Error catastrophe

Error catastrophe is a term used to describe the extinction of an organism(often in the context of microorganismssuch as viruses) as a result of excessive RNAmutations.

Many viruses deliberately 'make mistakes' (or mutate) during replication in order to increase biodiversityamong its population and to help subvert the ability of a mammal's immune system to recognise it in a subsequent infection. The more mutations (mistakes) the virus makes during replication, the more likely it is to avoid recognition by the immune system and the more diverse its population will be (see the article on biodiversityfor an explanation of the evolutionary advantages of this). However if it makes too many mutations it may lose some of its biological features which have evolved to its advantage over millions of years; or indeed it may quickly become a completely different organism altogether.

The question arises: how many mutations can be made during each replication before the population of viruses begins to lose self-identity?

A basic mathematical model

Consider a virus which has a genetic identity modeled by a string of ones and zeros (eg 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

<math>\dot{x}_j = \sum_i a_i Q_{ij} x_i</math>

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 (ie 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; let Q be the probability of a virus in the first group mutating to a member of the second group 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:

<math>

\begin{cases} \dot{x} = & a(1-Q)x + bRy \\ \dot{y} = & aQx + b(1-R)y \\ \end{cases} </math>

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

<math>

\begin{cases} \dot{x} = & a(1-Q)x \\ \dot{y} = & aQx + by \\ \end{cases} </math>

Then setting z = x/y we have

<math>

\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} </math>.

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

<math> z(\infty) = \frac{a(1-Q)-b}{aQ} </math>

(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. ie if and only if:

<math> z(\infty) > 0 \iff a(1-Q)-b >0 \iff (1-Q) > b/a .</math>

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

<math> z(\infty) > 0 \iff (1-Q) = (1-q)^L > 1-s </math>

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

<math>L \ln{(1-q)} \approx -Lq > \ln{(1-s)} \approx -s</math>

reducing the condition to:

<math> Lq < s </math>

RNA viruseswhich replicate close to the error threshold have a genome size of order 10⁴ base pairs. Human DNAis about 3.3 billion (10⁹) base units long. This means that the replication mechanism for DNA must be orders of magnitudemore accurate than for RNA.

Applications of the theory

Some viruses such as polioor hepatitis Coperate very close to the critical mutation rate (ie 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.

The result introduces a catch-22mystery for biologists: 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.

See also

• Haldane's Dilemma

External links

• Error catastrophe and antiviral strategy
• Applications of error catastrophe to the persistance of GM crops
• The Orgel's Error Catastrophe Theory of Aging and Longevity

This article is licensed under the GNU Free Documentation License.
It uses material from the http://en.wikipedia.org/wiki/Error+catastrophe Wikipedia article Error catastrophe.