Error threshold (evolution)
Encyclopedia
The error threshold is a concept in the study of evolutionary biology and population genetics
concerned with the origins of life, in particular of very early life, before the advent of DNA
. The first self-replicating molecules were probably small ribozyme
-like RNA
molecules. These molecules consist of strings of base pairs or "digits", and their order is a code that directs how the molecule interacts with its environment. All replication is subject to mutation error. During the replication process, each digit has a certain probability of being replaced by some other digit, which changes the way the molecule interacts with its environment, and may increase or decrease its fitness, or ability to reproduce, in that environment.
It was noted by Manfred Eigen
in his 1971 paper (Eigen 1971) that this mutation process places a limit on the number of digits a molecule may have. If a molecule exceeds this critical size, the effect of the mutations become overwhelming and a runaway mutation process will destroy the information in subsequent generations of the molecule. The error threshold is also controlled by the fitness landscape
for the molecules. Molecules that differ only by a few mutations may be thought of as "close" to each other, while those that differ by many mutations are distant from each other. Molecules that are very fit, and likely to reproduce, have a "high" fitness, those less fit have "low" fitness. These ideas of proximity and height form the intuitive concept of the "fitness landscape". If a particular sequence and its neighbors have a high fitness, they will form a quasispecies
and will be able to support longer sequence lengths than a fit sequence with few fit neighbors, or a less fit neighborhood of sequences. Also, it was noted by Wilke (Wilke 2005) that the error threshold concept does not apply in portions of the landscape where there are lethal mutations, in which the induced mutation yields zero fitness and prohibits the molecule from reproducing.
The concept of this critical mutation rate, or "error threshold" is crucial to understanding "Eigen's paradox," which is discussed in the next section.
in his 1971 paper (Eigen 1971). Simply stated, Eigen's paradox amounts to the following:
This is a chicken-or-egg
kind of a paradox, with an even more difficult solution. Which came first, the large genome or the error correction enzymes? A number of solutions to this paradox have been proposed:
copies, where 
. This molecule is called the "master sequence". The other seven sequences are less fit; they each produce only 1 copy per replication. The replication of each of the three digits is done with a mutation rate of μ. In other words, at every replication of a digit of a sequence, there is a probability 
that it will be erroneous; 0 will be replaced by 1 or vice versa. Let's ignore double mutations, and divide the eight molecules into three classes depending on their Hamming distance
from the master sequence:
Note that the number of sequences for distance d is just the binomial coefficient
for L=3, and that each sequence can be visualized as the vertex of an L=3 dimensional cube, with each edge of the cube specifying a mutation path in which the change Hamming distance is either zero or ±1. It can be seen that, for example, one third of the mutations of the [001] molecules will produce [000] molecules, while the other two thirds will produce the class 2 molecules [011] and [101]. We can now write the expression for the child populations 
of class i in terms of the parent populations 
.
where the fitness matrix w is given by:
where
is the probability that an entire molecule will be replicated successfully. The eigenvectors of the w matrix will yield the equilibrium population numbers for each class. For example, if the mutation rate μ is zero, we will have Q=1, and the equilibrium concentrations will be 
. The master sequence, being the fittest will be the only one to survive. If we have a replication fidelity of Q=0.95, then the equilibrium concentrations will be roughly 
. It can be seen that the master sequence is not as dominant. If we have a replication fidelity of Q=0, then the equilibrium concentrations will be roughly 
. This is almost an equal population of all sequences. (If we had perfectly equal population of all sequences, we would have populations of [1,3,3,1]/8.)
If we now go to the case where the number of base pairs is large, say L=100, we obtain behavior that resembles a phase transition. The plot below on the left shows a series of equilibrium concentrations divided by the binomial coefficient
.
(This multiplication will show the population for an individual sequence at that distance, and will yield a flat line for an equal distribution.) The selective advantage of the master sequence is set at a=1.05. The horizontal axis is the Hamming distance d . The various curves are for various total mutation rates
. It is seen that for low values of the total mutation rate, the population consists of a quasispecies gathered in the neighborhood of the master sequence. Above a total mutation rate of about 1-Q=0.05, the distribution quickly spreads out to populate all sequences equally. The plot below on the right shows the fractional population of the master sequence as a function of the total mutation rate. Again it is seen that below a critical mutation rate of about 1-Q=0.05, the master sequence contains most of the population, while above this rate, it contains only about 
of the total population.
