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Regulatory interaction networks are often studied on their dynamical side (existence of attractors, study of their stability). We focus here also on their robustness, that is their ability to offer the same spatiotemporal patterns and to resist to external perturbations such as losses of nodes or edges in the networks interactions architecture, changes in their environmental boundary conditions as well as changes in the update schedule (or updating mode) of the states of their elements (

“

^{st}Book, 10 A.D.).

The cell metabolism is regulated by interaction networks [

The interaction graphs associated to the regulatory interaction networks are inferred from the experimental data and from the literature [_{1} and C_{2}, from C, on

The definitions of attractors are numerous. We have chosen here a version available both for continuous and discrete cases and the only consistent with all encountered practical situations [

The

there is no set

there is no

An attractor

The simplest case of attractors is the fixed point called node [

An

A _{0}n_{0}

A graph is a

A

For a vertex of a digraph, the number of head endvertices (resp. tail endvertices) of edges adjacent to this vertex is called the ^{−}(^{+}^{−}^{+}(v) = 0_{v∈V}deg^{+}(v) = ∑_{v∈V}deg^{−}(v) =

The ^{2}

Let us denote by _{i}(t)_{i}(t))_{i∈R}; x(t)∈ Ω =^{n}_{i}(t)_{1},...,i_{k})_{i}(t+_{i}[x_{i1}(t),..., x_{ik}(t)]_{i}_{ij}(t)_{i}_{i}^{k}^{K}

A threshold Boolean automata network with _{ij}_{ij} corresponds to the interaction weight with which gene _{ij}_{i}_{i} ⇔ w_{ij} ≠^{n}_{i}^{th} activation threshold

In a random Kauffman Boolean network of size ^{K}^{c}

The notions of boundary and core of a regulatory interaction network come from graph theory. The notions of critical node and edge are more specific in biological applications. For defining correctly these notions, we consider a threshold Boolean automata network: its

An edge in a regulatory network is

It is the case in the example of the flowering regulatory network of ^{2} ≤ N ≤^{4}

The flowering regulatory network of Arabidopsis thaliana is sensitive to the change of updating mode, because it loses all the seven limit cycles observed in the parallel mode when we let it evolve with the sequential mode. The flowering network is also sensitive to the knockout of the gene EMF1 by a micro-RNA, which causes the loss of two fixed points among six [

A continuous potential differential equation on R^{n}^{n}^{n}_{i} ∈

_{k}(^{t}txA_{k}x)x_{k} + ^{t}xWx+Θx_{ijk})_{k} = (a_{ij})_{k}_{iii}_{ij})_{i})

Then the potential regulatory network associated to _{i}_{j,k}_{ijk}_{jik}_{jki}_{j}(t)x_{k}(t)_{j}(w_{ij}_{ji})x_{j}(t) + θ_{i}]_{ii}_{j ≠i}_{ijj}_{jij}_{jji}_{j}(t)_{i}(t + 1)_{i} + x_{i}(t))_{i} +x_{i}(t))

In the Boolean case, let suppose that ^{t}xWx + Bx_{ii}_{i}(t +_{i} + x_{i}(t))

A

_{i}(t)_{i}(t)_{i}(t)_{c}(t) = ∑_{i=0,...,n-1} Y_{i}(t)^{2}/2_{p}(t) = ∑_{i=0,...,n-1} Z_{i}(t)ΔT_{i}_{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:

Hence: _{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

A discrete system verifying the

Let us consider now a _{c}(t + n) = E_{c}(t)_{c}(t + 2n) = E_{c}(t)

A Kauffman Boolean network with only identity and negation functions [^{n}^{n}_{i}(t)_{i}(t)-_{i}(t)∈_{ii+1}_{ii}_{ii+1}_{ii}

Reciprocally, each threshold Boolean automata network, because its rule corresponds to a Boolean function from Ω = {0,1} ^{n}

Lemma 1 proves a relationship existing between the number of non-oriented edges defined on an undirected graph

_{m}^{m-k}2^{m-k}

From Lemma 1, the average indegree

Let us define now _{r}_{r}^{r}/^{r}/r^{r+1}^{r}^{−1}/^{2g–1}

There are ^{nd/2}_{n(n-1)/2}^{r}^{−1}/

The circuit direction is chosen to give the priority to the first forward post-control interaction over the backward one. If the control is exerted on a 3-circuit, whose undirected graph is given in _{+} = 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).

In 1949, M. Delbrück [

In 1993, S. Kauffman [

For

Let us consider now directed random networks and try to evaluate the number of attractors brought by its circuits. By multiplying the numbers of attractors of the disjoint circuits, we can get a lower bound of the total number of attractors in the whole random network. For example, if ^{3}^{r−1}/r

If the random oriented network contains ^{2}^{2}(^{n} = (^{n}

Because ^{n}^{nLog(1-2/n)}^{−2}^{2}^{S}^{S}. A_{n,2}_{n,2}

These results are consistent with the fact that about 3,000 boundary genes, acting as sources, control the network of 22,000 genes of the human genome: the proportion of boundary genes is about 1/7 in some large regulatory networks such as the cell cycle control network [^{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 (^{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.

If we consider that the boundary genes and micro-RNAs are always in state 1 of expression, the only possible changes in the attractors can come from an inhibition of the genes in state 1 of the positive circuits by the 700 micro-RNAs and the half of the 2,525 boundary genes non inhibited by the micro-RNAs: that corresponds to about 3,360 genes inhibited among the 19,000 not boundary nor isolated genes, that is 1/6 of these genes, ^{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 ^{p}^{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 [

The robustness of a regulatory network is defined by its ability to resist to exogenous or endogenous perturbations,

A threshold Boolean automata network rule can be implemented in 3 different updating protocols. Consider the example of a 3-switch (

- 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. [

- 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.

