Regulatory circuits in discrete dynamics Elisabeth Remy (IML) Ecole - - PowerPoint PPT Presentation
Regulatory circuits in discrete dynamics Elisabeth Remy (IML) Ecole thmatique Modlisation formelle de rseaux de rgulation biologique June 28, 2013 Modelling of biological networks Why? To gain better understanding To predict the
Elisabeth Remy (IML)
Ecole thématique Modélisation formelle de réseaux de régulation biologique
June 28, 2013
⊲ Gene regulation level: genetic networks
Qualitative/Quantitative Deterministic/Stochastic
Modelling of genetic regulations Focus on the logical modelling Analysis of regulatory feedback circuits Illustration: Segment polarity module Model reduction Hierarchical Transition Graphs
The pioneers
1950’s: genetic and biochemical analysis of bacteriophage induction and of the regulation of metabolic functions = ⇒ notion of regulatory gene, mRNA and operon structure
Jacob, Monod & Lwoff, 1961
. Jacob (1961); General conclusions: teleonomic mechanisms in cellular metabolism, growth and differentiation. Cold Spring Harbour. Symp. of Quantitative Biology 26 : 389-401
They are at the origin of
can be constructed from inductors/repressors
(1976))
Regulatory functions are well approxi- mated by Heavyside step functions → Boolean abstraction
Regulatory functions are well approxi- mated by Heavyside step functions → Boolean abstraction The logical formalism (S. Kauffman, 1969; R. Thomas, 1973)
Stuart Kauffman
the state of the system
fixed degree
possible following state)
X = f(x) Xi = fi(x) (logical function): indicates if gene i is currently transcribed xi (logical variable): current level of the functional product of gene i
the system
(f)
René Thomas (1973)
t xi fi(x) 1 1 dON dOF F
Activation
A B
Inhibition
A B
Activation
A B
xA = 1 ⇒ fB(x) = 1 xA = 0 ⇒ fB(x) = 0 fB(x) = xA Inhibition
A B
xA = 1 ⇒ fB(x) = 0 xA = 0 ⇒ fB(x) = 1 fB(x) = xA
A C B Rule for A Rule for B Rule for C
A C B Rule for A A is inhibited by itself: A if A (fA(x) = xA) Rule for B Rule for C
A C B Rule for A A is inhibited by itself: A if A (fA(x) = xA) Rule for B B is activated by C: B if C (fB(x) = xC) Rule for C
A C B Rule for A A is inhibited by itself: A if A (fA(x) = xA) Rule for B B is activated by C: B if C (fB(x) = xC) Rule for C C is activated by A and B
Regulatory graph(G, I, f)
A C B
2 2 1
i ∈ G, xi ∈ {0, . . . maxi}
interaction (i, j) is effective iff xi ≥ θi,j
fi : Πj∈G{0, . . . maxj} → {0, . . . maxi}
Extension to multivalued case is always possible Notations:
The State Transition Graph (S, E)
Given a Boolean dynamics f : {0, 1}n → {0, 1}n ֒ → state transition graph ? f : {0, 1}2 − → {0, 1}2 x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 0 1 0 1
−
1
−
1 0 0
+ +
01 1
+ −
1
−
1
The State Transition Graph (S, E)
Given a Boolean dynamics f : {0, 1}n → {0, 1}n ֒ → state transition graph ? f : {0, 1}2 − → {0, 1}2 x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 0 1 0 1
−
1
−
1 0 0
+ +
01 1
+ −
1
−
1
Updating rules
+ +
01 1
+ −
1
−
1 Synchronous updating
Updating rules
+ +
01 1
+ −
1
−
1 Synchronous updating
