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 individual characterized by a certain phenotypic behavior. Chemical: An ensemble of equal, identical molecules. complicated and loosely defined

An ensemble of “nearly” identical molecules? Preliminary understanding: a cluster of closely related but non-identical molecular species Definition

1970s Manfred Eigen and Peter Schuster Chemical Theory for the Origin of Life Assuming RNA as the first biological replicator – base-pairing Dynamics of chemical and spontaneous reproduction of RNA molecules Why quasi-species?

RNA replication Basis of all life Occur initially as spontaneous chemical reproduction of simple molecules at a very slow rate, subject to high error-rates. Why quasi-species?

Random event lead to mutations mismatching in base-pairing. The result: not an absolutely homogeneous population of RNA molecules, but a mixture of RNA molecules with different nucleotide sequences. Why quasi-species – Errors? ie. a QUASI-SPECIES

Selection molecules have different replication rates depending on their sequence (the faster, the fitter) Mutation offspring sequence differ from its parent in certain positions by ‘point mutation’ Chemical Kinetics

n different RNA sequences (length l) with population v 1, v 2, …, v n replication rates a 1, a 2, …, a n probability of replication of i results in j (i, j=1,2,…,n) Q ji Chemical Kinetics No error: Mutation:

Mathematical formulation (DE) population v 1, v 2, …, v n replication rates a 1, a 2, …, a n probability of replication of i results in j Q ji growth rate Chemical Kinetics

Rate of growth of one variant dependent on not only itself, but also all other variants In long run, no fixation of the fastest growing sequence. The population will reach an equilibrium which will contain a whole ensemble of mutants with different replication rates – quasi-species.

Quasi-species: the equilibrium distribution of sequences that is formed by this mutation and selection Quasi-species, not any individual mutant sequence, 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 is Hamming 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 determines Fitness landscape

Quasi-species: a small cloud in sequence space, wanders over the fitness landscape and search for peaks Evolution: distablization of the existing quasi- species upon change of fitness landscape – new peaks Hill-climbing under guidance of natural selection Mutations along the way is guided Quasi-species and Evolution

Error-free replication: evolution stops Error rate toooo high: population unable to maintain any genetic information, evolution impossible Error rate must be below a critical threshold value Error Threshold

Error rate (p): per base probability to make a mistake Mutation term H ij is the Hamming distance between variant i and j (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 v 1, the wild type, with replication rate a 1 its mutant distribution v 2 with a lower average replication rate a 2. q: the per base accuracy of replication ( q= 1- p). Prob(the whole sequence is replicated without error) = Error Threshold (Math again…)

( Neglecting the small probability that erroneous replication of a mutant gives rise to a wild-type sequence) Error Threshold (Math again…) the ratio converges to (consider )

 in order to maintain the wild type in the population Recall, there must be a critical q value where Error Threshold (Math again…)

A condition limiting the maximum length of the RNA sequence! ie.

An approximation for the upper genome length l that can be maintained by a given error rate Facts: Viral RNA replication (little proof-reading mechanism involved): p ≈ ; l ≈ 10 4 Human genome: p ≈ ; l ≈ 3x10 9 Error Threshold (Math again…)

Consider viral dynamics and basic reproductive ratio in a quasi-species concept Eliminate the fittest virus mutants by increasing the mutation rate with a drug Drive the whole virus population to extinction by 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 each individual sequences; Matrix contains the replication rate and mutation probabilities (unspecific degradation or dilution flow )is any function of that 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 population structure of the quasi-species - each mutant has a frequency can be understood as the fitness of the quasi-species Some fancier Mathematics

A Brief Review Quasi-species – produced by errors in the self-replication of molecules; a well-defined (eqm) distribution of mutants generated by mutation-selection process; target of 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 as eigenvalue and eigenvector of the selection-mutation matrix

The End Questions?