Entropy as a Robustness Marker in Genetic Regulatory Networks

Abstract

Genetic regulatory networks have evolved by complexifying their control systems with numerous effectors (inhibitors and activators). That is, for example, the case for the double inhibition by microRNAs and circular RNAs, which introduce a ubiquitous double brake control reducing in general the number of attractors of the complex genetic networks (e.g., by destroying positive regulation circuits), in which complexity indices are the number of nodes, their connectivity, the number of strong connected components and the size of their interaction graph. The stability and robustness of the networks correspond to their ability to respectively recover from dynamical and structural disturbances the same asymptotic trajectories, and hence the same number and nature of their attractors. The complexity of the dynamics is quantified here using the notion of attractor entropy: it describes the way the invariant measure of the dynamics is spread over the state space. The stability (robustness) is characterized by the rate at which the system returns to its equilibrium trajectories (invariant measure) after a dynamical (structural) perturbation. The mathematical relationships between the indices of complexity, stability and robustness are presented in case of Markov chains related to threshold Boolean random regulatory networks updated with a Hopfield-like rule. The entropy of the invariant measure of a network as well as the Kolmogorov-Sinaï entropy of the Markov transition matrix ruling its random dynamics can be considered complexity, stability and robustness indices; and it is possible to exploit the links between these notions to characterize the resilience of a biological system with respect to endogenous or exogenous perturbations. The example of the genetic network controlling the kinin-kallikrein system involved in a pathology called angioedema shows the practical interest of the present approach of the complexity and robustness in two cases, its physiological normal and pathological, abnormal, dynamical behaviors.

1. Introduction

The relationships between the notions of complexity, stability and robustness in genetic regulatory networks, as described in [1,2,3], are based on works from the eighties about Boolean automata [4,5,6,7,8,9] on the mathematical characterization of the stability in random dynamical systems. These studies about stability are based on theories of stochastic attractors (called confiners in [10,11]) and large deviations [12,13,14], and on an index of complexity—the evolutionary or dynamic entropy, introduced by L. Demetrius [15,16,17,18,19,20] and complementary to other complexity indices used in biological networks, such as number of nodes, connectivity (i.e., ratio between edge and node numbers) [21,22,23], number of strong connected components of their interaction graph [9], entropy of transformations with invariant measure [24,25], etc. The notion of stability (with respect to robustness) is quantified by the rate at which the system returns to its steady state (with respect to the threshold distance below which the system remains close to its former dynamics in terms of number and nature of attractors) after exogenous and/or endogenous perturbations.

If we restrict the study to discrete dynamical systems, many authors since Rochlin [24,25] have used the notion of entropy for quantifying the stability and robustness of dynamical processes, such as inference in Bayesian networks [26]; symbolic extension of a smooth interval map [27]; symbolic complexity of substitution dynamics on a set of sequences in a finite alphabet [28,29]; algorithmic complexity (called also Kolmogorov-Chaitin complexity) of binary strings involved in mutation dynamics [30]; and eventually, reconstruction and stability of Boolean dynamics of genetic regulatory networks following the René Thomas’ logic rules [31,32,33,34].

In the present paper, we will focus on the study of robustness of genetic regulatory networks driven by Hopfield’ stochastic rule [35], by using the Kolmogorov-Sinaï entropy of the Markov process underlying the state transition dynamics [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. In Section 2, we define the concepts underlying the relationships between complexity, stability and robustness in the Markov framework of genetic threshold Boolean random regulatory networks. We give in Section 3 some rigorous results concerning their dynamics, and eventually, in Section 4, we calculate the evolutionary entropy for a particular network controlling the kinin-kallikrein system in physiological and pathological conditions, and we give in Section 5 some perspectives of the present work.

2. Materials and Methods

2.1. Definitions

The evolutionary (or dynamic) entropy is related to the network dynamics, and can be identified to the Kolmogorov-Sinaï entropy in a Markovian dynamics. Based on the configuration of its attractor (in deterministic case) or confiner (in random case) landscape in the network state space, it is related to the richness of the network attractor or confiner landscape [11]. The literature on network entropies is abundant [19,20,21,22,51,52,53,54,55,56,57,58,59,60,61,62] and concerns both discrete or continuous dynamical systems, which share common mathematical concepts, such as attractor, attraction basin, Jacobian interaction graph, stability and robustness. We take as the definition of an attractor, that given in [63,64], available for both continuous and discrete cases. More specific definitions of the concept of attractor adapted for network dynamics are given in [65,66,67,68].

In the state space E⊂$ℝ$ⁿ provided with a dynamic flow φ and a distance d, {φ(a,t)}_t∊T denotes the trajectory starting at state a and being in state φ(a,t) at time t, where φ is a flow on ExT. L(a), the limit set of the trajectory starting in a, is made of the accumulation points of the trajectory {φ(a,t)}_t∊T, when the time t tends to infinity in the time set T (either discrete, equal to $\mathbb{N}$ or continuous, equal to $ℝ$):

$$\mathrm{L}\left(\mathrm{a}\right)=\{\mathrm{y}∊\mathrm{E};\forall \mathsf{\epsilon }>0, \forall \mathrm{t}\in \mathrm{T}, \exists \mathrm{s}(\mathsf{\epsilon },\mathrm{t})>\mathrm{t},\mathrm{such}\mathrm{as}\mathrm{d}(\phi (\mathrm{a},\mathrm{s}(\mathsf{\epsilon },\mathrm{t}\left)\right),\mathrm{y})<\mathsf{\epsilon }\},$$

where s(ε,t) is a time (> t) at which the trajectory is in a ball of radius ε centered in y.

