Complex dynamics in a cross-catalytic self-replication mechanism

Abstract

The authors consider a minimal cross-catalytic self-replicating system of only two cross-catalytic templates that mimics the R3C ligase ribozyme system of Dong-Eu and Joyce [Chem. Biol. 11, 1505 (2004)]. This system displays considerably more complex dynamics than its self-replicating counterpart. In particular, the authors discuss the Poincare-Andronov-Hopf bifurcation, canard transitions, excitability, and hysteresis that yield birhythmicity between simple and complex oscillations.

Self-regulation in a minimal model of chemical self-replication

“…Notice that the equation's order depends on m and n, and that a positive real solution cannot be obtained for r ≥ 1. The latter observation is consistent with our previous analysis [22][23][24][25][26][27][28][29][30], and it means that the influx of reagents is greater that the outflux, leading to a divergence at r = 1, and complex or negative real solutions for r > 1. Second, we consider some of the possible cases of repressed or activated regulation by considering values of m, n, and β.…”

“…In the next section, we consider dimensionless parameter values for k u , and K that we have used in other studies [23][24][25][26][27][28][29][30]. Our parameter selection is based on our analysis of chemical self-replication, where parameters are determined by fitting experimental data published by Paul and Joyce [46] and Lincoln and Joyce [47] for self-replicating ribozymes.…”

“…In this section, we first consider the model of chemical self-replication [21] that we have studied in the past [22][23][24][25][26][27][28][29][30]. After we describe the model, we include a particular regulatory mechanism, which has been considered by others to model genetic regulation [15][16][17][18][19].…”

“…The diagram in Fig. 4 had been studied at length elsewhere [22][23][24][25][26][27][28][29][30], and corresponds to the case of the chemical pool approximation. As usual, the thin light line represents unstable steady states and the darker line represents stable solutions.…”

Extending the zero-derivative principle for slow–fast dynamical systems

“…Such systems can be written in the general form (1) dz dt = F (z, ε), z ∈ R N , where 0 < ε 1 is a small parameter. In this paper, we wish to study slow-fast systems in the general form given by equation (1) assuming that, for small enough ε and after some "manipulations" (change of time and/or variables), one can rewrite this system in the form of a slow-fast dynamical system with an explicit splitting of timescales, that is, dx dt = f (x, y, ε), x ∈ R n f , (2a) dy dt = εg(x, y, ε), y ∈ R ns , (2b)…”