Reaction Hypergraphs and Influence Graphs

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Diagrams used with Biologists

Diagram of Ciliberto et al.’s model of P53/Mdm2 DNA repair system [cc05]

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Biological Diagrams on Computers

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Systems Biology Graphical Notation SBGN

Complexation rule A + B => A-B

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Reaction Hypergraphs and Influence Graphs

k1*[A] for A =[C]=> B.

k2*[B]*[D] for B+D => E.

rule_1

B

C A

rule_2

E D

Reaction Hypergraph:

• Rules linking several reactants and products

• Bipartite graph representation of hypergraph Molecule and reaction vertices (Petri net)

• Model reductions as subgraph epimorphisms delete/merge of molecule/reaction vertices infer model reductions in biomodels.net repository talk tomorrow 15.00 Computational Theory track

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Reaction Hypergraphs and Influence Graphs

k1*[A] for A =[C]=> B. A →B, C⁺ →B, C⁺ →A, ...⁻ k2*[B]*[D] for B+D => E.

rule_1

B

C A

rule_2 D

A

B

D C

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Ren´ e Thomas’s Conditions on Influence Graphs

Introduced to reason about gene regulatory networks [Thomas 73, 81] :

• The existence of positive circuits in the influence graph is a necessary condition for multistationarity (e.g. cell differentiation).

proved for :

ODE systems [Snoussi 89] ... [Soul´e 03]

Boolean networks [R´emy Ruet Thieffry 05] ...

Discrete networks [Richard 06] ...

• The existence of negative circuits is a necessary condition for oscillations (e.g. homeostasis).

ODE systems [Snoussi 89]

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Influence Graph from the Differential Semantics

k1 for _ => A

k2*[A] for A => _

k3*[A]*[B] for A + B => C

˙

xA = k1 − k2 ∗ xA − k3 ∗ xA ∗ xB

˙

x_B = −k3 ∗ x_A ∗ x_B

˙

x_C = k3 ∗ x_A ∗ x_B

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Influence Graph from the Differential Semantics

k1 for _ => A

k2*[A] for A => _

k3*[A]*[B] for A + B => C

˙

xA = k1 − k2 ∗ xA − k3 ∗ xA ∗ xB

˙

x_B = −k3 ∗ x_A ∗ x_B

˙

x_C = k3 ∗ x_A ∗ x_B Definition 1 The differential semantics of a reaction model

R = {ei for li => ri}i=1,...,n

is the ODE system dx_k/dt = ˙x_k = Pn

i=1(r_i(x_k) − l_i(x_k)) ∗ e_i

where r_i(x_k) (resp. l_i) is the stoichiometric coefficient of x_k in the right (resp. left) hand side of rule i.

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Influence Graph from the Differential Semantics

k1 for _ => A

k2*[A] for A => _

k3*[A]*[B] for A + B => C

˙

xA = k1 − k2 ∗ xA − k3 ∗ xA ∗ xB

˙

x_B = −k3 ∗ x_A ∗ x_B

˙

x_C = k3 ∗ x_A ∗ x_B Definition 1 The differential semantics of a reaction model

R = {ei for li => ri}i=1,...,n

is the ODE system dx_k/dt = ˙x_k = Pn

i=1(r_i(x_k) − l_i(x_k)) ∗ e_i

where r_i(x_k) (resp. l_i) is the stoichiometric coefficient of x_k in the right (resp. left) hand side of rule i.

The Jacobian matrix J is formed of the partial derivatives Jij = ∂x˙i/∂xj

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Differential Influence Graph (DIG)

Definition 2 The differential influence graph (DIG) of a reaction model R is the graph of molecules with two kinds of edges:

DIG(R) = {A →B⁺ | ∂x˙_B/∂x_A > 0 in some point of the phase space}

∪{A →B⁻ | ∂x˙B/∂xA < 0 in some point of the phase space}

A →B⁺ positive influence of molecule A on molecule B A →B⁻ negative influence of molecule A on molecule B

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Differential Influence Graph (DIG)

DIG(R) = {A →B⁺ | ∂x˙_B/∂x_A > 0 in some point of the phase space}

A →B⁺ positive influence of molecule A on molecule B A →B⁻ negative influence of molecule A on molecule B k1 for _ => A

k2*[A] for A => _

k3*[A]*[B] for A + B => C

˙

x_A = k1 − k2 ∗ x_A − k3 ∗ x_A ∗ x_B

˙

xB = −k3 ∗ xA ∗ xB

˙

xC = k3 ∗ xA ∗ xB

DIG = ?