Considering a subset A of E, L(A) is the union of all limit sets L(a), for a belonging to A:

$$\mathrm{L}\left(\mathrm{A}\right)={\cup }_{\mathrm{a}\in \mathrm{A}}\mathrm{L}\left(\mathrm{a}\right)$$

Conversely, B(A) is the set of all initial conditions (outside A), whose limit set L(a) is included in A (Figure 1). B(A) = {y∊E\A; L(y)⊂A)} is called the attraction basin of A.

An attractor A verifies the three following conditions (Figure 1 Top left):

(i)

A = LoB(A).

(ii)

There is no set A’ containing strictly A and shadow-connected to A.

(iii)

There is no set A” strictly contained in A verifying (i) and (ii).

The definition of the “shadow-connectivity” between subsets F₁ and F₂ of E, lies on the fact that there exists a “shadow-trajectory” between F₁ and F₂ (cf. Figure 1, where F₁ = {x} and F₂ = {y}). The notion of “shadow-trajectory” has been defined first in [69]: for any ε > 0, there is an integer n(ε), times t_(1,ε),…, t_(n(ε),ε) and states y_(1,ε),…, y_(n(ε),ε) = y, such that all distances between successive states of the shadow-trajectory are less than ε:

$$d\left(\phi \left(\mathrm{x},{\mathrm{t}}_{\left(1,\mathsf{\epsilon }\right)}\right),y_{\left(1,\epsilon \right)}\right)\le \mathsf{\epsilon },\dots , d\left(\phi \left({\mathrm{y}}_{\left(k-1,\mathsf{\epsilon }\right)},{\mathrm{t}}_{\left(k,\mathsf{\epsilon }\right)}\right),y_{\left(k,\epsilon \right)}\right)\le \mathsf{\epsilon },\dots ,d\left(\phi \left({\mathrm{y}}_{\left(n\left(\epsilon \right)-1, \mathsf{\epsilon }\right)},{\mathrm{t}}_{\left(n\left(\mathsf{\epsilon }\right),\mathsf{\epsilon }\right)}\right),\mathrm{y})\le \mathsf{\epsilon }\right)$$

For all known dynamical systems, the above definition of attractor remains available, and is meaningful for any discrete or continuous time set T, when the dynamical system is autonomous (with respect to the time), contrary to the other definitions proposed since the seventies, such as in [70,71,72], which are all less general.

2.2. Hopfield Dynamics

Let consider now discrete Hopfield dynamics [35], defined in a genetic regulatory network N having n genes, where x_i(t) = 1, if the gene i is expressing its protein; else x_i(t) = 0. The probability that x_i(t) = 1, knowing the states at time t in a neighborhood V_i of i, is given by:

$$\mathrm{P}\left(\left\{x_i\left(t\right)=1\right\}|\left\{x_j\left(t-1\right),j\in V_i\right\}\right)=\frac{e^{\left({{\displaystyle \sum }}_{j\in V_i}{w}_{ij} x_j\left(t-1\right)-\mathsf{\theta }\right)/T}}{1+e^{\left({{\displaystyle \sum }}_{j\in V_i}{w}_{ij} x_j\left(t-1\right)-\mathsf{\theta }\right)/T}}$$

(1)

w_ij is a parameter representing the influence or weight (inhibitory, if w_ij < 0; activating, if w_ij > 0 and null, if w_ij = 0) the gene j from the neighborhood V_i ={j; w_ij ≠ 0} exerts on the gene i, and which corresponds to the notion of external field in statistical mechanics and here to the minimal value of activation for interaction potential: ${{\displaystyle \sum }}_{j\in V_i}{w}_{ij}{x}_j$.

The thermodynamic parameter, temperature T, allows for introducing a certain degree of randomness in the transition rule. In particular, if T = 0, the rule is practically deterministic:

$$x_i\left(t\right)=1, if {{\displaystyle \sum }}_{j\in V_i}{w}_{ij}{x}_j\left(t-1\right)>\mathsf{\theta }; x_i\left(t\right)=0, if {{\displaystyle \sum }}_{j\in V_i}{w}_{ij}{x}_j\left(t-1\right)<\mathsf{\theta }$$

$$\iff x_i\left(t\right)=H({{\displaystyle \sum }}_{j\in V_i}{w}_{ij}{x}_j\left(t-1\right)-\mathsf{\theta })$$

(2)

and

$$\mathrm{P}\left(\left\{x_i\left(t\right)=1\right\}|\left\{x_j\left(t-1\right),j\in V_i\right\}\right)=\frac{1}{2},\mathrm{if}{{\displaystyle \sum }}_{j\in V_i}{w}_{ij}{x}_j\left(t-1\right)=\mathsf{\theta }$$

where H is the Heaviside function.

Once this rule defined, three incidence matrices of important graphs characterizing the network dynamics can be defined:

-

Interaction matrix W: Its coefficient w_ij corresponds to the action of the gene j on the gene i. W is analogue to a discrete Jacobian matrix. Signed Jacobian matrix A is defined as follows: α_ij=1, if w_ij > 0, α_ij = −1, if w_ij > 0 and α_ij=0, if w_ij=0. Its associated directed graph is the interaction graph G;

-

Updating matrix S: s_ij = 1, if j is updated before or with i, else s_ij = 0;

-

Flow matrix Φ: ϕ_bc = 1, where b and c belong to E, if and only if b = φ(c,1); else ϕ_bc = 0.

