what is a
QUASI-SPECIES
By Ye Dan U062281A
USC3002 Picturing the World through Mathematics
Definition
Quasi-: widely-used prefix to indicate“almost”, “seemingly”, “nearly” etc.
Species: ?
Biological: A class of individualcharacterized by a certainphenotypic behavior.
Chemical: An ensemble of equal,identical molecules.
complicated andloosely defined
An ensemble of “nearly” identicalmolecules?
Preliminary understanding: a cluster ofclosely related but non-identical molecularspecies
Definition
1970s Manfred Eigen and PeterSchuster Chemical Theory for theOrigin of Life
Assuming RNA as the first biologicalreplicator – base-pairing
Dynamics of chemical and spontaneousreproduction of RNA molecules
Why quasi-species?
RNA replication
Basis of all life
Occur initially asspontaneous chemicalreproduction of simplemolecules at a very slowrate, subject to higherror-rates.
Why quasi-species?
Random event lead to mutations
mismatching in base-pairing.
The result: not an absolutelyhomogeneous population of RNA molecules ,but a mixture of RNA molecules withdifferent nucleotide sequences.
Why quasi-species – Errors?
ie. a QUASI-SPECIES
Selection
molecules have different replication ratesdepending on their sequence (the faster, thefitter)
Mutation
offspring sequence differ from its parent incertain positions by ‘point mutation’
Chemical Kinetics
n different RNA sequences (length l) withpopulation v1, v2, …, vn
replication rates a1, a2, …, an
probability of replication of i results in j (i,j=1,2,…,n) Qji
Chemical Kinetics
No error:
Mutation:
Mathematical formulation (DE)
population v1, v2, …, vn
replication rates a1, a2, …, an
probability of replication of i results in j Qji
growth rate
Chemical Kinetics
Rate of growth of one variant dependent on notonly itself, but also all other variants
In long run, no fixation of the fastest growingsequence. The population will reach an equilibriumwhich will contain a whole ensemble of mutants withdifferent replication rates – quasi-species.
Quasi-species: the equilibrium distributionof sequences that is formed by thismutation and selection
Quasi-species, not any individual mutantsequence, is the target of selection
Guided mutation
(A more precise) Definition
Sequence Space & Fitness Landscape
Given a length, all possible variants
Distance between two sequences isHamming distance
No. of dimension = length of the sequence
4 possibilities in each dimension: A, T, C, G
One more dimension:reproduction rate ie. Fitness
Selection pressure determinesFitness landscape
Quasi-species: a small cloud in sequence space,wanders over the fitness landscape and search forpeaks
Evolution: distablization of the existing quasi-species upon change of fitness landscape – newpeaks
Hill-climbing under guidance of naturalselection
Mutations along the way is guided
Quasi-species and Evolution
Error-free replication: evolution stops
Error rate toooo high: population unable tomaintain any genetic information,evolution impossible
Error rate must be below a critical thresholdvalue
Error Threshold
Error rate (p): per base probability to make amistake
Mutation term
Hij is the Hamming distance between variant i andj (no. of bases in which the two strains differ)
Error-free replication:
Error Threshold
Assume a population of length l consists of
a fast replicating variant v1, the wild type, withreplication rate a1
its mutant distribution v2 with a lower averagereplication rate a2.
q: the per base accuracy of replication ( q=1- p).
Prob(the whole sequence is replicatedwithout error) =
Error Threshold (Math again…)
(Neglecting the small probability that erroneousreplication of a mutant gives rise to a wild-typesequence)
Error Threshold (Math again…)
the ratio converges to
(consider )
in order tomaintain the wild type in the population
Recall , there must be a critical qvalue where
Error Threshold (Math again…)
Error Threshold (Math again…)
A condition limitingthe maximum lengthof the RNA sequence!
ie.
An approximation for the upper genomelength l that can be maintained by a givenerror rate
Facts:
Viral RNA replication (little proof-readingmechanism involved): p ≈ 10-4; l ≈ 104
Human genome: p ≈ 10-9; l ≈ 3x109
Error Threshold (Math again…)
Consider viral dynamics and basicreproductive ratio in a quasi-species concept
Eliminate the fittest virus mutants byincreasing the mutation rate with a drug
Drive the whole virus population to extinctionby further increase of mutation rate
App. On Viral Quasi-species
Consider the standard equation for a dynamic(bacteria/viral) population
Vector represents the population sizes of eachindividual sequences;
Matrix contains the replication rate andmutation probabilities
(unspecific degradation or dilution flow )is any function ofthat keeps the total population in a constant size. It can be
Some fancier Mathematics
Equilibrium of ,
Largest Eigenvalue : max. average replication rate
Eigenvector (corresponding to ): the quasi-species
Normalize , describes the exact populationstructure of the quasi-species - each mutant has afrequency
can be understood as the fitness of the quasi-species
Some fancier Mathematics
A Brief Review
Quasi-species – produced by errors in the self-replicationof molecules; a well-defined (eqm) distribution ofmutants generated by mutation-selection process; targetof selection
Chemical kinetics; Mathematical framework
The fitness landscape, and the implication on evolution
Error threshold and application
Fitness and exact structure of the quasi-species aseigenvalue and eigenvector of the selection-mutationmatrix
The End
Questions?