MasterMPRIC2-19ComputationalMethodsinSystemsandSyntheticBiology ReactionHypergraphsandInﬂuenceGraphs

(1)

Interaction Diagrams Influence Graphs DIG SIG Examples Increasing kinetics Antagonists

Reaction Hypergraphs and Influence Graphs

Master MPRI C2-19

Computational Methods in Systems and Synthetic Biology

Fran¸ cois Fages

Inria Saclay - Ile de France, Lifeware team, France

(2)

Diagrams used with Biologists

Diagram of Ciliberto et al.’s model of P53/Mdm2 DNA repair

system with reaction activators and reaction inhibitors

(3)

Biological Diagrams on Computers

(4)

Systems Biology Graphical Notation SBGN

Example

Complexation rule A + B => A-B

(5)

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

(6)

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

E D

A

B

E D C

(7)

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

(8)

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

(9)

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

(10)

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

(11)

Influence Graph from the Differential Semantics

k1 for _ => A k2*[A] for A => _

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

˙

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

˙

x _B = −k3 ∗ x _A ∗ x _B

˙

x _C = k 3 ∗ x _A ∗ x _B

Definition

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

where v i = r i − l i is the stoichiometric change vector of reaction i .

Positive and negative influences are given by the Jacobian matrix

J _ij = ∂ x ˙ _i /∂x _j

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Influence Graph from the Differential Semantics

k1 for _ => A k2*[A] for A => _

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

˙

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

˙

x _B = −k3 ∗ x _A ∗ x _B

˙

x _C = k 3 ∗ x _A ∗ x _B Definition

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

where v i = r i − l i is the stoichiometric change vector of reaction i . Positive and negative influences are given by

the Jacobian matrix

J _ij = ∂ x ˙ _i /∂x _j

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Influence Graph from the Differential Semantics

k1 for _ => A k2*[A] for A => _

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

˙

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

˙

x _B = −k3 ∗ x _A ∗ x _B

˙

x _C = k 3 ∗ x _A ∗ x _B Definition

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

where v i = r i − l i is the stoichiometric change vector of reaction i .

Positive and negative influences are given by the Jacobian matrix

J ij = ∂ x ˙ i /∂x j

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Differential Influence Graph (DIG)

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

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}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

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

˙

x B = −k3 ∗ x A ∗ x B

˙

x _C = k3 ∗ x _A ∗ x _B DIG = {A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

Is symbolic computation necessary ?

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Differential Influence Graph (DIG)

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

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}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

k1 for _ => A k2*[A] for A => _

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

˙

x _A = k 1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

˙

x B = −k3 ∗ x A ∗ x B

˙

x _C = k3 ∗ x _A ∗ x _B DIG =

{A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

Is symbolic computation necessary ?

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Differential Influence Graph (DIG)

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

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}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

k1 for _ => A k2*[A] for A => _

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

˙

x _A = k 1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

˙

x B = −k3 ∗ x A ∗ x B

˙

x _C = k3 ∗ x _A ∗ x _B DIG = {A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

Is symbolic computation necessary ?

(17)

Differential Influence Graph (DIG)

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

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}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

k1 for _ => A k2*[A] for A => _

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

˙

x _A = k 1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

˙

x B = −k3 ∗ x A ∗ x B

˙

x _C = k3 ∗ x _A ∗ x _B DIG = {A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

Is symbolic computation necessary ?

(18)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

(19)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {

B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

(20)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {

B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

(21)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {

C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

(22)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {

A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

(23)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

}

Proposition

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

(24)

Influence Graph from the Stoichiometry (SIG)

Definition

The stoichiometric influence graph (SIG) of a reaction model R is

defined by

SIG(R) = {A →B ⁺ | ∃(f _i for l i ⇒ r i ) ∈ R, l _i (A) > 0 and v _i (B) > 0}

∪{A →B ⁻ | ∃(f _i for l _i ⇒ r _i ) ∈ R, l i (A) > 0 and v i (B) < 0}

SIG ({ =[B]=> A}) = {B →A}

⁺

SIG ({A =[B]=> }) = {B →A, A

⁻

→A}

⁻

SIG ({A =[C]=> B }) = {C →A, A

⁻

→A, A

⁻

→B, C

⁺

→B}

⁺

SIG ({A + B => C}) = {A →C, B

⁺

→C, A

⁺

→B, B

⁻

→A, A

⁻

→A, B

⁻

→B,

⁻

} Proposition

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

Stoichiometric Influence Graph:

computed in 0.2 sec.