The matrices involved in the network dynamics are constant or depend on time t [41,42]:

-

Concerning W, the dependence is called the Hebbian dynamics: if the two vectors {x_i(s)}_s<t and {x_j(s)}_s<t have a correlation coefficient j(t)≠0, then the dependence is expressed through the equation: w_ij(t + 1) = w_ij(t) + h_ij(t), with h > 0, corresponding to a reinforcement of the absolute value of interactions w_ij(t) having succeeded to increase the xi(s)’s, if w_ij(t) is positive, and conversely to decrease the x_i(s)’s, if w_ij(t) is negative.

-

Concerning S, the updating can be state dependent or not. If all s_ij equal one, the updating schedule is called parallel; if there exists a sequence of indices of the n nodes of the network, i₁,…, such as s_ikik+1 = 1, the other s_ij’s being equal to 0, the updating is called sequential. Between these two extreme schedules, the updating modes are called block-sequential, i.e., they are parallel in a block, the blocks being updated sequentially. A more realistic schedule in genetic and metabolic networks is called block parallel [50]: These networks are composed of blocks made of genes sequentially updated, these blocks being updated in parallel (Figure 2 Top middle). Some interactions between these blocks can exist (i.e., there are w_ij ≠ 0, with i and j belonging to 2 different blocks), but because the block sizes are different, the time at which the first attractor state is reached in a block is not necessarily synchronized with the corresponding times in other blocks: the synchronization expected as asymptotic behavior of the dynamics depends on the intra-block as well as on the inter-block interactions, which explains that states of genes in a block serving as the clock for the network are highly dependent on states of genes in other blocks connected to them (e.g., acting as transcription factors of the clock genes).

-

Concerning Φ, the transition operator is Markov, and hence autonomous in time.

Figure 2 Top shows the three important graphs of the network dynamics; i.e., the graphs having as incidence matrices respectively, the interaction W (left), updating S (middle) and trajectory (right) matrices. If the dynamics are ruled by a deterministic Hopfield network, with temperature equal to 0 and interaction weights w_ij, which are equal to 1, −1 or 0, their attractor is a limit-cycle given on Figure 2 (top right), showing two intricate rhythms, one of Period 4 corresponding to the inner clock dynamics, embedded in a rhythm of period 12, the rhythm of the whole network. Figure 2 (Bottom) shows other updating graphs, less realistic than the block-parallel for representing the clock action.

3. Results

3.1. Attractor Entropy and Isochronal Entropy, Indices of Complexity and Synchronizability

The attractor entropy E_attractor is a measure of heterogeneity of the attractor landscape on the state space E of a dynamical system. It is defined by the quantity:

$$E_{attractor}=-{{\displaystyle \sum }}_{k=1,m}ABRS\left(A_k\right)lo{g}_2\left[ABRS\left(A_k\right)\right],$$

(3)

where ABRS(A_k) is equal to the attraction basin relative size of the kth attractor A_k among the m attractors of the network dynamics, divided by 2ⁿ, n being the number of genes of the network. Let consider the case of a continuous network, called the Wilson-Cowan oscillator, close to the Hopfield system if τ_x = τ_y = 1/ε, w_xx = w_xy = w_yx = w_yy = λ/ε, 0<ε<<1 [73], which describes the activities x and y of two populations of interacting genes as follows:

$$\frac{\mathrm{d}x}{\mathrm{d}t}=-\frac{x}{{\tau }_x}+\tan \mathrm{h}\left(w_{xx}x-\mathsf{\theta }\right)-\tan \mathrm{h}\left(w_{xy}y-\mathsf{\theta }\right)$$

$$\frac{\mathrm{d}y}{\mathrm{d}t}=-\frac{y}{{\tau }_y}+\tan \mathrm{h}\left(w_{yx}x-\mathsf{\theta }\right)+\tan \mathrm{h}\left(w_{yy}y-\mathsf{\theta }\right).$$

(4)

The variables x and y represent the proteinogenic activities of two populations of genes, whose interaction is described through a quasi-sigmoid function parametrized by λ, controlling its stiffness and accounting for the response to a signal coming from the other gene (whose expressed protein acts as a transcription factor, inhibitor or activator, on the first gene). This response is highly non-linear when λ is large, becoming similar to a Hopfield-like response.

The parameters τ_x and τ_y refer to the influence power of the populations of genes x and y. If θ = 0, τ_x = τ_y = τ, the Wilson-Cowan oscillator presents a Hopf bifurcation when λτ crosses the value 1 [74], moving from dynamics with only one stable fixed point to dynamics with a repulsor at the origin of the state space E, surrounded by an attractor limit cycle C of period T. Let us consider now the point φ(O,h) of C reached after a time (or phase) h on C from a point chosen as origin O = φ(O,0) (Figure 3): there is a bijective map b between C and the phase interval [0, T]. The isochron I_h of phase h is the set of points x of B(C) such that d(φ(x,t), φ(h,t)) tends to 0, when t tends to infinity. I_h extends the map b to all the points of B(C). Let denote φ_h the flow φ restrained to I_hxT_d, where T_d = {kT}_k∈ℕ, such as:

$$\begin{gathered} \forall x\in \mathrm{E},\forall t\in {\mathrm{T}}_{\mathrm{d}},\mathrm{then}{\phi }_h\left(x,t\right)=\phi \left(x,t\right)\in I_h,\mathrm{if}h=li{m}_{t\in {\mathrm{T}}_{\mathrm{d}}; t\to \infty }\phi \left(x,t\right) \\ \mathrm{and}\mathrm{B}\left(\mathrm{C}\right)={\cup }_{h\in \mathrm{C}}{I}_h \end{gathered}$$

