Enzymatic networks and toric steady states

Enzymatic networks and toric steady states Mercedes Pérez Millán∗ and Alicia Dickenstein Dto. de Matemática–FCEyN–Universidad de Buenos Aires Dto. de Cs. Exactas–CBC–Universidad de Buenos Aires IMAS-CONICET Buenos Aires – Argentina SIAM AG13, August 2, 2013 Aim I To find a graphical method for detecting toric steady states. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 2 / 17 Aim I To find a graphical method for detecting toric steady states. We will construct some graphs from the reaction network and give some sufficient conditions on these graphs to guarantee toric steady states. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 2 / 17 What are toric steady states? (Enzymatic) Chemical Reaction Network k1 k3 S + E Y1 → P + E k2 k4 −→ Polynomial dynamical system mass action kinetics k 6 P + F Y2 → S +F k5 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states dx = f (x) dt SIAM AG13, August 2, 2013 3 / 17 What are toric steady states? (Enzymatic) Chemical Reaction Network k1 k3 S + E Y1 → P + E k2 k4 −→ Polynomial dynamical system mass action kinetics k 6 P + F Y2 → S +F k5 dx = f (x) dt Steady states They are the nonnegative zeros of a set of polynomial equations, f1 (x) = 0, · · · , fs (x) = 0. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 3 / 17 What are toric steady states? (Enzymatic) Chemical Reaction Network k1 k3 S + E Y1 → P + E k2 k4 −→ Polynomial dynamical system mass action kinetics k 6 P + F Y2 → S +F k5 dx = f (x) dt Steady states They are the nonnegative zeros of a set of polynomial equations, f1 (x) = 0, · · · , fs (x) = 0. Toric steady states We say the system has toric steady states if the steady state ideal is a binomial ideal and it admits nonnegative zeros. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 3 / 17 Why do we want toric steady states? If the system has toric steady states, then I the steady states can be explicitly parametrized by monomials (or shown to be empty). M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 4 / 17 Why do we want toric steady states? If the system has toric steady states, then I the steady states can be explicitly parametrized by monomials (or shown to be empty). I there are necessary and sufficient conditons that allow to decide about multistationarity and they take the form of linear inequality systems. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 4 / 17 Known example We showed in [–,Dickenstein,Shiu,Conradi(2012)] that the system associated to the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism has toric steady states E S0 E F M. Pérez Millán, A. Dickenstein F (UBA) ... S2 S1 E E F Enzymatic networks and toric s. states Sn F SIAM AG13, August 2, 2013 5 / 17 Are there more? We want to find more “seeable” examples. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 6 / 17 Are there more? We want to find more “seeable” examples. Toric steady states ⇒ rational parametrization of steady states. M. Thomson and J. Gunawardena. J. Theor. Biol. 261, (2009), pp. 626–636. