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Compositional Assemblies Behave Similarly to Quasispecies Model

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Compositional Assemblies Behave Similarly to Quasispecies Model Compositional Assemblies Behave Similarly to Quasispecies Model Renan Gross, Omer Markovitch and Doron Lancet Department of Molecular Genetics, Weizmann Institute of Science, Israel Group meeting, 01/10/2013 Introduction Quasispecies are a cloud of genotypes that appear in a population at mutation-selection balance. (J.J Bull et al, PLoS 2005) - Theoretical model (equations and assumptions), with experimental support by RNA viruses. - Usually applied when mutation rates are high. - GARD composomes replicate with relatively low fidelity (high mutation rate). Could they show similar dynamic behavior? 2 Quasispecies model • Basically a population model • n different genotypes / identities 3 Quasispecies model • Basically a population model • n different genotypes / identities Reactor 4 Quasispecies model • Basically a population model • n different genotypes / identities • Their relative concentration in the environment is denoted by Reactor the fraction xi. 푛 ෍ 푥푖 = 1 푖=1 5 Quasispecies model • Basically a population model • n different genotypes / identities • Their relative concentration in the environment is denoted by Reactor the fraction xi. 푛 • Goal: To find out how xi behave as a function of time. ෍ 푥푖 = 1 푖=1 6 Quasispecies model • Each one replicates at a certain rate – how many offspring it has per unit time. • Some replicate faster than others. • This is called the replication rate, denoted Ai 푛푢푚푏푒푟 표푓 표푓푓푠푝푟푖푛푔 • 퐴 = 7 푖 푡푖푚푒 Quasispecies model • However, replication is not exact. Sometimes, the offspring is of another genotype. • The chance that a genotype j replicates into genotype i is denoted Qij. Q11 = 0.5 Q21 = 0.2 Q31 = 0.2 Q41 = 0.1 8 Quasispecies model • We can put everything in a matrix, called the transition matrix, Q. From ퟎ. ퟓ 0.1 0 0.7 To 0.2 ퟎ. ퟓ 0 0.2 0.2 0.2 ퟏ 0 0.1 0.2 0 ퟎ. ퟏ • The main diagonal is faithful self replication. 9 Quasispecies model • How can we find out Q and A? • Assume that we are dealing with RNA sequences of length v: – There is a single genotype with highest replication rate: the master sequence 10 Quasispecies model • How can we find out Q and A? • Assume that we are dealing with RNA sequences of length v: – There is a single genotype with highest q 1 replication rate: the master sequence 1 – Single digit replication: 0 ≤ q ≤ 1 1-q 0 • What is the chance for error-less replication? 11 Quasispecies model • How can we find out Q and A? • Assume that we are dealing with RNA sequences of length v: – There is a single genotype with highest q 1 replication rate: the master sequence 1 – Single digit replication: 0 ≤ q ≤ 1 1-q 0 • What is the chance for error-less replication? – All mutations of the master sequence replicate slower according to the Hamming distance. • Effectively, many genotypes are grouped together. • Q and A are built only from q and v. 12 Quasispecies model 13 Quasispecies model • q = 1 14 Quasispecies model • q = 1 exact replication 15 Quasispecies model • q = 1 exact replication • q = 0 16 Quasispecies model • q = 1 exact replication • q = 0 exact complimentary replication 17 Quasispecies model • q = 1 exact replication • q = 0 exact complimentary replication • q = 0.5 18 Quasispecies model • q = 1 exact replication • q = 0 exact complimentary replication all data is lost. • q = 0.5 19 Quasispecies model • q = 1 exact replication • q = 0 exact complimentary replication all data is lost. • q = 0.5 • Starting at q = 1, lowering it results in loss of the master sequence as the most frequent genotype. – (requires lack of back mutation) ERROR CATASTROPHE • RNA viruses may be fought by bringing them to error catastrophe. 20 Quasispecies model • Under constant population assumptions, the transition matrix Q and the replication rates A are all that are needed in order to find out the concentration dynamics. 21 Quasispecies model • Under constant population assumptions, the transition matrix Q and the replication rates A are all that are needed in order to find out the concentration dynamics. • The Eigen equation (after Manfred Eigen): 푑푥 푖 = 퐴 푄 − 퐸෨ 푡 푥 + ෍ 퐴 푄 푥 푑푡 푖 푖푖 푖 푖 푖푗 푗 푗≠푖 22 Quasispecies model • Under constant population assumptions, the transition matrix Q and the replication rates A are all that are needed in order to find out the concentration dynamics. • The Eigen equation (after Manfred Eigen): 푑푥 푖 = 퐴 푄 − 퐸෨ 푡 푥 + ෍ 퐴 푄 푥 푑푡 푖 푖푖 푖 푖 푖푗 푗 푗≠푖 • Where E is the “average excess rate” 푛 퐸෨ 푡 = ෍ 퐴푖푥푖 푖=1 23 Quasispecies model • Q and A can be combined into one matrix W, which tells how much of each genotype is produced per unit time. 