From minimal signed circuits to the dynamics of Boolean regulatory networks
CNRS, Institut de Mathématiques de Luminy, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France
*To whom correspondence should be addressed.
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ABSTRACT |
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It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the two minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the three stable states corresponding to the Th0, Th1 and Th2 cells.
Contact: ruet{at}iml.univ-mrs.fr
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1 INTRODUCTION |
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The activity of a biological cell is to a large extent controlled by genetic regulation, an interactive process usually represented by graphs called genetic regulatory networks: in these graphs, vertices denote genes or regulatory products (e.g. RNA and proteins) and edges denote regulatory interactions between these genes or their products (de Jong, 2002; Schlitt and Brazma, 2007; Thieffry, 2007). These regulatory interactions are further directed and signed (+1 or –1) to denote activatory versus inhibitory effects.
In order to relate regulatory networks to relevant dynamical properties, biologists often use them as a basis to generate dy-namical models, using either a differential framework or a discrete framework (de Jong, 2002). The biological pertinence of the model considered is then evaluated by comparing numerical simulations with experimental observations, for instance, biochemical character-iz-ations of cellular states, phenotypes of genetic mutants, etc.
Since the computational complexity of these simulations is, in general, exponentially increasing with the size of the network, some mathematical properties could fruitfully help in controlling the space of necessary simulations. In the early 1980s, the biologist R. Thomas proposed two simple rules relating the structure of regulatory networks to their dynamical properties (Thomas, 1981):
- a necessary condition for multistability (i.e. the existence of several stable fixed points in the dynamics) is the existence of a positive circuit in the regulatory network (the sign of a circuit being defined as the product of the signs of its edges);
- a necessary condition for the existence of an attractive cycle in the dynamics is the existence of a negative circuit.
{0,1}n, and a signed directed graph G(f)(x) is associated to each state of the system x
{0,1}n. This graph corresponds to a local notion of regulatory graph (as in Soulé (2003) for instance), and is mathematically defined by means of the discrete Jacobian matrix J(f)(x) (Robert, 1986). The required definitions are recalled in Section 2. While these results provide graphic conditions which are necessary to observe some dynamical properties, they do not give sufficient conditions at all, while biologists often acknowledge certain positive or negative circuits as responsible for some dynamical behaviour (Thomas, 1973; Thomas, et al., 1995). In the very specific case of discrete isolated circuits however, i.e. when the regulatory graph G(f)(x) does not depend on the state x and consists in a circuit, Remy et al. (2003) provide an extensive analysis of the dynamics, recalled in Section 3.
In the present article, we show that the presence of certain positive or negative circuits in a local graph G(f)(x) suffices for the observation of the corresponding dynamical property (multistability or a restricted version of homeostasis). More precisely, the crucial property that C has to meet is to be globally minimal, i.e. minimal as a circuit in the global graph G(f)=
x{0,1}nG(f)(x) obtained by taking the union of all local graphs. In Section 4, we define a restricted form of fixed points and attractive cycles for each set I of genes, and we show that if C is a globally minimal positive (resp. negative) circuit with vertex set {k1,...,kp}, then a suitably defined restriction of f to {k1,...,kp} has two fixed points (resp. an attractive cycle). These results provide:
- a non-trivial converse to Thomas’ rules in the discrete framework,
- a natural approach to the question of modularity of regulatory networks, namely: given pieces of a network for which the dynamics is known, how do they combine to produce a global (more complex) behaviour? Our results on the effect of specific functional circuits in a network gives insights into this line of research.
