# Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

“In nova fert animus mutatas dicere formas corpora... Unus erat toto naturae vultus in orbe, Quem dixere chaos: rudis indigestaque moles, Nec quidquam, nisi pondus iners, congestaque eodem non bene junctarum discordia semina rerum…”. I want to speak about bodies changed into new forms... Nature appeared the same throughout the whole world, What we call chaos: a raw confused mass, Nothing but inert matter, badly combined discordant atoms of things, confused in the one place...(Ovide, Metamorphoses, 1^{st}Book, 10 A.D.).

## 1. Introduction

_{1}and C

_{2}, from C, on Figure 3 top). Interaction graphs architecture contains simple motifs (as those from Alon’s work in genetic networks [24] and from metabolic and physiologic networks [25,26]) with two to five nodes. Some examples of motifs are given in Figure 3 and systematically studied in [27]. Among these motifs, we will use, in the following, the negative (respectively positive) circuits, i.e., closed paths between nodes having an odd (respectively even) number of inhibitions like in the triple negative and quintuple positive circuits, in the 3-switch (three nodes fully connected with only inhibitions between them) and in the negative and positive regulons, the most simple motifs having both one positive and one negative circuit. When there is no circuit inside the motifs, we call coherent (resp. incoherent) feed-forward double path a couple of paths with same initial and final nodes, and same (resp. different) global signs.

## 2. Preliminary: Notations and Definitions

#### 2.1. Definition of an Attractor and of Its Basin

- A is a fixed set for the composed set operator LoB: A = L(B(A)),
- there is no set C ⊃ Ā, C ≠ Ā, verifying i),
- there is no D ⊂ A, D ≠ A, verifying (i) and (ii).

#### 2.2. Degree, Connectivity and Connectedness

#### 2.2.1. Undirected Graph

#### 2.2.2. Regular Graph

_{0}n, for some small positive constant c

_{0}, then with probability tending to 1 as n tends to infinity, G is r-edge-connected.

#### 2.2.3. Weighted and Signed Graph

#### 2.2.4. Directed Graph

#### 2.2.5. Indegree and Outdegree

^{−}(v) (resp. deg

^{+}(v)). A vertex v with deg

^{−}(v) = 0 is called a source, as it is the origin of each of its incident edges. Similarly, a vertex v with deg

^{+}(v) = 0 is called a sink. The degree sum formula states: ∑

_{v∈V}deg

^{+}(v) = ∑

_{v∈V}deg

^{−}(v) = |E|. Hence the average indegree is equal to the average outdegree.

#### 2.2.6. Connectedness and Connectivity in Graphs

^{2}and, in an undirected graph without loops, to C = 2|E|/[|V| (|V| - 1)]. Gardner and Ashby [4] have studied the relation between the probability of stability and the connectedness of large linear dynamical systems observed in biology [57,58]. We can notice that the connectivity in the Kauffman’s sense [41] is the average indegree, which is also equal in directed graphs to c = C|V|; c is set to a constant, however, it is possible to let it be random, chosen under various distributions [50,51], with average connectivity c.

#### 2.3. Kauffman Boolean Networks

_{i}(t) of the gene i equals 0 (resp. 1) if the gene is inactive (resp. active) at time t. We denote the whole configuration of the network at time t by: x(t) = (x

_{i}(t))

_{i∈R}; x(t)∈ Ω = {0,1}

^{n}, where Ω is the set of all possible Boolean configurations on R. A Kauffman Boolean network consists of n interacting elements whose states x

_{i}(t) (i from 1 to n) are binary variables [41–51]. Each gene i is connected to k other genes (i

_{1},...,i

_{k}) and its state is updated according to a specific rule x

_{i}(t+1) = F

_{i}[x

_{i1}(t),..., x

_{ik}(t)], where F

_{i}is a Boolean function, and the x

_{ij}(t) are the states of the units connected to i, which may or may not include i itself. The Boolean function F

_{i}is represented by a truth table that lists its outputs for each set of inputs values. For each F

_{i}with k variables, there are K = 2

^{k}possible sets of inputs values, yielding 2

^{K}different possible functions.

