Dynamical Systems in Biology

Transcription

1 Dynamical Systems in Biology Hal Smith A R I Z O N A S T A T E U N I V E R S I T Y H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

2 Outline 1 What s special about dynamical systems arising in biology? 2 Systems with Defined Feedback Relations 3 Cooperative Systems 4 Competitive systems in R 3 5 Monotone Cyclic Feedback Systems 6 Conjectures of R. Thomas 7 Summary and References H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

3 Dynamical Systems Theory x i = f i (x 1, x 2,, x n ), 1 i n Goal: characterize the asymptotic behavior, as t ± of every solution, or, of almost every solution, or, of almost every solution of almost every f, or, ω-limit set of solution x(t): {y : y = lim m x(t m), t m } Goal Reached for n = 1, 2; Little known for n 3. Restrict to special systems: Hamiltonian, Gradient,etc. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

4 Dynamical Systems Theory x i = f i (x 1, x 2,, x n ), 1 i n Goal: characterize the asymptotic behavior, as t ± of every solution, or, of almost every solution, or, of almost every solution of almost every f, or, ω-limit set of solution x(t): {y : y = lim m x(t m), t m } Goal Reached for n = 1, 2; Little known for n 3. Restrict to special systems: Hamiltonian, Gradient,etc. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

5 What s special about dynamical systems arising in biology? What s special about systems arising in biology? x i = f i (x 1, x 2,, x n ) = f i (x), 1 i n Variables tend to be positive: x 0, i.e., all x i 0. Theorem: Assume: 1 solutions of initial value problems with x(0) 0 exist and are unique. 2 i, x 0 : x i = 0 f i (x) 0. Then x(0) 0 x(t) 0, t > 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

6 What s special about dynamical systems arising in biology? What s special about systems arising in biology? x i = f i (x 1, x 2,, x n ) = f i (x), 1 i n Variables tend to be positive: x 0, i.e., all x i 0. Theorem: Assume: 1 solutions of initial value problems with x(0) 0 exist and are unique. 2 i, x 0 : x i = 0 f i (x) 0. Then x(0) 0 x(t) 0, t > 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

7 Systems with Defined Feedback Relations Systems with Defined Feedback Relations Positivity may lead to defined feedback relations among variables: Rate of change of x i depends positively or negatively on x j, j i: Either "x j activates x i " or "x j inhibits x i " x, x, f i (x) 0, i j f i (x) 0, i j Can we exploit DFR to circumscribe asymptotic behavior of solutions? H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

8 Systems with Defined Feedback Relations Systems with Defined Feedback Relations Positivity may lead to defined feedback relations among variables: Rate of change of x i depends positively or negatively on x j, j i: Either "x j activates x i " or "x j inhibits x i " x, x, f i (x) 0, i j f i (x) 0, i j Can we exploit DFR to circumscribe asymptotic behavior of solutions? H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

9 Systems with Defined Feedback Relations Signed Influence Directed Graph" for DFR System 1 Vertices: the dependent variables x i. 2 + directed edge from x j to x i if f i (x) 0. x j + x i 3 directed edge from x j to x i if f i (x) 0. x j x i 4 No directed edge from x j to x i if f i (x) 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

10 Systems with Defined Feedback Relations DFR and the Jacobian Df(x) Prey-Predator: [ + ] x 1 x 2 + [ Competition: ] x 1 x 2 H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

11 Systems with Defined Feedback Relations Repressilator with 2 genes x i = [protein] product of gene i y i = [mrna] of gene i. x i 1 represses transcription of y i : x i = β i (y i x i ) y i = α i f i (x i 1 ) y i, i = 1, 2, mod 2 where α i,β i > 0 and f i > 0 satisfies f i < 0. y + 1 x 1 y 2 x 2 + H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

12 Systems with Defined Feedback Relations HIV-T-cell dynamics T = T cell density in blood I = infected T cell-density V = HIV virus density N = # virus released T = δ αt + pt(1 T ) kvt T max I = kvt βi V = βni γv kvt T I + + V H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

