Open Markov processes and reaction networks

Transcription

1 Open Markov processes and reaction networks Blake S. Pollard Department of Physics and Astronomy University of California, Riverside Thesis Defense June 1, 2017 Committee: Prof. John C. Baez (Chair) Prof. Shan-Wen Tsai Prof. Nathaniel Gabor June 1, / 50

2 Compositional modeling of open systems June 1, / 50

3 Compositional modeling of open systems Consider the set of coupled differential equations represented by the labelled graph: dp dt = Hp H = p(t) = A(t) B(t) C(t) A 9 B C June 1, / 50

4 Compositional modeling of open systems Given smooth functions of time I(t) R and O(t) R, we can write down an open dynamical system: Ȧ = 9A + 7C + I(t) Ḃ = 9A 2B + 5C Ċ = 2B 12C O(t) I(t) A 9 B C O(t) June 1, / 50

5 Compositional modeling of open systems Let s couple this system with another such open system: I(t) A 9 B C O(t) I (t) C 7 9 D O (t) Ȧ = 9A + 7C + I(t) Ḃ = 9A 2B + 5C Ċ = 2B 12C O(t) C = 7C + 9D + I (t) Ḋ = 7C 9D O (t) June 1, / 50

6 Compositional modeling of open systems When the outflow O(t) matches the inflow I (t), we can compose the systems by identifying C(t) and C (t) and adding their time derivatives: I(t) A 9 B C 7 9 D O (t) Ȧ = 9A + 7C + I(t) Ḃ = 9A 2B + 5C Ċ = 2B 19C + 9D Ḋ = 7C 9D O (t) June 1, / 50

7 Some Papers Much of what I ll discuss can be found in: John C. Baez and Blake S. Pollard, A compositional framework for reaction networks, submitted. John C. Baez, Brendan Fong and Blake S. Pollard, A compositional framework for Markov processes, Journal of Mathematical Physics. Blake S. Pollard, Open Markov processes: A compositional perspective on non-equilibrium steady states in biology, Entropy. Blake S. Pollard, A Second Law for open Markov processes, Open Systems and Information Dynamics. June 1, / 50

8 Idea: View open systems as morphisms in categories June 1, / 50

9 Idea: View open systems as morphisms in categories A category C consists of a collection of objects X, Y... and a collection of morphisms f : X Y... closed under an associative composition operation fg X f Y g Z gh h A together with identity morphisms 1 X : X X satisfying the left/right identity laws 1 X f X 1 Y Y June 1, / 50

10 Open systems as morphisms in a category We can think of open systems as morphisms in a category. X f Y Composition corresponds to connecting systems. X f g Z Y June 1, / 50

11 Open systems as morphisms in a category We can think of open systems as morphisms in a category. X f Y Composition corresponds to connecting systems. X fg Z June 1, / 50

12 What types of open systems do we consider? We study systems which admit a graphical syntax. In my thesis, I focus on two classes of systems: Markov processes and reaction networks. Markov processes specify systems of linear differential equations and can be represented using directed, labelled graphs. Reaction networks specify systems of polynomial differential equations and can be represented by certain bipartite graphs, commonly known as Petri nets. A C 9 A 5 7 τ 2 B B C June 1, / 50

13 Reaction networks Definition A reaction network with rates (S, T, s, t, r) consists of: a finite set S a finite set T functions s, t : T N S a function r : T (0, ). We call the elements of S species, those of N S complexes, and those of T transitions. Any transition τ T has a source s(τ), a target t(τ), and a rate constant r(τ). If s(τ) = κ and t(τ) = κ we write τ : κ κ. A B α C June 1, / 50

14 Open reaction networks Open reaction networks are generalizations of reaction networks in which certain species are labelled as input and output species. R : X Y X A B α C D Y 4 5 June 1, / 50

15 Composition of open reaction networks Consider another open reaction network R : Y Z Y 4 5 E β F Z 6 June 1, / 50

16 Composition of open reaction networks To compose R : X Y and R : Y Z we first combine them X A B Y C 4 α E β F 5 D Z 6 June 1, / 50

17 Composition of open reaction networks Then, we identify any species which are in the image of the same point in Y X Z 1 A 2 α C β F 6 3 B This gives a new open reaction network RR : X Z. June 1, / 50