#### 2.4. Threshold Boolean Automata Networks

_{ij}which gives the intensity of the influence that gene j has on gene j and, consequently, of the role played by gene j through the protein it expresses, which can repress or induce the expression of gene i. Being given an arbitrary regulatory network, we associate to it an interaction matrix W of order n-n, whose general coefficient w

_{ij}corresponds to the interaction weight with which gene j acts on gene i. More precisely, the coefficient w

_{ij}can be positive or negative depending on the fact that gene j tends to respectively activate or inhibit gene i, and is null if j has no influence on i. Let V

_{i}be the neighbourhood of gene i, i.e., the set of genes j having an influence on i (j∈V

_{i}⇔ w

_{ij}≠0) and H the interaction potential:

^{n}and θ

_{i}is the i

^{th}activation threshold, i.e., the interaction potential to be overtaken for activating i. The threshold Boolean automata rule [52–54] is:

#### 2.5. Attractors in Kauffman Boolean Networks and Threshold Boolean Automata Networks

^{K}, where K = 2

^{c}possible random Boolean functions determined by the c (on average) neighbours of any node. For c = 1/2 and for the Boolean functions identity and negation chosen with the same probability, there are N = O(√(n)) states belonging to the attractors [41–45] and the number of possible limit cycles of length L has been recently proven to be in general exponential according to L [44–49]. In a random threshold Boolean automata network, with c =2 (i.e., with on average 2 neighbours acting on the same gene with weights randomly chosen at values 1 or −1), we conjecture that the number N of attractors in the sequential updating mode (each vertex being sequentially updated following a deterministic or random ergodic order) verifies [53,54]:

## 3. Notions of Boundary, Core, Critical Node and Critical Edge of a Regulatory Interaction Network

#### 3.1. Boundary and Core

#### 3.2. Critical Node and Critical Edge

^{2}≤ N ≤ 2

^{4}.

## 4. Theoretical Complements

#### 4.1. Potential Regulatory Networks

^{n}is defined by:

^{n}. In the same way, a potential regulatory network on the discrete state space E is defined by [64,65]:

^{n}), h is the identity, P a polynomial with integer coefficients and ∀i = 1,...,n, Δx

_{i}∈ {−1,0,1}. Then (5) is the discrete equivalent of (3):

**Example:**in the Boolean case, if P(x) = ∑

_{k}(

^{t}txA

_{k}x)x

_{k}+

^{t}xWx+Θx, where A = (a

_{ijk}) is an interaction tensor, with A

_{k}= (a

_{ij})

_{k}as marginal matrices and a

_{iii}= 0, W = (w

_{ij}) an interaction matrix and Θ = (θ

_{i}) a threshold vector, we have for the partial derivatives of P:

_{i}= −[∑

_{j,k}(a

_{ijk}+ a

_{jik}+a

_{jki})x

_{j}(t)x

_{k}(t)+∑

_{j}(w

_{ij}+ w

_{ji})x

_{j}(t) + θ

_{i}]/[1+w

_{ii}+ ∑

_{j ≠i}(a

_{ijj}+a

_{jij}+a

_{jji})x

_{j}(t)] and x

_{i}(t + 1) = G(Δx

_{i}+ x

_{i}(t)) = G(-ΔP/Δx

_{i}+x

_{i}(t)), where G is the Heaviside function. From (6) we derive:

^{t}xWx + Bx, with w

_{ii}= 1, and each sub-matrix on any subset J of indices in {1,...,n} of W is non positive. Then P decreases on the trajectories of the potential automaton defined by x

_{i}(t + 1) = G(-ΔP/Δx

_{i}+ x

_{i}(t)) for any mode of implementation of the dynamics (sequential, block sequential and parallel). This network is a threshold Boolean automata neural network whose stable fixed configurations (Figure 8) correspond to the minima of P [64].

#### 4.2. Hamiltonian Networks

_{i}(t) can be interpreted as the rate at which the system changes its state x

_{i}(t) at node i and time t and T

_{i}(t) as the mean local state over the time interval [t,t + 1]. Then let us define the Hamiltonian energy H equal to the kinetic energy: E

_{c}(t) = ∑

_{i=0,...,n-1}Y

_{i}(t)

^{2}/2, to which we substract the potential energy defined by: ΔE

_{p}(t) = ∑

_{i=0,...,n-1}Z

_{i}(t)ΔT

_{i}, where Z

_{i}(t) = ΔY

_{i}/Δt =Y

_{i}(t + 1)-Y

_{i}(t) = x

_{i}(t + 2)-2x

_{i}(t + 1) + x

_{i}(t) plays the role of an acceleration. We have:

_{c}(t+1)-E

_{c}(t)+E

_{p}(t)-E

_{p}(t+1) = ∑

_{i=0,...,n-}

_{1}[Y

_{i}(t+1)

^{2}-Y

_{i}(t)