13 Cooperative Systems Cooperative Systems A system is cooperative if f i (x) 0, i j. It is cooperative and irreducible if the Jacobian matrix is irreducible. Component-wise partial order: 1 x y i, x i y i. 2 x < y x y x y. 3 x y i, x i < y i. x 2 x y A Cooperative Irreducible system is strongly order-preserving in forward time: x(0) < x(0) x(t) x(t), t > 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31 x 1

14 Cooperative Systems Cooperative Systems A system is cooperative if f i (x) 0, i j. It is cooperative and irreducible if the Jacobian matrix is irreducible. Component-wise partial order: 1 x y i, x i y i. 2 x < y x y x y. 3 x y i, x i < y i. x 2 x y A Cooperative Irreducible system is strongly order-preserving in forward time: x(0) < x(0) x(t) x(t), t > 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31 x 1

15 Cooperative Systems Cooperative Systems A system is cooperative if f i (x) 0, i j. It is cooperative and irreducible if the Jacobian matrix is irreducible. Component-wise partial order: 1 x y i, x i y i. 2 x < y x y x y. 3 x y i, x i < y i. x 2 x y A Cooperative Irreducible system is strongly order-preserving in forward time: x(0) < x(0) x(t) x(t), t > 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31 x 1

16 Cooperative Systems Behavior of Cooperative Irreducible Systems Theorem [Non-ordering of Limit Sets] No pair of points of ω(x) are related by <. Theorem [Limit Set Dichotomy] If x < y then either (a) ω(x) < ω(y), or (b) ω(x) = ω(y) Equilibria. Theorem (M.W.Hirsch): For almost every x(0), the solution x(t) converges to equilibrium. Theorem: an attracting periodic orbit. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

17 Cooperative Systems Behavior of Cooperative Irreducible Systems Theorem [Non-ordering of Limit Sets] No pair of points of ω(x) are related by <. Theorem [Limit Set Dichotomy] If x < y then either (a) ω(x) < ω(y), or (b) ω(x) = ω(y) Equilibria. Theorem (M.W.Hirsch): For almost every x(0), the solution x(t) converges to equilibrium. Theorem: an attracting periodic orbit. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

18 Competitive Systems Cooperative Systems A system is competitive if f i (x) 0, i j. x = f(x) is competitive x = f(x) is cooperative. Time reversal flips sign on every arrow! Theorem (Hirsch): The flow on a limit set of an n-dimensional competitive or cooperative system is topologically equivalent to the flow of a general (n 1)-dimensional system on a compact invariant set. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

19 Competitive Systems Cooperative Systems A system is competitive if f i (x) 0, i j. x = f(x) is competitive x = f(x) is cooperative. Time reversal flips sign on every arrow! Theorem (Hirsch): The flow on a limit set of an n-dimensional competitive or cooperative system is topologically equivalent to the flow of a general (n 1)-dimensional system on a compact invariant set. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

20 Cooperative Systems A Generalization of Cooperative Systems Suppose x = f(x) can be decomposed x = (x 1, x 2 ) R k R n k x 1 = f 1 (x 1, x 2 ) x 2 = f 2 (x 1, x 2 ) diagonal blocks f i x i (x) have nonnegative off-diagonal entries. off-diagonal blocks f i (x) 0 have nonpositive entries. Jacobian = Components cluster into two subgroups. positive within-group interactions, negative between-group interactions. Then change of variables y = (y 1, y 2 ) = (x 1, x 2 ) yields a cooperative system in the usual sense! H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

21 Cooperative Systems A Generalization of Cooperative Systems Suppose x = f(x) can be decomposed x = (x 1, x 2 ) R k R n k x 1 = f 1 (x 1, x 2 ) x 2 = f 2 (x 1, x 2 ) diagonal blocks f i x i (x) have nonnegative off-diagonal entries. off-diagonal blocks f i (x) 0 have nonpositive entries. Jacobian = Components cluster into two subgroups. positive within-group interactions, negative between-group interactions. Then change of variables y = (y 1, y 2 ) = (x 1, x 2 ) yields a cooperative system in the usual sense! H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