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Differential Influence Graph (DIG)

DIG(R) = {A →B⁺ | ∂x˙B/∂xA > 0 in some point of the phase space}

A →B⁺ positive influence of molecule A on molecule B A →B⁻ negative influence of molecule A on molecule B k1 for _ => A

k2*[A] for A => _

k3*[A]*[B] for A + B => C

˙

x_A = k1 − k2 ∗ x_A − k3 ∗ x_A ∗ x_B

˙

xB = −k3 ∗ xA ∗ xB

˙

xC = k3 ∗ xA ∗ xB

DIG = {A→A, B⁻ →A, A⁻ →B, B⁻ →B, A⁻ →C, B⁺ →C⁺ } Not always immediate to compute

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Influence Graph from the Stoichiometry (SIG)

Definition 3 The structural (stoichiometric) influence graph (SIG) of a reaction model R is defined by

SIG(R) = {A →B⁺ | ∃(e_i for l_i ⇒ r_i) ∈ R,

li(A) > 0 and ri(B) − li(B) > 0}

∪{A →B⁻ | ∃(e_i for l_i ⇒ r_i) ∈ R,

l_i(A) > 0 and r_i(B) − l_i(B) < 0}

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Influence Graph from the Stoichiometry (SIG)

li(A) > 0 and ri(B) − li(B) > 0}

∪{A →B⁻ | ∃(e_i for l_i ⇒ r_i) ∈ R,

l_i(A) > 0 and r_i(B) − l_i(B) < 0}

SI G({ =[B]=> A}) = {B →A}⁺

SI G({A =[B]=> }) = {B →A, A⁻ →A}⁻

SI G({A =[C]=> B }) = {C →A, A⁻ →A, A⁻ →B, C⁺ →B}⁺

SI G({A + B => C}) = {A →C, B⁺ →C, A⁺ →B, B⁻ →A, A⁻ →A, B⁻ →B,⁻ }

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Influence Graph from the Stoichiometry (SIG)

li(A) > 0 and ri(B) − li(B) > 0}

∪{A →B⁻ | ∃(e_i for l_i ⇒ r_i) ∈ R,

l_i(A) > 0 and r_i(B) − l_i(B) < 0}

SI G({ =[B]=> A}) = {B →A}⁺

SI G({A =[B]=> }) = {B →A, A⁻ →A}⁻

SI G({A =[C]=> B }) = {C →A, A⁻ →A, A⁻ →B, C⁺ →B}⁺

SI G({A + B => C}) = {A →C, B⁺ →C, A⁺ →B, B⁻ →A, A⁻ →A, B⁻ →B,⁻ }

Proposition 4 The SIG of n reaction rules is computable in O(n) time

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The SIG of Kohn’s Map of the Mammalian Cell Cycle

Reaction model:

500 variables

800 reaction rules

Struct. Influence Graph:

computed in 0.2 sec.

1231 positive influences 1089 negative influences

no molecule is at the same time an activator and an inhibitor of a same target molecule

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MAPK Signalling Cascade

Purely directional “cascade” of reactions: no negative feedback

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MAPK Signalling Cascade

Purely directional “cascade” of reactions: no negative feedback sustained oscillations observed [Qiao et al. 07]

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MAPK Signalling Cascade

Purely directional “cascade” of reactions: no negative feedback sustained oscillations observed [Qiao et al. 07]

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MAPK Reaction and Influence Graphs

RAF

RAF-RAFK

RAFK RAF~{p1}

RAFPH

RAFPH-RAF~{p1}

MEK

MEK-RAF~{p1}

MEK~{p1}

MEK~{p1}-RAF~{p1}

MEKPH MEKPH-MEK~{p1}

MEK~{p1,p2}

MEKPH-MEK~{p1,p2}

MAPK

MAPK-MEK~{p1,p2}

MAPK~{p1}

MAPK~{p1}-MEK~{p1,p2}

MAPKPH

MAPKPH-MAPK~{p1}

MAPK~{p1,p2}

MAPKPH-MAPK~{p1,p2}

Negative feedback in the structural influence graph Inhibition by sequestration [Sepulchre et al. 08]

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MAPK Reaction and Influence Graphs

RAF

RAF-RAFK

RAFK RAF~{p1}

RAFPH

RAFPH-RAF~{p1}

MEK

MEK-RAF~{p1}

MEK~{p1}

MEK~{p1}-RAF~{p1}

MEKPH MEKPH-MEK~{p1}

MEK~{p1,p2}

MEKPH-MEK~{p1,p2}

MAPK MAPK~{p1}

MAPK~{p1}-MEK~{p1,p2}

MAPKPH

MAPKPH-MAPK~{p1}

MAPK~{p1,p2}

MAPKPH-MAPK~{p1,p2}

Negative feedback in the structural influence graph Inhibition by sequestration [Sepulchre et al. 08]

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Increasing Kinetics

Definition 5 In a reaction model R ={e_i for l_i=>r_i | i ∈ I}, we say that a kinetic expression e_i is increasing iff for all molecules x_k we have

1. ∂e_i/∂x_k ≥ 0 in all points of the phase space,

2. li(xk) > 0 whenever ∂ei/∂xk > 0 in some point of the phase space.