^{2})]/2-∑

_{i=0,...,n-1}Z

_{i}(t)ΔT

_{i}= ∑

_{i=0,...,n-1}[(Y

_{i}(t+1)+Y

_{i}(t))(Y

_{i}(t+1)-Y

_{i}(t))/2-Z

_{i}(t)ΔT

_{i}] = ∑

_{i=0,...,n-1}[(T

_{i}(t+1)-T

_{i}(t))ΔY

_{i-}Z

_{i}(t)ΔT

_{i}] = 0, and we can write:

_{c}(t + n) = E

_{c}(t) (resp. E

_{c}(t + 2n) = E

_{c}(t)), and hence, the quantity:

#### 4.3. Relationships between Kauffman Boolean and Threshold Boolean Automata Networks

^{n}, can be considered as a threshold Boolean automata network with states y(t)∈{−1,1}

^{n}by considering variables y

_{i}(t) = 2x

_{i}(t)- 1, where x

_{i}(t)∈{0,1}. It suffices to define thresholds as 0 and weights as follows: for identity function, w

_{ii+1}> w

_{ii}> 0 and for the negation function, -w

_{ii+1}> w

_{ii}> 0.

^{n}to {0,1}, can be expressed in terms of Kauffman Boolean networks.

#### 4.4. Relationships between Undirected and Directed Graphs

**Lemma 1:**for any undirected graph G having m non oriented edges, the mean number of oriented edges we can define on G from the non oriented configuration is equal to 4m/3.

**Proof:**Let us note <O> the mean number of oriented edges we can construct from a configuration of m non oriented edges; then, if exactly k from the m non oriented edges are decomposed into two oriented opposite connections, we have C

_{m}

^{m-k}2

^{m-k}different ways to dispatch the not double connections into the (m-k) other non oriented edges; we can write:

#### 4.5. Circuits

_{r}, the number of non-oriented circuits of length r in a random undirected d-regular graph of order n. In [68–70], it is shown that the random variables X

_{r}, for 3 ≤ r ≤ g, are asymptotically distributed as independent Poisson variables with means equal to (d-1)

^{r}/2r=(3c–2)

^{r}/r2

^{r+1}(2

^{r}

^{−1}/r, if c=2), with d = d(n) and g = g(n) allowed to increase with n, provided that: (d-1)

^{2g–1}= o(n), hence we have:

^{nd/2}

_{n(n-1)/2}random undirected d-regular graphs of order n, each bringing on average 2

^{r}

^{−1}/r non-oriented circuits of length r, if the average in-degree c = 2, and each undirected edge of a non-oriented circuit of length r gives birth to 1 (counted twice in case of a loop) or 2 signed oriented out-coming edges, hence giving at most two oriented circuits. The circuit direction is supposed to be imposed by the circuit nodes controlled by a boundary node: in the human genome, about 2,600 sources plus about 700 micro-RNAs (706 presently known after http://www.microrna.org and 800 expected [71]) are controlling 19,000 not boundary nor isolated genes [72], corresponding to one control among 4 not boundary nor isolated genes (if each micro-RNA is supposed to have on average three specific target genes), hence there is, on average, about 1 control per circuit of length 5, 4 or 3 (it is the case for the circuits in Figures 17 and 18).

_{+}= 4/7 (probability to have a sign +) and p

_{-}= 3/7, the oriented graph is given in (b) if the control by a boundary gene is positive, and in (c) and (d) if the control by a micro-RNA or a boundary gene is negative (depending on the sign of the first post-control forward interaction).

#### 4.6. Attractors Counting in Real Regulatory Networks

^{3}, then the mean number of oriented circuits of length r (for 3 ≤r ≤5) is equal to 2

^{r−1}/r≈ 16/5 ≈ 3, if r = 5, each bringing on average at most 6 different attractors in parallel updating mode [67]: 8 for positive and 4 for negative circuits (Figure 10) and 1.5 in sequential mode (2 fixed configurations for positive and 1 limit cycle for negative circuits). For r = 4 (resp. 3), the mean number of oriented circuits is equal to 2 (resp. 1) and the mean number of attractors is at most (6 + 2)/2 = 4 (resp. (4 + 2)/2 = 3). For r = 3, the attractors are given in Figure 9. For r = 2, the mean number of oriented circuits is equal to n(4/n) = 4, that is the number of the nodes of the network times the probability to have a reverse edge from the successor nodes of each node (whose mean number is equal to 2 in the case of connectivity 2). The number of attractors is 3 in the parallel mode of updating in the case of a positive circuit and 1 in the case of a negative circuit (Figure 10).