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 6 / 17 Are there more? We want to find more “seeable” examples. Toric steady states ⇒ rational parametrization of steady states. M. Thomson and J. Gunawardena. J. Theor. Biol. 261, (2009), pp. 626–636. I They ask for {Enzymes} ∩ {Substrates} = ∅ consider cascades. I We want more than a rational parametrization: we want binomials. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states we want to SIAM AG13, August 2, 2013 6 / 17 Running example k1 k S000 + E1 Y1 →3 S100 + E1 k2 k9 k5 k7 k k 14 S010 + F2 Y5 → S000 + F2 k10 k13 k18 k17 k 20 S001 + F3 Y7 → S000 + F3 S000 + E3 Y6 → S001 + E3 k16 k21 k6 k12 k 11 S000 + E2 Y4 → S010 + E2 k15 k4 S100 + F1 Y2 Y3 →8 S0000 + F1 k19 k24 k 23 S + S100 Y8 → P + S100 k 26 P + F4 Y9 → S + F4 k22 k25 S100 S100 S P F4 E2 S010 (UBA) F1 E3 S000 F2 M. Pérez Millán, A. Dickenstein E1 S001 F3 Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 7 / 17 Running example k1 k S000 + E1 Y1 →3 S100 + E1 k2 k9 k5 k7 k k 14 S010 + F2 Y5 → S000 + F2 k10 k13 k18 k17 k 20 S001 + F3 Y7 → S000 + F3 S000 + E3 Y6 → S001 + E3 k16 k21 k6 k12 k 11 S000 + E2 Y4 → S010 + E2 k15 k4 S100 + F1 Y2 Y3 →8 S0000 + F1 k19 k24 k 23 S + S100 Y8 → P + S100 k 26 P + F4 Y9 → S + F4 k22 k25 S100 S100 S P F4 E2 S010 (UBA) F1 E3 S000 F2 M. Pérez Millán, A. Dickenstein E1 S001 F3 Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 7 / 17 Intermediate species The equations: dy1 /dt = k1 s000 e1 − (k2 + k3 )y1 dy2 /dt = k4 s100 f1 − (k5 + k6 )y2 + k7 y3 dy3 /dt = k6 y2 − (k7 + k8 )y3 dy4 /dt = k9 s000 e2 − (k10 + k11 )y4 dy5 /dt = k12 s010 f2 − (k13 + k14 )y5 dy6 /dt = k15 s000 e3 − (k16 + k17 )y6 dy7 /dt = k18 s001 f3 − (k19 + k20 )y7 dy8 /dt = k21 s.s100 − (k22 + k23 )y8 dy9 /dt = k24 pf4 − (k25 + k26 )y9 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 8 / 17 Intermediate species The equations: 0 = k1 s000 e1 − (k2 + k3 )y1 0 = k4 s100 f1 − (k5 + k6 )y2 + k7 y3 0 = k6 y2 − (k7 + k8 )y3 0 = k9 s000 e2 − (k10 + k11 )y4 0 = k12 s010 f2 − (k13 + k14 )y5 0 = k15 s000 e3 − (k16 + k17 )y6 0 = k18 s001 f3 − (k19 + k20 )y7 0 = k21 s.s100 − (k22 + k23 )y8 0 = k24 pf4 − (k25 + k26 )y9 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 8 / 17 Intermediate species All of them are binomials except for... 0 = k1 s000 e1 − (k2 + k3 )y1 0 = k4 s100 f1 − (k5 + k6 )y2 + k7 y3 0 = k6 y2 − (k7 + k8 )y3 0 = k9 s000 e2 − (k10 + k11 )y4 0 = k12 s010 f2 − (k13 + k14 )y5 0 = k15 s000 e3 − (k16 + k17 )y6 0 = k18 s001 f3 − (k19 + k20 )y7 0 = k21 s.s100 − (k22 + k23 )y8 0 = k24 pf4 − (k25 + k26 )y9 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 8 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k5 k7 k8 k1 k S000 + E1 Y1 →3 S100 + E1 k2 E1 k1 s000 Y1 k2 + k3 k6 Y3 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k5 k7 k8 k k2 E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 M. Pérez Millán, A. Dickenstein k1 S000 + E1 Y1 →3 S100 + E1 (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k7 k k2 E1 k5 k8 k1 S000 + E1 Y1 →3 S100 + E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where Condition I: Strongly connected. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k7 k k2 E1 k5 k8 k1 S000 + E1 Y1 →3 S100 + E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R Condition I: Strongly connected. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k7 k k2 E1 k5 k8 k1 S000 + E1 Y1 →3 S100 + E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R Condition I: Strongly connected. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k7 k k2 E1 k5 k8 k1 S000 + E1 Y1 →3 S100 + E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R Condition I: Strongly connected. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k7 k k2 E1 k5 k8 k1 S000 + E1 Y1 →3 S100 + E1 k1 s000 Y1 k2 + k3 k6 Immediate binomial. Y3 Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R ρ2 = k8 k4 s100 + k7 k4 s100 Condition I: Strongly connected. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k5 k7 k8 k1 k S000 + E1 Y1 →3 S100 + E1 k2 E1 k1 s000 Y1 k2 + k3 k6 Y3 Immediate binomial. Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R ρ2 = (k8 k4 + k7 k4 )s100 ←monomial Condition I: Strongly connected. Condition II: Only one directed path from the enzyme to the each intermediate. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species k4 k6 k5 k7 k S100 +F1 Y2 Y3 →8 S0000 +F1 k4 s100 F1 Y2 k5 k7 k8 k1 k S000 + E1 Y1 →3 S100 + E1 k2 E1 k1 s000 Y1 k2 + k3 k6 Y3 Immediate binomial. Binomials at s.s.: ρ2 f1 − ρ1 y2 = 0 ρ3 f1 − ρ1 y3 = 0, where ρ1 = k5 k8 + k7 k5 + k6 k8 ∈ R ρ2 = (k8 k4 + k7 k4 )s100 ←monomial ρ3 = k6 k4 s100 ←monomial Condition I: Strongly connected. Condition II: Only one directed path from the enzyme to the each intermediate. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 9 / 17 Intermediate species dyi /dt = 0 1≤i ≤9 ⇐⇒ ρ1 e1 − ρe1 y1 = 0 ρ2 f1 − ρf1 y2 = 0 ρ3 f1 − ρf1 y3 = 0 ρ4 e2 − ρe2 y4 = 0 ρ5 f2 − ρf2 y5 = 0 ρ6 e3 − ρe3 y6 = 0 ρ7 f3 − ρf3 y7 = 0 ρ8 s100 − ρs1000 y8 = 0 ρ9 f4 − ρf4 y9 = 0 | {z } binomials We can solve for yi : y2 = M. Pérez Millán, A. Dickenstein ρ2 λ2 s100 f1 = f1 = ξ2 s100 f1 , ρf1 ρf1 (UBA) Enzymatic networks and toric s. states ξ2 ∈ R SIAM AG13, August 2, 2013 10 / 17 Enzymes and Substrates de1 /dt = −dy1 /dt de2 /dt = −dy4 /dt de3 /dt = −dy6 /dt df1 /dt df2 /dt df3 /dt df4 /dt = −(dy2 /dt + dy3 /dt) = −dy5 /dt = −dy7 /dt = −dy9 /dt ds100 /dt = − dy8 /dt + k3 y1 − k4 s100 f1 + k5 y2 ds000 /dt = − k1 s000 e1 + k2 y1 + k8 y3 − k9 s000 e2 + k10 y4 + k14 y5 − − k15 s000 e3 + k16 y6 + k20 y7 ds010 /dt = − k12 s010 f2 + k13 y5 + k11 y4 ds001 /dt = − k18 s001 f3 + k19 y7 + k17 y6 ds/dt = − k21 s.s100 + k22 y8 + k26 y9 dp/dt = − k24 pf4 + k25 y9 + k23 y8 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 11 / 17 Enzymes and Substrates ( (( de( =( −dy (( 1 /dt 1 /dt ( ( (( de( =( −dy (( 2 /dt 4 /dt ( ( (( de( =( −dy (( 3 /dt 6 /dt ( (((( (( ( ( df1 /dt = −(dy /dt + dy3 /dt) ( 2 ( (((((((( df /dt = −dy /dt ( 5 (2( ( (( ( ( df /dt = −dy /dt ( 3 7 ( ( ( ( ( /dt df4( /dt =( −dy (( 9 ( ds100 /dt = −dy 8 /dt + k3 y1 − k4 s100 f1 + k5 y2 ds000 /dt = − k1 s000 e1 + k2 y1 + k8 y3 − k9 s000 e2 + k10 y4 + k14 y5 − − k15 s000 e3 + k16 y6 + k20 y7 ds010 /dt = − k12 s010 f2 + k13 y5 + k11 y4 ds001 /dt = − k18 s001 f3 + k19 y7 + k17 y6 ds/dt = − k21 s.s100 + k22 y8 + k26 y9 dp/dt = − k24 pf4 + k25 y9 + k23 y8 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 11 / 17 Enzymes and Substrates