The point h is an attractor for φ_h, whose basin is I_h\{h}. Let us decompose now the time interval [0,T] in m equidistant sub-intervals [kT/m, (k+1)T/m[, for k = 0,…,m−1. Let denote φ_k the flow φ restrained to A_k∪B_k x T_d, with A_k = F([kT/m, (k+1)T/m]) as attractor, having as attraction basin the set B_k = ∪_{h∊F([kT/m, (k+1)T/m[)} I_h, with B(C) = ∪_k=0,m−1B_k. The isochronal entropy of degree m is related to the m A_k’s subsets of of C=∪_k=0,m−1A_k, each A_k being the attractor for the dynamics φ_k:

$$E_{attractor}^m=-{{\displaystyle \sum }}_{\mathrm{k}=1,m}ABRS\left({\mathrm{A}}_{\mathrm{k}}\right)lo{g}_2\left[ABRS\left({\mathrm{A}}_{\mathrm{k}}\right)\right]$$

(5)

If isochron basins have all the same size, then E^m_attractor = log₂m. If not, E^m_attractor reflects the spatial heterogeneity of the asymptotic behavior of the network dynamics: the basins B_k can be large (when the flow φ is rapid inside) or small (when the flow φ is slow inside). In the example of the Wilson-Cowan oscillator with τ_x = τ_y = 1, if λ is growing from 0 to infinite, then E^m_attractor is increasing from 0 to log₂m [75].

Let us fix now the values of the parameters: τ_x = τ_y = τ = 1 and λ = 1.1, close to the bifurcation in the parameter space (λτ ≈ 1) and consider the isochron landscape of the Wilson-Cowan oscillator (Figure 3). The maximum phase shift observed after an instantaneous translation of the limit cycle of the Wilson-Cowan oscillator is inversely proportional to the intensity of this perturbation (Figure 3 Thumbnail), which means that a population of oscillators must be perturbed by a sufficiently intense stimulus to be synchronized. Indeed, isochrons form spirals, which diverge from one another far from the limit cycle and the synchronization, are observed if the translation of the limit cycle falls between two isochrons having close phases f₁ and f₂. These isochrons can be reached by translating the limit cycle C sufficiently far from its initial position [75,76,77]: it is possible if attraction basins are sufficiently wide, and hence, if the isochronal entropy E^m_attractor is sufficiently large. Hence, if E_attractor can serve as index of complexity of attractor landscape, E^m_attractor can serve more precisely as index of synchronizability, which is particularly interesting in the study of biological clocks and the entrainment of biological rhythms.

3.2. Energy, Frustration and Dynamic Entropy

We define first the functions energy U and frustration F for a genetic network N with n genes in interaction [1,2,15,16,17,18,19,20,21,22,23,36,37,38,39,40,41,42,43,44,78]:

$$\forall x\in \mathrm{E},U\left(x\right)={{\displaystyle \sum }}_{i,j=1,n}{\alpha }_{ij}{x}_i{x}_j=Q_{+}\left(N\right)-F\left(x\right),$$

(6)

where x is a configuration of gene expression (x_i = 1, if the gene i is expressed, and x_i = 0, if not), E denotes the set of all configurations of gene expression; that is, for a Boolean network, the hypercube {0,1}ⁿ, and α_ij = sign(w_ij) is the sign of the interaction weight w_ij, which quantifies the influence of the gene j on the gene i: α_ij = −1 (with respect to +1), if j is an inhibitor (with respect to activator) of the expression of i, and α_ij = 0, if j exerts no influence on i. Q₊(N) is equal to the number of positive edges of the interaction graph G of the network N having n genes, whose incidence matrix is A = (α_ij)_{i,j = 1,n}. F(x) denotes the global frustration of x; i.e., the number of pairs of genes (i,j) for which state values x_i and x_j are contradictory with the sign α_ij of the influence of j on i:

$$F\left(x\right)={{\displaystyle \sum }}_{i,j=1,n}{F}_{ij}\left(x\right),$$

(7)

where F_ij is the local frustration of the pair (i,j) defined by:

F_ij(x) = 1, if {α_ij = 1 ∧ {{x_j = 1 ∧ x_i = 0}∨{x_j = 0 ∧ x_i = 1}} ∨ {α_ij = −1 ∧ {{x_j = 1 ∧ x_i = 1}∨{x_j = 0 ∧ x_i = 0}}, else F_ij(x) = 0.

We choose for the dynamics of the network the Hopfield rule (1). It corresponds to a Markov operator defined on the state space E provided with a sequential updating schedule for the expression of the genes in a predefined order [1,2]. Let M = (M_xy) denote its Markov matrix, giving the transition probability (defined by the rule (1) and this sequential updating mode) to go from the configuration x to the configuration y in E, and μ = {µ_x = μ({x})}_x∊E denote its stationary distribution on E. The dynamic entropy E is then defined as the Kolmogorov-Sinaï entropy of M by the classical formula:

$$E=-{{\displaystyle \sum }}_{x,y\in \mathrm{E}}{\mu }_x{M}_{xy}lo{g}_2\left(M_{xy}\right)$$

(8)

In the sequential updating mode defined by the order of the nodes equal to the integer order on N={1,n}, by denoting I = {1,…,i − 1}, N\I = {i,…,n} and X (with respect to Y) the set of the indices i such that x_i = 1 (with respect to y_i = 1), we have for the general coefficient of the transition matrix M:

$$\begin{gathered} M_{xy}={{\displaystyle \prod }}_{i=1,n}[P\left(\left\{y_i=1\right\}|\left\{y_j, j\in I;x_k, k\in N\backslash I\right\}\right){\mathbb{I}}_{\left\{y_i=1\right\}} \\ +P\left(\left\{y_i=0\right\}|\left\{y_j,j\in I;x_k,k\in N\backslash I\right\}\right){\mathbb{I}}_{\left\{y_i=0\right\}}] \end{gathered}$$