It can be seen that there is a sharp transition at a value of 1-Q just a bit larger than 0.05. For mutation rates above this value, the population of the master sequence drops to practically zero. Above this value, it dominates.
In the limit as L approaches infinity, the system does in fact have a phase transition at a critical value of Q:
. One could think of the overall mutation rate (1-Q) as a sort of "temperature", which "melts" the fidelity of the molecular sequences above the critical "temperature" of 
. For faithful replication to occur, the information must be "frozen" into the genome.
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
concerned with the origins of life, in particular of very early life, before the advent of 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...
. The first self-replicating molecules were probably small ribozyme
Ribozyme
A ribozyme is an RNA molecule with a well defined tertiary structure that enables it to catalyze a chemical reaction. Ribozyme means ribonucleic acid enzyme. It may also be called an RNA enzyme or catalytic RNA. Many natural ribozymes catalyze either the hydrolysis of one of their own...
-like RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....
molecules. These molecules consist of strings of base pairs or "digits", and their order is a code that directs how the molecule interacts with its environment. All replication is subject to mutation error. During the replication process, each digit has a certain probability of being replaced by some other digit, which changes the way the molecule interacts with its environment, and may increase or decrease its fitness, or ability to reproduce, in that environment.
It was noted by Manfred Eigen
Manfred Eigen
Manfred Eigen is a German biophysical chemist who won the 1967 Nobel Prize in Chemistry for work on measuring fast chemical reactions.-Career:...
in his 1971 paper (Eigen 1971) that this mutation process places a limit on the number of digits a molecule may have. If a molecule exceeds this critical size, the effect of the mutations become overwhelming and a runaway mutation process will destroy the information in subsequent generations of the molecule. The error threshold is also controlled by the fitness landscape
Fitness landscape
In evolutionary biology, fitness landscapes or adaptive landscapes are used to visualize the relationship between genotypes and reproductive success. It is assumed that every genotype has a well-defined replication rate . This fitness is the "height" of the landscape...
for the molecules. Molecules that differ only by a few mutations may be thought of as "close" to each other, while those that differ by many mutations are distant from each other. Molecules that are very fit, and likely to reproduce, have a "high" fitness, those less fit have "low" fitness. These ideas of proximity and height form the intuitive concept of the "fitness landscape". If a particular sequence and its neighbors have a high fitness, they will form a quasispecies
Quasispecies model
The quasispecies model is a description of the process of the Darwinian evolution of certain self-replicating entities within the framework of physical chemistry...
and will be able to support longer sequence lengths than a fit sequence with few fit neighbors, or a less fit neighborhood of sequences. Also, it was noted by Wilke (Wilke 2005) that the error threshold concept does not apply in portions of the landscape where there are lethal mutations, in which the induced mutation yields zero fitness and prohibits the molecule from reproducing.
The concept of this critical mutation rate, or "error threshold" is crucial to understanding "Eigen's paradox," which is discussed in the next section.
Eigen's Paradox
Eigen's paradox is one of the most intractable puzzles in the study of the origins of life. It is thought that the error threshold concept described above limits the size of self replicating molecules to perhaps a few hundred digits, yet almost all life on earth requires much longer molecules to encode their genetic information. This problem is handled in living cells by enzymes that repair mutations, allowing the encoding molecules to reach sizes on the order of millions of base pairs. These large molecules must, of course, encode the very enzymes that repair them, and herein lies Eigen's paradox, first put forth by Manfred EigenManfred Eigen
Manfred Eigen is a German biophysical chemist who won the 1967 Nobel Prize in Chemistry for work on measuring fast chemical reactions.-Career:...
in his 1971 paper (Eigen 1971). Simply stated, Eigen's paradox amounts to the following:
- Without error correction enzymes, the maximum size of a replicating molecule is about 100 base pairs.
- For a replicating molecule to encode error correction enzymes, it must be substantially larger than 100 bases.