^{2}independent choices - n for each of the n nodes - the mean value of the total number of interactions is equal to n

^{2}(2/n) = 2n, that is the expected number of interactions; hence, the probability of having no input at is given by p = 1-2/n, and the probability of having no input for a node after n independent choices is p

^{n}= (1-2/n)

^{n}. Then the mean number of sources is:

^{n}= e

^{nLog(1-2/n)}≈ e

^{−2}≈ 1/7.4, then S is approximatively equal to n/7.4, if n is sufficiently large. By using the same argument, the mean number of sinks L is equal to S, and the number I of isolated nodes is equal to about n/(7.4)

^{2}[81]. To conclude, the sources, if their 2

^{S}states are possible, bring 2

^{S}. A

_{n,2}attractors, if A

_{n,2}is the number of attractors brought by the n-(2S-I) nodes having at least one in and one out interaction. These nodes belong to a random network having a connectivity (in the Kauffman’s sense) equal to c′ = 2n/[n-(2S-I)] ≈ 2.8n. Then, for a fixed initial configuration of the sources, the calculation of attractors due to circuits inside these n-(2S-I) nodes can be made by using the previous Sections. The mean number of tree structures can also be calculated following [61], but these structures do not bring new attractors, except those brought by the sources and circuits connected to these trees.

^{2}≈ 2,973-402 = 2,571 sources or boundary genes, an amount of about 700 human micro-RNAs, which corresponds to the real size of the repositories ( http://microrna.sanger.ac.uk; http://www.microrna.org) and to their expected size [79]. They control about 19,000 not boundary nor isolated genes [80], which corresponds to one control for every four not boundary nor isolated genes (each micro-RNA being supposed to have about three specific targets and hence to inhibit about 245 boundary genes), hence there is, on average, about 1 control per circuits of length 5, 4 or 3 (it is the case for the circuits in the Figures 17 and 18). A micro-RNA can hybridize a mRNA stopping the ribosomal elongation, then constituting a post-transcriptional control we can assimilate as coming from a new inhibitory source, i.e., a boundary node acting negatively (rarely positively [81]) on the gene transcribed in this mRNA. The n-(G + L-I) nodes having at least one in and one out interaction are about 22,000-5,544 = 16,456 and they form about three circuits of length 5, 2 of length 4 and one of length 3. They bring 2

^{3}attractors in the sequential mode and 8

^{1.5}.4

^{1.5}6

^{1}.2

^{1}.4

^{0.5}.2

^{0.5}= 6.10

^{3}attractors in the parallel mode. These numbers have been obtained by multiplying the number of the attractors brought by each positive or negative circuits, considered as disjoint and in equal average number.

^{0.5}.4

^{0.5}.6

^{0.5}.2

^{0.5}.4

^{0.25}.2

^{0.25}.3

^{2/3}= 3.6 × 2

^{4.25}≈ 69; this number 69 is in fact a majorant due to the Jensen’s inequality applied to the concave function f(x)=x

^{p}, where p < 1, the numbers of attractors, that is 8, 4, 6, 2 and 3, coming from the Figure 10 and the only non negligible circuits being of length less than 5, because of the Bollobas results [68]. The mean number of attractors allowed for the controlled circuits is in the same way equal at most to 8

^{0.5}.4

^{0.25}.6

^{0.25}≈ 4 × 1.56 ≈ 6.24, because the negative control acts efficiently only on genes in state 1, these genes concerning only the half of the positive controlled circuits. Then we can observe on average at most 69 × 6.24 ≈ 430 attractors, which is in qualitative agreement with the number of observed cell attractors in human (the number of the different types of cell differentiation in human is between 178 [82] and 411 [83]), if we suppose that the Boolean rules used in circuits of regulatory genetic networks are only made of identity and negation functions and that the genes are expressed synchronously (parallel updating). The pure sequential mode (8 attractors) is unable to explain the richness observed, hence we must propose a certain degree of synchrony in the gene expresssion, as well as new rules more sophisticated than the threshold Boolean automata one (e.g., rules in which the interaction potential is non-linear in the state variables [61]).

## 5. Robustness

- - if the nodes are sequentially visited by the updating process, the system has 6 fixed configurations, with state 1 (resp. 0) at one node and state 0 (resp. 1) at the others. Such a system having only fixed configurations is potential in the sense of the Section 4.1. [64], because the discrete velocity of the dynamics is equal to the gradient of a Lyapunov function (it is for example more generally the case in a n-switch when the interaction weights are symmetrical),
- - if the nodes are synchronously updated, we have one limit-cycle of order 2 (made of the full 0 and full 1 configurations) and 6 fixed configurations (corresponding to those of the sequential updating). Such a discrete system is Hamiltonian in the sense of Section 4.2.,
- - in the intermediary case, called block-sequential, in which we update first a node, and then synchronously the two others, we have the same attractors as in the sequential case.