22 Cooperative Systems A Generalization of Cooperative Systems Suppose x = f(x) can be decomposed x = (x 1, x 2 ) R k R n k x 1 = f 1 (x 1, x 2 ) x 2 = f 2 (x 1, x 2 ) diagonal blocks f i x i (x) have nonnegative off-diagonal entries. off-diagonal blocks f i (x) 0 have nonpositive entries. Jacobian = Components cluster into two subgroups. positive within-group interactions, negative between-group interactions. Then change of variables y = (y 1, y 2 ) = (x 1, x 2 ) yields a cooperative system in the usual sense! H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

23 Cooperative Systems Repressilator with 2 genes is cooperative x i = [protein] product of gene i y i = [mrna] of gene i. x i 1 represses transcription of y i : x i = β i (y i x i ) y i = α i f i (x i 1 ) y i, i = 1, 2, mod 2 where α i,β i > 0 and f i > 0 satisfies f i < 0. Jacobian = Gardner et al, Construction of a genetic toggle switch in E. coli", Nature(403),2000. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

24 Cooperative Systems Dynamics of Repressilator Equilibria u = (x 1, y 1, x 2, y 2 ) are in 1-to-1 correspondence with fixed points of increasing map g α 2 f 2 α 1 f 1 2 Fixed Points of g g(x2) x2 Theorem: If g has no degenerate fixed points, odd number of equilibria u 1, u 2,, u 2m+1 indexed by increasing values of x 2. u 2i+1 are stable, u 2i are unstable. If B(u i ) denotes the basin of attraction of u i, then odd i B(u i ) is open and dense in R 4 +. u 1 is globally attracting if m = 0. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

25 Cooperative Systems Competition in Two Patches yields cooperative system x 1 = x 1 (r 1 a 1 x 1 b 1 y 1 ) + d(x 2 x 1 ) x 2 = x 2 (r 2 a 2 x 2 b 2 y 2 ) + d(x 1 x 2 ) y 1 = y 1 (s 1 c 1 x 1 d 1 y 1 ) + D(y 2 y 1 ) y 2 = x 1 (s 2 c 2 x 2 d 1 y 2 ) + D(y 1 y 2 ) Jacobian = Jiang & Liang, Quart. Appl. Math. 64 (2006): System is cooperative and competitive! Has 2 D dynamics-a limit set containing no equilibrium is periodic orbit. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

26 Cooperative Systems Is my system x = f(x) Cooperative? Competitive? For Cooperative: 1 DFR system: no sign change of f i (x), i j. 2 sign symmetry: f i (x) f j x i (y) 0, i j, x, y. (no predator-prey-like relations.) 3 every loop in undirected*, signed, incidence graph for Jacobian ( f i (x)) i,j has even # of negative edges. *just drop arrows on signed influence directed graph. For Competitive: 1 as above 2 as above 3 as above except even # of positive edges in every loop. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

27 Cooperative Systems Is my system x = f(x) Cooperative? Competitive? For Cooperative: 1 DFR system: no sign change of f i (x), i j. 2 sign symmetry: f i (x) f j x i (y) 0, i j, x, y. (no predator-prey-like relations.) 3 every loop in undirected*, signed, incidence graph for Jacobian ( f i (x)) i,j has even # of negative edges. *just drop arrows on signed influence directed graph. For Competitive: 1 as above 2 as above 3 as above except even # of positive edges in every loop. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

28 Cooperative Systems Is my system x = f(x) Cooperative? Competitive? For Cooperative: 1 DFR system: no sign change of f i (x), i j. 2 sign symmetry: f i (x) f j x i (y) 0, i j, x, y. (no predator-prey-like relations.) 3 every loop in undirected*, signed, incidence graph for Jacobian ( f i (x)) i,j has even # of negative edges. *just drop arrows on signed influence directed graph. For Competitive: 1 as above 2 as above 3 as above except even # of positive edges in every loop. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

29 Cooperative Systems Is my system x = f(x) Cooperative? Competitive? For Cooperative: 1 DFR system: no sign change of f i (x), i j. 2 sign symmetry: f i (x) f j x i (y) 0, i j, x, y. (no predator-prey-like relations.) 3 every loop in undirected*, signed, incidence graph for Jacobian ( f i (x)) i,j has even # of negative edges. *just drop arrows on signed influence directed graph. For Competitive: 1 as above 2 as above 3 as above except even # of positive edges in every loop. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