This is a chicken-or-egg
The chicken or the egg
The Chicken or the egg causality dilemma is commonly stated as "which came first, the chicken or the egg?" To ancient philosophers, the question about the first chicken or egg also evoked the questions of how life and the universe in general began....
kind of a paradox, with an even more difficult solution. Which came first, the large genome or the error correction enzymes? A number of solutions to this paradox have been proposed:
- Stochastic corrector model (Szathmáry & Smith, 1995). In this proposed solution, a number of primitive molecules of say, two different types, are associated with each other in some way, perhaps by a capsule or "cell wall". If their reproductive success is enhanced by having, say, equal numbers in each cell, and reproduction occurs by division in which each of various types of molecules are randomly distributed among the "children", the process of selection will promote such equal representation in the cells, even though one of the molecules may have a selective advantage over the other.
- Relaxed error threshold (Kun et al., 2005) - Studies of actual ribozymes indicate that the mutation rate can be substantially less than first expected - on the order of 0.001 per base pair per replication. This may allow sequence lengths of the order of 7-8 thousand base pairs, sufficient to incorporate rudimentary error correction enzymes.
A simple mathematical model illustrating the error threshold
Consider a 3-digit molecule [A,B,C] where A, B, and C can take on the values 0 and 1. There are eight such sequences ([000], [001], [010], [011], [100], [101], [110], and [111]). Let's say that the [000] molecule is the most fit; upon each replication it produces an average of copies, where . This molecule is called the "master sequence". The other seven sequences are less fit; they each produce only 1 copy per replication. The replication of each of the three digits is done with a mutation rate of μ. In other words, at every replication of a digit of a sequence, there is a probability that it will be erroneous; 0 will be replaced by 1 or vice versa. Let's ignore double mutations, and divide the eight molecules into three classes depending on their Hamming distanceHamming distance
In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...
from the master sequence:
| Hamming distance |
Sequence(s) |
| 0 | [000] |
| 1 | [001] [010] [100] |
| 2 | [110] [101] [011] |
| 3 | [111] |
Note that the number of sequences for distance d is just the binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
for L=3, and that each sequence can be visualized as the vertex of an L=3 dimensional cube, with each edge of the cube specifying a mutation path in which the change Hamming distance is either zero or ±1. It can be seen that, for example, one third of the mutations of the [001] molecules will produce [000] molecules, while the other two thirds will produce the class 2 molecules [011] and [101]. We can now write the expression for the child populations of class i in terms of the parent populations .where the fitness matrix w is given by:
where
is the probability that an entire molecule will be replicated successfully. The eigenvectors of the w matrix will yield the equilibrium population numbers for each class. For example, if the mutation rate μ is zero, we will have Q=1, and the equilibrium concentrations will be . The master sequence, being the fittest will be the only one to survive. If we have a replication fidelity of Q=0.95, then the equilibrium concentrations will be roughly . It can be seen that the master sequence is not as dominant. If we have a replication fidelity of Q=0, then the equilibrium concentrations will be roughly . This is almost an equal population of all sequences. (If we had perfectly equal population of all sequences, we would have populations of [1,3,3,1]/8.)If we now go to the case where the number of base pairs is large, say L=100, we obtain behavior that resembles a phase transition. The plot below on the left shows a series of equilibrium concentrations divided by the binomial coefficient
.(This multiplication will show the population for an individual sequence at that distance, and will yield a flat line for an equal distribution.) The selective advantage of the master sequence is set at a=1.05. The horizontal axis is the Hamming distance d . The various curves are for various total mutation rates
. It is seen that for low values of the total mutation rate, the population consists of a quasispecies gathered in the neighborhood of the master sequence. Above a total mutation rate of about 1-Q=0.05, the distribution quickly spreads out to populate all sequences equally. The plot below on the right shows the fractional population of the master sequence as a function of the total mutation rate. Again it is seen that below a critical mutation rate of about 1-Q=0.05, the master sequence contains most of the population, while above this rate, it contains only about of the total population.It can be seen that there is a sharp transition at a value of 1-Q just a bit larger than 0.05. For mutation rates above this value, the population of the master sequence drops to practically zero. Above this value, it dominates.
In the limit as L approaches infinity, the system does in fact have a phase transition at a critical value of Q:
. One could think of the overall mutation rate (1-Q) as a sort of "temperature", which "melts" the fidelity of the molecular sequences above the critical "temperature" of . For faithful replication to occur, the information must be "frozen" into the genome.