Thermodynamics of unicellular life: Entropy production rate as function of the balanced growth rate

Maarten J. Droste; Maaike Remeijer; Robert Planqué; Frank J. Bruggeman

doi:10.1101/2024.07.13.603363

Abstract

In isothermal chemical reaction networks, reaction rates depend solely on the reactant concentrations setting their thermodynamic driving force. Living cells can, in addition, alter reaction rates in their enzyme-catalysed networks by changing enzyme concentrations. This gives them control over their metabolic activities, as function of conditions. Thermodynamics dictates that the steady-state entropy production rate (EPR) of an isothermal chemical reaction network rises with its reaction rates. Here we ask whether microbial cells that change their metabolism as function of growth rate can break this relation by shifting to a metabolism with a lower thermodynamic driving force at faster growth.

We address this problem by focussing on balanced microbial growth in chemostats. Since the driving force can then be determined and the growth rate can be set, chemostats allow for the calculation of the (specific) EPR.

First we prove that the EPR of a steady-state chemical reaction network rises with its driving force. Next, we study an example metabolic network with enzyme-catalysed reactions to illustrate that maximisation of specific flux can indeed lead to selection of a pathway with a lower driving force.

Following this idea, we investigate microbes that change their metabolic network responsible for catabolism from an energetically-efficient mode to a less efficient mode as function of their growth rate. This happens for instance during a shift from complete degradation of glucose at slow growth to partial degradation at fast growth. If partial degradation liberates less free energy, fast growth can occur at a reduced driving force and possibly a reduced EPR. We analyse these metabolic shifts using three models for chemostat cultivation of the yeast Saccharomyces cerevisiae that are calibrated with experimental data. We also derive a criterion to predict when EPR drops after a metabolic switch that generalises to other organisms. Both analyses gave however inconclusive results, as current experimental evidence proved insufficient. We indicate which experiments are required to get a better understanding of the behaviour of the EPR during metabolic shifts in unicellular organisms.

Competing Interest Statement

The authors have declared no competing interest.

Footnotes

https://github.com/MaartenJDroste/Entropy-Production-Rate

5. Glossary

b: Biomass concentration in the chemostat vessel.
: (Equilibrium) Concentration of chemical compound C_i. D: Dilution rate of a chemostat.
D_c: Critical growth/dilution rate after which a shift in metabolic strategies occurs.
D_f: Dilution rate at which an organism has replaced its metabolic strategy at slow growth completely by its strategy at fast growth.
D_max: Maximal dilution rate in a chemostat, above which wash-out of cells is faster than growth. For .
e_j: Concentration of enzyme j catalysing reaction j.
E_i: Elementary flux mode i of a metabolic network.
e_T: The concentration of all enzymes in a metabolic network summed together.
f_j(c^′): Saturation function of enzyme j catalysing reaction j.
j: Vector containing the number of moles of reactants consumed and/or produced in the reactions, obtained from normalising the flux vector Forward/backward (catalytic) rate constants of reaction j.
K_eq,j: Equilibrium constant of reaction j, directly related to its standard Gibbs energy dissipation.
: Monod saturation or affinity constant for carbon source S_c in the chemostat.
N: Stoichiometric matrix of a chemical reaction network with entries n_ij.
N^′: Stoichiometric matrix extended with stoichiometric coefficients of externally fixed substrate and product concentrations.
p_l: Concentration of product P_l in the chemostat vessel.
q_i/B: Uptake or excretion rate of compound C_i per mole biomass per hour.
S_c: Carbon source, usually limiting growth.
s_k: Concentration of substrate S_k in the chemostat vessel. s_R,k: Concentration of substrate S_k in the reservoir medium. R: The universal gas constant, 8.314 J/(mol K).
T: Temperature of the system.
v_j: Rate of reaction j.
: Forward/backward reaction rates.
X: The thermodynamic driving force of a reaction or pathway per mole, which equals minus the Gibbs free energy potential.
Y_i/B: Yield in moles of metabolite C_i per mole biomass.
α_i: Conic coefficients for a flux vector decomposition in terms of the EFMs of the reaction network.
α(D): Mixing function representing the fraction of resources invested in fermentation.
Δμ: The Gibbs free energy potential per mole of a reaction or pathway.
: The (biological) standard Gibbs free energy potential per mole of a reaction or pathway.
λ: Growth rate of a microbial culture.
λ_max: Maximal growth rate (during batch cultivation).
: Gibbs free energy per mole of compound C_i.
: Standard Gibbs free energy per mole of compound C_i.
Φ: Entropy production rate (EPR).
ϕ: Specific entropy production rate (sEPR) scaled with temperature.
ϕ_Approx: Approximation of the sEPR that neglects concentration effects.

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