  Cells generally convert external nutrient resources to support metabolismand
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 non-equilibrium
thermodynamics. The minimal entropy production per unit cell growth is found to
be achieved at a non-zero nutrient uptake rate, rather than at a quasi-static
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.