(9)

and for the invariant measure, the classical Gibbs measure µ [1,2]:

$$\forall x\in \mathrm{E}, {\mu }_x=\frac{{\mathrm{e}}^{\left({{\displaystyle \sum }}_{i,j=1,n}{w}_{ij}{x}_i{x}_j-\theta \right)/T}}{{{\displaystyle \sum }}_{y\in \mathrm{E}}{\mathrm{e}}^{\left({{\displaystyle \sum }}_{i,j=1,n}{w}_{ij}{y}_i{y}_j-\mathsf{\theta }\right)/\mathrm{T}}}$$

(10)

When T = 0, the invariant measure μ is concentrated on the m (≤2ⁿ) deterministic attractors A_k of size a_k of the network dynamics (e.g., two fixed points and a limit cycle on Figure 4 Top) and we have:

$$E=-{{\displaystyle \sum }}_{x,y\in \mathrm{E}}{\mu }_x{M}_{xy}lo{g}_2\left(M_{xy}\right)=0,$$

(11)

because each line of M is reduced to one coefficient equal to one, the others being equal to 0.

When +∞ > T > 0, if μ becomes scattered uniformly on the attraction basins of the m deterministic attractors supposed to be fixed points of the network dynamics (cf. Figure 4 Middle), they are transiently the final classes of the Markov transition matrix related to the dynamics (9), and are denoted A_k of size card(A_k) = a_k. Then we have, when this scattering is observed, the approximate formula:

$$\begin{gathered} E=-{{\displaystyle \sum }}_{x,y\in \mathrm{E} }{\mu }_x{M}_{xy}lo{g}_2{M}_{xy }\approx -{{\displaystyle \sum }}_{k=1,m}\frac{a_k}{2^n}lo{g}_2\left(\frac{1}{2^n}\right) \\ =lo{g}_2\left(2^n\right)+{{\displaystyle \sum }}_{k=1,m}\frac{a_k}{2^n}lo{g}_2\left(\frac{a_k}{2^n}\right) \end{gathered}$$

which can be written as:

$$E\approx lo{g}_2\left(2^n\right)-{{\displaystyle \sum }}_{k=1,m}ABRS\left(A_k\right)lo{g}_2\left[ABRS\left(A_k\right)\right]=lo{g}_2\left(2^n\right)-E_{attractor}$$

(12)

where ABRS(A) = Σ_x∊B(A)∪A μ_x = μ(B(A)∪A).

When T tends to +∞, μ tends to be scattered uniformly over E (cf. Figure 4 Bottom) and E = log₂2ⁿ = n. More generally, we have:

$$\begin{gathered} E=-{{\displaystyle \sum }}_{x,y\in \mathrm{E} }{\mu }_x{M}_{xy}lo{g}_2{M}_{xy }=-{{\displaystyle \sum }}_{x,y\in \mathrm{E} }{\mu }_x{M}_{xy}lo{g}_2({\mu }_x{M}_{xy })+{\mu }_{x}lo{g}_2{\mu }_x \\ =E_{\nu }-E_{\mu } \end{gathered}$$

(13)

where E_v and E_µ denote, respectively, the entropy of the joint measure $\nu ={\mu }_x{M}_{xy}$ on E² and the entropy of the invariant measure µ, which can be estimated by E_attractor, if attractors are fixed points and by E^m_attractor if the attractor is a limit-cycle.

When T = +∞, and v and μ are scattered uniformly, respectively, on E² and E, and we have: E = log₂ ((2ⁿ)²)−log₂2ⁿ = n.

When T = 0, if v is concentrated on a set of states in E² and µ is concentrated on the projection of this set of states on E, then they have the same entropy and E = 0. In between (when +∞ > T > 0), Equation (12) constitutes an estimator of E, if T is s large enough.

3.3. Dynamic Entropy, Index of Robustness

The problem of robustness of a biological network controlling important functions, such as morphogenesis, memory, cell division, etc., has been often considered for 80 years [79,80,81] under different names (structural stability, resilience, bifurcation, etc.), but is still pertinent [82,83].

The dynamic entropy E serves as index of robustness, being related (as Kolmogorov-Sinaï entropy) to the ability a system has to return to its equilibrium after exogenous or endogeneous perturbations [23,43].

By considering a Hopfield rule with all non-zero interaction weights w_ij having the same absolute value c, we will study the robustness of the network in response to the variations of c, by proving the following Propositions [39,42]:

Proposition 1.

Let us consider a deterministic Hopfield Boolean network, which is a circuit sequentially or synchronously updated with constant absolute value c for its non-zero interaction weights. Then, its dynamics are conservative, keeping constant on the trajectories the Hamiltonian function L defined by:

$$\mathrm{L}\left(x\left(t\right)\right)={{\displaystyle \sum }}_{i=1,n}\frac{{(x_i\left(t\right)-x_i\left(t-1\right))}^2}{2}={{\displaystyle \sum }}_{i=1,n}\frac{H{(w_{i\left(i-1\right)modn}{x}_{i-1}\left(t-1\right)-x_i\left(t-1\right))}^2}{2}$$

(14)

whereHdenotes the classical Heaviside function. L(x(t)) is the total discrete kinetic energy, equal to the half of the global dynamic frustrationF(x(t)) = ∑_i=1,nF_{i,(i−1)modn} (x(t)), where F_{i,(i−1)modn} is the local dynamic frustration defined between nodes (i−1) and i as follows:

F_i,(i−1)(x(t)) = 1, if {sign(w_i(i−1)) = 1 ∧x_i(t) ≠ x_i−1(t − 1)} ∨ {sign(w_i(i−1)) = −1 ∧ x_i(t) = x_i−1(t − 1)}, else F_i,(i−1)(x(t)) = 0.