30 Cooperative Systems 2-gene repressilator is competitive & cooperative x i = [protein] product of gene i y i = [mrna] of gene i. x i 1 represses transcription of y i : x i = β i (y i x i ) y i = α i f i (x i 1 ) y i, i = 1, 2, mod 2 where α i,β i > 0 and f i > 0 satisfies f i < 0. even number of + and even number of edges! y + 1 x 1 y 2 x 2 + H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

31 Cooperative Systems MAPK intra-cellular signaling Cascade is cooperative x y 1 = = v 2x k 2 + x + v 0u + v 1 v 6 (y tot y 1 y 3 ) k 6 + (y tot y 1 y 3 ) v 3xy 1 k 3 + y 1 y 3 = v 4x(y tot y 1 y 3 ) k 4 + (y tot y 1 y 3 ) v 5y 3 k 5 + y 3 z 1 = v 10(z tot z 1 z 3 ) k 4 + (z tot z 1 z 3 ) v 7y 3 z 1 k 7 + z 1 z 2 = v 8y 3 (z tot z 1 z 3 ) k 4 + (z tot z 1 z 3 ) v 9z 3 k 9 + z 3 input u x + y 1 y 3 + z 1 z 3 output Molecular Systems Biology & Control, E. Sontag H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

32 Competitive systems in R 3 Competitive Systems in R 3 Corollary[Hirsch]: A Limit set containing no equilibria is a periodic orbit Influence graph must have even # of + edges in each loop. x 3 x 3 + x 1 x 2 + x 1 x 2 H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

33 Competitive systems in R 3 Classical Lotka-Volterra Competition x 1 = x 1 (r 1 c 11 x 1 c 12 x 2 c 13 x 3 ) x 2 = x 2 (r 2 c 21 x 1 c 22 x 2 c 23 x 3 ) x 3 = x 3 (r 3 c 31 x 1 c 32 x 2 c 33 x 3 ) Hirsch s Carrying Simplex" M.L. Zeeman: 33 dynamically distinct phase portraits How many limit cycles? Three! Hofbauer & So, Gyllenberg & Ping Yan H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

34 Competitive systems in R 3 Examples with one negative edge (term in red) classical S E I R predator-prey with stage structure S E I = µ µs σis = σis (µ + γ)e = γe (µ + ρ)i x = x(r ax) bxy 2 /(1 + cx) y 1 = kbxy 2 /(1 + cx) (m + d 1 )y 1 y 2 = my 1 d 2 y 2 Goldbeter s model for Mitotic Oscillator C C = v i v d X K d + C k dc M C 1 M = V M1 K c + C K M V M 2 K 2 + M X = V M3 M 1 X K X V X 4 K 4 + X H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

35 Competitive systems in R 3 Examples with one negative edge (term in red) classical S E I R predator-prey with stage structure S E I = µ µs σis = σis (µ + γ)e = γe (µ + ρ)i x = x(r ax) bxy 2 /(1 + cx) y 1 = kbxy 2 /(1 + cx) (m + d 1 )y 1 y 2 = my 1 d 2 y 2 Goldbeter s model for Mitotic Oscillator C C = v i v d X K d + C k dc M C 1 M = V M1 K c + C K M V M 2 K 2 + M X = V M3 M 1 X K X V X 4 K 4 + X H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

36 Competitive systems in R 3 Examples with one negative edge (term in red) classical S E I R predator-prey with stage structure S E I = µ µs σis = σis (µ + γ)e = γe (µ + ρ)i x = x(r ax) bxy 2 /(1 + cx) y 1 = kbxy 2 /(1 + cx) (m + d 1 )y 1 y 2 = my 1 d 2 y 2 Goldbeter s model for Mitotic Oscillator C C = v i v d X K d + C k dc M C 1 M = V M1 K c + C K M V M 2 K 2 + M X = V M3 M 1 X K X V X 4 K 4 + X H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