18 Decorated cospan categories We utilize an approach to the categorical modeling of open systems, due to Brendan Fong, called decorated cospans. A cospan in any category C is a diagram of the form i S o X Y. To open a system built on some finite set S, we specify a pair of functions i : X S and o : Y S specifying the inputs and outputs of the system. The apex S of the cospan is decorated by some additional data. For RxNet, this data is that of a reaction network with rates on S. June 1, / 50

19 The category of open reaction networks Definition An open reaction network R : X Y consists of a cospan of finite sets i S o X Y. together with a reaction network R = (S, T, s, t, r) on S. Theorem ( Baez, P. ) There is a category RxNet whose objects are finite sets and whose morphisms are isomorphism classes of open reaction networks. June 1, / 50

20 Functors A functor is a map between categories which respects composition and preserves identities. Given categories C and D, a functor F : C D sends objects to objects and morphisms to morphisms such that: F (fg) = F (f )F (g) F (1 x ) = 1 F (x) June 1, / 50

21 Functors for studying behaviors of open systems F : OpenSys Behavior F f F g = F fg June 1, / 50

22 What types of behaviors do we consider for these systems? For my Oral, I discussed non-equilibrium steady states of open Markov processes using a variational principle. Not all non-equilibrium steady states of open Markov processes obey a variational principle. Today I ll describe compositional approaches to capturing the dynamical and steady state behaviors of an arbitrary open reaction network without recourse to a variational principle. June 1, / 50

23 The rate equation A reaction network with rates specifies a set of coupled, non-linear differential equations called its rate equation: da(t) dt db(t) dt = r(α)a(t)b(t) = r(α)a(t)b(t) A B α C dc(t) dt = 2r(α)A(t)B(t) June 1, / 50

24 The rate equation Given a reaction network with rates R = (S, T, s, t, r), with species set S = {1, 2,..., S }, let us denote a vector of concentrations of each species by c = (c 1, c 2,..., c S ) R S. Concentrations are non-negative. Introducing the notation c s(τ) = σ S c sσ(τ) σ, we can write the rate equation of a general reaction network obeying mass-action kinetics as dc dt = r(τ) ( t(τ) s(τ) ) c s(τ). τ T June 1, / 50

25 The rate equation Given a reaction network R = (S, T, s, t, r), we can define a vector field v(c) = r(τ) ( t(τ) s(τ) ) c s(τ) τ T generating the time evolution of the concentrations c R S via dc dt = v(c). For mass-action kinetics, the vector field v : R S R S is polynomial in the concentrations. June 1, / 50

26 A category of open dynamical systems Definition An open dynamical system D : X Y on S consists of a cospan of finite sets S together with a polynomial vector field v on R S. Theorem (Baez, P.) X i There is a category Dynam where objects are finite sets and morphisms are isomorphism classes of open dynamical systems. o Y June 1, / 50

27 The gray-boxing functor X Z 1 A E 5 2 α C β 3 B F 6 Theorem (Baez, P.) There is a functor : RxNet Dynam sending an open reaction network to its corresponding open dynamical system. June 1, / 50

28 The gray-boxing functor (R : X Y ) X A B α C Y 4 v A = r(α)a(t)b(t) v B = r(α)a(t)b(t) v C = 2r(α)A(t)B(t) June 1, / 50

29 The gray-boxing functor (R : Y Z) Y 4 D β E F Z 5 6 v D = r(β)d(t) v E = r(β)d(t) v F = r(β)d(t) June 1, / 50

30 The gray-boxing functor (R : X Y ) (R : Y Z) X A B α C Y 4 D β E F Z 5 6 v A = r(α)ab v B = r(α)ab v C = 2r(α)AB v D = r(β)d v E = r(β)d v F = r(β)d June 1, / 50

31 The gray-boxing functor (R : X Y ) (R : Y Z) X Z 1 A E 5 2 α C β 3 B F 6 v A = r(α)ab v B = r(α)ab v C + v D = 2r(α)AB r(β)d and C = D v E = r(β)d v F = r(β)d June 1, / 50