The Proposition 1 still holds if the network is a circuit whose transition functions are Boolean identity or negation, a circuit on which it is easy to calculate the global frustration F and to show that it characterizes attractors by remaining constant along them and decreasing inside their attraction basin like in discrete gradient dynamics (Figure 5 Left) [84].

Let consider now the quantity ∂E/∂c, which denotes the derivative of the dynamic entropy E with respect to the common absolute value c. ∂E/∂c can be considered as the capacity the robustness parameter E has to resist to environmental variations of the network weights and we have from formula (13): ∂E/∂c = ∂E_v/∂c − ∂E_µ/∂c. Then we can prove the following result [29,42]:

Proposition 2.

If the updating mode of the Hopfield network is sequential, we have:

∂E_µ/∂c = −cVar_µ(F),

(15)

where the variance Var_µ is taken for the invariant Gibbs measure µ defined by:

$$\forall x\in \mathrm{E}, {\mu }_x=\frac{{\mathrm{e}}^{\left({{\displaystyle \sum }}_{i,j=1,n}{w}_{ij}{x}_i{x}_j-\mathsf{\theta }\right)/T}}{{{\displaystyle \sum }}_{y\in \mathrm{E}}{\mathrm{e}}^{\left({{\displaystyle \sum }}_{i,j=1,n}{w}_{ij}{y}_i{y}_j-\mathsf{\theta }\right)/\mathrm{T}}}$$

Proof: 

We have:

E_µ = −Σ_x∊E μ_xlog₂µ_x, then ∂E /∂c = −Σ_x∊E ∂μ_x/∂c log₂µ_x − Σ_x∊E μ_x ∂log₂µ_x/∂c,

where ∂μ_x/∂c = ∂[exp((Σ_i,j=1,n cα_ijx_ix_j − θ)/T)/Z]/∂c; N_i is the neighborhood of i made of the nodes j such as α_ij ≠ 0, Z = Σ_y∊E exp((Σ_i,j=1,n cα_ijy_jy_j − θ)/T) and ∂Z/∂c = Σ_y∊E (Σ_i,j=1,n α_ijy_iy_j/T)Zμ_y. Then:

∂μ_x/∂c =[∂[exp((Σ_i,j=1,n cα_ijx_ix_j-θ)/T)/∂c]/Z − exp((Σ_i,j=1,n cα_ijx_ix_j-θ)/T)(∂Z/∂c)/Z²
= (Σ_i,j=1,n α_ijx_ix_j/T)μ_x − Σ_y∊E (Σ_i,j=1,n α_ijy_iy_j/T)μ_yμ_x,

which implies:

∂μ_x/∂c log₂µ_x=[c(Σ_i,j=1,n α_ijx_ix_j/T)²μ_x − Σ_y∊E c(Σ_i,j=1,n α_ijy_iy_j/T)(Σ_i,j=1,n α_ijx_ix_j/T)μ_yμ_x]/Log2 − ∂μ_x/∂c log₂Z

and

μ_x ∂log₂µ_x/∂c = (∂μ_x/∂c)/Log2

Hence, we have:

∂E_µ/∂c = −Σ_x∊E ∂μ_x/∂c log₂µ_x − Σ_x∊E μ_x ∂log₂µ_x/∂c

= −c[E_µ(U²) − (E_µ(U))²]/Log2 + Σ_x∊E [∂μ_x/∂c log₂Z − ∂µ_x/∂c]/Log2,

but Σ_x∊E ∂μ_x/∂c = ∂(Σ_x∊E μ_x)/∂c = 0, therefore ∂E_µ/∂c = −cVar_µ(U) = −cVar_µ(F) □

Let us define now the local cross-frustration G_ij(x,y) = 1: if α_ij = 1, x_iy_j = 0 or α_ij = −1, x_iy_j = 1, and G_ij(x,y) = 0 elsewhere, the global cross-frustration is G(x,y) = Σ_i,j=1,n G_ij(x,y). We can remark than G(x,x) = F(x). The local V(x,y) and conditional V(x) cross-energy functions are respectively defined as:

V(x,y) = Σ_i,j=1,nα_ijx_iy_j/T = Q₊(N) – G(x,y) and V(x) = Σ_y∊EV(x,y).

By considering the conditional entropy E_x = −Σ_y∊E M_xy log₂M_xy, where, in the parallel updating mode:

$$\begin{gathered} M_{xy}={{\displaystyle \prod }}_{i=1,n}[P\left(\left\{y_i=1\right\}|\left\{\mathrm{x}\right\}\right){\mathbb{I}}_{\left\{{\mathrm{y}}_{\mathrm{i}}=1\right\}}+P\left(\left\{y_i=0\right\}|\left\{\mathrm{x}\right\}\right){\mathbb{I}}_{\left\{{\mathrm{y}}_{\mathrm{i}}=0\right\}}] \\ =\exp (({\mathsf{\Sigma }}_{i,j=1,n}c{\alpha }_{ij}{x}_i{y}_j-\mathsf{\theta })/T)/{\mathsf{\Sigma }}_{y∊\mathrm{E}}\exp (({\mathsf{\Sigma }}_{i,j=1,n}c{\alpha }_{ij}{x}_i{y}_j-\mathsf{\theta })/T) \end{gathered}$$

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and the invariant measure µ_x equals:

μ_x = Σ_y∊E exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T)/Z, with Z = Σ_x,y∊E exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T)

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Then, the following Propositions holds:

Proposition 3.

