4.1 Phylogeny and Evolution

35

d Subscript normal upper H Baseline left parenthesis bold s comma bold s Superscript prime Baseline right parenthesis equals sigma summation Underscript i equals 1 Overscript upper N Endscripts StartFraction left parenthesis s Subscript i Baseline minus s prime Subscript i right parenthesis squared Over 4 EndFraction commadH(s, s^,) =

N

Σ

i=1

(si −s^,

i⁾²

4

,

(4.5)

and the overlap between two genomes bold ss and bold s primes^, is given by the related parameter

omega left parenthesis bold s comma bold s Superscript prime Baseline right parenthesis equals StartFraction 1 Over upper N EndFraction sigma summation Underscript i equals 1 Overscript upper N Endscripts s Subscript i Baseline s Subscript i Superscript prime Baseline equals 1 minus StartFraction 2 d Subscript normal upper H Baseline left parenthesis bold s comma bold s Superscript prime Baseline right parenthesis Over upper N EndFraction periodω(s, s^,) = ¹

N

N

Σ

i=1

sis^,

i ^{= 1 −2dH(s, s,)}

N

.

(4.6)

omegaω is an order parameter analogous to magnetization in a ferromagnet. If the mutation

rate is higher than an error rate threshold, then the population is distributed uniformly

over the whole genotype space (“wandering” régime) and the average overlaptilde 1 divided by upper N∼1/N

(see Sect. 14.7.2); below the threshold, the population lies a finite distance away from

the fittest genotype and omega tilde 1 minus script upper O left parenthesis 1 divided by upper N right parenthesisω ∼1 −O(1/N). ¹⁷ Intermediate between these two cases

(none and maximal epistatic interactions) are the rugged landcapes studied by Kauff-

man (1984). ¹⁸ More realistic models need to include changing fitness landscapes,

resulting from interactions between species—competition (one species inhibits the

increase of another), exploitation (A inhibits B but B stimulates A), or mutualism

(one species stimulates the increase of another; i.e., coevolution).

As presented, the models deal with asexual reproduction. Sex introduces compli-

cations but can, in principle, be handled within the general framework.

These models concern microevolution (the evolving units are individuals); if the

evolving units are species or larger units such as families, then one may speak of

macroevolution. There has been particular interest in modelling mass extinctions,

which may follow a power law (i.e., the number nn of extinguished families tilde n Superscript gamma∼n^γ,

withgammaγ equal to aboutnegative 2−2 according to current estimates). Bak and Sneppen (1993) ¹⁹

invented a model for the macroevolution of biological units (such as species) in

which each unit is assigned a fitness upper FF, defined as the barrier height for mutation

into another unit. At each iteration, the species with the lowest barrier is mutated—

implying assigned a new fitness, chosen at random from a finite range of values.

The mean fitness of the ecosystem rises inexorably to the maximum value, but if

the species interact and a number of neighbours are also mutated, regardless of their

fitnesses (this simulates the effect of, say, the extinction of a certain species of grass

on the animals feeding exclusively on that grass ²⁰), the ecosystem evolves such that

almost all species have fitnesses above a critical threshold; that is, the model shows

self-organized criticality. Avalanches of mutations can be identified and their size

follows a power law distribution, albeit withgamma tilde negative 1γ ∼−1. Hence, there have been various

attempts to modify the model to bring the value of the exponent closer to the value

(negative 2−2) believed to be characteristic of Earth’s prehistory. ²¹

17 See Peliti (1996) for a comprehensive treatment.

18 Cf. Sect. 12.2; see Jongeling (1996) for a critique.

19 See also Flyvbjerg et al. (1995).

20 For example, the takahe feeds almost exclusively on snow grass.

21 Newman (1996).