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Quasispecies model
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==Mathematical description== A simple mathematical model for a quasispecies is as follows:<ref name=eigen1989 /> let there be <math>S</math> possible sequences and let there be <math>n_i</math> organisms with sequence ''i''. Let's say that each of these organisms asexually gives rise to <math>A_i</math> offspring. Some are duplicates of their parent, having sequence ''i'', but some are mutant and have some other sequence. Let the mutation rate <math>q_{ij}</math> correspond to the [[probability]] that a ''j'' type parent will produce an ''i'' type organism. Then the [[Expected value|expected]] fraction of offspring generated by ''j'' type organisms that would be ''i'' type organisms is <math>w_{ij}=A_j q_{ij}</math>, where <math>\sum_i q_{ij}=1</math>. Then the total number of ''i''-type organisms after the first round of reproduction, given as <math>n'_i</math>, is :<math>n'_i=\sum_j w_{ij}n_j</math> Sometimes a death rate term <math>D_i</math> is included so that: :<math>w_{ij}=A_j q_{ij}-D_i\delta_{ij}</math> where <math>\delta_{ij}</math> is equal to 1 when i=j and is zero otherwise. Note that the ''n-th'' generation can be found by just taking the ''n-th'' power of '''W''' substituting it in place of '''W''' in the above formula. This is just a [[system of linear equations]]. The usual way to solve such a system is to first [[Diagonalizable matrix|diagonalize]] the '''W''' matrix. Its diagonal entries will be [[eigenvalues]] corresponding to certain linear combinations of certain subsets of sequences which will be [[eigenvectors]] of the '''W''' matrix. These subsets of sequences are the quasispecies. Assuming that the matrix '''W''' is a [[primitive matrix]] ([[Irreducibility (mathematics)|irreducible]] and [[Perron–Frobenius theorem|aperiodic]]), then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium.<ref>{{cite book|title=Phase Portraits of Linear Systems|last=S Tseng|first=Zachary|year=2008}}</ref> === Note about primitive matrices === '''W''' being primitive means that for some integer <math> n > 0 </math>, that the <math>n^{th} </math> 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.{{citation needed|date=August 2013}} ===Alternative formulations=== The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state <math>n'_i</math> and the old state <math>n_i</math> to be the state change over one moment of time, then we can state that the time derivative of <math>n_i</math> is given by this difference, <math>\dot{n}_i = n'_i-n_i</math> we can write: :<math>\dot{n}_i=\sum_j w_{ij}n_j-n_i</math> The quasispecies equations are usually expressed in terms of concentrations <math>x_i</math> where :<math>x_i\ \stackrel{\mathrm{def}}{=}\ \frac{n_i}{\sum_j n_j}</math>. :<math>x'_i\ \stackrel{\mathrm{def}}{=}\ \frac{n'_i}{\sum_j n'_j}</math>. The above equations for the quasispecies then become for the discrete version: :<math>x'_i = \frac{\sum_j w_{ij}x_j}{\sum_{ij} w_{ij}x_j}</math> or, for the continuum version: :<math>\dot{x}_i =\sum_j w_{ij}x_j-x_i\sum_{ij}w_{ij}x_j.</math> ===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, <math>1-k</math> replicas of themselves, and <math>k</math> of each of the other two types, where <math>0\le k\le 1</math>. The '''W''' matrix is then: :<math>\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} </math>. The diagonalized matrix is: :<math>\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} </math>. And the eigenvectors corresponding to these eigenvalues are: :{| class="wikitable" |- ![[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 <math>1+k</math> is more than unity. For the n-th generation, the corresponding eigenvalue will be <math>(1+k)^n</math> 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).{{citation needed|date=August 2013}}
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