Error catastrophe
Encyclopedia
Error catastrophe is a term used to describe the extinction of an organism
(often in the context of microorganism
s 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?
Due to the mutations resulting from erroneous replication, there exist up to 2L distinct strains derived from the parent virus. Let xi denote the concentration of strain i; let ai denote the rate at which strain i reproduces; and let Qij denote the probability of a virus of strain i mutating to strain j.
Then the rate of change of concentration xj is given by
At this point, we make a mathematical idealisation: we pick the fittest strain (the one with the greatest reproduction rate aj) and assume that it is unique (ie that the chosen aj satisfies aj > ai 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:
We are particularly interested in the case where L is very large, so we may safely neglect R and instead consider:
Then setting z = x/y we have
.
Assuming z achieves a steady concentration over time, z settles down to satisfy
(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:
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
Taking a logarithm on both sides and approximating for small q and s one gets
reducing the condition to:
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.
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.
(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.
Organism
In biology, an organism is any contiguous living system . In at least some form, all organisms are capable of response to stimuli, reproduction, growth and development, and maintenance of homoeostasis as a stable whole.An organism may either be unicellular or, as in the case of humans, comprise...
(often in the context of microorganism
Microorganism
A microorganism or microbe is a microscopic organism that comprises either a single cell , cell clusters, or no cell at all...
s such as viruses) as a result of excessive RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....
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
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
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
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
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 2L distinct strains derived from the parent virus. Let xi denote the concentration of strain i; let ai denote the rate at which strain i reproduces; and let Qij denote the probability of a virus of strain i mutating to strain j.
Then the rate of change of concentration xj is given by
At this point, we make a mathematical idealisation: we pick the fittest strain (the one with the greatest reproduction rate aj) and assume that it is unique (ie that the chosen aj satisfies aj > ai 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:
We are particularly interested in the case where L is very large, so we may safely neglect R and instead consider:
Then setting z = x/y we have
.Assuming z achieves a steady concentration over time, z settles down to satisfy
(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:
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
Taking a logarithm on both sides and approximating for small q and s one gets
reducing the condition to:
RNA viruses which replicate close to the error threshold have a genome size of order 104 base pairs. Human DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...
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 CHepatitis C
Hepatitis C is an infectious disease primarily affecting the liver, caused by the hepatitis C virus . The infection is often asymptomatic, but chronic infection can lead to scarring of the liver and ultimately to cirrhosis, which is generally apparent after many years...
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
Catch-22 (logic)
A Catch-22, coined by Joseph Heller in his novel Catch-22, is a logical paradox arising from a situation in which an individual needs something that can only be acquired with an action that will lead him to that very situation he is already in; therefore, the acquisition of this thing becomes...
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 AccelerationViral decay acceleration
Viral decay acceleration is a therapeutic strategy which increases the mutation frequency of a virus toward an error catastrophe threshold.Viruses evolve at rates approximately one million times faster than the human genome...
(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.