+ +
01 1
+ −
1
−
1 Asynchronous updating
Updating rules
+ +
01 1
+ −
1
−
1 Synchronous updating
+ +
01 1
+ −
1
−
1 Asynchronous updating
+ +
01 1
+ −
1
−
1 With priorities
priority, synchronous
priority, asynchronous
Regulatory graph(G, I, f)
A C B
2 2 1
The asynchronous STG (S, E)
000 001 010 011 100 101 110 111 200 201 210 211
i ∈ G, xi ∈ {0, . . . maxi}
interaction (i, j) is effective iff xi ≥ θi,j
fi : Πj∈G{0, . . . maxj} → {0, . . . maxi}
yi = xi + |fi(x)−xi|
fi(x)−xi
∀j = i yj = xj
Given a Boolean dynamics f : ֒ → How to get the Regulatory graph G(f) ? x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 0 1 0 1
−
1
−
1 0 0 G0 G1
Given a Boolean dynamics f : ֒ → How to get the Regulatory graph G(f) ? x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 → 0 1 0 1
−
1
−
1 0 0 G0 G1
Given a Boolean dynamics f : ֒ → How to get the Regulatory graph G(f) ? x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 → 0 1 0 1
−
1
−
1 0 0 G0 G1
G(f)(x): i → j ∈ G(f)(x) with sign ε iff fj(xi) = fj(x), with ε = + if xi = fj(x) and ε = − if xi = fj(x)
Given a Boolean dynamics f : ֒ → How to get the Regulatory graph G(f) ? x0 x1 f0(x) f1(x)
+ +
1 1 1
+
1 1 0 1 0 1
−
1
−
1 0 0 G0 G1
G(f)(x): i → j ∈ G(f)(x) with sign ε iff fj(xi) = fj(x), with ε = + if xi = fj(x) and ε = − if xi = fj(x)
x G(f)(x)
Given a Boolean asynchronous dynamics, represented by the state transition graph (V, E). ֒ → Local regulatory graph G(x)
Given a Boolean asynchronous dynamics, represented by the state transition graph (V, E). ֒ → Local regulatory graph G(x)
ε
→ j if ∃x s.t. (x, xj) ∈ E and (xi, xi,j) ∈ E with ε = + if xi = xj and ε = − if xi = xj
Given a Boolean asynchronous dynamics, represented by the state transition graph (V, E). ֒ → Local regulatory graph G(x)
ε
→ j if ∃x s.t. (x, xj) ∈ E and (xi, xi,j) ∈ E with ε = + if xi = xj and ε = − if xi = xj x xj xi xi,j x xj xi xi,j
Given a Boolean asynchronous dynamics, represented by the state transition graph (V, E). ֒ → Local regulatory graph G(x)
ε
→ j if ∃x s.t. (x, xj) ∈ E and (xi, xi,j) ∈ E with ε = + if xi = xj and ε = − if xi = xj
+
:
Given a Boolean asynchronous dynamics, represented by the state transition graph (V, E). ֒ → Local regulatory graph G(x)
ε
→ j if ∃x s.t. (x, xj) ∈ E and (xi, xi,j) ∈ E with ε = + if xi = xj and ε = − if xi = xj
+
:
−
:
Given a Boolean dynamics f : {0, 1}n → {0, 1}n ֒ → Asynchronous state transition graph (V, E) : (x, xi) ∈ E iff xi = fi(x) Definitions:
cycle
Identify, in huge STG (2n states), asymptotical behaviours (attractors), properties along trajectories, impact of perturbations...
Identify, in huge STG (2n states), asymptotical behaviours (attractors), properties along trajectories, impact of perturbations... Deduce properties from the model, without constructing its dynamics
Reduce the size of the model
Reduce the size of the dynamics (STG), yet keeping properties of interest
. Monteiro, C. Chaouiya (2012) AISC, Springer,
3 pioneer articles
transcriptional regulators. Nature 403 : 335-338.
switch in Escherichia Coli. Nature 403 : 339-342.
