Model The ABPRE Extinction results Survival case Branching within branching: a general model for host-parasite co-evolution Gerold Alsmeyer (joint work with S¨ oren Gr¨ ottrup) May 15, 2017 Gerold Alsmeyer Host-parasite co-evolution 1 of 26
Model The ABPRE Extinction results Survival case Model 1 The associated branching process in random environment 2 (ABPRE) Extinction results 3 Limit theorems in the case of survival 4 Gerold Alsmeyer Host-parasite co-evolution 2 of 26
Model The ABPRE Extinction results Survival case Model description The basic ingredients a cell population (the hosts) a population of parasites colonizing the cells The basic assumptions cells form an ordinary Galton-Watson tree (GWT) parasites sitting in different cells multiply and share their progeny into daughter cells independently, but for parasites hosted by the same cell v , offspring numbers and sharing of progeny are conditionally independent given the number of daughter cells of v Gerold Alsmeyer Host-parasite co-evolution 3 of 26
Model The ABPRE Extinction results Survival case Notational details: the cell population Ulam-Harris tree V = � n ≥ 0 N n with N 0 = { ∅ } . cell population: a GWT T = � n ∈ N 0 T n ⊂ V with T 0 = { ∅ } and T n := { v 1 ... v n ∈ V | v 1 ... v n − 1 ∈ T n − 1 and 1 ≤ v n ≤ N v 1 ... v n − 1 } , where N v denotes the number of daughter cells of v. the N v , v ∈ V are iid with common law ( p k ) k ≥ 0 , the offspring distribution of cells, having finite mean ν = ∑ k ≥ 1 kp k . the number of cells process: T n = ∑ v ∈ T n − 1 N v (a GWP). Gerold Alsmeyer Host-parasite co-evolution 4 of 26
Model The ABPRE Extinction results Survival case Details: the parasites the number of parasites in cell v are denoted by Z v . the number of parasites process is then defined by Z n := ∑ Z v , n ∈ N 0 . v ∈ T n the set of contaminated cells: T ∗ n = { v ∈ T n : Z v > 0 } . the number of contaminated cells: T ∗ n = # T ∗ n . cell counts with a specific number of parasites: � � T n := T n , 0 , T n , 1 , T n , 2 ,... , where T n , k gives the number of cells in generation n hosting k parasites. Gerold Alsmeyer Host-parasite co-evolution 5 of 26
Model The ABPRE Extinction results Survival case Z ∅ =1 Z 1 =2 Z 2 =4 Z 3 =1 Z 32 =5 Z 33 =2 Z 11 =3 Z 12 =1 Z 31 =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure: A typical realization of the first three generations of a BwBP . Gerold Alsmeyer Host-parasite co-evolution 6 of 26
Model The ABPRE Extinction results Survival case Reproduction of parasites To describe reproduction of parasites consider those hosted by a cell v ∈ T and suppose that Z v = z ≥ 1 and that v has k daughter cells, labeled v1 ,..., v k , thus N v = k . For i = 1 ,..., z , let � � X ( • , k ) X ( 1 , k ) ,..., X ( k , k ) := i , v i , v i , v � X ( 1 , k ) ,..., X ( k , k ) � be iid copies of a random vector X ( • , k ) := with arbitrary law on N k 0 and independent of any other occurring rv’s. Then, given N v = k , and for i = 1 ,..., Z v = z , equals the number of offspring of the i th parasite in v X ( j , k ) i , v that is shared into daughter cell v j . Gerold Alsmeyer Host-parasite co-evolution 7 of 26
Model The ABPRE Extinction results Survival case Model parameters and basic assumptions µ j , k = E X ( j , k ) γ = E Z 1 = ∑ k ≥ 1 p k ∑ k j = 1 µ j , k , the mean number of offspring per parasite, is supposed to be positive and finite, thus µ j , k < ∞ for all j ≤ k and P ( N v = 0 ) < 1. We further assume p 1 = P ( N v = 1 ) < 1, P ( Z 1 = 1 ) < 1, and a positive chance for more than one parasite to be shared in the same daughter cell: p k P ( X ( j , k ) ≥ 2 ) > 0 for at least one ( j , k ) , 1 ≤ j ≤ k . Gerold Alsmeyer Host-parasite co-evolution 8 of 26
Model The ABPRE Extinction results Survival case A very short list of earlier contributions M. Kimmel. Quasistationarity in a branching model of division-within-division. In Classical and modern branching processes (Minneapolis, MN, 1994) , volume 84 of IMA Vol. Math. Appl. , pages 157–164. Springer, New York, 1997. V. Bansaye. Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. , 18(3):967–996, 2008. G. Alsmeyer and S. Gr¨ ottrup. A host-parasite model for a two-type cell population. Adv. in Appl. Probab. , 45(3):719–741, 2013. Gerold Alsmeyer Host-parasite co-evolution 9 of 26