In parallel updating mode, we have: ∂E_x/∂c = −cVar_xG, where the variance Var_x is taken for the conditional measure {M_xy}_y∊E.

Proof: 

It is the same proof as for Proposition 2 by replacing F with G. □

Let remark now that dynamic entropy E is related to the conditional entropy E_x as follows:

$$E = {{\displaystyle \sum }}_{x\in \mathrm{E}}{\mu }_x{E}_x$$

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Proposition 4.

In parallel updating mode, we have: ∂E/∂c = -cVar_vG + Cov_v(J,G), were Var_v and Cov_v are respectively the variance and the covariance taken for the random variables G(x,y) and J(x,y) = log₂M_xy and for the joint measure v =μ_xM_xy.

Proof: 

We have: ∂E/∂c = Σ_x∊E (μ_x ∂E_x/∂c + E_x ∂μ_x/∂c). Hence, from Proposition 3, we get:

∂E/∂c = −cVar_vG + Σ_x∊E E_x ∂μ_x/∂c,

With ∂μ_x/∂c = [Σ_y∊Ω[∂exp((Σ_i,j=1,n cα_ijx_iy_j-θ)/T)/∂c]/Z − [Σ_y∊E exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T](∂Z/∂c)/Z², where Z = Σ_x,y∊E exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T) and ∂Z/∂c = Σ_{x,y∊E; i,j=1,n} α_ijx_iy_j/T)exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T)

Then, we have:

∂μ_x/∂c = Σ_{y∊E; i,j=1,n} α_ijx_iy_j exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T)/ZT − μ_xΣ_{x,y∊E; i,j=1,n} α_ijx_iy_j exp((Σ_i,j=1,n cα_ijx_iy_j − θ)/T)/ZT
= μ_x (E_x(G) – E_v(G))

and Σ_x∊E E_x ∂μ_x/∂c = Σ_x∊E μ_x E_x E_x(G) – E E_µ(G) = E_v(JG) – E_v(J)E_v(G).

Finally, we get: ∂E/∂c = −cVar_vG + Cov_v(J,G) □

We have shown in the Propositions above a direct link between the sensitivity to c of entropies E_µ, E_x and E, and the variability of frustrations of the network; e.g., if the network is sequentially updated, E_µ decreases when the variance of the global frustration F increases, because of Equation (15): ∂E_µ/∂c = −cVar_µF [39].

Remarks:

(1)

The Proposition 2 still holds when we replace c, the absolute value of weights w_ij, by the temperature T, and we have: ∂E_µ/∂T = Var_µ(F)/T³.

(2)

The Proposition 2 shows that there is a close relationship between the robustness (or structural stability) with respect to variations of the parameter c and the global dynamic frustration F. This global dynamic frustration F is in general easy to calculate. For the configuration x of the genetic network controlling the flowering of Arabidopsis thaliana [37], there is only one frustrated pair; hence, F(x) = 2 (Figure 5 Right). More generally, all the thermodynamic functions introduced here are calculable in theory and could serve to quantifying the trajectory (Lyapunov) stability and the structural stability. If the number of states of a discrete state space E is too large, or if integrals on a continuous state space E are too difficult to calculate, it is possible to use Monte Carlo procedures for getting the values of the variance of the global frustration and of the dynamic entropy.

3.4. Entropy Centrality of Nodes

There are four classical types of centrality in an interaction graph G (Figure 2 Top). The betweenness centrality [85] is the first type and is defined for a node k as follows:

$$C_k^b={\displaystyle \sum }_{i\ne j\ne k\in G}\frac{{\beta }_{ij}\left(k\right)}{{\beta }_k},$$

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where $\beta$_ij(k) is the number of shortest paths going from j to i through k, and $\beta$_k = Σ_{I ≠ j∊G} $\beta$_ij(k).

The degree centrality is the second type of classical centrality. It is defined from the notions of in-, out- or total-degree of a node i, which correspond respectively to the number of arrows of the interaction graph G having i respectively as end, start, or as both. For example, the in-degree centrality is defined by:

$$C_i^{in-deg}=\frac{{{\displaystyle \sum }}_{j=1,n}\left|a_{ij}\right|}{n-1}$$

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where a_ij denotes the general coefficient of the signed incidence matrix A of the graph G.

The closeness centrality is the third type of classical centrality. Closeness, as the inverse of the farness, is defined by considering the inverse of an average distance between i and j, over all nodes j of the interaction graph G:

$$C_i^{clo}=\frac{n-1}{{{\displaystyle \sum }}_{j=1,n}l\left(i,j\right)}$$

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where the distance chosen here is l(i,j), the length of the closest path between i and j.

The eigen-centrality or spectral centrality is the last classical centrality. It takes into account the fact that neighbors of a node i are possibly highly connected to the interaction graph, considering that connections to (possibly few) highly connected nodes contribute to the centrality of i as much as connections to (possibly numerous) weakly connected nodes. Hence, the eigenvector centrality C_i^eigen of the gene i measures more the global influence of i on the whole network and verifies [86]:

$$C_i^{eigen}=\lambda {{\displaystyle \sum }}_{j=1,n} C_j^{eigen}$$

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where λ is the greatest eigenvalue of the incidence matrix of the interaction graph G.