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2 BOOLEAN DYNAMICS AND DISCRETE JACOBIAN MATRICES |
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2.1 Notations
Let us start with preliminary notations. For β
{0,1}, we define 
by 
=1 and 
=0. Let n be a positive integer. For x{0,1}n and I
{1,...,n}, 
I{0,1}n is defined by: |
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{i} is denoted by i. The distance d:{0,1}n x {0,1}n{0,1,...,n} is the Hamming distance: d(x,y) is the number of i{1,...,n} such that xi
yi. Suppose 0
kn, and I is a k-element subset of {1,...,n}. Then each x{0,1}n generates an affine k-dimensional subspace x[[I]] of {0,1}n=
defined by: |
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2.2 Dynamics
In the context of genetic regulatory networks, we are interested in the evolution of the system consisting of n genes, which are denoted by the integers 1,...,n. We consider {0,1}n as the set of states of this dynamical system. Given a state x=(x1,..., xn){0,1}n, xi denotes the (discretized) expression level of gene i. These expression levels are either 0 (when the gene product is considered absent or inactive) or 1 (when the gene product is present and active).
In discrete models, a dynamics is a binary relation R which we assume to be irreflexive: R gives the rule for updating a state, i.e. it is the set of pairs of states (x,y) such that state x can lead to state y. In particular, a stable state is a state x such that for no y, (x,y)R.
In the context considered in this article (genetic networks), it is not realistic to assume a simultaneous update of all variables. Indeed, the Boolean dynamical systems we are interested in can be seen as discretizations of piecewise-linear differential systems (de Jong, 2002; Glass and Kauffman, 1973); Soulé, 2006; Thomas, 1981), and for these systems, the set of trajectories meeting more than one threshold hyperplane at a time has measure 0. We shall therefore consider asynchronous dynamics, i.e. relations R such that:
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i for some i. Clearly, the asynchronous dynamics encompasses, among many others, the realistic trajectories, and a more refined analysis would take into account, e.g. delays and probabilistic issues. Such an asynchronous dynamics R may be non-deterministic (it needs not be a function), but even then, it is possible and convenient to represent it by a map f:{0,1}n{0,1}n with coordinate functions f1,...,fn, defined by:
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{1,...,n}, an I-fixed point is an x such that fi(x)=xi for all iI, i.e. the coordinates in I are fixed under f.
Given such a map f, the corresponding asynchronous dynamics is defined in a straightforward way, and for each x{0,1}n and i=1,...,n, fi(x) denotes the value to which xi, the expression level of gene i, tends when the system is in state x.
For instance, the asynchronous dynamics corresponding to the map f:{0,1}2{0,1}2 defined by 
is illustrated in Figure 1.
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, which are more standard in biological literature than +1 A trajectory in the dynamics is a sequence of states (x1, ...,xr) such that for each i=1,...,r–1, (xi,xi+1)
R, and a cycle is a trajectory of the form (x1,...,xr,x1) with r
2. We shall be especially interested in a specific class of cycles which correspond to periodic oscillations: a cycle (x1,...,xr,x1) is said to be attractive when no trajectory may leave it, i.e. for all i=1,...,r, d(xi, f(xi))=1. More generally, if I{1,...,n}, a cycle (x1,...,xr,x1) is said to be I-attractive when for all i=1,...,r, by considering indices modulo r:- the only coordinate
(i) such that 
belongs to I,
- the set J such that
is disjoint from I.
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2.3 Discrete Jacobian matrices and signed directed graphs
Given f:{0,1}n
{0,1}n, we attach to each x{0,1}n its discrete Jacobian matrix J(f)(x) as defined in (Robert, 1986): J(f)(x) is the n x n matrix with (i,j) entry |
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{0,1}n and x{0,1}n, define |
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) such that G has an edge with sign from i to j. Such a triple will be denoted by 
.
A circuit in a signed graph G is a non-empty sequence
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For instance, in the example of Figure 1 corresponding to 
, it is easy to check that the Jacobian matrix associated to any state x is therefore given by:
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2. Therefore, the graph G(x) at any state consists in a circuit between 1 and 2, hence a {1,2}-circuit. Since x1f2(x) and x2f1(x), the two edges are negative and the circuit is: |
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2.4 Functionality
The signed directed graph G(f)(x) attached to each state x encompasses a subset of the regulatory interactions found in the complete regulatory network. These graphs are analogous to the local interaction graphs considered in Soulé (2003) for instance. Consequently, in our discrete framework, a regulatory interaction and its sign may depend on the context, i.e. on the state of the system, in particular on the values of co-regulators acting on the same target. By taking unions of graphs on states x, we lose some details on the regulatory network and recover more global notions, closer to the objects usually manipulated by biologists: let G(f)=x{0,1}n G(f)(x) be the graph with a positive (resp. negative) edge from j to i when there exists x{0,1}n such that G(f)(x) contains a positive (resp. negative) edge from j to i. Note that G(f) may have both a positive and a negative edge between two given vertices.