**Lemma 2**: When a limit cycle occurs in the dynamics of a block, then the whole network dynamics has only limit cycles as attractors. Reciprocally, if the global network dynamics has a limit cycle, there is necessarily at least one block having at least one limit cycle as attractor.

**Lemma 3**: If the global network dynamics has a limit cycle of length m, then m ≤ Π

_{i∈C}m

_{i}, where C is the set of the blocks having at least a limit cycle as attractor, m

_{i}being the maximal length of limit cycles of the block i.

- - those for which the cycles disappear when we are going down in the hierarchy from the synchronous to the sequential modes (behaviour “Down”),
- - those for which the cycles disappear when we are going up in the hierarchy from the sequential modes to the synchronous one (behaviour “Up”),
- - those not corresponding to any previous behaviour, for which the cycles occur and disappear inside the hierarchy without clear rule (behaviour “None”).

## 6. Examples of Robust and Non-Robust Networks

#### 6.1. Neuron and Plant Morphogenesis

_{i}denotes the concentration of the morphogen i, we suppose that all the inhibitions of the n-switch are expressed through a Hill competitive term in the following differential equations [83], where the cooperativity c is supposed to be strictly greater than 1, k is a catabolic constant and σan enzymatic Vmax (Figure 13 left):

_{i}=(X

_{i})

^{1/2}, we can check that if we define the potential P by:

_{i}= Π

_{k=1,n;k≠i}(1 + c

_{k}X

_{k})

^{n}/(1 + e

_{k}X

_{k})

^{n}and e

_{k}<< c

_{k}< 1, which corresponds to an allosteric inhibition of X

_{i}by the X

_{k}’s, for k ≠ i. If e

_{k}= 0, by changing the variables Y

_{i}= (X

_{i})

^{1/2}and considering the potential P defined by: $P({Y}_{1},\mathrm{\dots}{Y}_{n})={\sum}_{j=1,\mathrm{\dots},n}\hspace{0.17em}{k}_{j}{Y}_{j}^{2}/4-(\sigma /4n)\mathit{\text{Log}}({\sum}_{j=1,\dots ,n}\hspace{0.17em}{(1+{Y}_{j}^{2})}^{n}+{\prod}_{j=1,\dots ,n}\hspace{0.17em}{(1+{c}_{j}{Y}_{j}^{2})}^{n})$, then the new differential equations can be written like (13) as:

#### 6.2. Cardio-Respiratory Physiologic Regulation

#### 6.3. Glycolytic/Oxidative Coupling

#### 6.3.1. The Glycolysis

_{1}, oscillations of all the metabolites of the glycolysis, with a period of several minutes. Let us apply it to neurons and astrocytes: the mitochondrial shuttles for NADH are less active in astrocytes than in neurons: that causes a flow of lactate from astrocytes to neurons [101,102] and this difference in NADH transportation efficiency is provoked by a weaker efficacy of the translocase (ANT) and ATPase enzymes less optimally located inside the inner mitochondrial membrane [103].

_{1}, x

_{2}, x

_{3}and x

_{4}the concentrations of respectively the successive main metabolites of the glycolysis: glucose, glyceraldehyde-3-P, 1,3-biphospho-glycerate and phospho-enol-pyruvate. We assume that steps E2 and E3 of the glycolysis (summarised in Figure 15 a) are Michaelian and reversible, the enzymatic complex E1 includes the allosteric irreversible kinetics of the phospho-fructo-kinase PFK with a cooperativity n (see [104–108] for its complex kinetics), and both pyruvatekinase (E4) and dehydrogenases of the complex E5 are irreversible. Then, consider the differential system (S) ruling the glycolysis and the pentose pathway until the ribulose-5-P:

_{i}’s are ruled by the system (S′):

_{i}[109] defined by:

_{i}’s are large, the gradient of P dominates and the system has a principal potential part [110], this part giving then the direction of the flow. Conversely, when the gradient of P vanishes, the part made of the partial derivatives of the H

_{i}’s dominates.

#### 6.3.2. Control Strength

_{i}}

_{i=1,4}, between two parts, one gradient part dissipating the potential P and the other part made of the partial derivatives of the H

_{i}’s ; that allows, when the attractor is a limit cycle - which is the case if we add fructose-2,6-diphosphate (F26P

_{2}) or ADP (see Figures 11 a, 13 left and 14 right) as activator of the complex E

_{1}[99,108] - to share the parameters of the system (S’) into 2 sets: the parameters appearing exclusively in P modulating the mean value and the amplitude, the parameters appearing exclusively in the H

_{i}’s modulating more the frequency and those appearing both in P and in the H

_{i}’s.

