The evolution of sexual reproduction as a repair mechanism part II. Mathematical treatment of the wheel model and its significance for real systems

Abstract

The dynamics of populations of self-replicating, hierarchically structured individuals, exposed to accidents which destroy their sub-units, is analyzed mathematically, specifically with regard to the roles of redundancy and sexual repair. The following points emerge from this analysis: 1. A population of individuals with redundant sub-structure has no intrinsic steady-state point; it tends to either zero or infinity depending on a critical accident rate alpha c. 2. Increased redundancy renders populations less accident prone initially, but population decline is steeper if alpha is greater than a fixed value alpha d. 3. Periodic, sexual repair at system-specific intervals prevents continuous decline and stabilizes the population insofar as it will now oscillate between two fixed population levels. 4. The stabilizing sexual interval increases with increased complexity provided this is accompanied by appropriate levels of redundancy. 5. The model closely simulates the dynamics of heterosis effects. 6. Repair fitness is a population fitness: the chance of an individual being repaired is a function of the statistical make-up of the population as a whole at the particular period. Populations living at alpha greater than alpha c either engage in sexual repair at the appropriate time or they die out. 7. The mathematical properties of the model illustrate mechanisms which possibly played a role in the evolution of a mortal soma in relation to sexual reproduction.

The pattern of population growth as a function of redundancy and repair

“…It is biologically significant that each population oscillation is characterized by a "youth phase" during which more individuals are reproduced than die, and by an "aged phase", during which more individual~ die than are reproduced. Defining the individuals as cells, the possible significance of this feature for the evolution of mortal, somatic tissues, and for the mode of growth in Ciliata, is discussed in Williams and Walker (1978). Their solutions lead to the same conclusions with regard to redundancy and repair as the solutions of eqs.…”

“…A decade ago Walker (1978) and Williams and Walker (1978) found in their statistical models of sexual and asexual reproduction that populations of individuals with redundant substructure could not be stabilized by any possible rate of accidents to the subunits of individuals; populations either kept growing or they died out. In the following we investigate the general dynamics of populations as a function of redundancy and repair in the substructure of their individuals, with (binary) fission as the only mode of reproduction.…”

The mechanical properties of proteins determine the laws of evolutionary change

“…In this case invariant phenotype would be the result of efficient DNA-repair which accelerates homozygosity of these phenotypes which are already favoured by selection (Walker, 1978;Williams and Walker, 1978).…”

“…This must lead to precarious chances of survival in the course of evolution and I am convinced that this pityless law is the reason for the evolution of sexual reproduction in connection with diploidy, redundancy and DNA-repair. The process creates, accelerates and maintains homozygosity of essential sites (Walker, 1978;Williams and Walker, 1978).…”

“…Evidence for the reality of DNA repair in connection with sexual reproduction and diploidy was given and discussed in extenso by Walker (1978) and Williams and Walker (1978).…”