All of these works ֒ → report the conception and construction of synthetic networks which induce amazing behaviors of the cells ֒ → rely on feedback regulatory circuits mathematical modelisation
1 2 3 4 n
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1}
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Starting at state (
−
1, 1, 1,
+
0) Asynchronous dynamics (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Activating gene 4 Asynchronous dynamics → Fixed point (1, 1, 1, 1) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Starting again at (
−
1, 1, 1,
+
0) Asynchronous dynamics (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Disactivating gene 1 Asynchronous dynamics → (0,
−
1, 1,
+
0) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Disactivating gene 2 Asynchronous dynamics → (0, 0,
−
1,
+
0) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Activating gene 4 Asynchronous dynamics → (
+
0, 0,
−
1, 1) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Disactivating gene 3 Asynchronous dynamics → (
+
0, 0, 0,
−
1) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Disactivating gene 4 Asynchronous dynamics → Fixed point (0, 0, 0, 0) (one update at a time)
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} Asynchronous dynamics (one update at a time) Multistationarity
1 2 3 4 n
ε1 εn
Simple modelling:
a unique interaction ⇒ Boolean case
if εi = +1 1 − xi if εi = −1
Complete description of the structure of the synchronous state transition graph: ֒ → Disconnected elementary cycles ֒ → Staged structure:
1111 0000 1101 1001 0001 1011 0011 1000 0111 1100 0010 1110 0110 0100 1010 0101
From the synchronous state transition graph: Complete description of the structure of the asynchronous state transition graph: ֒ → Connected graph ֒ → Staged structure: at each stage k, each state has exactly k successors
The state transition graph
1111 0000 1101 1001 0001 1011 0011 1000 0111 1100 0010 1110 0110 0100 1010 0101
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 Starting at state (
−
1,
−
1, 1,
+
0) x1, x2, x3, x4 ∈ {0, 1} x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 Activating gene 4 x1, x2, x3, x4 ∈ {0, 1} → (1,
−
1, 1, 1) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (1, 0,
−
1, 1) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (1, 0, 0,
−
1) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (
−
1, 0, 0, 0) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (0,
+
0, 0, 0) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (0, 1,
+
0, 0) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (0, 1, 1,
+
0) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → (
+
0, 1, 1, 1) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} → return to (1,
−
1, 1, 1) x : negation
1 2 4 3
f x1 x2 x3 x4 = x4 x1 x2 x3 x1, x2, x3, x4 ∈ {0, 1} x : negation One (trap) cycle: homeostasis
The state transition graph
1110 1100 1000 1111 0000 0111 0011 0001 1010 0010 0110 1011 0100 0100 1101 0101
Theorem Let f be an asynchronous Boolean dynamics such that G(f) is an isolated circuit. Then,
and has no stable state.
Flower graphs
a d h b c
Flower graphs
a d h b c A flower-graph
such that E = {h
εh i
→ i | i = h}∪{i
εi h
→ h | i = h}
Flower graphs
a d h b c A flower-graph
such that E = {h
εh i
→ i | i = h}∪{i
εi h
→ h | i = h}
Number of stable states?
We define the state v by:
+
→ i 1 if h
−
→ i a d h b c
֒ → v and v are the only potential stable states
֒ → v and v are the only potential stable states a d h b c fh(v)? fh(v)?
֒ → v and v are the only potential stable states a d h b c fh(v)? fh(v)? Let E0 = {y ∈ {0, 1}C | yh = 0 and fh(y) = 0} E1 = {y ∈ {0, 1}C | yh = 1 and fh(y) = 1}
We define the state u by:
+
→ h 1 if i
−
→ h a d h b c
We have
We have
We have
֒ → v and v are the only potential stable states
!" !" #$" #%"
no stable state
!" !" #$" #%"
2 stable states
!" !" #$" #%" !" !" #$" #%"
1 stable state
then v = u and v = u ⇒ 2 stable states
then v = u and v = u ⇒ 2 stable states
then v = u and v = u ⇒ no stable state
a) and at least one positive circuit then v = ua and v = uC\a. Suppose there is no stable state, ua ∈ E0 and uC\a ∈ E1. Then, uD ∈ E0 iff a ∈ D → a unique regulator of h, a contradiction. ⇒ there exists at least one stable state
a) and at least one positive circuit then v = ua and v = uC\a. Suppose there is no stable state, ua ∈ E0 and uC\a ∈ E1. Then, uD ∈ E0 iff a ∈ D → a unique regulator of h, a contradiction. ⇒ there exists at least one stable state
Hub graphs
t w p a h b f d m n c
t w p a h b f d m n c
Given a hub-graph, there exists a flower graph
1- Transformation Tb, with component b such that: b has a unique input a and O(b) ∩ O(a) = ∅ or {a} a b c
s s′
⇒ b a c
s ss′
2- Transformation Pb: elimination of the output b
Iterations of transformation T·
t w p a h b f d m n c t a w p d h f c b n m
Iterations of transformation P·
t a w p d h f c b n m a d h b c
Theorem Let f be an asynchronous Boolean dynamics. If G(f) is a hub-graph then f contains at most two stable states. In the particular case f does contain two stable states, they are complementary. More precisely:
states;
positive circuit, then f contains at least one stable state;
negative circuit, then f contains at most one stable state.
