Concepts, challenges, and successes in modeling thermodynamics of metabolism

Abstract

The modeling of the chemical reactions involved in metabolism is a daunting task. Ideally, the modeling of metabolism would use kinetic simulations, but these simulations require knowledge of the thousands of rate constants involved in the reactions. The measurement of rate constants is very labor intensive, and hence rate constants for most enzymatic reactions are not available. Consequently, constraint-based flux modeling has been the method of choice because it does not require the use of the rate constants of the law of mass action. However, this convenience also limits the predictive power of constraint-based approaches in that the law of mass action is used only as a constraint, making it difficult to predict metabolite levels or energy requirements of pathways. An alternative to both of these approaches is to model metabolism using simulations of states rather than simulations of reactions, in which the state is defined as the set of all metabolite counts or concentrations. While kinetic simulations model reactions based on the likelihood of the reaction derived from the law of mass action, states are modeled based on likelihood ratios of mass action. Both approaches provide information on the energy requirements of metabolic reactions and pathways. However, modeling states rather than reactions has the advantage that the parameters needed to model states (chemical potentials) are much easier to determine than the parameters needed to model reactions (rate constants). Herein, we discuss recent results, assumptions, and issues in using simulations of state to model metabolism.

Keywords: adaptation; biological; fluctuation theory; metabolism; molecular motors; simulations; statistical thermodynamics; tricarboxylic acid cycle.

Figure 1

The tricarboxylic acid cycle (TCA) from E. coli. The enzymes catalyzing the reactions are shown in italics, the co-factors are shown tangentially to each respective reaction, and the reaction intermediates are shown in line with the cyclic reaction arrows indicating direction of the cycle for E. coli. Q and QH₂ are electron acceptor/donator pairs and are entry points to the electron transfer chain.

Figure 2

Thermodynamic profile of the TCA cycle from E. coli (Cannon, 2014). Eq. 4 was used to calculate the change in entropy ΔS, energy ΔE, and the log of the (unnormalized) mass density Δ𝒜. Because the probability mass density consists of a combinatorial coefficient that is represented by the entropy term and an energy-based (Boltzmann) probability, there is energy–entropy compensation throughout the cycle. Δ𝒜 changes smoothly across the reaction pathway indicating that the concentrations of the metabolites are close to optimal, likely because the concentrations were taken from an experimental measurement of E. coli metabolite levels.

Figure 3

Comparison of the thermodynamic profiles of the TCA cycles of E. coli, Synechococcus sp. PCC 7002 and Chlorobium tepidum. The free energy profile of the TCA cycle for each organism reflects its environmental niche (see Discussion).

Figure 4

(Top) The cumulative free energy profile of the E. coli TCA cycle as a function of the total concentration of the reaction intermediates. Although carbon can enter the cycle as acetyl-coa and leave as CO₂, the total number of intermediates is constrained by the overall reaction (see text). The concentrations used are 1-fold, 0.1-fold, 0.01-fold, and 0.001-fold of those reported by Bennett et al. (2009) for exponential growth on glucose. (Bottom) The distribution of reaction intermediates as a function of total concentration.