The four centralities above can be very different (Figure 6), but each has a specific interest: (i) betweenness centrality relates to the global connectivity with all nodes of the network, (ii) degree centralities correspond to the local connectivity with only nearest nodes, (iii) closeness centrality measures the proximity with other nodes using a distance on the interaction graph and (iv) eigen-centrality quantifies the ability of a node to be connected to nodes, possibly only a few, but with a high connectivity; hence, identifying important hub-relays controlling a wide sub-network.

As a complement of the classical centralities, we introduce now a new notion of centrality, called entropy centrality, taking into account the heterogeneity of the distribution of states of the neighbors of a node i, and not only its connectivity in the graph, and highlighting the symmetric sub-graphs having equi-distributed states in the neighborhoods of their nodes (Figure 6):

C_i^entropy = −Σ_{k = 1,s} v_k log₂v_k,

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where v_k denotes the kth frequency among s (s = 2 in Boolean case) in the histogram of the values of states observed in the neighborhood V_i of the node i, set of the nodes out- or in-linked to the node i.

4. Application to a Real Genetic Network

The genetic regulatory network controlling kinins and kallikrein described on Figure 7 corresponds to genes involved in a familial disease, angioedema, for which there are normal profiles of genetic expression and pathologic ones, particularly in the hereditary form of the disease. By studying the connectivity of the nodes of the interaction graph of the network, we can select four critical genes having a high connectivity power with high total degree (≥5) and entropy centrality: these genes are FAK1, SP1, F12 and SERPING1. Surprisingly, the last three of them are the most involved in the etiology of the angioedema, being susceptible of mutations and/or hormonal interactions, favoring the appearance of the disease. Four are represented circled in blue in the network of Figure 7 and three are in a blue line on Figure 8.

E_attractor can be evaluated by the quantity − Σ_{k = 1,m} ABRS(A_k)log₂[ABRS(A_k)], where ABRS(A_k) is equal to the size of the attraction basin of the k^th attractor A_k among the m attractors of the network dynamics, divided by 2ⁿ, where n is the number of genes of the studied network N (equal to 21 in the present case, by supposing that circular RNAs are absent). Then, in the present example, we have E_attractor ≈ 0.4520. Hence, we have: E ≈ log₂2ⁿ − E_attractor ≈ 21 − 0.4520 ≈ 20.55, which corresponds to a high value of the dynamic entropy, and consequently to a high structural stability of the genetic network N controlling the kinin- kallikrein system.

We can notice on Figure 8 Top that the physiologic attractor is a fixed point (PF3) whose attraction basin is made of 93.43% of the possible states of the network. The pathologic attractor corresponding to the hereditary form of the disease called angioedema is the fixed point PF10, at which there is the absence of the gene SERPING1 and the presence of the genes KLKB1 coding for kallikrein B1 and KNG coding for kininogen. The attraction basin of PF10 has a relative size equal about to 5‰ of the possible states of the network, and the observed prevalence of hereditary angioedema in the general population is reported to range from 1:10,000 to 1:150,000, without major sex or ethnic differences [87,88].

A last remark concerns the number of attractors (11 fixed points and no periodic attractor on Figure 8). A previous work [40] gives an algebraic formula allowing for the calculation of the number of attractors of a Boolean network; in their Jacobian interaction graph were two tangential circuits (

Table 1. Total number of attractors (here 8 in red) of two tangential circuits at the intersection of a line fixing the length L of a negative circuit (here 4) and a column fixing the length R of positive circuit (here 8) tangential to the negative one.

	1	2	3	4	5	6	7	8	9
1	1	2	2	3	3	5	5	8	10
2	1	1	2	3	3	4	5	8	10
3	1	2	1	3	3	6	5	8	8
4	1	1	2	1	3	4	5	11	10
5	1	2	2	3	1	5	5	8	10
6	1	1	1	3	3	1	5	8	8
7	1	2	2	3	3	5	1	8	10
8	1	1	2	1	3	4	5	1	10
9	1	2	1	3	3	6	5	8	1

): one positive of length right, (involved in the richness in attractors, as predicted by [32,33,34], and one negative of length four (responsible of the trajectory stability, as predicted by [36,89]). This predicted number (11) is the same as that calculated from the simulation of all trajectories from all possible initial conditions summarized in Figure 8, and more results both theoretical and applied to real Boolean networks can be found in more recent literature [46,47,48,49,90] showing a large spectrum of possible applications in genetic, metabolic or social Boolean networks.

5. Conclusions and Perspectives

The notion of dynamic entropy is common to both discrete and continuous dynamical systems and can be applied to numerous situations in biological modeling in order to interpret the dynamical behavior of complex systems having several elements in interaction, leading to many observed trajectories until their ultimate asymptotic behavior, via the attractors. Biological regulatory networks are currently widely used to design the mechanisms of these interactions at different scales, from genes and cells [68,91,92] to individuals [46,93]. Between these extreme molecular and population levels, identical functions can be concerned from different species: from paramecia, a plant-like Bidens pilosus to humans as memory function [49,94] or motion control [95,96,97]. Because the corresponding regulatory systems are often random, future work will be done in the framework of general random systems [98]—in which the notions of entropy, stochastic attractor (confiner) and invariant measure can help to interpret and classify the observed data in many regulatory networks (neural, genetic, metabolic, social, etc.)—and in biomedical fields for practical applications (see for example [61,99]).