Didier-R., Soumis
Consequence: the dynamics of such flower graph (more than two positive and negative circuits) may contain 0,1 or 2 stables states....! a d h b c
Thomas’rules (1981):
regulatory graph.
regulatory graph.
f has at least two stable states ⇒ there exits a positive circuit in G(f) Proof: Let x and y = x∆(x,y) two stable states
f has at least two stable states ⇒ there exits a positive circuit in G(f) Proof: Let x and y = x∆(x,y) two stable states
f has at least two stable states ⇒ there exits a positive circuit in G(f) Proof: Let x and y = x∆(x,y) two stable states
f has at least two stable states ⇒ there exits a positive circuit in G(f) Proof: Let x and y = x∆(x,y) two stable states
We have (x, xi) ∈ E (stable state), and (xD, xD∪i) ∈ E. There exists D′ ⊆ D s.t., for j ∈ D′ (xD′\j, x(D′\j)∪i) ∈ E and (xD′, xD′∪i) ∈ E. Hence, j
ε
→ i. As xi = yi, we can deduce that εi = −1 if xi = xj
f has at least two stable states ⇒ there exits a positive circuit in G(f) Proof: Let x and y = x∆(x,y) two stable states
We have (x, xi) ∈ E (stable state), and (xD, xD∪i) ∈ E. There exists D′ ⊆ D s.t., for j ∈ D′ (xD′\j, x(D′\j)∪i) ∈ E and (xD′, xD′∪i) ∈ E. Hence, j
ε
→ i. As xi = yi, we can deduce that εi = −1 if xi = xj ⇒ ∀i ∈ ∆(x, y), ∃p(i) ∈ ∆(x, y) such that p(i)
εi
→ i , with εi = −1 if xi = xp(i)
l s.t.pj(i) = pj+l(i) and pj(i), . . . pj+l−1(i) are all differents ⇒ circuit pj(m), . . . pj+l−1(i), with an even number of negative edges, (as i
−
→ j if xpj(i) = xpj+1(i)). ⇒ For each i ∈ ∆(x, y), there exists j ∈ ∆(x, y) occuring in a positive circuit involving only components of ∆(x, y), and a path from i to j.
Theorem: f has two stable states x and y = ⇒ for each i ∈ ∆(x, y) there exists j ∈ ∆(x, y) occuring in a positive circuit involving only components of ∆(x, y), and a path from j to i. In particular the regulatory graph contains a positive circuit involving a subset of ∆(x, y)
Didier, R. Discrete Applied Math., 160 : 2147-57, 2012.
