1 Specific features of dynamics of large systems with inheritance and microevolution Alexander Gorban, ETH Zurich & Institute of Computational Modeling Russian Academy of Sciences M a t e r i a l s

2 Contents Gauses Principle; Kinetics of Radon measures with conservation of supports; Description of stable steady-states; Description of limit sets; Efficiency of selection (discreteness of limit distributions); Example: Cells divisions autosynchronization; Conclusion and outlook. M a t e r i a l s

3 Gauses Principle Equations of Populations Dynamics : dN i /dt=k i (N,Q)N i, dQ/dt= F(N,Q); N i - the biomass of ith population, Q - the state of non-living environment. M a t e r i a l s

4 Coexistence I={i 1,i 2,…, i n }- the numbers of stationary coexisting populations k i (N,Q)=0 for i from I. Let k depend on N,Q only through a system of m factors f j (N,Q): k i (N,Q)=q i (f 1 (N,Q),…, f m (N,Q)). Then generically n m The Gauses exclusion principle M a t e r i a l s

5 Infinite system: a simple example N(x,t), x [a,b], dN(x,t)/dt=k(x,{N(x,t)}) N(x,t), k(x,{N(x,t)})=q(x)- a b p(x)N(x,t)dx. q(x)>0, p(x) >0 If N(x,0)>0 then N(x,t) c (x-x 0 ), where x 0 =argmax q(x) ( the point of global maximum of q(x)) M a t e r i a l s

6 Dynamics of distribution density (in simple example) x0x0 q(x) N(x,t) x N(x,0) M a t e r i a l s

7 X, C(X), M(X) Let X be a compact (metric) space without isolated points. Let C(X) be the space of real-valued continuous functions on X with the topology of uniform convergence on X, Let M(X)=C*(X) be the space of Radon measures on X, µ (f)=[ µ,f]. The space M(X) is endowed with the weak topology of the dual space: µ i µ* if [ µ i,f] [ µ*,f] for every f C(X). M a t e r i a l s

8 dµ /dt = k µ ×µ (1) The phase space will be a compact set of bounded positive measures M N (X): for N>0 M N (X)={ µ M + (X) | [ µ,1] = N]}. (M + (X) is the cone of nonnegative measures). The reproduction coefficient for each distribution µ M N (X) is a function k µ C(X) The map µ k (M N (X) C(X)) is assumed to be continuous. Systems with inheritance M a t e r i a l s

9 Inheritance is the conservation of the support Def. suppµ is the minimal closed subset in X with the property: if f(x) = 0 on suppµ, then [ µ,f]=0. Prop. If µ (t) is a solution of the equation (1), then the support of µ (t) does not depend on t. M a t e r i a l s

10 Asymptotically stable steady-state distributions M a t e r i a l s Only states with a discrete (hence finite) support can be asymptotically stable steady- state distributions for the system (1): N i µ *= i N i i (x), where i (x) is an unit mass at the point x i.

11 Conditions of external stability The external stability is the stability with respect to a one-point expansion of the support. A necessary condition for the external stability of µ * is: k µ *(x) 0 for all x X A sufficient condition is: k µ *(x)<0 for x X, x supp µ *, k µ *(x)=0 for x X, x supp µ * M a t e r i a l s

12 Conditions of external stability M a t e r i a l s k (x) x supp

13 The average reproduction coefficient M a t e r i a l s Let µ (t) be a solution of (1): µ (t)= µ (0)exp( 0 t k µ ( ) d ), k µ t be the average value of k µ ( ) on [0,t] : k µ t = 1 / t 0 t k µ ( ) d ; then µ (t)= µ (0) exp(t k µ t )

14 Supports of -limit distributions M a t e r i a l s The phase space M N (X) is compact. The space conv(k(M N (X))) is compact too. If µ * is the -limit point of the solution µ (t), then there is a sequences of times t i such that µ (t i ) µ *. k µ t i conv(k(M N (X))). { k µ t i }has a limit point k*(x). k*(x)=0, if x supp µ *; k*(x) 0, if x supp µ (0).

15 How much points of zero global maximum can have a continuous function? Almost always one function has only one point of global maximum, and corresponding maximum value is not 0. For a generic n-parameter family of functions, there may exist stably a function with n-1 points of global maximum and with zero value of this maximum. M a t e r i a l s

16 Existence in one-parametric family of functions a zero point of global maximum (a) and two global maxima (b) M a t e r i a l s

17 Completely thin sets A set Y in a Banach space Q iscompletely thin, if for any compact set K in Q and arbitrary positive >0 there exists a vector q in Q, such that q < and K+q does not intersect Y. Examples: in finite dimensions only, for infinite-dimensional Q - compact sets, closed subspaces of infinite codimension. M a t e r i a l s

18 Typically a support of limit distribution is negligible Systems (1) are parameterized by continuous maps µ µ k. Denote by Q the space of these maps M N (X) C(X) with the topology of uniform convergence on M N (X). It is a Banach space. For almost all* ) systems (1) the support of any -limit distribution is nowhere dense in X, and has the Lebesgue measure 0 for Euclidean space X. * ) The set of exclusions is completely thin. M a t e r i a l s

19 Typically a support of limit distribution is nearly finite Let n >0, n 0. The following statement is almost always true for the system (1). Let the support of initial distribution be the whole X. Then the support of any -limit distribution µ * is nearly finite, meaning that it is approximated by finite sets faster than n 0: there is a number N, such that for n>N there exists a finite set S n of n elements such that dist(S n,supp µ *)< n, where dist is the Hausdorff distance. M a t e r i a l s

20 Cell division autosynchronization M a t e r i a l s T is the cell circle time, [0,T) is the phase of the cell state, =t(modT), (NT)d is a measure, the distributions of cells with respect to phase at the moment NT. ( N T+T)= = (NT) exp[k T k 1 ( - ) (NT)d ].

21 Instability of uniform distributions 0 =k 0 / 0 T k 1 ( )d. The linearized near 0 equation: (NT+T)=A[ (NT)]= = (NT)- 0 0 T k 1 ( - ) (NT)d. e 0 =1, e sn =sin(2 n /T), e cn =cos(2 n /T), = [ n e sn + n e cn ], k 1 ( )=b [a n e sn +b n e cn ]. (A): 0 =1- 0 b 0 T, n =1- 0 b n T/2 i 0 a n T/2. M a t e r i a l s

22 Stable and unstable regions 0 b n T/2 0 a n T/2 n <1 n >1 M a t e r i a l s The set of functions k 1 ( ) with all b n 0 is nowhere dense in C([0,T]), L 2 ([0,T]), etc., moreover, it is completely thin set. The uniform distribution is almost always unstable

23 Waves of autosynchronization M a t e r i a l s 0 b 1 T/2 0 a 1 T/2 Stable uniform distribution Stable waves with non-zero velocity v b/( a) Waves with velocity v N 1/N 0 v k 1 ( )=b 0 + a 1 e s1 +b 1 e c1, b 0 > (a 1 2 +b 1 2 ).

24 How to apply the theory to discussion on biological optimality? 1. The only source of specific optimality in biology is the dynamics of systems with inheritance and the principle of the maximum of the average reproduction coefficient. 2. To apply these principles to reality we should: a) determine, what are the inherited units in our problem; b) produce a dynamical model (1) or its modification; c) describe the limit behavior of this system via the optimality principles. M a t e r i a l s