Quasispecies model
Encyclopedia
The quasispecies model is a description of the process of the Darwinian evolution
of certain self-replicating
entities within the framework of physical chemistry. Put simply, a quasispecies is a large group or cloud of related genotype
s that exist in an environment of high mutation rate, where a large fraction of offspring are expected to contain one or more mutations relative to the parent. This is in contrast to a species
, which from an evolutionary perspective is a more-or-less stable single genotype, most of the offspring of which will be genetically accurate copies.
It is useful mainly in providing a qualitative understanding of the evolutionary processes of self-replicating macromolecules such as RNA
or DNA
or simple asexual organisms such as bacteria or virus
es (see also viral quasispecies
), and is helpful in explaining something of the early stages of the origin of life. Quantitative predictions based on this model are difficult because the parameters that serve as its input are hard to obtain from actual biological systems. The quasispecies model was put forward by Manfred Eigen
and Peter Schuster
based on initial work done by Eigen.
Some organisms or genotypes, however, may exist in circumstances of low fidelity, where most descendants contain one or more mutations. A group of such genotypes are constantly changing, so discussions of which single genotype is the most fit become meaningless. Importantly, if many closely related genotypes are only one mutation away from each other, then genotypes in the group can mutate back and forth into each other. For example, with one mutation per generation, a child of the sequence AGGT could be AGTT, and a grandchild could be AGGT again. Thus we can envision a cloud of related genotypes that are rapidly mutating, with sequences going back and forth among different points in the cloud. Though the proper definition is mathematical, that cloud, roughly speaking, is a quasispecies.
Quasispecies behavior exists for large numbers of individuals existing at a certain (high) range of mutation rates. Essentially all species on earth, apart from groups of inbreeding mammals and self-cloning plant populations, are quasispecies.
.
In a quasispecies, however, mutations are ubiquitous and so the fitness of an individual genotype becomes meaningless: if one particular mutation generates a boost in reproductive success it can't amount to much because that genotype's offspring are unlikely to be accurate copies with the same properties. Instead, what matters is the connectedness of the cloud. For example, the sequence AGGT has 12 (3+3+3+3) possible single point mutants, AGGA, AGGG and so on). If 10 of those mutants are viable genotypes that may reproduce (and some of whose offspring or grandchilden may mutate back into AGGT again), we would consider that sequence a well-connected node in the cloud. If instead only two of those mutants are viable, the rest being lethal mutations, then that sequence is poorly connected and most of its descendants will not reproduce. The analogue of fitness for a quasispecies is the tendency of nearby relatives within the cloud to be well-connected, meaning that more of the mutant descendants will be viable and give rise to further descendants within the cloud.
When the fitness of a single genotype becomes meaningless because of the high rate of mutations, the cloud as a whole or quasispecies becomes the natural unit of selection.
and sometimes single genes or molecules within the genomes of other organisms.
In the quasispecies model, mutation
s occur through errors made in the process of copying already existing sequences. Further, selection
arises because different types of sequences tend to replicate at different rates, which leads to the suppression of sequences that replicate more slowly in favor of sequences that replicate faster. However, the quasispecies model does not predict the ultimate extinction of all but the fastest replicating sequence. Although the sequences that replicate more slowly cannot sustain their abundance level by themselves, they are constantly replenished as sequences that replicate faster mutate into them. At equilibrium, removal of slowly replicating sequences due to decay or outflow is balanced by replenishing, so that even relatively slowly replicating sequences can remain present in finite abundance.
Due to the ongoing production of mutant sequences, selection does not act on single sequences, but on mutational "clouds" of closely related sequences, referred to as quasispecies. In other words, the evolutionary success of a particular sequence depends not only on its own replication rate, but also on the replication rates of the mutant sequences it produces, and on the replication rates of the sequences of which it is a mutant. As a consequence, the sequence that replicates fastest may even disappear completely in selection-mutation equilibrium, in favor of more slowly replicating sequences that are part of a quasispecies with a higher average growth rate. Mutational clouds as predicted by the quasispecies model have been observed in RNA viruses and in in vitro RNA replication.
The mutation rate and the general fitness of the molecular sequences and their neighbors is crucial to the formation of a quasispecies. If the mutation rate is zero, there is no exchange by mutation, and each sequence is its own species. If the mutation rate is too high, exceeding what is known as the error threshold
, the quasispecies will break down and be dispersed over the entire range of available sequences.
possible sequences and let there be 
organisms with sequence i. Let's say that each of these organisms asexually gives rise to 
offspring. Some are duplicates of their parent, having sequence i, but some are mutant and have some other sequence. Let the mutation rate 
correspond to the probability
that a j type parent will produce an i type organism. Then the expected
number of i type organisms produced by any j type parent is
,
where
.