37 Competitive systems in R 3 Virus dynamics model is competitive T I V = δ αt + pt(t max T)/T max kvt = kvt βi = βni γv kvt 2000 Periodic Virus Dynamics T* T,T*,V T V time Smith & de Leenheer, SIAM Appl. Math. 63 (2003) H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

38 Monotone Cyclic Feedback Systems Monotone Cyclic Feedback Systems single-loop DFR system: where x 1 = f 1 (x 1, x n ) x 2 = f 2 (x 2, x 1 ). x n = f n (x n, x n 1 ) δ 2 x 1 x 2 δ 1 δ 3 δ i f i > 0, δ i {,+} x i 1 δ 4 x 4 x 3 Mallet-Paret & H.S., Poincaré-Bendixson theorem for MCFS, J. Dynam. & Diff. Eqns. (2) 1990 H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

39 Monotone Cyclic Feedback Systems Monotone Cyclic Feedback Systems Theorem: A limit set L of a bounded solution of a MCFS is either: an equilibrium a periodic orbit a set of equilibria and orbits connecting them. Moreover, Π i : R n R 2 defined by Π i x = (x i, x i 1 ) is injective on L. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

40 Monotone Cyclic Feedback Systems Repressilator with n genes is a MCFS! x i 1 represses transcription of y i : where f i 0 satisfy f i < 0. x i = β i (y i x i ) y i = α i f i (x i 1 ) y i, i = 1, 2, 3 mod 3 y + 1 x 1 y + 2 x 2 y + 3 x 3 H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

41 Monotone Cyclic Feedback Systems Results for n-gene repressilator Theorem: 1 if n even, then almost all orbits converge. 2 n odd, then stable periodic orbits may exist. Hofbauer et. al., A generalized model of the repressilator. J. Math. Biol. 53 (2006) H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

42 Conjectures of R. Thomas Basic Definitions R. Thomas* formulated conjectures about the possible dynamics a system could have based on its signed directed influence graph. Below are some key points required to formulate his conjectures. 1 Partial derivatives may have different signs in different regions of phase space! The graph G depends on the point x in phase space: G = G(x). 2 G includes signed self-loops i i with + sign if f i x i (x) > 0 and - sign if partial derivative is negative. 3 G is a directed graph (edges have direction). 4 A circuit in G is a sequence of distinct vertices i 1, i 2,, i p so that there is an edge from i k to i k+1, 1 k < p, and from i p to i 1. 5 the sign of a circuit is the product of signs of its edges. *On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillation, Springer Ser. Synergetics 9, , 1981 H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

43 Conjectures of R. Thomas Thomas s Conjectures Conjecture 1: A positive circuit (even # negative edges) in G(x), for some x, is a necessary condition for multistationarity (more than one equilibrium). Conjecture 2: A negative circuit of length at least two is a necessary condition for stable periodicity. Conjecture 3: Chaotic dynamics requires both a positive and a negative circuit. More recent reference: Kaufman, Soul, Thomas, A new necessary condition on interaction graphs for multistationarity, T. Theor. Biol. 248 (2007), H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

44 Conjectures of R. Thomas Examples 1 predator-prey system has a negative circuit so cannot have two coexistence equilibria but can have stable periodic solutions. 2 competitive system has positive circuit so can have multistationarity but not stable periodic solutions. H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31

45 Summary and References References for competitive and cooperative systems Monotone Dynamical Systems, H.S. & M. Hirsch, Handbook of Differential Equations, Ordinary Differential Equations ( volume 2), eds. A.Canada, P.Drabek, A.Fonda, Elsevier, , Monotone systems, a mini-review, H.S. & M.W. Hirsch, in Positive Systems. Proceedings of the First Multidisciplinary Symposium on Positive Systems (POSTA 2003). Luca Benvenuti, Alberto De Santis and Lorenzo Farina (Eds.) Lecture Notes on Control and Information Sciences vol. 294, Springer Verlag, Heidelberg, Monotone Dynamical Systems: an introduction to the theory of competitive and cooperative systems, Amer. Math. Soc. Surveys and Monograghs, 41, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Review 30, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, with J.Mallet-Paret, J. Dynamics and Diff. Eqns., 2, 1990, H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, / 31