32 The gray-boxing functor (RR : X Z) X Z 1 A E 5 2 α C β 3 B F 6 v A = r(α)ab v B = r(α)ab v C = 2r(α)AB r(β)c v E = r(β)c v F = r(β)c June 1, / 50

33 The open rate equation Let I : R R X and O : R R Y be arbitrary smooth functions of time specifying the inflows and outflows. X A B α C Y 4 da(t) dt db(t) dt dc(t) dt = r(α)a(t)b(t) + I 1 (t) = r(α)a(t)b(t) + I 2 (t) + I 3 (t) = 2r(α)A(t)B(t) O 4 (t) June 1, / 50

34 The open rate equation Given an open dynamical system together with specified inflows I R X and outflows O R X, we define the pushforward i : R X R S by and define o : R Y R S by i (I) σ = o (O) σ = {x:i(x)=σ} {y:o(y)=σ} I x O y. We can then write down the open rate equation as dc(t) dt = v(c(t)) + i (I(t)) o (O(t)). June 1, / 50

35 Steady states A steady state solution of the open rate equation is a concentration vector c R S such that dc dt = 0. From the open rate equation we see that this implies dc dt = v(c) + i (I) o (O) v(c) = o (O) i (I). This imposes relations among the steady state concentrations and flows along the boundary. June 1, / 50

36 Semialgebraic relations A semialgebraic subspace of a vector space is a one defined in terms of polynomials and inequalities. June 1, / 50

37 Semialgebraic relations A semialgebraic subspace of a vector space is a one defined in terms of polynomials and inequalities. A semialgebraic relation A: U V is a semialgebraic subspace A U V. June 1, / 50

38 Semialgebraic relations A semialgebraic subspace of a vector space is a one defined in terms of polynomials and inequalities. A semialgebraic relation A: U V is a semialgebraic subspace A U V. There is a category SemiAlgRel where objects are real vector spaces V and morphisms are semialgebraic relations. June 1, / 50

39 Semialgebraic relations A semialgebraic subspace of a vector space is a one defined in terms of polynomials and inequalities. A semialgebraic relation A: U V is a semialgebraic subspace A U V. There is a category SemiAlgRel where objects are real vector spaces V and morphisms are semialgebraic relations. Composition of relations requires that they agree on their overlap. June 1, / 50

40 Semialgebraic relations A semialgebraic subspace of a vector space is a one defined in terms of polynomials and inequalities. A semialgebraic relation A: U V is a semialgebraic subspace A U V. There is a category SemiAlgRel where objects are real vector spaces V and morphisms are semialgebraic relations. Composition of relations requires that they agree on their overlap. Given semialgebraic relations A: U V and B : V W, their composite AB : U W is given by AB = {(u, w): v V with (u, v) A and (v, w) B}. June 1, / 50

41 Steady state behavior We characterize the steady state behavior of an open reaction network in terms of the semialgebraic relation imposed between inputs and outputs. X A B α C Y 4 da(t) dt db(t) dt dc(t) dt = r(α)a(t)b(t) + I 1 (t) = r(α)a(t)b(t) + I 2 (t) + I 3 (t) = 2r(α)A(t)B(t) O 4 (t) June 1, / 50

42 Steady state behavior c X = (c 1, c 2, c 3 ) R 3, I X = (I 1, I 2, I 3 ) R X c Y = c 4 R Y, O Y = O 4 R Y X A B α C Y 4 such that (c X, I X, c Y, O Y ) R X R X R Y R Y I 1 = r(α)ab I 2 + I 3 = r(α)ab O 4 = 2r(α)AB June 1, / 50

43 The black-box functor Theorem (Baez, P.) There is a functor : Dynam SemiAlgRel sending an open dynamical system to the semialgebraic relation characterizing its steady state boundary concentrations and flows. June 1, / 50

44 The black-box functor Theorem (Baez, P.) There is a functor : Dynam SemiAlgRel sending an open dynamical system to the semialgebraic relation characterizing its steady state boundary concentrations and flows. Theorem (Baez, P.) Composing the gray-boxing and black-boxing functors gives a functor RxNet Dynam SemiAlgRel sending an open reaction network to the subspace of possible steady state boundary concentrations and flows. June 1, / 50