f has a trap cycle ⇒ there exits a negative circuit in G(f) Proof: (x1, . . . , xr) with strategy ϕ is a trap cycle ⇒ f(xi) = xiϕ(i) = xi+1
(xiϕi, xiϕi,ϕi+1) ∈ E: ϕi
εi
→ ϕi+1 with εi = −1 if xi
ϕi = xi+1 ϕi+1
000 100 010 110 001 101 011 111 Cycle (000, 001, 011, 111, 101, 100) Strategy ϕ = (3, 2, 1, 3, 2, 1) (000, 000
2) ∈ E, (000 3, 000 2,3) ∈ E :
3
+
− → 2
existence of a bridge (xk, xk+l), i.e. such that ϕk = ϕk+l and {ϕk+1, · · · , ϕk+l−1} contains once all the genes of the strategy except ϕk
⇒ Circuit, with negative sign
Theorem: If f has a trap cycle C = x1 ϕ1 → x2 ϕ2 → · · ·
ϕp−1
→ xp ϕp → x1 , then the regulatory graph contains a negative circuit with vertices ϕ1, . . . , ϕp
The loop is functional, i.e. it actually generates homeostasis if it is a negative loop and multistationarity if it is a positive loop
E.H. Snoussi and R. Thomas, Bull.Math.Biol, 1995
A C B 000 001 010 011 100 101 110 111 The negative circuit is functional only in the absence of gene A Context of functionality of circuit C (Φ(f)(C)): set of constraints on the expression levels of regulators Φ(f)(C) = {xA = 0}
Illustration with p53-Mdm2 network
W.Abou-Jaoude, DA.Ouattara, M.Kaufman From structure to dynamics: frequency tuning in the p53-Mdm2 network I. Logical approach. J Theor Biol 258(4):561-77
Mdm2cyt Mdm2nuc P53
2
2 circuits:
between P53 and nuclear Mdm2)
components
Illustration with p53-Mdm2 network
Case 1:Weak decay rate of Mdm2nuc: Absence of P53 OR Presence
Mdm2cyt Mdm2nuc P53
2
000 110 100 010 001 101 111 011 200 201 210 211 One stable state and one non-attractive cycle : Coexistence of cells in resting state and cells showing sustained P53 oscillations
Illustration with p53-Mdm2 network
Case 1:Weak decay rate of Mdm2nuc: Absence of P53 OR Presence
Mdm2cyt Mdm2nuc P53
2
000 110 100 010 001 101 111 011 200 201 210 211 One stable state and one non-attractive cycle : Coexistence of cells in resting state and cells showing sustained P53 oscillations Both circuits are functional
Illustration with p53-Mdm2 network
Case 2: Intermediate decay rate of Mdm2nuc: Presence of Mdm2cyt is sufficient for expression of Mdm2nuc Mdm2cyt Mdm2nuc P53
2
One attractive cycle 000 110 100 010 001 101 111 011 200 201 210 211
Illustration with p53-Mdm2 network
Case 2: Intermediate decay rate of Mdm2nuc: Presence of Mdm2cyt is sufficient for expression of Mdm2nuc Mdm2cyt Mdm2nuc P53
2
The negative circuit is functional 000 110 100 010 001 101 111 011 200 201 210 211
Illustration with p53-Mdm2 network
Case 3:High decay rate of Mdm2nuc:Absence of P53 AND presence
Mdm2cyt Mdm2nuc P53
2
000 110 100 010 001 101 111 011 200 201 210 211 One stable state : P53 is steadily high
Illustration with p53-Mdm2 network
Case 3:High decay rate of Mdm2nuc:Absence of P53 AND presence
Mdm2cyt Mdm2nuc P53
2
000 110 100 010 001 101 111 011 200 201 210 211 Context of the positive circuit: xMdm2cyt = 1. Negatif circuit not functional
x belongs to the context of functionality
f1(x) = f1(xn)
interactions are functional
intersection of all the interaction contexts
1 2 3 4 n
If C belongs to the local graph G(f)(x), then x ∈ Φ(f)(C)
Globally minimal circuit: circuit in some local G(f)(x) which is minimal (no shortcut) in G(f). I-subcube, I ⊆ {1, · · · , n} xI = {y ∈ {0, 1}n such that yj = xj for all j ∈ I}
Theorem
Let f : {0, 1}n → {0, 1}n, x, and suppose that G(f)(x) contains a circuit C = k1
ε1
→ k2
ε2
→ · · ·
εp−1
→ kp
εp
→ k1 which is globally minimal. Then Φ(f)(C) ⊇ xk1, . . . , kp and:
R.-Ruet, Bioinformatics, 2008