1231 positive influences 1089 negative influences

There is no tuple of molecules (A,B) with both A →B and A ⁺ →B. ⁻

(26)

MAPK Signalling Cascade

No negative feedback reaction Negative circuit in the SIG

multistability [Kholodenko 06]

but also sustained oscillations [Qiao 07]

Inhibition by complexation (sequestration of the upper-level kinase)

What is the relationship between the SIG and the DIG ?

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MAPK Signalling Cascade

No negative feedback reaction Negative circuit in the SIG

multistability [Kholodenko 06]

but also sustained oscillations [Qiao 07]

Inhibition by complexation (sequestration of the upper-level kinase)

What is the relationship between the SIG and the DIG ?

(28)

Increasing Kinetics

Definition

In a reaction model R ={f _i for l _i =>r _i | i ∈ I }, we say that a kinetic expression f i is increasing iff for all molecules x k we have

1

∂f _i /∂x _k ≥ 0 in all points of the phase space,

2

l _i (x _k ) > 0 whenever ∂f _i /∂x _k > 0 in some point of the phase space.

Proposition

The mass action law kinetics, f = k ∗ Πx i l

i

, Michaelis Menten and Hill’s kinetics f = V _m ∗ x _s ⁿ /(K _m ⁿ + x _s ⁿ ) are increasing.

Negative Hill kinetics f i = V m /(K m n + x s n ) is not increasing

(29)

Increasing Kinetics

Definition

In a reaction model R ={f _i for l _i =>r _i | i ∈ I }, we say that a kinetic expression f i is increasing iff for all molecules x k we have

1

∂f _i /∂x _k ≥ 0 in all points of the phase space,

2

l _i (x _k ) > 0 whenever ∂f _i /∂x _k > 0 in some point of the phase space.

Proposition

The mass action law kinetics, f = k ∗ Πx i l

i

, Michaelis Menten and Hill’s kinetics f = V _m ∗ x _s ⁿ /(K _m ⁿ + x _s ⁿ ) are increasing.

Negative Hill kinetics f i = V m /(K m n + x s n ) is not increasing

(30)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R). If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B).

⁻

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(31)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R). If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B).

⁻

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(32)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R).

If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B).

⁻

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(33)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R).

If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B).

⁻

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(34)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R).

If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B

⁻

).

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(35)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R).

If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B

⁻

).

DIG6= SIG for

{k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

˙

x _A = (k ₂ − k ₁ ) ∗ x _A can be made positive, null or negative.

(36)

DIG⊆SIG

Theorem

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

Proof.

If (A →B)

⁺

∈ DIG (R) then ∂ x ˙

B

/∂x

A

> 0 in some point of the phase space. Hence there exists a term in the differential semantics, of the form v

i

(B) ∗ f

i

with ∂f

i

/∂ x

A

of the same sign as v

i

(B ).

Let us suppose that v

i

(B) > 0, then ∂f

i

/∂x

A

> 0 and, since f

i

is increasing, we get that l

_i

(A) > 0 and thus that (A →B)

⁺

∈ SIG (R).

If on the contrary v

i

(B ) < 0, then ∂f

i

/∂x

A

< 0, which is not possible for an increasing kinetics.

The proof is symmetrical for (A →B

⁻

).

DIG6= SIG for {k ₁ ∗ A for A => , k 2 ∗ A for A => 2 ∗ A}

(37)

Strongly Increasing Kinetics

Definition

In a reaction model R ={f _i for l _i =>r _i | i ∈ I }, a kinetic

expression f _i is strongly increasing iff for all molecules x _k we have

1

∂f _i /∂x _k ≥ 0 in all points of the phase space,

2

l _i (x _k ) > 0 if and only if there exists a point in the phase space s.t. ∂f _i /∂x _k > 0

Proposition

Mass action law, Michaelis Menten, and Hill kinetics are strongly

increasing.

(38)

Strongly Increasing Kinetics

Definition

In a reaction model R ={f _i for l _i =>r _i | i ∈ I }, a kinetic

expression f _i is strongly increasing iff for all molecules x _k we have

1

∂f _i /∂x _k ≥ 0 in all points of the phase space,

2

l _i (x _k ) > 0 if and only if there exists a point in the phase space s.t. ∂f _i /∂x _k > 0

Proposition

Mass action law, Michaelis Menten, and Hill kinetics are strongly

increasing.

(39)

Equivalence Theorem

Theorem

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).

Corollary

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

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.

(40)

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.

In a 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,

• one mutual activation Cdc25 <-> MPF.

(41)

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.

In a 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,

• one mutual activation Cdc25 <-> MPF.

(42)

Reaction Inhibitors

In Ciliberto et al.’s Model of P53/Mdm2 [CNT05cc]

P53 inhibits the phosphorylation of Mdm2

k1*Mdm2/(k2+P53) for Mdm2 => Mdm2p

(43)

Generalized Reaction Rules with Antagonists

f for l /a => r reaction rule with antagonists a.