This discussion motivates the following definition of the functionality context of a signed edge e: intuitively the set of states at which e is effective or functional (Remy et al., 2006). The functionality context of a circuit is then a notion of particular significance (as we shall see in Section 4). It is defined in the obvious way as follows:
DEFINITION 1
(Functionality context) Let f:{0,1}n(f)(e) of e is the set of x
.
Clearly, x(f)(C) if C is a circuit of G(f)(x).
2.5 Globally minimal circuits
We shall be interested in a specific kind of circuits in regulatory graphs, namely circuits C occurring in some G(f)(x), with the additional property that the global graph G(f) has no other edge between vertices of C than the edges of C itself.
DEFINITION 2
(Minimal circuit). Letbe a directed graph. The set of circuits of
X2. A circuit C is then said to be minimal when it is minimal for this order.
DEFINITION 3
(Globally minimal circuit) Let f:{0,1}n
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3 ISOLATED CIRCUITS |
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We reformulate the following result proved in Remy et al. (2003). According to the definition of the asynchronous dynamics, see (1), this result determines the dynamics of an isolated circuit, i.e. a regulatory graph constantly equal to a circuit.
THEOREM 1.
If f:{0,1}nthen for any x
if and only if(–1)xi–1+xi
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4 GLOBALLY MINIMAL CIRCUITS |
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Let us start with some notations. If
is a face of {0,1}n, let 
:{0,1}n be the projection onto the affine subspace (identified with {0,1}I), i.e. (y)i=yi for any iI, and let 
:{0,1}n be inclusion map of into {0,1}n, i.e. |
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does not depend on the choice of x such that 
, and that 
is the identity. The following lemma, an equivalent simpler reformulation of Lemma 1 in Remy and Ruet (2007), is a commutation property between the Jacobian and projection (or restriction).
LEMMA 1.
If f:{0,1}nis a face of {0,1}n and y
PROOF.
Let i,jSimilarly, (
if and only if
Moreover, since i
Consequently, signed edges in G(
strictionI are the same.
Then we show that the presence of a globally minimal circuit C has some important consequences on the dynamics restricted to the coordinates involved in C. Essentially, it enables to consider C as an isolated circuit. 
THEOREM 2.
Let f:{0,1}nwhich is globally minimal. Let
. Then
PROOF.
Let us first prove that{ki}
and
where indices are considered modulo p. Now, if j=i, it is straightforward that the signed edge (kj,kj+1,
On the other hand, if j
and
(2) therefore:
and G(f)(
and the sign of this edge is
ki
Let us now prove that the dynamics of
We are now in position to combine Theorems 1 and 2 and delineate the dynamical properties implied by a globally minimal circuit.
THEOREM 3.
Under the hypotheses of Theorem 2, if C is positive, then f has two {k1,...,kp}-fixed points; and if C is negative, then f has a {k1,...,kp}-attractive cycle.
PROOF.
If C is positive, by Theorems 1 and 2,Of course, P(0) and P(1) are fixed points of f because, by the positivity of C, P(
)k1
If C is negative, by Theorems 1 and 2, it is easy to check that
Hence, the image
of this cycle under
The global minimality hypothesis in Theorems 2 and 3 cannot be simply avoided. For instance, the dynamics corresponding to the map f:{0,1}2{0,1}2 defined by 
gives rise to a globally minimal positive circuit and indeed has two fixed points (0,1) and (1,0) (Fig. 1), whereas the perturbated dynamics corresponding to 
has a single fixed point (1,0): the {1,2}-circuit is no more globally minimal, it is perturbated by the negative loop on 1 (Fig. 3).
