Error catastrophe

Consider a virus which has a genetic identity modeled by a string of ones and zeros (e.g. 11010001011101....). Suppose that the string has fixed length L and that during replication the virus copies each digit one by one, making a mistake with probability q independently of all other digits.

Due to the mutations resulting from erroneous replication, there exist up to 2^L distinct strains derived from the parent virus. Let x_i denote the concentration of strain i; let a_i denote the rate at which strain i reproduces; and let Q_ij denote the probability of a virus of strain i mutating to strain j.

Then the rate of change of concentration x_j is given by

${\displaystyle {\dot {x}}_{j}=\sum _{i}a_{i}Q_{ij}x_{i}}$

At this point, we make a mathematical idealisation: we pick the fittest strain (the one with the greatest reproduction rate a_j) and assume that it is unique (i.e. that the chosen a_j satisfies a_j > a_i for all i); and we then group the remaining strains into a single group. Let the concentrations of the two groups be x , y with reproduction rates a>b, respectively; let Q be the probability of a virus in the first group (x) mutating to a member of the second group (y) and let R be the probability of a member of the second group returning to the first (via an unlikely and very specific mutation). The equations governing the development of the populations are:

${\displaystyle {\begin{cases}{\dot {x}}=&a(1-Q)x+bRy\\{\dot {y}}=&aQx+b(1-R)y\\\end{cases}}}$

We are particularly interested in the case where L is very large, so we may safely neglect R and instead consider:

${\displaystyle {\begin{cases}{\dot {x}}=&a(1-Q)x\\{\dot {y}}=&aQx+by\\\end{cases}}}$

Then setting z = x/y we have

${\displaystyle {\begin{matrix}{\frac {\partial z}{\partial t}}&=&{\frac {{\dot {x}}y-x{\dot {y}}}{y^{2}}}\\&&\\&=&{\frac {a(1-Q)xy-x(aQx+by)}{y^{2}}}\\&&\\&=&a(1-Q)z-(aQz^{2}+bz)\\&&\\&=&z(a(1-Q)-aQz-b)\\\end{matrix}}}$.

Assuming z achieves a steady concentration over time, z settles down to satisfy

${\displaystyle z(\infty )={\frac {a(1-Q)-b}{aQ}}}$

(which is deduced by setting the derivative of z with respect to time to zero).

So the important question is under what parameter values does the original population persist (continue to exist)? The population persists if and only if the steady state value of z is strictly positive. i.e. if and only if:

${\displaystyle z(\infty )>0\iff a(1-Q)-b>0\iff (1-Q)>b/a.}$

This result is more popularly expressed in terms of the ratio of a:b and the error rate q of individual digits: set b/a = (1-s), then the condition becomes

${\displaystyle z(\infty )>0\iff (1-Q)=(1-q)^{L}>1-s}$

Taking a logarithm on both sides and approximating for small q and s one gets

${\displaystyle L\ln {(1-q)}\approx -Lq>\ln {(1-s)}\approx -s}$

reducing the condition to:

${\displaystyle Lq<s}$

RNA viruses which replicate close to the error threshold have a genome size of order 10⁴ (10000) base pairs. Human DNA is about 3.3 billion (10⁹) base units long. This means that the replication mechanism for human DNA must be orders of magnitude more accurate than for the RNA of RNA viruses.

Some viruses such as polio or hepatitis C operate very close to the critical mutation rate (i.e. the largest q that L will allow). Drugs have been created to increase the mutation rate of the viruses in order to push them over the critical boundary so that they lose self-identity. However, given the criticism of the basic assumption of the mathematical model, this approach is problematic.[8]

The result introduces a Catch-22 mystery for biologists, Eigen's paradox: in general, large genomes are required for accurate replication (high replication rates are achieved by the help of enzymes), but a large genome requires a high accuracy rate q to persist. Which comes first and how does it happen? An illustration of the difficulty involved is L can only be 100 if q' is 0.99 - a very small string length in terms of genes.

• Error threshold
• Extinction vortex
• Mutational meltdown

• Error catastrophe and antiviral strategy
• Examining the theory of error catastrophe