Definition

The generalized stoichiometric influence graph (GSIG) is the graph: {A →B

⁻

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M,

l

_i

(A) > 0 and v

_i

(B) < 0}

∪{A →B

⁻

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, a

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, l

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M, a

i

(A) > 0 and v

i

(B) < 0}

SIG(A=[/I]=>B})={A →B, I ⁺ →B, I ⁻ →A, A ⁺ →A} ⁻

(44)

Generalized Reaction Rules with Antagonists

f for l /a => r reaction rule with antagonists a.

Definition

The generalized stoichiometric influence graph (GSIG) is the graph:

{A →B

⁻

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M, l

_i

(A) > 0 and v

_i

(B) < 0}

∪{A →B

⁻

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, a

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, l

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M, a

i

(A) > 0 and v

i

(B) < 0}

SIG(A=[/I]=>B})={A →B, I ⁺ →B, I ⁻ →A, A ⁺ →A} ⁻

(45)

Generalized Reaction Rules with Antagonists

f for l /a => r reaction rule with antagonists a.

Definition

The generalized stoichiometric influence graph (GSIG) is the graph:

{A →B

⁻

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M, l

_i

(A) > 0 and v

_i

(B) < 0}

∪{A →B

⁻

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, a

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

i

for l

i

=[/a

i

]=> r

i

) ∈ M, l

i

(A) > 0 and v

i

(B) > 0}

∪{A →B

⁺

| ∃(f

_i

for l

_i

=[/a

_i

]=> r

_i

) ∈ M, a

i

(A) > 0 and v

i

(B) < 0}

SIG(A=[/I]=>B})={A →B, I ⁺ →B, I ⁻ →A, A ⁺ →A} ⁻

(46)

Compatible Kinetics with Antagonists

Definition

In a rule f for l =[/a]=> r , a kinetic expression f is (strongly) compatible iff for all molecules x _k we have

l(x _k ) > 0 if(f) there exists a point in the phase space such that

∂f /∂x k > 0,

a(x _k ) > 0 if(f) there exists a point in the phase space such that

∂f /∂x _k < 0.

Example k1*Mdm2/(k2+P53) for Mdm2 =[/P53]=> Mdm2p

Theorem

For any generalized reaction model R with a compatible kinetics,

DIG(R)⊆GSIG(R).

(47)

Compatible Kinetics with Antagonists

Definition

In a rule f for l =[/a]=> r , a kinetic expression f is (strongly) compatible iff for all molecules x _k we have

l(x _k ) > 0 if(f) there exists a point in the phase space such that

∂f /∂x k > 0,

a(x _k ) > 0 if(f) there exists a point in the phase space such that

∂f /∂x _k < 0.

Example k1*Mdm2/(k2+P53) for Mdm2 =[/P53]=> Mdm2p Theorem

For any generalized reaction model R with a compatible kinetics,

DIG(R)⊆GSIG(R).

(48)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG.

By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

_i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

i

(y) < 0, f

i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(49)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

_i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

i

(y) < 0, f

i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(50)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

.

Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

i

(y) < 0, f

i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(51)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

i

(y) < 0, f

i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(52)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

i

(y) < 0, f

i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(53)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

_i

(y) < 0, f

_i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z) > 0, i.e. x →

⁺

y is in the DIG.

The proof for an arc x →

⁻

y in the SIG is symmetrical.

(54)

Equivalence Theorem for Reactions with Antagonists

Theorem

For any generalized reaction model R with a strongly compatible kinetics, and a GSIG containing no positive−negative pair, DIG(R)=GSIG(R).

We just have to prove that the SIG is a subgraph of the DIG.

Let us consider an arc x →

⁺

y in the SIG. By definition there exists a reaction i with either v

i

(y ) > 0 and r

i

(x) > 0, or v

i

(y) < 0 and a

i

(x) > 0.

Since the reaction is well-formed, we have either v

i

(y) > 0 and

∂f

_i

/∂ x(~ z) > 0, or v

_i

(y ) < 0 and ∂f

_i

/∂x (~ z ) < 0, for some ~ z ∈ R

^s+

. Now, if v

_i

(y) > 0 then f

_i

occurs in ˙ y with a positive sign.

Since ∂f

_i

/∂x (~ z ) > 0 and there is no conflict in the SIG, we thus get

∂ y ˙ /∂x(~ z ) > 0, i.e. x →

⁺

y is in the DIG.

Similarly, if v

_i

(y) < 0, f

_i

occurs in ˙ y with a negative sign and

∂f

i

/∂ x(~ z) < 0, hence ∂ y/∂ ˙ x(~ z) > 0, i.e. x →

⁺

y is in the DIG.

