Quasispecies model

Quasispecies, fitness, and evolutionary selection

In a species, though reproduction may be mostly accurate, periodic mutations will give rise to one or more competing genotypes. If a mutation results in greater replication and survival, the mutant genotype may out-compete the parent genotype and come to dominate the species. Thus, the individual genotypes (or species) may be seen as the units on which selection acts and biologists will often speak of a single genotype's fitness.^[7]

In a quasispecies, however, mutations are ubiquitous and so the fitness of an individual genotype becomes meaningless: if one particular mutation generates a boost in reproductive success, it can't amount to much because that genotype's offspring are unlikely to be accurate copies with the same properties. Instead, what matters is the connectedness of the cloud. For example, the sequence AGGT has 12 (3+3+3+3) possible single point mutants AGGA, AGGG, and so on. If 10 of those mutants are viable genotypes that may reproduce (and some of whose offspring or grandchildren may mutate back into AGGT again), we would consider that sequence a well-connected node in the cloud. If instead only two of those mutants are viable, the rest being lethal mutations, then that sequence is poorly connected and most of its descendants will not reproduce. The analog of fitness for a quasispecies is the tendency of nearby relatives within the cloud to be well-connected, meaning that more of the mutant descendants will be viable and give rise to further descendants within the cloud.^[8]

When the fitness of a single genotype becomes meaningless because of the high rate of mutations, the cloud as a whole or quasispecies becomes the natural unit of selection.

Application to biological research

Quasispecies represents the evolution of high-mutation-rate viruses such as HIV and sometimes single genes or molecules within the genomes of other organisms.^[9][10][11] Quasispecies models have also been proposed by Jose Fontanari and Emmanuel David Tannenbaum to model the evolution of sexual reproduction.^[12] Quasispecies was also shown in compositional replicators (based on the Gard model for abiogenesis)^[13] and was also suggested to be applicable to describe cell's replication, which amongst other things requires the maintenance and evolution of the internal composition of the parent and bud.

Note about primitive matrices

W being primitive means that for some integer ${\displaystyle n>0}$, that the ${\displaystyle n^{th}}$ power of W is > 0, i.e. all the entries are positive. If W is primitive then each type can, through a sequence of mutations (i.e. powers of W) mutate into all the other types after some number of generations. W is not primitive if it is periodic, where the population can perpetually cycle through different disjoint sets of compositions, or if it is reducible, where the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below.

Alternative formulations

The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state ${\displaystyle n'_{i}}$ and the old state ${\displaystyle n_{i}}$ to be the state change over one moment of time, then we can state that the time derivative of ${\displaystyle n_{i}}$ is given by this difference, ${\displaystyle {\dot {n}}_{i}=n'_{i}-n_{i}}$ we can write:

${\displaystyle {\dot {n}}_{i}=\sum _{j}w_{ij}n_{j}-n_{i}}$

The quasispecies equations are usually expressed in terms of concentrations ${\displaystyle x_{i}}$ where

${\displaystyle x_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {n_{i}}{\sum _{j}n_{j}}}}$.

${\displaystyle x'_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {n'_{i}}{\sum _{j}n'_{j}}}}$.

The above equations for the quasispecies then become for the discrete version:

${\displaystyle x'_{i}={\frac {\sum _{j}w_{ij}x_{j}}{\sum _{ij}w_{ij}x_{j}}}}$

or, for the continuum version:

${\displaystyle {\dot {x}}_{i}=\sum _{j}w_{ij}x_{j}-x_{i}\sum _{ij}w_{ij}x_{j}.}$

Simple example

The quasispecies concept can be illustrated by a simple system consisting of 4 sequences. Sequences [0,0], [0,1], [1,0], and [1,1] are numbered 1, 2, 3, and 4, respectively. Let's say the [0,0] sequence never mutates and always produces a single offspring. Let's say the other 3 sequences all produce, on average, ${\displaystyle 1-k}$ replicas of themselves, and ${\displaystyle k}$ of each of the other two types, where ${\displaystyle 0\leq k\leq 1}$. The W matrix is then:

${\displaystyle \mathbf {W} ={\begin{bmatrix}1&0&0&0\\0&1-k&k&k\\0&k&1-k&k\\0&k&k&1-k\end{bmatrix}}}$.

The diagonalized matrix is:

${\displaystyle \mathbf {W'} ={\begin{bmatrix}1-2k&0&0&0\\0&1-2k&0&0\\0&0&1&0\\0&0&0&1+k\end{bmatrix}}}$.

And the eigenvectors corresponding to these eigenvalues are:

Eigenvalue	Eigenvector
1-2k	[0,-1,0,1]
1-2k	[0,-1,1,0]
1	[1,0,0,0]
1+k	[0,1,1,1]

Only the eigenvalue ${\displaystyle 1+k}$ is more than unity. For the n-th generation, the corresponding eigenvalue will be ${\displaystyle (1+k)^{n}}$ and so will increase without bound as time goes by. This eigenvalue corresponds to the eigenvector [0,1,1,1], which represents the quasispecies consisting of sequences 2, 3, and 4, which will be present in equal numbers after a very long time. Since all population numbers must be positive, the first two quasispecies are not legitimate. The third quasispecies consists of only the non-mutating sequence 1. It's seen that even though sequence 1 is the most fit in the sense that it reproduces more of itself than any other sequence, the quasispecies consisting of the other three sequences will eventually dominate (assuming that the initial population was not homogeneous of the sequence 1 type).