_{1}is reached the first among the other x*

_{k}’s, then (S’) becomes (S″):

_{4}, are modulating the localisation of the fixed point, hence the values of the stationary concentrations of the glycolytic metabolites (cf. [109] for a more general approach of the potential-Hamiltonian decomposition).

_{1}and J

_{2}. Then from the system (S) we can calculate the sharing parameter α (which regulates the pentose pathway and the low glycolysis dispatching) from the steady-state equations equalizing the in and outflows at each step. By determining the stationary state x̄* = {x

_{i}}

_{i=1,4}, we have:

_{ik}exerted by the metabolite x

_{i}on the flux Φ

_{k}of the k

^{th}step by: C

_{ik}= ∂LogΔΦ

_{k}/∂LogΔx

_{i}[105,111,112] and we have:

_{ik}(resp. A

_{ik}) to quantify the control of the period τ

_{k}(resp. amplitude Amp

_{k}) of the k

^{th}step flux by x

_{i}[112–114]: T

_{ik}=∂Logτ

_{k}/∂Logx

_{i}and A

_{ik}= ∂LogAmp

_{k}/∂Logx

_{i}. If ξ is the eigenvalue of the Jacobian matrix of the differential system for which the stationary state has bifurcated in a limit cycle (Hopf bifurcation), then τ

_{k}= 2π/Imξ, and we have in the 2-dimensional case, if for example dx

_{1}/d t= −∂P/∂x

_{1}+∂H/∂x

_{2}, dx

_{2}/dt=-∂P/∂x

_{2}-∂H/∂x

_{1}:

^{2}P/∂x

_{1}

^{2}+ ∂

^{2}P/∂x

_{2}

^{2}is the Laplacian of P and C(P) = ∂

^{2}P/∂

_{1}

^{2}∂

^{2}P/∂x

_{2}

^{2}- (∂

^{2}P/∂x

_{1}∂x

_{2})

^{2}is the mean Gaussian curvature of the surface P, both taken at the stationary state of the differential system.

_{max}of the PFK is diminished in astrocytes due to a lack of ATP, the oscillatory behaviour is less frequent (Figure 16 left) and the production rate of lactate from pyruvate is more important than in neuron, creating a flux of lactate to neurons (Figure 17 top). The neurons consume the lactate coming from the extracellular space, partially replenished by the astrocytes production. That gives to neurons an ATP level higher than in astrocytes, an extra-pyruvate production from lactate, and an extra-oxygen and glucose consumption theoretically predicted and experimentally observed [101,102]. ANT and ATPase concentrations are in human under the negative control of two micro-RNAs, micro-RNA 151 and micro-RNA 34 acting as boundary controller nodes ( http://microrna.sanger.ac.uk) by diminishing the glycolytic/oxidative system efficiency.

#### 6.4. Cell Cycle Control

#### 6.5. Feather Morphogenesis

## 7. Perspectives and Conclusions

^{Log(r(k))Log(Log(r(k)))}, where C is a positive constant and the directed graph diameter is equal to 2r(k) at step k. Then, if the indegree of the interaction graph comes from the initial value 2, it loses this indegree not too rapidly and goes to a small-world structure [137], like those observed in the metabolic regulatory networks [1].

^{r(r+1)/2}/(r + 1)! [68,138–142]. All these trees controlled by their root bring each only one attractor. Following [143], if we consider for example the network of the copper biolixiviation by Thiobacillus ferrooxidans [144,145], we have 354 genes and 534 interactions: the prediction for the numbers of isolated genes and 2-trees are 354/(7.4)

^{2}= 6.5 and 3

^{3}/3! = 4.5, and the observations show 8 isolated genes and 6 2-trees, in qualitative agreement with the hypothesis of randomness. This result emphasizes the role of circuits in the differentiation, because positive ones are alone to bring multiple fixed points and be present since the origin of life [131]. A last open problem consists in making explicit the dynamical consequences of the transcription regulation of micro-RNAs, by boundary genes like p53 [117]: this regulation enlarges the classical boundary reduced to sources, by introducing 2-circuits in it.

