Invariant Manifolds of models from Population Genetics Belgin Seymenoğlu October 9, 2018
What did I investigate? I investigated continuous-time models in Population Genetics. I proved that each model has an invariant manifold in its phase plot...... as long as the parameters satisfy certain conditions.
A Brief History of Population Genetics 1886: Mendel discovered genes, and that they come in different variants, or alleles. 1909: Janssens discovered genetic recombination. Early 1900s: Fisher, Haldane and Wright devised the first models in Population Genetics by combining statistics and Mendel s results. 1956: Kimura derived a continuous-time selection-recombination model. 1976: Nagylaki and Crow proposed the following model...
Model 1: The Nagylaki-Crow model Nagylaki and Crow s model focuses on a single gene with two alleles. All individuals in the population are diploid. Real world example: Cystic fibrosis. Let C and c be the normal and cystic fibrosis alleles respectively. Then there are three genotypes... Normal : CC, x (1) Carriers : Cc, z (2) Cystic fibrosis : cc, y (3)
The model The Nagylaki-Crow model is as follows: ẋ = F 11 x 2 + F 12 xz + 1 4 F 22z 2 D 1 x xϕ ż = F 12 xz + 2F 13 xy + F 23 yz + 1 2 F 22z 2 D 2 z zϕ ẏ = F 33 y 2 + F 23 yz + 1 4 F 22z 2 D 3 y yϕ,
... where: Fertilities are F ij for the paired genotypes i and j. Death rates are D i for genotype i. Mean fitness is ϕ = F 11 x 2 + 2F 12 xz + F 22 z 2 + 2F 23 yz + 2F 13 xy + F 33 y 2 D 1 x D 2 z D 3 y.
A manifold in the model? 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
A manifold in the model? Conjecture: The Nagylaki-Crow model always has an invariant manifold. The manifold has been shown to exist by assuming additive fertilities F ij = f i + f j and D 2 = 1 2 (D 1 + D 3 ). Aim: Drop those assumptions!
Rewriting the model It would be very difficult to prove that the manifold exists using the original ODEs...... so we use a new coordinate system w = 2x z, t = 2y, z = 0 iff w, t =, z Unfortunately the new phase space is unbounded, but this is a minor drawback.
A graphical proof 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
A graphical proof 12 10 8 6 4 2 0 0 2 4 6 8 10 12
Conditions for my proof My proof requires some conditions to make it work: One condition ensures all the black curves in the sequence are decreasing functions D 2 < D 3 + F 12. (4) Meanwhile the other constraint forces all of the curves to stay convex F 11 D 1 D 3 F 33. Also, F 13 > 0 guarantees that the limit we found is indeed invariant.
Remarks 1. We don t need all of D 2 min(d 1 + F 23, D 3 + F 12 ), so we don t need the model to be competitive in the proof. 2. The proof still works if you swap condition (4) with D 2 < D 1 + F 23.
Is the manifold... Smooth? Convex? Unique? Globally attracting?
Example: The manifold is not unique
Example: The manifold is not convex 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
A blow-up of the previous example 1.00 0.98 0.96 0.94 0.92 0.90 0.00 0.02 0.04 0.06 0.08 0.10
Conclusion for the first model I conjectured that the Nagylaki-Crow model always has an invariant manifold. By rewriting this model in different coordinates, I proved that the manifold does indeed exist given three conditions are satisfied. For that case, the manifold is both decreasing and convex.
Open questions When is the manifold unique? Globally attracting? Smooth? Can we relax the existence conditions? What happens for the three-allele model?
Model 2: The Selection-Recombination model This model focuses on two genes, each with two alleles. All individuals in the population are diploid. Example: Pea plants. Genes for... Flower colour: F (purple), f (white). Height: T (tall), t(short). Then there are sixteen genotypes...
The Selection-Recombination model... but we will only look at the gamete genotypes (four of them!)
The Selection-Recombination model Label the gamete genotypes ft, F t, ft and F T as 1, 2, 3, 4.
The Selection-Recombination model Label the gamete genotypes ft, F t, ft and F T as 1, 2, 3, 4.
Recombination Exchange of genetic material between two chromosomes. Responsible for variation among offspring. Happens when the chromosomes cross over during meiosis (cell division that produces gametes).
The model The Selection-Recombination model is as follows:... where, if x = (x 1, x 2, x 3, x 4 ), x 1 = x 1 (m 1 m) rd x 2 = x 2 (m 2 m) + rd x 3 = x 3 (m 3 m) + rd x 4 = x 4 (m 4 m) rd m i = (W x) i for i = 1, 2, 3, 4. Mean fitness m = x T W x. Coefficient of linkage disequilibrium D = x 1 x 4 x 2 x 3. Recombination rate 0 < r 1 2.
A manifold in the model?
A manifold in the model?
The manifold does not always exist!
A manifold in the model? Question: When does the Selection-Recombination model have the Quasilinkage Equilibirum (QLE) manifold? The manifold has been previously shown to exist in the case of weak selection: W = I + sa, where A ij < 1 and s r. Aim: Drop those assumptions!
Assumption for the fitnesses 0 < s 1, 0 k h 1 s, h 0.
Graphical proof Start with the Wright manifold (D = 0)...
Graphical proof... and let it evolve in time according to the flow...
Graphical proof We want to show the surface is Lipschitz for all time...
Graphical proof Suffices to prove the normal stays in the cone for all time...
Graphical proof But the evolving surface is always bounded in a tetrahedron...
Conditions for existence of the manifold Conditions needed for the manifold to exist: One condition ensures the normal does stay in the cone r(1 ks) s (2 max(1, h) k). (5) The QLE manifold is globally attracting if 9 > s(2 + 7h + k). (6)
Conclusions for the model I conjectured that selection-recombination model has an invariant manifold for a wide range of parameter values...... but it may not always exist! I found explicit conditions for existence of the manifold.
Open questions Geometry of the manifold, especially curvature. Is it a saddle surface? Smoothness of the manifold? Can we relax the existence conditions? What happens in the limiting case r = 0, i.e. no recombination?