy of reaction can be incorporated into the model constraints by requiring that any forward reaction must satisfy:

The reverse inequality must be true for any reverse reaction. Here, R is the gas constant, T is the temperature of the system, N is the number of species in the model, ni is the stoichiometric coefficient of the species for this reaction, and xi is the concentration of the species in the system. This additional constraint further shrinks the design space in the same way as the mass balance constraint. Now the only viable solutions to the model are ones that both satisfy a mass balance of the whole system and predict that any reaction flux is spontaneous.

Including thermodynamics in a model opens some intriguing applications for metabolic models. Flux variance analysis is still possible for these models, which evaluates the feasible range of fluxes that still allow for growth. However, including thermodynamics shows the physiological range of metabolite concentrations that governs any reaction. Increasing or decreasing the concentration of a metabolite can fundamentally alter the overall flow of the model, redirecting flux through different pathways depending on the concentration. This control over the model may be used to identify of points of regulation in a biological system[4]. It can also be used to predict the feasibility of an engineered pathway in an organism and what the upper concentration limit of production through that pathway might be[5].

In some cases, an unconstrained model may predict closed-loop cycles of some reactions. These loops may not generate any benefit to the biomass optimization function, but they satisfy the optimization mathematically. However, such a cycle at steady state violates the second law of thermodynamics. For a closed loop to exist at steady state, the metabolites and reactions would all have to be at equilibrium. Therefore, the flux through each of those loop reactions must be zero. Having flux through a closed-loop implies that entropy is decreasing, thus violating the second law. Adding in thermodynamic constraints can effectively break up these infeasible loops by removing them from the allowable solution space[6].

Thermodynamic constrains alone do not govern whether a model is “good” or not. Even a model which incorporates thermodynamics but does not accurately annotate the reactions inside of a biological system can predict unrealistic results. However, thermodynamics do go a long way towards grounding a model in reality and having it validated by experimental results.

Concept Questions

1. How does a flux balance model determine the metabolite concentrations in its solution?

   The model predicts a steady state solution, which indicates that metabolite concentrations are not changing under the conditions of the solution. This is only possible when the total input reaction flux is balanced by the total output reaction flux. When this condition is satisfied for every metabolite, then the solution is accepted.

2. How can a reaction favorability change from forward to reverse in a flux balance model?

   Reaction favorability is based on Gibb’s free energy of reaction, which changes depending on the concentration of the reactants and products. As the FBA model searches for a steady state solution, the metabolite concentrations are adjusted. Changing these values changes the thermodynamics of every reaction they’re involved in. Thus reaction favorability can switch between forward and reverse as the metabolite concentration changes.Points of regulation are often tied to the concentrations of target metabolites. Changes in regulation can cause a build-up or consumption of certain metabolites. If a change in concentration affects the reversibility of a particular reaction, it could cause flux to shuttle through other competing pathways or through its own pathway. Therefore, looking at the thermodynamics of reactions in an FBA solution can indicate whether or not a particular reaction has changed its reversibility. This change indicates that the cell can manipulate competing pathways by regulating enzyme production at this point.

3. How can points of regulation be identified in a flux balance model?References

   1. Orth, J., I. Thiele, and B. Palsson. “What is flux balance analysis?” 2010. Nature Biotechnology, 28(3):245-248.

   2. Schilling, C., J. Edwards, D. Letscher, and B. Palsson. “Combining Pathway Analysis with Flux Balance Analysis for the Comprehensive Study of Metabolic Systems.” 2000. Biotechnology and Bioengineering, 71(4):286-306.

   3. Kummel, A., S. Panke, and M. Heinemann. “Systematic assignment of thermodynamic constraints in metabolic network models.” BMC Bioinformatics, 7:512.

   4. Henry, C., L. Broadbelt, and V. Hatzimanikatis. “Thermodynamics-Based Metabolic Flux Analysis.” 2007. Biophysical Journal, 92(5):1792-1805.

   5. Ataman, M. and V. Hatzimanikatis. “Heading in the right direction: thermodynamics-based network analysis and pathway engineering.” Current Opinion in Biotechnology, 36:176-182.

   Schellenberger, J., N. Lewis, and B. Palsson. “Elimination of Thermodynamically Infeasible Loops in Steady-State Metabolic Models.” Biophysical Journal, 100:544-553.