## Acknowledgments

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**Figure 1.**Systems biology circuit with (a) phenotype observation (spina bifida), (b) pedigree inquiry (proving the familiar origin, with affected in black and healthy carriers in bicolor), (c), (d) genomic and proteomic data (karyotypes, databases and DNA chips), (e) cell imaging and concatenation of these elements into a (f) regulatory interaction network and (g) mathematical model allowing the simulation in silico.

**Figure 2.**Some contributors to the kinetic equations ruling the regulatory interaction networks dynamics (top from left to right): M. Menten, A. Hill, M. Delbrück, R. Thomas, J. Thiéry; some contributors to the theory of dynamical systems (bottom from left to right): H. Poincaré, A.M. Lyapunov, G.D. Birkhoff, J. von Neumann, R. Thom.

**Figure 3.**Example of some motifs observed in regulatory interaction networks (inhibitions are in red and activations in blue): non-directed correlation (1) and directed interaction (2) network, coherent feed-forward double path (3), triple negative and quintuple positive circuits (4), 3-switch (5), negative (6) and positive regulons (7).

**Figure 4.**Definition of an attractor and its basin in the case of a Boolean regulatory network; the initial condition is indicated in blue disks and the final attracting conditions for the majority rule after synchronous iterations (the state of a node is 1 if activating neighbours number is equal to or more than those of inhibiting ones) are given outside the disks (left); attractor A and its basin B(A) (right).

**Figure 5.**Different categories of attractors: trajectories landscapes for (a) a node, (b) a saddle, (c) a centre, (d) a focus and (e) a limit cycle, in the Poincaré’s terminology.

**Figure 6.**From realistic biological regulatory networks (a & b) to regular networks (c & d) with a reduction of the strongly connected components (scc) from 3 to 1; the boundary (respectively core) of the networks is indicated in dark (respectively light) brown nodes.

**Figure 7.**(a) Regulatory network controlling the flowering of Arabidopsis thaliana [52]. Sources are coloured in orange, a micro-RNA acting on one of them. Green (resp. blue) arrows indicate the location of the positive (resp. negative) circuits. (b) Phenotypic expressions with some of the different possible attractors, with the symbols Sepals (S), Petals (P), Stamens (St) and Carpels (C): SPStC, SStC, SSt, SPC. (c) Simplified network with conservation of attractors. (d) Attractors, with genes in the following order: EMF1, TFL1, LFY, AP1, CAL, LUG, UFO, BFU, AG, AP3, PI, SUP.

**Figure 8.**Potential networks with Δx = -gradP (left) and with a Lyapunov function L decreasing on its trajectories (right). The potential network has a fixed point x

_{s}at the projection of the P sink and presents an identity between the discrete velocity Δx/Δt and the opposite of the gradient of P (left); on the contrary, there is only identity between the projection of the P sink and the fixed point localization in the case of a Lyapunov function L decreasing along the trajectories (right).

**Figure 9.**(a) undirected 3-circuit; (b) directed negative 3-circuit with positive control by a boundary gene; (c) directed positive 3-circuit with negative control by a micro-RNA or a boundary gene followed by a positive interaction; (d) directed positive 3-circuit with negative control by a micro-RNA or a boundary gene followed by a negative interaction; (e) attractors observed for the 3-circuit (c).

**Figure 10.**Top: Number of limit cycles of period p in parallel mode for a positive circuit of length n. Bottom: Number of limit cycles of period p for a negative circuit of length n. The last line of these tables gives the total number of attractor for circuits of size n.

**Figure 12.**(a) Neuron/Astrocyte regulatory embryonic 2-switch (left). Co-evolution of the neuronal (red) and glial (green) tissue (middle). Neuron/Astrocyte adult network (right) (b) n-switch interaction graph (left). Potential P associated to a 2-switch (ν = 1, c = σ = 2, a

_{i}=.1), with 2 stable minima on which vanishes either X

_{1}or X

_{2}(right). (c) Hard representation of the surface P (after [65]) (left). 2-switch plant growth (right).

**Figure 14.**Left: Bulbar cardio-respiratory centre, with inspiratory I and expiratory E neurons, cardio-moderator CM, peripheral sinusal pace-maker S. Right: (a) Periodic dynamics of I and CM (3-harmonic signal) when they are uncoupled or weakly coupled without noise, (b) respiratory sinusal arrhythmia with noise, (c) pathological entrainment when they are strongly coupled without noise and (d) with noise.

