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## what is a

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**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**Definition**• An ensemble of “nearly” identical molecules? • Preliminary understanding: a cluster of closely related but non-identical molecular species**Why quasi-species?**• 1970s Manfred Eigen and Peter Schuster Chemical Theory for the Origin of Life • Assuming RNAas the first biological replicator – base-pairing • Dynamics of chemicaland 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 – Errors?**• Random event lead to mutations • mismatchingin base-pairing. • The result: not an absolutely homogeneouspopulation of RNA molecules , but a mixture of RNA molecules with different nucleotide sequences. ie. a QUASI-SPECIES**Chemical Kinetics**• 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 v1, v2, …, vn • replication rates a1, a2, …, an • probability of replication of iresults in j (i, j=1,2,…,n) Qji No error: Mutation:**Chemical Kinetics**• Mathematical formulation (DE) • population v1, v2, …, vn • replication rates a1, a2, …, an • probability of replication of iresults in j Qji • growth rate**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.**(A more precise) Definition**• 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**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 and Evolution**• Quasi-species: a small cloud in sequence space, wanders over the fitness landscape and search for peaks • Evolution: distablizationof the existing quasi-species upon change of fitness landscape – new peaks • Hill-climbing under guidance of natural selection • Mutationsalong the way is guided**Error Threshold**• 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 Hij is the Hamming distance between variant i and j (no. of bases in which the two strains differ) • Error-free replication:**Error Threshold (Math again…)**• Assume a population of length l consists of • a fast replicating variant v1, the wild type, with replication rate a1 • its mutant distribution v2 with a lower average replication rate a2. • 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) the ratio converges to (consider )**Error Threshold (Math again…)**• 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.**Error Threshold (Math again…)**• 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 ≈ 10-4; l ≈ 104 • Human genome: p ≈ 10-9; l ≈ 3x109**App. On Viral Quasi-species**• 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**Some fancier Mathematics**• 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**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?