푾 = 푸 ⋅ 푑푖푎푔(푨) 24 Examples: 1 0.001 0.001 0.01 0.1 2 0.01 0.01 푾 = 0.001 0.1 3 0.01 0.001 0.001 0.001 4 • Initial conditions: x1 = 1, all others are 0. 25 Examples: 1 0.001 0.001 0.01 0.1 2 0.01 0.01 푾 = 0.001 0.1 3 0.01 0.001 0.001 0.001 4 • Initial conditions: x1 = 1, all others are 0. 26 Examples: 1 0.001 0.001 1 0.1 2 0.01 1 푾 = 0.001 0.1 3 1 0.001 0.001 0.001 4 • Initial conditions: x1 = 1, all others are 0. 27 Examples: 1 0.001 0.001 1 0.1 2 0.01 1 푾 = 0.001 0.1 3 1 0.001 0.001 0.001 4 • Initial conditions: x1 = 1, all others are 0. 28 • The most frequent genotype is not necessarily the one with highest Ai. • The steady state population distribution is called the quasispecies. – That means, a vertical slice 29 GARD model (Graded Autocatalysis Replication Domain) RNA world Lipid world DNA / RNA / Polymers Sequence covalent bonds 30 GARD model (Graded Autocatalysis Replication Domain) RNA world Lipid world Assemblies / Clusters / DNA / RNA / Polymers Vesicles / Membranes Sequence Composition covalent bonds non-covalent bonds Segre and Lancet, EMBO Reports 1 (2000) 31 GARD model (Graded Autocatalysis Replication Domain) Synthetic chemistry Kinetic model Catalytic network (b) of rate-enhancement values Segre, Ben-Eli and Lancet, Proc. Natl. Acad. Sci. 97 (2000) GARD model (Graded Autocatalysis Replication Domain) Synthetic chemistry Kinetic model Catalytic network (b) of rate-enhancement values Rate enhancement dn NG n i k N k n 1 b j i 1...N Molecular repertoire f i b i ij G dt j1 N Segre, Ben-Eli and Lancet, Proc. Natl. Acad. Sci. 97 (2000) GARD model (Graded Autocatalysis Replication Domain) Synthetic chemistry Kinetic model Homeostatic growth Catalytic network (b) of rate-enhancement values b Fission / Split Rate enhancement dn NG n i k N k n 1 b j i 1...N Molecular repertoire f i b i ij G dt j1 N Segre, Ben-Eli and Lancet, Proc. Natl. Acad. Sci. 97 (2000) GARD model (Graded Autocatalysis Replication Domain) Following a single lineage. 35 GARD model (Graded Autocatalysis Replication Domain) Following a single lineage. Similarityng=30; split=1.5; seed=361 ‘carpet’ 1000 1 800 0.8 600 0.6 400 0.4 Generation 200 0.2 Compositional Similarity 0 200 400 600 800 1000 Generation 36 GARD model (Graded Autocatalysis Replication Domain) Following a single lineage. Similarityng=30; split=1.5; seed=361 ‘carpet’ 1000 1 800 0.8 600 0.6 400 0.4 Generation 200 0.2 Compositional Similarity 0 200 400 600 800 1000 Generation Composome (compositional genome): a faithfully replicating composition/assembly. Compotype (composome type): a collection of similar composomes. Molecular Compotype: the center of mass of the compotype cloud, treated as a molecular assembly. 37 GARD model (Graded Autocatalysis Replication Domain) • Idea: if we ignore the stochasticity inherent in the model, then errorless-replication occurs according to beta matrix eigenvectors. dn NG n i k N k n 1 b j i 1...N f i b i ij G dt j1 N 38 Moment Algebraux • Multiplying a matrix by a vector gives another vector • 푨 ⋅ 푥Ԧ = 푦Ԧ 푎 푎 푥 푎 푥 + 푎 푥 푦 • 11 12 ⋅ 1 = 11 1 12 2 = 1 푎21 푎22 푥2 푎21푥1 + 푎22푥2 푦2 • An eigenvector is a vector 푥Ԧ such that: • 푨 ⋅ 푥Ԧ = 휆푥Ԧ – l is called the eigenvalue. – l may be complex, and so may the eigenvector. 39 Moment Algebraux • The Perron-Frobenius Theorem: – A matrix with strictly positive entries contains a maximal real eigenvalue. – Its eigenvector is real and non-negative. In fact, it’s the only one with this property. • As molecular assemblies must contain real non- negative number of molecules, this looks interesting. 40 GARD and Quasispecies • Do GARD population dynamics behave like the quasispecies model? • What we would like to do: – For each assembly, experimentally find out the transition frequencies and replication rates – In other words: find Q and A. • Problem: 199 – N = 100, n = 100 There are possible G max 100 assemblies ( ~4 ⋅ 1058) 41 GARD and Quasispecies • Solution: group some assemblies together and treat them as one genotype. • We decided to group together by distances from the eigenvector. N space (simplified to 2d) 2 3 G 2 Molecule 1 Eigenvector Molecule 1 42 GARD and Quasispecies • Problem: how do we sample such a large space? • 30000 assemblies are randomly generated. 43 GARD and Quasispecies • Problem: how do we sample such a large space? • 30000 assemblies are randomly generated. – By filling up assemblies until they reach nmax. • Assemblies generated this way are far from the target. 43 GARD and Quasispecies • Problem: how do we sample such a large space? • 30000 assemblies are randomly generated.
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