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It is not true either that the localized dynamics predicted by the above results leads necessarily to the corresponding global behaviour. In particular, the presence of a globally minimal positive circuit does not imply the existence of disjoint stable subspaces in general. This can be seen by considering the map h(x)=(1,x1
x2). The positive circuit consisting in a loop on 2 is globally minimal and its functionality context is given by x1=0. The dynamics, which is given in Figure 4, has two 1-fixed points (0,0) and (0,1), but the only global fixed point of h is (1,1): the positive loop on 2 acts as a ‘partial separator’ between the subspaces x2=0 and x2=1. A natural question is therefore, to understand more precisely under which conditions these modules combine to produce global separators and global differentiation.
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5 APPLICATION |
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We present here a biological illustration and then apply the results proved in the previous section.
We consider the network involved in the control of the Th-lymphocyte differentiation. The vertebrate immune system contains various cell populations. Among B and T lymphocytes, CD4+ T-helper lymphocytes can further differentiate into Th1 or Th2 cells, which respectively enable cell-mediated immunity and humoral responses. Th1 and Th2 cells can be distinguished according to their pattern of cytokine secretion. Immune responses biased towards the Th1 phenotype result in auto-immune diseases, while enhanced Th2 responses originate allergic reactions (Agnello et al., 2003; Murphy and Reiner, 2002). Various mathematical models have been proposed for the differentiation, activation and proliferation of Th-lymphocytes. Many of them were focusing on interactions between immunological cell populations at a macroscopic level (Bergmann et al., 2001; Yates et al., 2000; Yates et al., 2007). Other model analyses aim at understanding the mechanism of the generation of antibody and T-cell receptorsdiversity, as well as the dynamical properties of the large networks defined by the interactions between cytokines (Krueger et al., 2002) or between immunoglobulins see, e.g.(Weisbuch et al., 1990). We consider here a very simplified Boolean modelling of this Th-lymphocyte differentiation already presented in (Remy et al., 2006), which involves 12 regulatory components (Fig. 5). Other regulatory graphs using the same discrete modelling (Boolean or multivalued) have been proposed (Mendoza and Xenarios, 2006; Naldi et al., 2007).
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It has been shown (Mendoza, 2006) that the system can reach the three stable states given in Table 1. The first stable state s1 corresponds to the virgin Th-cells (Th0), whereas the second and third ones s2, s3 correspond, respectively, to Th2 and Th1 differentiated lymphocytes.
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5.1 Functional circuits
The regulatory graph represented in Figure 5 contains 18 circuits. Only four of them are functional, in the sense of Definition 1. Among these functional circuits, three are positive:
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{0,1}12 be the map corresponding to the asynchronous dynamics (not shown here for sake of space). The graph G(f) represented Figure 5 is the union of all the local graphs G(f)(x) for x{0,1}12. Only C1, C2 and C4 are globally minimal, C3 is not because of the loop C2. Let us compute the functionality contexts of these circuits.- Circuit C1 is functional when Tbet, STAT1 and SOCS1 are not expressed, therefore (f)(C1)={x| xTbet=xSTAT1=xSOCS1=0}.
- Circuit C2 (self-regulation of Tbet) is functional when STAT1 and GATA3 are not expressed, i.e. (f)(C2)={x| xGATA3=xSTAT1=0}.
- The non-globally minimal circuit C3 is functional when STAT6 and STAT1 are expressed, i.e. (f)(C3)={x| xSTAT6=xSTAT1=1}.
- Finally, the negative circuit C4 is functional when Tbet is not expressed and IFN
expressed, i.e. (f)(C4)={x| xIFN=1, xTbet=0}.
