Minimal Replicator Theory I: Parabolic Versus Exponential Growth

Abstract

The paper describes an analytical treatment of artificial self-replicating systems. All artificial self-replicating systems known today are minimal replicators in the sense that their kinetic behavior can be rationalized by a common, minimal reaction model which is outlined in the introduction. In the second section, empirical rate equations are introduced which have proven useful for the evaluation of experimental concentration-time profiles. The third section begins with an discussion of reaction models which have been described earlier to explain the autocatalytic synthesis of self-replicating template molecules. It is followed by an analytical treatment of the minimal reaction model: A + B + C⇌ABC→ C ₂;⇌ 2C, where C is a self-complementary template molecule, A and B its precursor molecules, ABC a termolecular complex, and C ₂ a template duplex. It is assumed that the irreversible formation of C ₂ from ABC is the rate limiting step and that the total template concentration is small as compared to its precursors. The analytical expressions derived allow us to estimate the rate and autocatalytic reaction order for synthetic self-replicating systems from the elementary rate and equilibria constants involved. Three limit growth laws for minimal self-replicating systems—termed as parabolic, weak exponential, and strong exponential—can be distinguished. The following section deals with the influence of temperature. Strong exponential growth is to be expected for low temperatures, whereas weak exponential growth should occur at high temperatures. Parabolic growth is expected for average temperatures. Depending on the activation energy of the irreversible step as well as on the enthalpies of the formation of ABC and C ₂, the maximum of the autocatalytic rate occurs either at the temperature of the transition from strong exponential to parabolic growth, or, at the temperature of transition from parabolic to weak exponential growth, or, at an average temperature. The analytical results from the treatment of the above minimal reaction model are then compared to results from more realistic models. In particular, it is shown that the formation of a complex AB from A and B makes it difficult to observe strong exponential growth which otherwise might be found at low temperatures.