From this particular network of size 3, the 3-switch, we conjecture more generally that the cyclic behaviour can appear when the updating rule goes from the sequential to the parallel mode and not in the inverse way. The statistics done on the set of the all possible (188,968) networks of size 3 shows that this conjecture is almost everywhere true, but it fails for 0.3% of them [

_{i∈C} m_{i}, where _{i}

The proofs of Lemmas 2 and 3 are obvious. The statistical behaviour of networks of size 3 needs a complete simulation of all their dynamics; we are only interested by networks of size 3 having some specific dynamical features: for certain updating modes, they have both fixed configurations and at least one limit cycle, the limit cycles disappearing completely for other iterations rules, where there are only fixed configurations. Among the 188,968 possible networks of size 3, corresponding to all the configurations of interactions and thresholds, only 34,947 (18.5%) have this dynamical behaviour. If we try to understand how limit cycles are disappearing when the iteration mode changes, frequently this disappearance occurs by passing from synchronous to sequential updating, in a hierarchy of updating modes, defined by the hierarchy of blocks based on the pre-order inclusion. We observe in the simulations three different possible behaviours for these 34947 size 3 networks:

- 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”).

Among the 34,947 simulated networks of size 3 having limit cycles, the dispatching into the 3 possible behaviours follows the repartition below.

This repartition confirms that there is practically no network (only 0.31%) for which the cycles are present in the sequential updating modes and disappear in the synchronous one.

Let us now consider the 108 networks with an “Up” behaviour, in order to try to find inside them a differentiation character. For that, let us calculate the maximal length of their limit cycles. For the 34,947 studied networks, this maximal length is equal to 6. The table below gives the repartition of the networks of size 3 as function of their maximal limit cycle length (the column “Total” corresponds to the repartition of the maximal lengths for the 34,947 simulated networks), showing the diversity in the response of networks of size 3 to the changes of the updating mode (cf. [

Both brain and plants are built during their morphogenesis under the control of regulatory systems called _{i}

By doing the change of variables _{i}=(X_{i})^{1/2}

The new differential system is then a gradient system with P as associated potential, whose minima are just located at the fixed points of the _{i} = Π_{k=1,n;k≠i}(1 + c_{k}X_{k})^{n}/(1 + e_{k}X_{k})^{n}_{k} << c_{k} <_{i}_{k}_{k} = 0_{i} = (X_{i})^{1/2} and considering the potential

The applications of the

The bulbar vegetative regulation of the cardio-respiratory system is made of three main neuronal populations [

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 [

Let us define now by _{1}_{2}, x_{3}_{4}

Let us consider the change of variables:
_{i}

Let us consider now the potential _{i}_{i}_{i}’s dominates.

Let us share the velocity field _{i}}_{i=1,4}_{i}_{2}) or ADP (see Figures 11 a, 13 left and 14 right) as activator of the complex E_{1} [_{i}_{i}

When the attractor is a fixed point, the first enzymatic complex E1 has a stable stationary state defined by
_{1}_{k}

In this case, parameters appearing only in _{4}

Let us suppose now that we measure the outflows _{1}_{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 _{i}}_{i=1,4}

Hence, we can calculate

When all the fluxes of a metabolic system have reached their stable stationary value, then we can define the notion of control strength _{ik}_{i}_{k}^{th} step by: _{ik}_{k}/∂LogΔx_{i}

The control molecules can be enzymes [Figure 11(b)] or metabolites and the

When oscillations occur, we can use the variables _{ik}_{ik}_{k}_{k}^{th} step flux by _{i}_{ik}=∂Logτ_{k}/∂Logx_{i}_{ik} = ∂LogAmp_{k}/∂Log_{i}. If _{k} =_{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}^{2}P/∂_{1}^{2}∂^{2}P/∂x_{2}^{2}^{2} P/∂x_{1} ∂x_{2}^{2}

When the apparent _{max}

The cell cycle is sensitive to the control by two micro-RNAs whose one (micro-RNA 34) has p53 as positive transcription factor, involved in selective cytotoxicity of intracellular amyloid [

The feather morphogenesis network has been well studied in chicken [

We have studied in this paper the robustness of regulatory interaction networks, especially the influence of boundaries in threshold Boolean automata networks. The sensitivity (or absence of robustness) in real biological networks appears dominant in the case of inhibitory actions exerted by micro-RNAs and we have noticed it in both cases of the cell cycle control and the feather morphogenesis regulation networks. We have also remarked the dependence on the updating mode [^{Log(r(k))Log(Log(r(k)))}

More, let us consider the notion of r-^{r(r+1)/2}/(r +^{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 [

To conclude, all the predictions from the theoretical models have to be falsifiable by the empirical observations, reinforcing the interest of the methodological approach, summarised here shortly in the framework of the systems biology by using the tools of the dynamical systems and complex systems theories, and applied in various biological fields, from genetic and metabolic to physiologic control.

We are indebted to S. Cadau and N. Glade for many helpful discussions. This work has been supported by the EC NoE VPH.

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

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.

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).

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

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.

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.

(a) Regulatory network controlling the flowering of

Potential networks with Δ_{s}

(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).

Top: Number of limit cycles of period

(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) _{i} =._{1}_{2}

Potential (left) and Hamiltonian (right) systems.

Left: Bulbar cardio-respiratory centre, with inspiratory

(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.

Left. Parametric instability region reduced by taking into account the fraction of metabolites fixed to PFK when its _{max}_{1}_{R,F6P}/K_{R,ADP}_{max}

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}.

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).

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,

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.

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

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

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

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

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

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

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

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