Entropy production of a steady-growth cell with catalytic reactions

Cells generally convert external nutrient resources to support metabolism and growth. Understanding the thermodynamic efficiency of this conversion is essential to determine the general characteristics of cellular growth. Using a simple protocell model with catalytic reaction dynamics to synthesize the necessary enzyme and membrane components from nutrients, the entropy production per unit-cell-volume growth is calculated analytically and numerically based on the rate equation for chemical kinetics and linear nonequilibrium thermodynamics. The minimal entropy production per unit-cell growth is found to be achieved at a nonzero nutrient uptake rate rather than at a quasistatic limit as in the standard Carnot engine. This difference appears because the equilibration mediated by the enzyme exists only within cells that grow through enzyme and membrane synthesis. Optimal nutrient uptake is also confirmed by protocell models with many chemical components synthesized through a catalytic reaction network. The possible relevance of the identified optimal uptake to optimal yield for cellular growth is also discussed.

Figure 1

Schematic representation of our three-component protocell model. Here N, M, and E denote nutrient, membrane precursor, and enzyme, respectively. The nutrient is taken up from the extracellular nutrient pool by diffusion, indicated by a blue arrow. All chemical reactions, indicated by black solid arrows, are reversible and catalyzed by the enzyme, as indicated by dashed arrows. Membrane precursors are transformed to the membrane as indicated by the green ring with some leaks. The membrane growth results in an increase in cell volume.

Figure 2

Logarithm of $\tilde{\eta }$ plotted as a function of scaled nutrient concentration $\tilde{r}$ and $\tilde{k}=k/l$, the ratio between two rate constants, with the color code given in the side bar. It is calculated from the solutions of Eq. (4). The parameter $\tilde{\kappa }$ is chosen to be $1.0$. For given $\tilde{k}$, there is an optimal nutrient concentration that gives the minimum $\tilde{\eta }$.

Figure 3

Logarithm of the rescaled entropy production rate per unit rescaled growth rate $\widetilde{{\sigma }_x}/\tilde{\lambda }$ and $\widetilde{{\sigma }_y}/\tilde{\lambda }$ for the enzyme and membrane precursor synthesis reactions, respectively, plotted as a function of the rescaled nutrient concentration $\tilde{r}$ and the ratio between rate constants $\tilde{k}$, computed by Eq. (4): (a) $\widetilde{{\sigma }_x}/\tilde{\lambda }$ for the enzyme producing reaction and (b) $\widetilde{{\sigma }_y}/\tilde{\lambda }$ for the membrane precursor producing reaction.

Figure 4

Entropy production plotted as a function of the external nutrient concentration $r_{\text{ext}}$ and the rate constant $k \left(=l\right)$, calculated from the fixed-point solution of Eq. (6): (a) the logarithm of total entropy production per unit-cell growth $\eta$ and (b) the logarithm of the entropy production per unit growth by material flow ${\sigma }_{\text{mf}}/\lambda$ and dilution ${\sigma }_d/\lambda$ only. The parameters are chosen to be ${\kappa }_x=1.0,{\kappa }_y=1.0,D=1.0,\varphi =1.0,\gamma =1.0$, and $l=k$.

Figure 5

Thermodynamic efficiency for the model (6) plotted as a function of the external nutrient concentration $r_{\text{ext}}$ and the rate constant $k \left(=l\right)$. The parameters are set as ${\mu }_r=0.0,D=1.0,\varphi =1.0,\gamma =1.0$, and ${\kappa }_x={\kappa }_y=1.0$. The standard concentrations are chosen to be ${10}^{-8}$.

Figure 6

Entropy production and deviation from equilibrium calculated from the steady-state solution of the multicomponent model (7) plotted as a function of the external nutrient concentration: (a) entropy production rate per growth rate $\eta$ and (b) Kullback-Leibler divergence of the steady-state distribution from the Boltzmann distribution defined in Eq. (8). The results of ten randomly generated networks are overlaid. The number of chemical species is set as 100, whereas the parameter $\varphi$ is chosen to be unity, and the ratio of the average number of reactions to the number of chemical species is set to 3.