Division Strategies Determine Growth and Viability of Growing-Dividing Autocatalytic Systems

Abstract

We present a geometric framework to study the growth-division dynamics of cells and protocells, and demonstrate that self-reproduction emerges only when a system's growth dynamics and division strategy are mutually compatible. Using several commonly used models (the linear Hinshelwood cycle and non-linear 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 diverges to infinity or collapses to zero over successive generations). In particular, we show that cellular trajectories in an N-dimensional phase space (where N is the number of distinct chemical species) shuttle between two N-1 dimensional surfaces - a division surface and a birth surface - and that the relationship between these surfaces and the growth trajectories determine something as fundamental as cellular homeostasis and self-reproduction. Geometrically visualizing bacterial growth and division uncovers, for the first time, the type of division processes that sustain or destroy cellular homeostasis in autocatalytic chemical systems, thereby offering strategies to stabilize or destabilize growing-dividing systems. This reveals that, in addition to autocatalysis of growth, division mechanisms are not passive bystanders but active determinants of a growing-dividing system's long-term fate. Our work thus provides a framework for further exploration of growing dividing systems that will aid in the design of self-reproducing synthetic cells.

Figures (2)

• Figure 1: Exponential or non-exponential balanced growth of the growing-dividing Hinshelwood 2-cycle with different division mechanisms. Parameter values: $k_1=1$, $k_2=25$. (a), (b) and (c) show trajectories as plots of $X$ and $Y$ versus $t$. (a) Stable exponential GDSS reached with division control variable $D=X$, division threshold $d=200$ and standard birth map $B(X,Y) = (X/2, Y/2)$. IC: $(X,Y) = (190,30)$. (b) Stable exponential GDSS reached with $D=Y/X^{1/3}$, $d=100$ and standard birth map. IC: (30,290). (c) Stable non-exponential GDSS reached with $D=X$, $d=200$ and a non-standard birth map $B(X,Y)=(20,Y/2)$. IC: (160,30). Notice that the growth trajectories become exponential (straight line segments in a semi-log plot) in (a) and (b) at the steady-state with the expected rate $\lambda = \sqrt{k_1k_2}=5$. But in (c) the steady-state trajectories are not exponential. (d), (e) and (f) plot the same trajectories in phase space for (a), (b) and (c) respectively. Orange lines with circles and blue arrows are growth trajectories. Orange lines with red arrows are instantaneous jumps at division. The division surface, $S_D$, is the red curve $D(X,Y)$=$d$. The birth surface, $S_B$ is the green curve. The blue line is the asymptotic growth trajectory, AGT. The growth-division trajectory starts at ${\bf z}_0$ and then passes through the sequence of points ${\bf z}^M_0$, ${\bf z}_1$, ${\bf z}^M_1$, ${\bf z}_2$,... which alternately lie on $S_D$ and $S_B$. The trajectory converges to the limit cycle ${\bf z}_\infty$, ${\bf z}^M_\infty$, ${\bf z}_\infty$, which is the GDSS. Note that for the standard birth map the GDSS lies on a segment of the AGT but for the non-standard birth map it does not.
• Figure 2: Division processes when the growing-dividing Hinshelwood cycle (Eq. \ref{['eq:xy-ACS']}) fails to reach a self-reproducing state. In both figures the division process is defined by the intensive division variable $D=X/V$ (where $V=X+Y$), division threshold $d$, and the non-standard birth map $B(X,Y) = (100,Y/2)$. $S_D$ is a straight line (red) passing through the origin. Conventions are the same as in Fig. \ref{['fig:xy-autocatalytic-1']}. IC: (X,Y) = (100,2000) for both plots. $k_1=1$, $k_2=25$. (a) $d=0.14$. The system progressively becomes smaller after every division, and asymptotically shrinks to zero. (b) $d=0.16$. For the IC considered, the system progressively becomes larger after every division and asymptotically blows up to infinity. In (a) $m_D > m_{max}$. ($m_D \equiv (1-d)/d = 6.14$, $m_{max} \equiv (2/\sqrt{3})m_{A} = 5.77$.) In (b) $m_{A} < m_D < m_{max}$. ($m_{A} =\sqrt{k_2/k_1}= 5$, $m_D = 5.25$, $m_{max} = 5.77$.)