Model The ABPRE Extinction results Survival case The notorious questions Extinction-explosion dichotomy for the number of parasites process ( Z n ) n ≥ 0 . Extinction-explosion dichotomy for the number of contaminated cells process ( T ∗ n ) n ≥ 0 { Z n → ∞ } = { T ∗ n → ∞ } ? Necessary and sufficient conditions for almost sure extinction of contaminated cells. Limit theorems for the relevant processes in the survival case (after normalization). Gerold Alsmeyer Host-parasite co-evolution 10 of 26
Model The ABPRE Extinction results Survival case The ABPRE The associated branching process in random environment (ABPRE) is obtained by picking an infinite random cell-line (spine) in a size-biased version of T : Gerold Alsmeyer Host-parasite co-evolution 11 of 26
Model The ABPRE Extinction results Survival case The construction (standard) Let ( T n , C n ) n ≥ 0 be a sequence of iid random vectors independent of ( N v ) v ∈ V and ( X ( • , k ) ) k ≥ 1 , i ≥ 1 , v ∈ V . i , v The law of T n equals the size-biasing of the law of the N v , i.e. P ( T n = k ) = kp k ν for each n ∈ N 0 and k ∈ N , and P ( C n = j | T n = k ) = 1 k for 1 ≤ j ≤ k , which means that C n has a uniform distribution on { 1 ,..., k } given T n = k . Gerold Alsmeyer Host-parasite co-evolution 12 of 26
Model The ABPRE Extinction results Survival case The ABPRE The random cell-line (spine) ( V n ) n ≥ 0 is then recursively defined by V 0 = ∅ and V n := V n − 1 C n − 1 for n ≥ 1. Then ∅ =: V 0 → V 1 → V 2 → ··· → V n → ... provides us with a random cell line in V (not picked uniformly) as depicted in the following picture. Gerold Alsmeyer Host-parasite co-evolution 13 of 26
Model The ABPRE Extinction results Survival case The ABPRE V 0 V 1 V 2 . . . . . . . . . V 3 . . . . . . . . . . . . . . . Gerold Alsmeyer Host-parasite co-evolution 14 of 26
Model The ABPRE Extinction results Survival case The number of parasites along the spine Regarding the structure of the number of parasites process along the spine ( Z V n ) n ≥ 0 , the following lemma is fundamental. Lemma Let ( Z ′ n ) n ≥ 0 be a BPRE with Z ∅ ancestors and iid environmental sequence Λ := (Λ n ) n ≥ 0 taking values in { L ( X ( j , k ) ) | 1 ≤ j ≤ k < ∞ } and such that � � = p k = 1 k · kp k Λ 0 = L ( X ( j , k ) ) = P ( C 0 = j , T 0 = k ) P ν ν for all 1 ≤ j ≤ k < ∞ . Then ( Z V n ) n ≥ 0 and ( Z ′ n ) n ≥ 0 are equal in law. Gerold Alsmeyer Host-parasite co-evolution 15 of 26
Model The ABPRE Extinction results Survival case Definition of the ABPRE The BPRE ( Z ′ n ) n ≥ 0 is now called the associated branching process in random environment (ABPRE). It is one of the major tools used in the study of the BwBP , and the following lemma provides a key relation between this process and its associated ABPRE. Gerold Alsmeyer Host-parasite co-evolution 16 of 26
Model The ABPRE Extinction results Survival case A key result Lemma For all n , k , z ∈ N 0 , � � = ν − n E z T n , k Z ′ n = k P z and � � = ν − n E z T ∗ Z ′ P z n > 0 n . Gerold Alsmeyer Host-parasite co-evolution 17 of 26
Model The ABPRE Extinction results Survival case Generating functions For n ∈ N and s ∈ [ 0 , 1 ] E ( s Z ′ n | Λ) = g Λ 0 ◦ ... ◦ g Λ n − 1 ( s ) is the quenched generating function of Z ′ n with iid g Λ n and g λ defined by g λ ( s ) := E ( s Z ′ 1 | Λ 0 = λ ) = ∑ λ n s n n ≥ 0 for any distribution λ = ( λ n ) n ≥ 0 on N 0 . Moreover, p k ν E X ( j , k ) = E Z 1 = γ 1 = ∑ E g ′ Λ 0 ( 1 ) = E Z ′ ν < ∞ , (1) ν 1 ≤ j ≤ k where γ = E Z 1 . Gerold Alsmeyer Host-parasite co-evolution 18 of 26
Model The ABPRE Extinction results Survival case Regimes of the ABPRE It is well-known that ( Z ′ n ) n ≥ 0 survives with positive probability iff E log g ′ E log − ( 1 − g Λ 0 ( 0 )) < ∞ . Λ 0 ( 1 ) > 0 and Recall that γ < ∞ is assumed and that there exists 1 ≤ j ≤ k < ∞ such that p k > 0 and P ( X ( j , k ) � = 1 ) > 0, which ensures that Λ 0 � = δ 1 with positive probability. The ABPRE is called supercritical, critical or subcritical if E log g ′ Λ 0 ( 1 ) > 0, = 0 or < 0, respectively. Gerold Alsmeyer Host-parasite co-evolution 19 of 26
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