Intra-cellular regulatory graph:
A proper modelling requires 6 inter-connected cells, here we only consider the 2 cells flanking the border
Analysis of two connected cells
W <-> autocrine Wg-pathway E <-> paracrine Wg-pathway
Analysis of two connected cells
wild-type pattern W E inverted wild-type pattern E W trivial pattern T T W <-> autocrine Wg-pathway E <-> paracrine Wg-pathway
Circuit analysis
Stable states:
symmetrical circuits only once)
Circuit analysis
"Intra-cellular" circuits:
֒ → ensures mutual exclusion En/Slp ֒ → enables the generation of the 2 Stable states EW and WE
Ci present;
֒ → action of Wg autocrine loop
present;
֒ → role of Slp in the maintenance of Wg
present, Ci[act] at level 1 and; Wg less than 1;
֒ → Functionality context not realistic in WT ⇒ no oscillations
Circuit analysis
"Inter-cellular" circuit:
symmetrical), when {Wg,Slp,Cirep}a and Enp absent, {Dsh,Slp,Ci}p present;
֒ → involves both Wg and Hh pathways ֒ → maintenance of Wg vs En expression ֒ → forces the right combination of stable states, at the right location, to specify the position of the forming segmental border
Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction
G0 G1 G3 G2
G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1)
Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction
G0 G1 G3 G2
G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1) G0 function revised
G0 G1 G2
G0 → 1 if G0 is present OR G2 is absent (condition for which g3 absent) f
r
G0(x) = (xG1 = 1)∨ (xG2 = 0)
Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction
G0 G1 G3 G2
G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1) G0 function revised
G0 G1 G2
G0 → 1 if G0 is present OR G2 is absent (condition for which g3 absent) f
r
G0(x) = (xG1 = 1)∨ (xG2 = 0)
Reduction amounts to consider that the component is faster
Emergence of new complex attractors indicate the existence of robust (transient) oscillations in the original model
is (generally) not conserved If an attractor is reachable in the reduced model, it is reachable in the original one
Transient oscillations φ(C1) = {x2 = 0} ∪ {x2 = 1, x3 = 0}
G1 G2 G3
φ(C2) = {x1 = 0}
000 010 001 011 002 102
+
01
−
2 100 110 101 111 1
−
12
Transient oscillations
G1 G2 G3
000 010 001 011 002 102
+
01
−
2 100 110 101 111 1
−
12
G3 faster → some transitions are lost
Transient oscillations
G1 G2 G3
000 010 001 011 002 102
+
01
−
2 100 110 101 111 1
−
12
G3 faster → some transitions are lost
G1 G2
01 00 10 11 102 010 002
+
01
−
2 110 1
−
12
Assuming that reduced are faster:
transient oscillations in the original STG;
reduced STG indicates a stability of the functionality context of the related negative circuit.
How to compact the dynamics?
STG SCC graph
SCC graph HTG σ(C) = {C′ ∈ Scc, C′ attractor or complex, s.t. C C′}
σ gathers trivial transient SCCs from which the same attractors and complex SCCs are reachable
A hierarchical representation of network dynamics
generate compressed representation
behaviours
decisions
A dedicated tool for the qualitative modelling and analysis of regulatory networks
http://gin.univ-mrs.fr/
STG too large...
Reduced nodes: {Dsh, Ci, Pka, Ptc, Nkd, Cirep}
֒ → The 3 stable states are reachable ֒ → Size of the basins of attraction to be linked to the size of functionality contexts
Reduction Reached stable states
∅ ?? STG too large ρ = {Dsh, Ci, Pka, Ptc, Nkd, Cirep} WE, EW, TT ρ ∪ {Ciact, Fz} WE, TT EW not reachable ρ ∪ {Slp} WE, TT EW not reachable
SCC: no sustained oscilla- tions