  All cells derive nutrition by absorbing some chemical and energy resources
from the environment; these resources are used by the cells to reproduce the
chemicals within them, which in turn leads to an increase in their volume. In
this study, we introduce a protocell model exhibiting catalytic reaction
dynamics, energy metabolism, and cell growth. Results of extensive simulations
of this model show the existence of four phases with regard to the rates of
both the influx of resources and the cell growth. These phases include an
active phase with high influx and high growth rates, an inefficient phase with
high influx but low growth rates, a quasi-static phase with low influx and low
growth rates, and a death phase with negative growth rate. A mean field model
well explains the transition among these phases as bifurcations. The
statistical distribution of the active phase is characterized by a power law
and that of the inefficient phase is characterized by a nearly equilibrium
distribution. We also discuss the relevance of the results of this study to
distinct states in the existing cells.
