The Good Symbiont

A self-reproducing cycle has the fundamental organization, A + X −→ 2A, and is autocatalytic, i.e. the products catalyze the formation of the products. The rate of increase of A is proportional to A, i.e. exponential. Asexual living entities often grow exponentially when resources are abundant, and decay exponentially when resources are scarce, according to autocatalytic kinetics. If two previously independently replicating autocatalytic entities can form a physical union that is still capable of autocatalysis but with a reduced decay rate, then the symbiosis can be viable in an environment in which resources have been depleted, even if the symbiont has a lower growth rate than either of its component particles. A good symbiont possesses the following features: i. low steric hindrance between components, ii. policing of defection or cheating by symbiont components. iii. low decay rate back to components. iv. absence of emergence of active sites susceptible to decay reactions. v. high rate of the final reproductive step. Failure to form stable symbiosis can result from deficits in any of these features, and is a problem central to the origin of both metabolism and template replication. 1 Recycling Autocatalytic Systems For persistence of the biosphere, the net ‘biosphere metabolism’ must be recycling [22]. Cells are autocatalytic systems, i.e. fluid automata composed of coupled autocatalytic cycles, at a stoichiometric level, that underly enzymatic catalysis [5]. Therefore we must explain how recycling autocatalytic systems evolved. This paper develops the work of G.A.M King on symbiosis of autocatalysts [1][2][3], by isolating 5 properties that a successful chemical symbiotic event requires. Autocatalytic cycles must not be confused with ‘autocatalytic sets’, or ‘reflexively autocatalytic systems’, capable of ‘collective autocatalysis’ [7] [8], nor with other models of higher-order replicators [10][9][11]. Autocatalytic cycles consist of stoichiometricly described entities, consisting of small intermediates. Chemically feasible autocatalytic cycles of chemicals exist, for example, the Formose cycle [12], a primitive form of reductive citric acid cycle [14], and the reductive citric acid cycle [15], but only the formose cycle ‘bioid’ has been implemented [13]. No reflexively autocatalytic system is known to exist unless superimposed upon an autocatalytic cycle. Even recent models of Kaufmann type autocatalytic sets ignore the problem of depleting side-reactions [16], which would turn otherwise neat autocatalytic sets into tar [17]. This problem must be faced head on when tackling autocatalytic cycles [18] [19]. ’Chemical’ symbiosis has been explored recently by Fontana and Buss who describe a ‘parasite route’ to organization, where parasite ‘autocatalytic cycles’ incompletely deplete a core cycle. However, the likelihood of such a mechanism cannot be addressed with λ-calculus alone[20]. The aim here is not to contribute to the very sophisticated work on symbiosis [21], but merely to confirm that symbiosis of autocatalytic cycles is what needs to be explained. Consider the simple recycling system containing an autocatalytic replicator in the top right of figure 1. A is the growing state and XA is the reproducing state of the autocatalyst. This translates into the differential equations below. X ′(t) = −k1A(t)X(t) + k′ 1XA(t) + r1DA(t) (1) A′(t) = −k1A(t)X(t) + k′ 1XA(t) + 2k2XA(t)− 2k′ 2A(t) − d1A(t) (2) XA′(t) = k1A(t)X(t)− k′ 1XA(t)− k2XA(t) + k′ 2A(t) (3) DA′(t) = d1A(t)− r1DA(t) (4) Figure 1 top left, shows that a stable equilibrium is reached in a recycling system. If the rate of recycling, r1, is increased then the constituents of the cycle persist at a higher concentration. There is a fixed point at