Error catastrophe

Error catastrophe is a term used to describe the extinction of an organism (often in the context of microorganisms such as viruses) as a result of excessive RNA mutations. The term specifically refers to the predictions of mathematical models similar to that described below, and not to an observed phenomenon.

Like every organism, viruses 'make mistakes' (or mutate) during replication. The resulting mutations increase biodiversity among the population and help subvert the ability of a host'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 biodiversity for an explanation of the selective advantages of this). However if it makes too many mutations it may lose some of its biological features which have evolved to its advantage, including its ability to reproduce at all.

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

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

$\dot{x}_j = \sum_i a_i Q_{ij} x_i$

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 (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, 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:

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

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

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

Then setting z = x/y we have

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

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

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

(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. ie if and only if:

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

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

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

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

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

reducing the condition to:

Lq < s

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

Applications of the theory

Some viruses such as polio or hepatitis C operate 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. However, given the criticism of the basic assumption of the mathematical model, this approach is problematic.

The result introduces a Catch-22 mystery 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.

KP-1461

Scientists have discovered an enzyme (A3G) that appears to mutate HIV to death, and supports the viability of Viral Decay Acceleration (VDA) as a therapeutic strategy. In this same context, researchers have identified a pharmaceutical agent, KP-1461 that is capable of inducing additional errors into the viral genome of HIV. It does this by being incorporated into a copy strand of viral DNA and mismatch base pairing. In this regard while KP-1461, a cytidine analog, should base pair with guanosine, the flexible molecular structure facilitates base pairing with adenosine as well. This ability to mismatch base pair results in an increase of guanosine (G) to adenosine (A) and A to G mutations in the viral genome. Over time these additional mutations accumulate until an error catastrophe threshold is breached and the viral population collapses. In this regard, in vitro studies have demonstrated that KP-1461 increases viral mutation frequency resulting in a progressive debilitation of the viral population, which eventually leads to extinction of the entire viral population.

Koronis Pharmaceuticals has completed phase 1a, 1b and 2a clinical studies demonstrating KP-1461 to be generally safe and well tolerated with no drug related Serious Adverse Events (SAE’s). Additionally, results from the limited duration Phase 2a clinical study provide a mechanistic proof of concept for this therapeutic strategy with the observation of a highly statistically significant increase in viral mutation frequency. Based on these recent results Koronis is currently in the process of refining the KP-1461 oral drug formulation and defining clinical protocols to support the next series of Phase 2 clinical studies.

Phase 2a trial

An open-label, multicenter, mechanism validation study to evaluate the Safety, Efficacy, and Tolerability of KP-1461 as Monotherapy for 124 Days in ARV-experienced, HIV-1-infected subjects. This study enrolled 27 of the 32 subjects targeted for the trial. Koronis stopped this trial prior to complete enrollment in order to conduct in vitro serial passage studies, which confirmed HIV ablation consistent with original in vitro studies. The trial closure was not requested or required by the FDA and was not related to any safety concerns or adverse events during the trial. KP-1461 was shown to be well tolerated with no drug related SAE’s. Furthermore, genetic analysis of patient viral isolates from this limited duration clinical study demonstrated highly statistically significant increases in viral mutation frequency. The observed increase in mutation frequency provides an in vivo mechanistic proof of concept for this therapeutic strategy and justifies the conduct of a longer duration phase 2 trial to determine the treatment duration required to see a clinically meaningful response in HIV infected patients.

Phase 1b trial

A double-blind, placebo-controlled, dose escalation study of the safety, tolerability, and pharmacokinetics of multiple oral doses of KP-1461 in HIV+ adults who have failed two or more highly active antiretroviral regimens (HAART). The study demonstrated that KP-1461 was well tolerated over 2 weeks of dosing. The study also showed that the highest dose level (3200 mg twice per day) was associated with a viral load decrease of 0.4 log that was statistically significant when compared to placebo.

Phase 1a trial

A double-blind, placebo-controlled dose escalation study of the safety tolerability, and pharmacokinetics of a single oral dose of KP-1461 in normal healthy adults. The study demonstrated that a single dose of KP-1461 was well tolerated and had acceptable pharmacokinetics.

See also

• Viral decay acceleration

External links

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