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Critical population and error threshold on the sharp peak by Raphael Cerf PDF

By Raphael Cerf

ISBN-10: 1470409674

ISBN-13: 9781470409678

The aim of this paintings is to suggest a finite inhabitants counterpart to Eigen's version, which includes stochastic results. the writer considers a Moran version describing the evolution of a inhabitants of dimension m of chromosomes of size over an alphabet of cardinality ?. The mutation likelihood consistent with locus is q. He bargains in basic terms with the pointy top panorama: the replication price is s>1 for the grasp series and 1 for the opposite sequences. He stories the equilibrium distribution of the method within the regime the place ? 8,m? 8,q?0, q?a?]0, 8[,m?a?[0, eight]

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Additional resources for Critical population and error threshold on the sharp peak landscape for a Moran model

Example text

CHAPTER 6 Lumping The state space of the process (Xt )t≥0 is huge, it has cardinality κ m . We will rely on a classical technique to reduce the state space called lumping (see the appendix). We consider here only the sharp peak landscape. In this situation, the fitness of a chromosome is a function of its distance to the master sequence. A close look at the mutation mechanism reveals that chromosomes which are at the same distance from the Master sequence are equivalent for the dynamics, hence they can be lumped together in order to build a simpler process on a reduced space.

4 can be rewritten as 1 + E τ0 Z0 = 1 δ0 m f i=1 i ν(i) = m f 1≤i≤m i 1 π(i − 1) . m γi Our goal is to estimate the asymptotic behavior of the right–hand quantity. 3. Let f : [0, 1] → R be a continuous non–decreasing function such that f (0) = 0. Let a ∈]0, +∞[. We have f lim 1≤i≤m ,m→∞ q→0, q→a i 1 π(i − 1) m γi E(τ0 | Z0 = 1) = f ρ∗ (a) . Proof. Throughout the proof, we write simply ρ∗ instead of ρ∗ (a). Let η > 0. For , m large enough and q small enough, we have ρ∗ − ρ MH (0, 0), MH (θ, 0) < η, ¨ CERF RAPHAEL 52 whence f 1≤i≤m i 1 π(i − 1) m γi = f 1≤i≤m |i/m−ρ∗ |≤η i 1 π(i − 1) + m γi f (ρ∗ + η) ≤ 1≤i≤m |i/m−ρ∗ |≤η i 1 π(i − 1) m γi f 1≤i≤m |i/m−ρ∗ |>η 1 π(i − 1) + γi f (1) 1≤i≤m |i/m−ρ∗ |>η 1 π(i − 1) γi ≤ f (ρ∗ + η)E(τ0 | Z0 = 1)+ m3 f (1) π (ρ∗ − η)m 1 − MH (0, 0) + π (ρ∗ + η)m .

We have the following explicit formula for the expected value of τ0 : m E(τ0 | Z0 = 1) = i=1 1 π(i − 1) . γi Let ν be the invariant probability measure of (Zn )n≥0 . We have the following explicit formula for ν: 1 , ν(0) = 1 + δ0 E(τ0 | Z0 = 1) δ0 π(i − 1) γi . ∀i ∈ { 1, . . 2. 2. For these two processes, we have the following explicit formula for the transition probabilities: γi = σi2 1 − MH (0, 0) + i(m − i) 1 − MH (θ, 0) , 1 ≤ i ≤ m, m(σi + m − i) δi = σi(m − i)MH (0, 0) + (m − i)2 MH (θ, 0) , 0 ≤ i ≤ m− 1.

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Critical population and error threshold on the sharp peak landscape for a Moran model by Raphael Cerf

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