Then the total number of i-type organisms after the first round of reproduction, given as
, is
Sometimes a death rate term
is included so that:
where
is equal to 1 when i=j and is zero otherwise. Note that the n-th generation can be found by just taking the n-th power of W substituting it in place of W in the above formula.
This is just a system of linear equations. The usual way to solve such a system is to first diagonalize
the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is irreducible, then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium.
, that the 
power of W is > 0, i.e. all the entries are positive. If W is irreducible then each type can, through a sequence of mutations (i.e. powers of W) mutate into all the other types. If W isn't irreducible, then the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below.
and the old state 
to be the state change over one moment of time, then we can state that the time derivative of 
is given by this difference, 
we can write:
The quasispecies equations are usually expressed in terms of concentrations
where
.
.
The above equations for the quasispecies then become for the discrete version:
or, for the continuum version:
replicas of themselves, and 
of each of the other two types, where 
. The W matrix is then:
The diagonalized matrix is
and the eigenvectors corresponding to these eigenvalues are:
Only the eigenvalue
is larger than unity. For the n-th generation, the corresponding eigenvalue will be 
and so will increase without bound as time goes by. This eigenvalue corresponds to the eigenvector [0,1,1,1], which represents the quasispecies consisting of sequences 2, 3, and 4, which will be present in equal numbers after a very long time. Since all population numbers must be positive, the first two quasispecies are not legitimate. The third quasispecies consists of only the non-mutating sequence 1. Its seen that even though sequence 1 is the most fit in the sense that it reproduces more of itself than any other sequence, the quasispecies consisting of the other three sequences will eventually dominate (assuming that the initial population was not homogeneous of the sequence 1 type).
----
Based on article from Nupedia
(http://www.nupedia.com/article/600/) by Claus O. Wilke, posted 2001-10-12.
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
of certain self-replicating
Self-replication
Self-replication is any behavior of a dynamical system that yields construction of an identical copy of that dynamical system. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction...
entities within the framework of physical chemistry. Put simply, a quasispecies is a large group or cloud of related genotype
Genotype
The genotype is the genetic makeup of a cell, an organism, or an individual usually with reference to a specific character under consideration...
s that exist in an environment of high mutation rate, where a large fraction of offspring are expected to contain one or more mutations relative to the parent. This is in contrast to a species
Species
In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...
, which from an evolutionary perspective is a more-or-less stable single genotype, most of the offspring of which will be genetically accurate copies.
It is useful mainly in providing a qualitative understanding of the evolutionary processes of self-replicating macromolecules such as RNA
RNA
Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....
or 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...
or simple asexual organisms such as bacteria or virus
Virus
A virus is a small infectious agent that can replicate only inside the living cells of organisms. Viruses infect all types of organisms, from animals and plants to bacteria and archaea...
es (see also viral quasispecies
Viral quasispecies
A viral quasispecies is a group of viruses related by a similar mutation or mutations, competing within a highly mutagenic environment. The theory predicts that a viral quasispecies at a low but evolutionarily neutral and highly connected region in the fitness landscape will outcompete a...
), and is helpful in explaining something of the early stages of the origin of life. Quantitative predictions based on this model are difficult because the parameters that serve as its input are hard to obtain from actual biological systems. The quasispecies model was put forward 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:...
and Peter Schuster
Peter Schuster
Peter K. Schuster is a renowned theoretical chemist, known for his work with the German Nobel Laureate Manfred Eigen in developing the quasispecies model...
based on initial work done by Eigen.
Simplified explanation
When evolutionary biologists describe competition between species, they generally assume that each species is a single genotype whose descendants are mostly accurate copies. (Such genotypes are said to have a high reproductive fidelity.) Evolutionarily, we are interested in the behavior and fitness of that one species or genotype over time.Some organisms or genotypes, however, may exist in circumstances of low fidelity, where most descendants contain one or more mutations. A group of such genotypes are constantly changing, so discussions of which single genotype is the most fit become meaningless. Importantly, if many closely related genotypes are only one mutation away from each other, then genotypes in the group can mutate back and forth into each other. For example, with one mutation per generation, a child of the sequence AGGT could be AGTT, and a grandchild could be AGGT again. Thus we can envision a cloud of related genotypes that are rapidly mutating, with sequences going back and forth among different points in the cloud. Though the proper definition is mathematical, that cloud, roughly speaking, is a quasispecies.