(55)

Conclusion I

The stoichiometric influence graph is an abstraction of the reaction hypergraph.

The differential influence graph is an abstraction of the ODE semantics (graph of signs of the Jacobian matrix)

Both graphs are identical for a very general class of reaction models (strongly increasing kinetics without conflicting pairs).

It may be easier to reason with influence models rather than reaction models.

Influence models may be sufficient to express “natural algorithms” at various scales [Chazelle 2012]

Hierarchy of semantics of influence systems ?

(56)

Conclusion I

The stoichiometric influence graph is an abstraction of the reaction hypergraph.

The differential influence graph is an abstraction of the ODE semantics (graph of signs of the Jacobian matrix)

Both graphs are identical for a very general class of reaction models (strongly increasing kinetics without conflicting pairs).

It may be easier to reason with influence models rather than reaction models.

Influence models may be sufficient to express “natural algorithms” at various scales [Chazelle 2012]

Hierarchy of semantics of influence systems ?

(57)

MasterMPRIC2-19ComputationalMethodsinSystemsandSyntheticBiology ReactionHypergraphsandInﬂuenceGraphs

Reaction Hypergraphs and Influence Graphs

Master MPRI C2-19

Computational Methods in Systems and Synthetic Biology

Fran¸ cois Fages

Inria Saclay - Ile de France, Lifeware team, France

Diagrams used with Biologists

Diagram of Ciliberto et al.’s model of P53/Mdm2 DNA repair

system with reaction activators and reaction inhibitors

Biological Diagrams on Computers

Systems Biology Graphical Notation SBGN

Example

Complexation rule A + B => A-B

Reaction Hypergraphs and Influence Graphs

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

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

Reaction Hypergraphs and Influence Graphs

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

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

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] and [Soliman 13] for reaction influence graphs

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).

proved for : ODE systems [Snoussi 89]

Influence Graph from the Differential Semantics

k1 for _ => A k2*[A] for A => _

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

˙

x A = k 1−k2 ∗x A −k3∗ x A ∗x B

˙

x B = −k3 ∗ x A ∗ x B

˙

x C = k 3 ∗ x A ∗ x B

Definition

The differential semantics of a reaction model R = {f i for l i => r i } i=1,...,n is the ODE system dx k /dt = ˙ x k = P n

i=1 v i (x k ) ∗ f i

where v i = r i − l i is the stoichiometric change vector of reaction i .

Positive and negative influences are given by the Jacobian matrix

J ij = ∂ x ˙ i /∂x j

Influence Graph from the Differential Semantics

k1 for _ => A k2*[A] for A => _

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

˙

x A = k 1−k2 ∗x A −k3∗ x A ∗x B

˙

x B = −k3 ∗ x A ∗ x B

˙

x C = k 3 ∗ x A ∗ x B Definition

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

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

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

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

x _B = −k3 ∗ x _A ∗ x _B

x _C = k 3 ∗ x _A ∗ x _B

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

J _ij = ∂ x ˙ _i /∂x _j

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

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

x _B = −k3 ∗ x _A ∗ x _B

x _C = k 3 ∗ x _A ∗ x _B Definition

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

J _ij = ∂ x ˙ _i /∂x _j

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

x A = k 1−k2 ∗x _A −k3∗ x A ∗x _B

x _B = −k3 ∗ x _A ∗ x _B

x _C = k 3 ∗ x _A ∗ x _B Definition

The differential semantics of a reaction model R = {f _i for l i => r i } _i=1,...,n is the ODE system dx _k /dt = ˙ x _k = P n

i=1 v _i (x _k ) ∗ f _i

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

DIG (R) = {A →B ⁺ | ∂ x ˙ _B /∂x _A > 0 in some point}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

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

x _A = k1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

x _C = k3 ∗ x _A ∗ x _B DIG = {A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

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

DIG (R) = {A →B ⁺ | ∂ x ˙ _B /∂x _A > 0 in some point}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

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

x _A = k 1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

x _C = k3 ∗ x _A ∗ x _B DIG =

{A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

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

DIG (R) = {A →B ⁺ | ∂ x ˙ _B /∂x _A > 0 in some point}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

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

x _A = k 1 − k2 ∗ x _A − k3 ∗ x _A ∗ x _B

x _C = k3 ∗ x _A ∗ x _B DIG = {A →A, ⁻ B →A, ⁻ A →B, ⁻ B →B, ⁻ A →C ⁺ , B →C ⁺ }

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

DIG (R) = {A →B ⁺ | ∂ x ˙ _B /∂x _A > 0 in some point}

∪{A →B ⁻ | ∂ x ˙ _B /∂x _A < 0 in some point}

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