All of them are binomials except for... ( (( de( =( −dy (( 1 /dt 1 /dt ( ( (( de( =( −dy (( 2 /dt 4 /dt ( ( (( de( =( −dy (( 3 /dt 6 /dt ( (((( (( ( ( df1 /dt = −(dy /dt + dy3 /dt) ( 2 ( (((((((( df /dt = −dy /dt ( 5 (2( ( (( ( ( df /dt = −dy /dt ( 3 7 ( ( ( ( ( /dt df4( /dt =( −dy (( 9 ( ds100 /dt = −dy 8 /dt + k3 ξ1 s000 e1 − (k4 − k5 ξ2 )s100 f1 ds000 /dt = − (k1 − k2 ξ1 )s000 e1 + k8 ξ3 s100 f1 − (k9 − k10 ξ4 )s000 e2 + + k14 ξ5 s010 f2 − (k15 − k16 ξ6 )s000 e3 + k20 ξ7 s001 f3 ds010 /dt = − (k12 − k13 ξ5 )s010 f2 + k11 ξ4 s000 e2 ds001 /dt = − (k18 − k19 ξ7 )s001 f3 + k17 ξ6 s000 e3 ds/dt = − (k21 − k22 ξ8 )s.s100 + k26 ξ9 pf4 dp/dt = − (k24 − k25 ξ9 )pf4 + k23 ξ8 s.s100 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 11 / 17 Enzymes and Substrates S100 S010 M. Pérez Millán, A. Dickenstein (UBA) S000 S P S001 Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 12 / 17 Enzymes and Substrates S100 S010 S000 S P S001 Condition III: multiple directed edges are not allowed M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 12 / 17 Enzymes and Substrates S100 S010 S000 S P S001 Condition III: multiple directed edges are not allowed Condition IV(a): reversible + forest M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 12 / 17 Enzymes and Substrates S100 S010 S000 S P S001 Condition III: multiple directed edges are not allowed Condition IV(a): reversible + forest M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 12 / 17 Enzymes and Substrates S100 S010 S000 S P S001 Condition III: multiple directed edges are not allowed Condition IV(a): reversible + forest Condition IV(b): outdegree, indegree ≤ 1 S0 → M. Pérez Millán, A. Dickenstein (UBA) S1 → S2 Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 12 / 17 Theorem Theorem Given an enzymatic chemical reaction network G = (V , R, ki ). Assume Conditions I and II hold for each digraph GT and Condition III holds for G , if the digraph GS satisfies either Condition IV(a) or Condition IV(b), then the mass action system arising from G has toric steady states. GS GT k4 s100 F1 k5 k8 k7 S100 Y2 k6 Y3 M. Pérez Millán, A. Dickenstein S010 S000 S P S001 ··· (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 13 / 17 More examples E. Feliu and C. Wiuf. Enzyme sharing as a cause of multistationarity in signaling systems. J. Roy. Soc. Interf., 9:71, (2012), pp. 1224–1232. M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 14 / 17 Another relevant example Cascades such as the MAPK/ERK pathway: E S0 S1 F1 P0 P1 F2 P2 F2 R0 R1 F3 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states R2 F3 SIAM AG13, August 2, 2013 15 / 17 GT and GS : GS : GT : k1 S0 E Y1 k4 S1 F1 Y2 k2 +k3 S0 S1 P0 P1 P2 R0 R1 R2 k5 +k6 k7 P 0 k10 P1 k8 +k9 k11 +k12 k13 P2 k16 P1 k14 +k15 k17 +k18 k19 R0 k22 R1 k20 +k21 k23 +k24 Y3 S1 Y4 Y5 F2 Y6 Y7 P2 Y8 k25 R2 k28 R1 k26 +k27 k29 +k30 Y9 F3 Y10 M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 16 / 17 Thanks for your attention! M. Pérez Millán, A. Dickenstein (UBA) Enzymatic networks and toric s. states SIAM AG13, August 2, 2013 17 / 17

Enzymatic networks and toric steady states

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