Quasispecies behavior exists for large numbers of individuals existing at a certain (high) range of mutation rates. Essentially all species on earth, apart from groups of inbreeding mammals and self-cloning plant populations, are quasispecies.
Quasispecies, fitness, and evolutionary selection
In a species, though reproduction may be mostly accurate, periodic mutations will give rise to one or more competing genotypes. If a mutation results in greater replication and survival, the mutant genotype may out-compete the parent genotype and come to dominate the species. Thus, the individual genotypes (or species) may be seen as the units on which selection acts and biologists will often speak of a single genotype's fitnessFitness (biology)
Fitness is a central idea in evolutionary theory. It can be defined either with respect to a genotype or to a phenotype in a given environment...
.
In a quasispecies, however, mutations are ubiquitous and so the fitness of an individual genotype becomes meaningless: if one particular mutation generates a boost in reproductive success it can't amount to much because that genotype's offspring are unlikely to be accurate copies with the same properties. Instead, what matters is the connectedness of the cloud. For example, the sequence AGGT has 12 (3+3+3+3) possible single point mutants, AGGA, AGGG and so on). If 10 of those mutants are viable genotypes that may reproduce (and some of whose offspring or grandchilden may mutate back into AGGT again), we would consider that sequence a well-connected node in the cloud. If instead only two of those mutants are viable, the rest being lethal mutations, then that sequence is poorly connected and most of its descendants will not reproduce. The analogue of fitness for a quasispecies is the tendency of nearby relatives within the cloud to be well-connected, meaning that more of the mutant descendants will be viable and give rise to further descendants within the cloud.
When the fitness of a single genotype becomes meaningless because of the high rate of mutations, the cloud as a whole or quasispecies becomes the natural unit of selection.
Application to biological research
While the applicability of the quasispecies model to real organisms is still a matter of debate in the scientific community, some researchers believe that it accurately represents the evolution of high-mutation-rate viruses such as HIVHIV
Human immunodeficiency virus is a lentivirus that causes acquired immunodeficiency syndrome , a condition in humans in which progressive failure of the immune system allows life-threatening opportunistic infections and cancers to thrive...
and sometimes single genes or molecules within the genomes of other organisms.
Formal background
The model rests on four assumptions:- The self-replicating entities can be represented as sequences composed of a small number of building blocks--for example, sequences of RNA consisting of the four bases adenineAdenineAdenine is a nucleobase with a variety of roles in biochemistry including cellular respiration, in the form of both the energy-rich adenosine triphosphate and the cofactors nicotinamide adenine dinucleotide and flavin adenine dinucleotide , and protein synthesis, as a chemical component of DNA...
, guanineGuanineGuanine is one of the four main nucleobases found in the nucleic acids DNA and RNA, the others being adenine, cytosine, and thymine . In DNA, guanine is paired with cytosine. With the formula C5H5N5O, guanine is a derivative of purine, consisting of a fused pyrimidine-imidazole ring system with...
, cytosineCytosineCytosine is one of the four main bases found in DNA and RNA, along with adenine, guanine, and thymine . It is a pyrimidine derivative, with a heterocyclic aromatic ring and two substituents attached . The nucleoside of cytosine is cytidine...
, and uracilUracilUracil is one of the four nucleobases in the nucleic acid of RNA that are represented by the letters A, G, C and U. The others are adenine, cytosine, and guanine. In RNA, uracil binds to adenine via two hydrogen bonds. In DNA, the uracil nucleobase is replaced by thymine.Uracil is a common and...
. - New sequences enter the system solely as the result of a copy process, either correct or erroneous, of other sequences that are already present.
- The substrates, or raw materials, necessary for ongoing replication are always present in sufficient quantity. Excess sequences are washed away in an outgoing flux.
- Sequences may decay into their building blocks. The probability of decay does not depend on the sequences' age; old sequences are just as likely to decay as young sequences.
In the quasispecies model, mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...
s occur through errors made in the process of copying already existing sequences. Further, selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....
arises because different types of sequences tend to replicate at different rates, which leads to the suppression of sequences that replicate more slowly in favor of sequences that replicate faster. However, the quasispecies model does not predict the ultimate extinction of all but the fastest replicating sequence. Although the sequences that replicate more slowly cannot sustain their abundance level by themselves, they are constantly replenished as sequences that replicate faster mutate into them. At equilibrium, removal of slowly replicating sequences due to decay or outflow is balanced by replenishing, so that even relatively slowly replicating sequences can remain present in finite abundance.