**Figure 15.**(a) The glycolysis and the pentose pathway. E1 denotes the four enzymes of the high glycolysis (hexokinase HK, phosphoglucose-isomerase PGI, phosphofructo-kinase PFK and aldolase ALDO), E2 denotes the glyceraldehyde-3P-dehydrogenase, E3 denotes the four enzymes of low glycolysis (phosphoglycerate-kinase, phosphoglycerate-mutase, enolase ENO and pyruvate-kinase PK), E4 denotes the pyruvate-kinase and E5 the 3 enzymes of the oxydative part of pentose pathway, glucose-6P-dehydrogenase G6PDH, 6P-glucono-lactonase and phosphogluconate-dehydrogenase PGDH, alternative to the phospho-transferase system PTS). (b) Control strengths on the glucose flux showing the influence of the main enzymatic steps.

**Figure 16.**Left. Parametric instability region reduced by taking into account the fraction of metabolites fixed to PFK when its V

_{max}is large, entry flux of fructose σ

_{1}is small and ratio ρ = K

_{R,F6P}/K

_{R,ADP}between association constants of Fructose-6-Phosphate and ADP to the active allosteric form R of the PFK is large. Right. Experimental evidence of oscillations in the large parametric instability region (V

_{max}large) [99].

**Figure 17.**Left. Connection between neurons and astrocytes glycolysis with lactate flux, controlled by ANT (green) and ATPase (red) inside the mitochondrial inner membrane (top). Right. PFK regulatory interaction network with the effectors (activators and inhibitors) of the PFK inside the enzymatic complex E

_{1}.

**Figure 18.**Cell cycle network (top left). Micro-RNA traductional regulation (top right). Some attractors (bottom left) in an arbitrary state space and list of all attractors of the cell cycle network, with fixed boundary conditions (micro-RNAs in state 1) and parallel updating mode (bottom right). ABRS denotes the percentage of initial states in an attractor basin. The involved genes are: p27, Cdk2, pCyCE_Cdk2, CyCE_Cdk2, micro-RNA 159, pCycA_Cdk2, CycA_Cdk2, Rbp-E2F, Rb-E2F, E2F, Rbp and Rb. Arbitrary representation of a part of the attraction basins with only 2 fixed points, A

_{1}and A

_{3}, and 2 limit cycles, A

_{2}and A

_{4}(bottom left).

**Figure 19.**Feather morphogenesis network (top right) connected to a double negative regulon with BMP-2 as inhibitor, BMP-7 as activator, and Follistatin as intermediary controller identified by specific dyes on the back of chicken ambryo (top left). Simulated attractors, where AD denotes the attractor basin diameter in Hamming distance with nodes in the order: miRNA 141, EphA3, p53, Vav3, Stk11, Wnt2, RhoA, Smad3, SrC, Id3, Cyclin D1, Zfhx3, Sox11, β-catenin, cMyc, and β-catenin/LEF/TCF/BL9/CBP (middle). Morphogenetic targets of the regulatory network (bottom).

**Table 1.**Repartition of the 34,947 networks of size three having limit cycles into three classes: Down (resp. Up) for which cycles disappear when the updating mode loses its synchrony (resp. sequentiality) and None for which cycles occur and disappear without clear rule.

Down | None | Up | Total |
---|---|---|---|

21,729 | 13,110 | 108 | 34,947 |

62.18% | 37.51% | 0.31% | 100% |

**Table 2.**Repartition of the 34947 networks of size 3 having limit cycles of length 2 to 6 into the 3 classes Down, None and Up.

Down | None | Up | Total | |
---|---|---|---|---|

2 | 86.19% | 70.53% | 37.04% | 80.16% |

3 | 8.28% | 20.93% | 62.96% | 13.20% |

4 | 4.59% | 6.71% | 0.00% | 5.37% |

5 | 0.70% | 1.83% | 0.00% | 1.12% |

6 | 0.24% | 0.00% | 0.00% | 0.15% |

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Demongeot, J.; Ben Amor, H.; Elena, A.; Gillois, P.; Noual, M.; Sené, S.
Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic. *Int. J. Mol. Sci.* **2009**, *10*, 4437-4473.
https://doi.org/10.3390/ijms10104437

**AMA Style**

Demongeot J, Ben Amor H, Elena A, Gillois P, Noual M, Sené S.
Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic. *International Journal of Molecular Sciences*. 2009; 10(10):4437-4473.
https://doi.org/10.3390/ijms10104437

**Chicago/Turabian Style**

Demongeot, Jacques, Hedi Ben Amor, Adrien Elena, Pierre Gillois, Mathilde Noual, and Sylvain Sené.
2009. "Robustness in Regulatory Interaction Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic and Genetic" *International Journal of Molecular Sciences* 10, no. 10: 4437-4473.
https://doi.org/10.3390/ijms10104437