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Increasing Kinetics

Proposition 6 The mass action law kinetics, e = k ∗ Πxili, Michaelis Menten and Hill’s kinetics e = Vm ∗ xsn/(Kmn

+ xsn) are increasing.

Negative Hill kinetics ei = Vm/(Kmn

+ xsn) is not increasing

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Increasing Kinetics

Proposition 6 The mass action law kinetics, e = k ∗ Πxili, Michaelis Menten and Hill’s kinetics e = Vm ∗ xsn/(Kmn

+ xsn) are increasing.

Negative Hill kinetics ei = Vm/(Kmn

+ xsn) is not increasing

Theorem 7 For any reaction model R with increasing kinetics, the DIG is a subgraph of the SIG: DIG(R) ⊆ SIG(R).

DIG(R)6= SIG(R) for R = {k₁ ∗ A for A => k₂ ∗ A for = [A] => A}

as ˙xA = (k2 − k1) ∗ xA can be made always positive, null or negative.

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Strongly Increasing Kinetics

Definition 8 In a reaction model R ={e_i for l_i=>r_i | i ∈ I}, a kinetic expression e_i is strongly increasing iff for all molecules x_k we have

2. li(xk) > 0 if and only if there exists a point in the phase space s.t.

∂e_i/∂x_k > 0

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Strongly Increasing Kinetics

∂e_i/∂x_k > 0

Proposition 9 Mass action law, Michaelis Menten, and Hill kinetics are strongly increasing.

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Strongly Increasing Kinetics

∂e_i/∂x_k > 0

Proposition 9 Mass action law, Michaelis Menten, and Hill kinetics are strongly increasing.

Lemma 10 Let R be a reaction model with strongly increasing kinetics.

If (A →B⁺ )∈ SIG(R) and (A →B⁻ )6∈ SIG(R) then (A →B⁺ )∈ DIG(R).

If (A →B⁻ )∈ SIG(R) and (A →B⁺ )6∈ SIG(R) then (A →B⁻ )∈ DIG(R).

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Equivalence Theorem

Main Theorem 11 Let R be a reaction model with strongly increasing kinetics and where no molecule is at the same time an activator and an inhibitor of the same target molecule, then SIG(R) = DIG(R).

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Equivalence Theorem

Corollary 12 The DIG of a reaction model is independent of the kinetic expressions as long as they are strongly increasing, if there is no

positive−negative influence pairs in the SIG.

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Equivalence Theorem

Corollary 12 The DIG of a reaction model is independent of the kinetic expressions as long as they are strongly increasing, if there is no

positive−negative influence pairs in the SIG.

Corollary 13 The DIG of a reaction model of n rules with strongly increasing kinetics is computable in time O(n) if there is no

positive−negative pair in the SIG.

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Cell Cycle Control Models

The SIG of Kohn’s map contains no positive−negative pair hence the DIGs of Kohn’s map are the same for any strongly increasing kinetics and any strictly positive parameter values.

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Cell Cycle Control Models

In smaller model [Tyson 91] the autoactivation rule pMPF =[MPF]=> MPF with MPF => pMPF

creates a pair

MPF →pPMF⁻ and MPF →pMPF.⁺

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Cell Cycle Control Models

In smaller model [Tyson 91] the autoactivation rule pMPF =[MPF]=> MPF with MPF => pMPF

creates a pair

MPF →pPMF⁻ and MPF →pMPF.⁺

In kohn’s map, this is decomposed in two positive circuits

• one mutual inhibition Wee1 |-| MPF,

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Conclusion

• ODE’s systems derived from reaction rules enjoy remarkable properties

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Conclusion

• ODE’s systems derived from reaction rules enjoy remarkable properties – The signs of the Jacobian matrix coefficients are essentially

independent of the kinetics

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Conclusion

– The differential influence graph is computable in linear time

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Conclusion

• Supports qualitative reasoning on the rule structure before simulation – circuit analyses in the influence graph (multistability and

oscillations)

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Conclusion

• Supports qualitative reasoning on the rule structure before simulation – circuit analyses in the influence graph (multistability and

oscillations)

• Supports writing reaction rules instead of directly ODEs.

– consistency checks: kinetic modifiers as reactants, catalysts or antagonists; parameter dimension analysis; concentration

conversions w.r.t. volume of locations,...

– model reductions/refinements as subgraph epimorphisms