Geometry of Growing-Dividing Autocatalytic Dynamical Systems

Abstract: We formalize the structure of a class of mathematical models of growing-dividing autocatalytic systems demonstrating that self-reproduction emerges only if the system's 'growth dynamics' and 'division strategy' are mutually compatible. Using various models in this class (the linear Hinshelwood cycle and nonlinear coarse-grained models of protocells and bacteria), we show that depending on the chosen division mechanism, the same chemical system can exhibit either (i) balanced exponential growth, (ii) balanced nonexponential growth, or (iii) system death (where the system either explodes to infinity or collapses to zero in successive generations). We identify the class of division processes that lead to these three outcomes, offering strategies to stabilize or destabilize growing-dividing systems. Our work provides a geometric framework to further explore growing-dividing systems and will aid in the design of self-reproducing synthetic cells.