45 Black-boxing (R : X Y ) X A B α C Y 4 da(t) dt db(t) dt dc(t) dt = r(α)a(t)b(t) + I 1 (t) = r(α)a(t)b(t) + I 2 (t) + I 3 (t) = 2r(α)A(t)B(t) O 4 (t) June 1, / 50

46 Black-boxing ( (R)): R X R X R Y R Y X A B α C Y 4 (c X, I X, c Y, O Y ) such that I 1 = r(α)ab I 2 + I 3 = r(α)ab O 4 = 2r(α)AB June 1, / 50

47 The gray-boxing functor (R : Y Z) Y 4 D β E F Z 5 6 dd(t) dt de(t) dt df (t) dt = r(β)d(t) + I 4 (t) = r(β)d(t) O 5 (t) = r(β)d(t) O 6 (t) June 1, / 50

48 Black-boxing ( (R )): R Y R Y R Z R Z Y 4 D β E F Z 5 6 (c Y, I Y, c Z, O Z ) I 4 = r(β)d O 5 = r(β)d O 6 = r(β)d June 1, / 50

49 Composing relations ( (R)) ( (R )) (c X, I X, c Y, O Y )(c Y, I Y, c Z, O Z ) I 1 = r(α)ab I 2 + I 3 = r(α)ab O 4 = 2r(α)AB I 4 = r(β)d O 5 = r(β)d O 6 = r(β)d June 1, / 50

50 Composing relations ( (R)) ( (R )) (c X, I X, c Y, O Y )(c Y, I Y, c Z, O Z ) I 1 = r(α)ab I 2 + I 3 = r(α)ab O 4 = 2r(α)AB I 4 = r(β)d O 5 = r(β)d O 6 = r(β)d C = D O 4 = I 4 June 1, / 50

51 Composing relations ( (R)) ( (R )) (c X, I X, c Y, O Y )(c Y, I Y, c Z, O Z ) I 1 = r(α)ab I 2 + I 3 = r(α)ab O 4 = 2r(α)AB I 4 = r(β)d O 5 = r(β)d O 6 = r(β)d C = D 2r(α)AB = r(β)d June 1, / 50

52 Composing relations ( (R)) ( (R )): R X R X R Z R Z (c X, I X, c Z, O Z ) I 1 = r(α)ab I 2 + I 3 = r(α)ab 2r(α)AB = r(β)c O 5 = r(β)c O 6 = r(β)c. June 1, / 50

53 Black-boxing (RR : X Y ) X Z 1 A E 5 2 α C β 3 B F 6 da dt = r(α)ab + I 1 db dt = r(α)ab + I 2 + I 3 dc dt = 2r(α)AB r(β)c de dt = r(β)c O 5 df dt = r(β)c O 6 June 1, / 50

54 Black-boxing ( (RR )): R X R X R Z R Z X Z 1 A E 5 2 α C β 3 B F 6 I 1 = r(α)ab I 2 + I 3 = r(α)ab 2r(α)AB = r(β)c O 5 = r(β)c O 6 = r(β)c. June 1, / 50

55 Conclusions The fact that black-boxing is accomplished via a functor means that one can compute the steady state behavior of a composite open reaction network by composing the semialgebraic relations characterizing the steady state behaviors of its constituent systems: ( (R)) ( (R )) = ( (RR )) This provides a compositional approach to studying both the dynamical and steady state behaviors of open reaction networks. June 1, / 50

56 Thank you! June 1, / 50

57 Composition in Dynam Given open dynamical systems D : X Y on S and D : Y Z on S i S S o i o X Y Z with vector fields v : R S R S and v : R S R S dynamical system DD : X Z on S + Y S to get an open j S + Y S j i S o i S o X Y Z we need to cook up a vector field v : R S+ Y S R S+ Y S. June 1, / 50

58 Composition in Dynam To get a vector field v : R S+ Y S R S+ Y S, first take the inclusion map [j, j ]: S + S S + Y S and define two maps, [j, j ] : R S+S R S+ Y S as [j, j ] (v + v ) σ = (v + v ) σ, {σ [j,j ](σ )=σ} and [j, j ] : R S+ Y S R S+S as [j, j ] (c ) = c [j, j ] with c R S+ Y S. We can then define our vector field via the expression v (c ) = [j, j ] (v + v )[j, j ] (c ). June 1, / 50