5.2 Analysis and comments
Let us consider the circuit C1. By Theorem 2, we know the structure of the states space of any face 
with x(f)(C1). Moreover, by Theorem 3, as C1 is positive, there are two {IL4R, STAT6, GATA3, IL4}-fixed points. Here, s1 and s2 belong to (f)(C1), and they differ only in the four coordinates which correspond to the four genes of C1: 
. These two local fixed points are also stable for the whole dynamics.
We can do the same type of analysis for the circuit C2. Theorem 2 gives the structure of the states space of any face 
with x(f)(C2). Moreover, as C2 is positive, there are two {Tbet}-fixed points (Theorem 3). Here again, two of the three global fixed points belong to the context of functionality of C2: s1, s2(f)(C2).
When Tbet is not expressed (e.g. by the indirect effect of a perturbation of IL4, as proposed in Mendoza and Xenarios (2006), GATA3 can be activated, and the circuit C1 is functional. Hence, the system reaches the differentiated state s2 (which represents Th2 cells). But if the expression of Tbet increases, for example, because the lymphokines IFN is transiently expressed, thenC1 is no more functional, but C2 is, and this self-regulation maintains Tbet expressed. Then, the system reaches the differentiated state Th1 (s3).
Concerning the negative circuit C4, by Theorems 2 and 3, we know that any face 
with x(f)(C4) has a {IFNR}, STAT1,SOCS1}-attractive cycle. In fact, the dynamics restricted to (f)(C4), i.e. the restriction of f to 
, contains an attractive cycle, where all the genes not in C4 and (f)(C4) are not expressed. The negative circuit C4 is functional when the lymphokine IFN is expressed and Tbet is not expressed. This functionality context is therefore fragile: as Tbet is an activator of IFN, the absence of Tbet implies that the expression level of IFN tends to 0, hence C4 should stay functional for a short time.
As it is proved in Section 4, when a circuit is considered in a state which belongs to its functionality context, then, letting only the variables of the circuit free, the structure of the dynamics is the same as the one of an isolated circuit. Hence, we have a precise local knowledge of the dynamics.
In this application, the functionality contexts of the three positive circuits cover all the phase space. Each positive circuit creates in its functionality context two basins of attraction, and finally, the whole space is divided into three basins corresponding to the three stable states (their maximal number is 23 in general (Aracena et al., 2004). Therefore, one of the challenges is now to be able to describe more precisely the position of the basins of attraction, where they separate and how they possibly connect each other.
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Even when the dynamics, i.e. the function f is known, the study of the phase space is not easy, and often not computationally feasible. The idea of getting as more information as possible on the dynamics from the structure of the regulatory graph—which is much smaller—is really attractive. The property of functionality of a circuit is well suited for this purpose. Indeed, the important role of the circuits on the dynamics of the systems is well-known, but the number of circuits in a typical regulatory graph is generally quite large. Fortunately, the circuits which have a real incidence on the dynamics are the functional ones, at least the globally minimal functional ones according to Section 4, and their number is much more accessible. For instance, in the illustration considered in Section 5, the model contains 18 circuits, but only 4 of them are functional. Actually, the proportion of functional circuits can be often much smaller in practice.
An interesting current line of research is therefore to decompose regulatory graphs into modules. The notion of modularity is not trivial, and the definition of a module is not straightforward, and, at least, not unique. This article leads to naturally define modules around the notion of globally minimal functional circuits.
While the example studied in Section 5 provides a good illustration of the potentiality of the method, the present work is clearly in progress, and many improvements can certainly be done. The generalization of our results to multivalued dynamics, as in Remy et al. (2006), requires a careful definition of regulatory graphs and functionality of circuits. The possibility of a sufficient condition on the Jacobian matrix of differential or piecewise-linear systems (Gouzé, 1998; Snoussi, 1998; Soulé, 2006) is worth exploring too.
On the other hand, relaxing the minimality constraint on circuits in Theorems 2 and 3 seems to require further work. The presence of a non-minimal functional circuit is indeed not so rare. For instance, the self-regulations of ‘clue-genes’, involved in functional circuits, should create this situation. Therefore, this constraint prevents us from analysing some important circuits.
Conflict of Interest: none declared.
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