Relating Bisimulations with Attractors in Boolean
Network Models
Daniel Oliveira Figueiredo
University of Aveiro
2016
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
1 / 20
Outline
Biological regulatory networks and models.
Boolean networks and attractors.
Complete bisimulations.
Comparing with other methods.
Conclusion and future work.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
2 / 20
Biological regulatory networks and models.
Preliminaries.
The dynamics of an intracellular are guided by the components within it.
These components can be diverse:
Gens;
mRNA;
Proteins;
...
Other components as molecules and secondary organelles can also be
considered.
To represent the dynamics of such systems, several kind of models can be
used.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
3 / 20
Biological regulatory networks and models.
Graph models
In these system, we consider a tuple (V , E ) in which V represents the set
of components and E ⊆ V × V × {+, −} represents each relation of
activation/inhibition.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
4 / 20
Biological regulatory networks and models.
Ordinary differential equations models.
Components – 1, 2, ..., n
Concentration (level of expression) of component i – State variable xi .
A positive regulation of i over a component j can be given by
xn
.
x n + θijn
A negative regulation of i over a component j can be given by 1 −
xn
.
x n + θijn
A ODE model is given by:
x 0 = F (x)
where each Fi (x) is the sum/product of all expressions that represents a
regulation over the component i.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
5 / 20
Biological regulatory networks and models.
Ordinary differential equations models.
From de previous example, we consider the variables a, t, s and ts.

n
θs,a


a0 = n

n

s + θs,a








ts n



s0 = n

n

ts + θts,s



an


0 =

t

n

an + θa,t









an
tn

0

ts = an + θn . t n + θn
a,ts
t,ts
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
6 / 20
Biological regulatory networks and models.
Boolean networks.
If we consider that there exists a threshold θi such that, for all j ∈ 1, ..., n:
θi = θij
then we can consider each xi as a boolean variable.
(
xi = 0, if xi < θi
xi = 1, if xi > θi
In this way, we can think the ODE model presented before as a boolean
models whose dynamics are defined by the following update equations:


a := ¬s



s := ts

t := a



ts := t ∧ a
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
7 / 20
Biological regulatory networks and models.
Boolean networks.
We represent each state as a vector (x1 , x2 , ..., xn ) ∈ Rn and thus, we can
construct a digraph in which the set of vertex is the set of states and the
edges represents possible transitions between states. These possible
transitions are induced by the boolean functions system.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
8 / 20
Boolean networks and attractors.
Steady states, orbits and connection with boolean networks.
We know that in ODE’s system model x 0 = F (x), the state of a system
evolves till eventually reach a steady state (in which F (x) = 0) or a
periodic orbit.
The existence and the study of such steady states/orbits are a main
concern when one is studying a biological regulatory network.
In boolean networks, the existence of steady states/orbits is signalized by
the existence of Strongly Connected Components (Subset of states in
which there is a path between any two states within it) whose do not
admit a outgoing transition (also known as terminals). In a biological
context, we call them attractors.
Boolean models studied more easily than systems composed of ODE.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
9 / 20
Boolean networks and attractors.
Attractors.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
10 / 20
Complete bisimulations
Definition (Bisimulation)
Let D = (V , E ) be a digraph. We say that an equivalence relation
S ⊆ V × V is a bisimulation when it verifies:
1
2
If (v , w ) ∈ S and (v , v 0 ) ∈ E then there exists w 0 ∈ V such that
(w , w 0 ) ∈ E and (v 0 , w 0 ) ∈ S;
If (v , w ) ∈ S and (w , w 0 ) ∈ E then there exists v 0 ∈ V such that
(v , v 0 ) ∈ E and (v 0 , w 0 ) ∈ S.
Definition (Complete bisimulation)
Let (V , E ) be a graph.
We say that B ⊆ V × V is a complete bisimulation if it is a bisimulation
and there exists B ⊆ V such that B = B × B.
We say that a complete bisimulation B is minimal if there is not any other
complete bisimulation B 0 such that B 0 ( B.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
11 / 20
Complete bisimulations
Lemma
Let (V , E ) be a graph, B ⊆ V and B = B × B a minimal complete
bisimulation. For any A ( B, ∃ a ∈ A, v ∈ B\A such that (a, v ) ∈ E .
Proof
Let us assume that there exists A ( B such that, for any a ∈ A, v ∈ B\A,
(a, v ) ∈
/ E.
In this case, we can easily verify that A × A is an equivalence relation since
all states of A are related. By hypothesis, for any (a, a0 ) ∈ A × A ⊆ B such
that (a, b) ∈ E , there exists some b 0 which verifies (a0 , b 0 ) ∈ E and
(b, b 0 ) ∈ B. Since for any a ∈ A, v ∈ B\A, (a, v ) ∈
/ E , we can conclude
that b, b 0 ∈ A and, therefore, (b, b 0 ) ∈ A × A. Thus, A × A is a complete
bisimulation and this contradicts the minimality of B.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
12 / 20
Complete bisimulations
Theorem
Let (V , E ) be a graph. B = B × B ⊆ V × V is a minimal complete
bisimulation ⇔ B is a terminal of V .
Proof
“⇐”
We now assume that B is a terminal of V . We can easily see that
B = B × B is a equivalence relation since all states are related. We
consider (u, v ) ∈ B and (u, u 0 ) ∈ E . Since B is a terminal, u 0 ∈ B and
∃v 0 ∈ B such that (v , v 0 ) ∈ E . Furthermore, (v , v 0 ) ∈ B, by definition and,
therefore, B is a complete bisimulation.
Let us assume that B is not minimal, i.e., there is a complete bisimulation
A := A × A ( B. Since B is terminal, it is possible to find a path from any
a ∈ A for any b ∈ B\A. Thus, ∃a0 ∈ A, b 0 ∈ B\A such that (a0 , b 0 ) ∈ E .
But this contradicts the fact of A being bisimulation because b 0 ∈
/ A.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
13 / 20
Complete bisimulations
“⇒”
We start by proving that if B is a minimal complete bisimulation, then
there exists a path between any two elements of B. We prove that B is a
terminal afterwards.
We consider u, v ∈ B. By the previous lemma, we know that there is a
transition (u, u1 ) ∈ E from {u} to B\{u}. if u1 = v , we are done.
Otherwise, using the lemma again, we know that there is a transition from
{u, u1 } to some u2 ∈ B\{u, u1 }. Here, either (u, u2 ) ∈ E or (u1 , u2 ) ∈ E .
In any case, there is a path from u to u2 . Again, if u2 =v we are done.
Otherwise, we can continue to apply this procedure till find a path
between u and v (this procedure will end in finite time since we are only
considering finite graphs). As u and v were arbitrary, we can conclude that
B is a Strongly Connected Component.
Finally, if u ∈ B and (u, v ) ∈ E , then (u, u) ∈ B and, by definition of
complete bisimulation, (v , v ) ∈ B. Then v ∈ B and, thus, B is a terminal.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
14 / 20
Complete bisimulation
Minimization of models
We recall the boolean model previously presented.
It is possible to verify that if
A = {0110, 0010, 1010, 1111,
0111, 1011, 0101, 0100, 0000,
1000}, then A×A is a minimal
complete bisimulation.
We can thus verify that A is an attractor.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
15 / 20
Complete bisimulations.
Minimization of models.
Moreover, for any digraph (V , E ) and a complete bisimulation B × B, we
can always extend this complete bisimulation to a larger bisimulation by
adding the pairs (v , v ) of vertices in V \B.
Since we consider each bisimulation as an equivalence relation, we can
obtain the “quotient digraph” if we consider each equivalence classes as a
vertex.
In general, larger bisimulations can be found. The bigger the bisimulation
found, the most reduced the resulting quotient digraph becomes.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
16 / 20
Complete bisimulations.
Minimization of models.
If we consider the extended bisimulation of the previous example, we
obtain:
Thus, we obtain a reduced digraph which preserves the attractor of the
original one.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
17 / 20
Comparing with other methods.
Hierarchical representations.
An idea already used to study biological regulatory networks are the
hierarchical representations. In these representation, the states of a
boolean network are sorted by their shortest distance to an attractor.
The major inconvenient of this process is that the attractors must be
previously known. Also, it is difficult to identify several features of the
system like, for example, cyclic behaviors.
The method we propose do not need to now the attractors to minimize
the model and only clusters states with identical behavior.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
18 / 20
Comparing with other methods.
Reduction of Strongly Conected Components.
This idea looks for all Strongly Connected Components(SCCs) and clusters
each one of them into one single state.
This method is widely used because it both preserves the attractors and
reduces the model significantly.
The disadvantages of using this method is that all SCCs (not only the
attractors) are clustered. Furthermore, no other states which do not
belong to an attractor are clustered.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
19 / 20
Conclusion.
Bisimulations can be used to obtain to minimize biological boolean models
and, guaranteeing some conditions, the methodology we presented
preserves the attractors. These results can pave the way for an
minimization algorithm.
We also evaluated the convenience of using this minimization methodology
when compared with other methods already used. It has both some
advantages and disadvantages. However, as seen, since it preserves the
attractors and, moreover, it can be combined with a modal logic, we
believe that this approach is worth.
In future, we want to come out with an axiomatization of such systems
which would allow us to formally prove diverse properties of them. Indeed,
since we can use modal logic along with bisimulations, we can combine
both ideas.
Daniel Oliveira Figueiredo (University of Aveiro)
Relating Bisimulations with Attractors in Boolean Network Models
2016
20 / 20