Survival of the Quasispecies

This Demonstration illustrates that if the mutation rate is in a certain range, natural selection selects a quasispecies (B+B') of slightly lower individual fitnesses over an individual (A) of higher fitness. This idealized genome has length four; a genotype is specified by a "word" of four base pairs of type "0" or "1" (compare the four types, A, T, C, G in DNA); a point mutation interchanges a "0" and "1". The 16 genotypes are shown in a 4×4 grid; the

and

coordinates are given by the first two and the last two base pairs, and the

coordinate is the relative population of the genotype.

Contributed by: Michael Rogers (Oxford College/Emory University)

Show Initialization Code

(6 lines omitted)

For given fitness values, there are two critical values

and

of the mutation rate

such that (1) if

, selection favors the fittest species A; (2) if

, selection favors the quasispecies with the best average fitness B+B'; and (3) if

, all species survive with similar relative abundance. For the default fitness values,

and

; see Snapshots 1, 2, and 3. At the mutation rate

, all species will have the same population 1/16, no matter what the fitnesses are; see Snapshot 4. At a mutation rate

≈1, some (mathematically) interesting things happen; see Snapshots 5 and 6. Biologically, however, such high mutation rates are unrealistic. Most life forms have low enough mutation rates that the product of the length of the genome and the mutation rate is less than 1. For example, the length of the human genome is about

and its mutation rate is about

, which gives a product of about 0.16.

Definitions. Quasispecies: a group of genotypes which differ by relatively few mutations; in this Demonstration, at most one mutation from a given individual (e.g., B+B' = 1111 + {0111, 1011, 1101, 1110}). Fitness: the exponential growth rate of the genotype in the absence of competing factors. Mutation rate: the probability of one "0" changing to a "1" or vice versa when the genotype replicates. Note that the slider values of the mutation rate and evolution time are mapped nonlinearly to the actual rate and time. The quasispecies equation: in this case,

, where

and

run over all the genotypes,

is the population of genotype

is the fitness of

is the mutation rate,

is the number of different "bits" between

and

, and

is the average fitness.

Assumptions: (1) Individual genotype populations are modeled as continuous variables representing the proportion of the total population. (2) Total population size remains constant. (3) The population is infinite, so that the probabilities of the mutations are the actual proportions of the genotype populations that mutate.

Manfred Egen and Peter Schuster developed the general quasispecies equation. Schuster and Börg Swetina first observed the selection changes from the fittest to the quasispecies to no selection. See M. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Cambridge, MA: Belknap Press, 2006.