Due to the ongoing production of mutant sequences, selection does not act on single sequences, but on mutational "clouds" of closely related sequences, referred to as quasispecies. In other words, the evolutionary success of a particular sequence depends not only on its own replication rate, but also on the replication rates of the mutant sequences it produces, and on the replication rates of the sequences of which it is a mutant. As a consequence, the sequence that replicates fastest may even disappear completely in selection-mutation equilibrium, in favor of more slowly replicating sequences that are part of a quasispecies with a higher average growth rate. Mutational clouds as predicted by the quasispecies model have been observed in RNA viruses and in in vitro RNA replication.
The mutation rate and the general fitness of the molecular sequences and their neighbors is crucial to the formation of a quasispecies. If the mutation rate is zero, there is no exchange by mutation, and each sequence is its own species. If the mutation rate is too high, exceeding what is known as the error threshold
Error threshold (evolution)
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...
, the quasispecies will break down and be dispersed over the entire range of available sequences.
Mathematical description
A simple mathematical model for a quasispecies is as follows: let there be possible sequences and let there be organisms with sequence i. Let's say that each of these organisms asexually gives rise to offspring. Some are duplicates of their parent, having sequence i, but some are mutant and have some other sequence. Let the mutation rate correspond to the probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that a j type parent will produce an i type organism. Then the expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
number of i type organisms produced by any j type parent is
,where
.Then the total number of i-type organisms after the first round of reproduction, given as
, isSometimes a death rate term
is included so that:where
is equal to 1 when i=j and is zero otherwise. Note that the n-th generation can be found by just taking the n-th power of W substituting it in place of W in the above formula.This is just a system of linear equations. The usual way to solve such a system is to first diagonalize
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is irreducible, then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate. The components of this eigenvector give the relative abundance of each sequence at equilibrium.
A note about irreducible
W being irreducible means that for some integer, that the power of W is > 0, i.e. all the entries are positive. If W is irreducible then each type can, through a sequence of mutations (i.e. powers of W) mutate into all the other types. If W isn't irreducible, then the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below.Alternative formulations
The quasispecies formulae may be expressed as a set of linear differential equations. If we consider the difference between the new state and the old state to be the state change over one moment of time, then we can state that the time derivative of is given by this difference, we can write:The quasispecies equations are usually expressed in terms of concentrations
where..The above equations for the quasispecies then become for the discrete version:
or, for the continuum version:
A simple example
The quasispecies concept can be illustrated by a simple system consisting of 4 sequences. Sequence 1 is [0,0], and sequences [0,1], [1,0] and [1,1] are numbered 2,3 and 4 respectively. Lets say the [0,0] sequence never mutates and always produces a single offspring. Lets say the other 3 sequences all produce, on average, replicas of themselves, and of each of the other two types, where . The W matrix is then:The diagonalized matrix is
and the eigenvectors corresponding to these eigenvalues are:
| Eigenvalue | Eigenvector |
| 1-2k | [0,-1,0,1] |
| 1-2k | [0,-1,1,0] |
| 1 | [1,0,0,0] |
| 1+k | [0,1,1,1] |
Only the eigenvalue
is larger than unity. For the n-th generation, the corresponding eigenvalue will be and so will increase without bound as time goes by. This eigenvalue corresponds to the eigenvector [0,1,1,1], which represents the quasispecies consisting of sequences 2, 3, and 4, which will be present in equal numbers after a very long time. Since all population numbers must be positive, the first two quasispecies are not legitimate. The third quasispecies consists of only the non-mutating sequence 1. Its seen that even though sequence 1 is the most fit in the sense that it reproduces more of itself than any other sequence, the quasispecies consisting of the other three sequences will eventually dominate (assuming that the initial population was not homogeneous of the sequence 1 type).Further reading
- Eigen, M., J. McCaskill and P. Schuster. "The Molecular Quasi-species." Advances in Chemical Physics 75 (1989): 149-263.
- Nowak, M. A. "What is a Quasi-species?" Trends in Ecology and Evolution 7 (1992): 118-121.
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Based on article from Nupedia
Nupedia
Nupedia was an English-language Web-based encyclopedia whose articles were written by experts and licensed as free content. It was founded by Jimmy Wales and underwritten by Bomis, with Larry Sanger as editor-in-chief...
(http://www.nupedia.com/article/600/) by Claus O. Wilke, posted 2001-10-12.