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1. Error threshold (evolution)
In evolutionary biology and population genetics, the error threshold (or critical mutation rate) is a
limit on the number of base pairs a self-replicating molecule may have before mutation will destroy
the information in subsequent generations of the molecule. The error threshold is crucial to
understanding "Eigen's paradox".
The error threshold is a concept in the origins of life (abiogenesis), in particular of very early life,
before the advent of DNA. It is postulated that the first self-replicating molecules might have been
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.
Fitness landscape
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.
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 Eigen 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.

2. 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:
 Stochastic corrector model (Szathmáry & Maynard 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
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 the death of molecules (the population will grow infinitely), and divide the eight
molecules into three classes depending on theirHamming distance from the master sequence:
Hamming
distance
Sequence(s)
0 [000]
1
[001]
[010]
[100]
2
[110]
[101]
[011]
3 [111]

3. 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 according to quasispecies model 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 and genetic advantage of a=1.05, then the equilibrium
concentrations will be roughly . It can be seen that the
master sequence is not as dominant, however, sequences with low Hamming distance
are in majority. If we have a replication fidelity of Q approaching 0, then the equilibrium
concentrations will be roughly . This is a population
with equal number of each of 8 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 aquasispecies 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.

4. Population numbers as a function of Hamming distanced and mutation rate (1-Q). The horizontal
axis d is the Hamming distance of the molecular sequences from the master sequence. The
vertical axis is the logarithm of population for any sequence at that distance divided by total
population (thus the division of nd by the binomial coefficient). The total number of digits per
sequence is L=100, and the master sequence has a selective advantage of a=1.05.
The population of the master sequence as a fraction of the total population (n) as a function
of overall mutation rate (1-Q). The total number of digits per sequence is L=100, and the
master sequence has a selective advantage of a=1.05. The "phase transition" is seen to
occur at roughly 1-Q=0.05.
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

5. 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.