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Chaotic Motifs in Gene Regulatory Networks Zhaoyang Zhang, Gang Hu Beijing Normal University I. Introduction (1) Complex networks In recent two decades, study of topology and dynamics of complex networks has attracted great attention (2) Applications l Complex networks in physical, chemical systems; l Complex networks in social systems, economic systems, finacial systems; l Complex networks in biological systems (Neural networks, Foodweb, Gene Regulatory Networks) (3) Gene Regulatory Networks (GRNs) (a) Model NOTEs l l l (b) Example of a three-node GRN l 1→2, Active regulation from node 1 to node 2 l 2→1, Repressive regulation from node 2 to node 1 l 3→3, Self-active regulation of node 3 Each node in the network has at least one input and one output Cross interaction — interaction between different nodes II. General disccussion (1) Rareness of chaotic GRNs Ø ODEs of GRNs are nonlinear and usually high-dimensional we expect N≥2: Self-sustained oscillations popular N≥3: Chaotic GRNs easily observed Ø Actual biological observations and numerical results Most GRNs → Stable steady states Much less GRNs → Self-sustained oscillations Up to now, no chaotic GRN biologically observed and extremely rare chaotic GRNs numerically explored For N=3, no chaotic GRN reported even for numerical simulations These results are welcome in biology l Stable steady states are suitable for various balances of biological processes l Much less sustained oscillations may be needed for some particular biological functions (i.e., cell devisions, circadian rhythms, glycolytic oscillations) l Chaos seems to be not good for healthy biological rhythm, and it is rare in observations However, how it can happen dynamically? (2) Primary analysis (a) Attractor distributions Most of GRNs with N≥3 are expected to show chaos with suitable parameters and initial conditions Tests of 3-node, 4-node, 5-node GRNs with different interaction structures (both cross and self regulations, active and repressive regulations), different parameters and different initial conditions Table 1. Attractor distributions of 3, 4, 5-node GRNs in tests for each type We found indeed: Most of GRNs tend to stable steady states Much less sustained oscillations Extremely rare chaotic motions Why? (b) Possible reason for the rareness of chaos We guess the monotonous behaviour of regulation may be the key reason, and the results shown in this talk are valid for all similar monotonous regulatory interactions Active regulation monotonously Repressive regulation monotonously increasing (K=0.10,h=3.0) decreasing (K=0.10,h=3.0) (c) How to obtain Sufficient samples of chaotic GRNs Analysis l m-p periodic motion imply chaos nearby l Oscillatory GRNs imply more probability of chaos Method Searching in oscillatory GRNs in Table 1 by changing parameters and initial conditions Table 2. Chaotic samples observed in oscillatory GRNs of Table 1 with different parameters and initial conditions These chaotic GRNs serve as samples for our study of chaos in GRNs III. Chaotic Motifs (1) Concept of chaotic motifs Motif l Extensively used in biological networks l Subgraphs appearing in some biological networks with certain particular functions, far more often than in randomized networks l Elementary building blocks carrying some key functions l All previous defined motifs carry simple functions Chaotic Motifs Ø Weak sense: Smallest and irreducible subnetworks of which each cross link is essential for producing chaos. Minimal and irreducible building blocks for chaotic GRNs Ø Strong sense: Among the above subnetworks, those appearing in chaotic GRNs with probabilities much higher than that in random networks (2) Example of chaotic motifs (a) Example 1: 3-node chaotic GRNs Ø In 3-node chaotic GRN D, all the cross interactions are irreducible. D is identified as a 3- node chaotic motif Ø A include the motif D and an additional link 3→1 Ø A and D produce very similar chaotic behaviors: similar road to chaos (B, E) and similar chaotic attractors (C, F) Ø The link 3→1 is not essential for chaos Fig. 1 Two 3-node chaotic GRNs (b) Example 2: 4-node chaotic GRNs Fig. 2 Three 4-node chaotic GRNs From A to B: Discarding 1→2, 3→2 From B to C: Discarding 1→3 In chaotic GRN C, all the cross interactions are irreducible. C is a 4-node chaotic motif Fig. 2 (c) Conditions for chaotic motifs Motifs — Defined by cross interactions NOTEs Feedback loop: A regulation chain from any given node can finally regulate itself Negative (positive) feedback loop (NFL/PFL): The number of repressive regulations on the loop is odd (even) PFL NFL Fig. 3 PFL and NFL Conditions for chaotic motifs (i) Two feedback loops Idea: chaos is the result of competition between different oscillatory modes (at least two), and all these modes should be constructed by loops One loop Two loops Three loops (ii) At least one of these two loops must be NFL Existence of at least one NFL is the necessary condition for self-sustained oscillations and also for chaos Zero NFL One NFL Two NFLs (ii) If there is only one NFL among the two loops, some nodes of the PFL must be not included by the NFL NFL is the only source for sustained oscillation, if all nodes of PFL are included in NFL, there will be no competition of two modes and thus no chaos Included Not (iv) If a motif includes one NFL and one PFL, the node jointly regulated by two neighbors should receive one positive and one negative regulations This condition is for strong conpetition between two loops Two types of joint regulations Single type (4) Statistics of chaotic motifs Minimal two-loop structures (MTLS) A subnetwork containing only two feedback loops, and at least one of the loops is NFL Three non-MTLSs MTLS All nodes of MTLSs are regulated by only one neighbor except a single node (center node) regulated by two neighbors 3-node networks: 19 MTLSs; 4-node networks: 86 MTLSs tests for each of 105 MTLSs Table 3. Numbers of tests showing chaos Extensive tests show that: (i)(ii)(iii)— Necessary conditions for chaos (i)(ii)(iii)(iv)— Sufficient conditions for chaos Those MTLSs with the center node regulated by one active and one repressive regulations can definitely produce chaos with proper parameters and initial conditions We can distinguish all the 105 MTLSs which satisfy all conditions (i)-(iii) into three types Type 1: Satisfy condition (iv) with a node regulated jointly by an active and repressive interactions; Type 2: Violate condition (iv), with two NFLs; Type 3: Violate condition (iv), with one NFL and one PFL Type 1: 74.17%; Type 2: 25.28%; Type 3: < 0.6% (5) Chaotic and nonchaotic MTLSs Chaotic MTLSs 12 chaotic motifs having the largest probabilities of chaos Fig. 4 There are two NFLs in the network; and 8/12 the center nodes are regulated by both repressive and active interactions Nonchaotic MTLSs l tests for each without any chaos observed l Advantage on rhythmic functions. Avoiding disturbances of chaos definitely l These building blocks are welcome biologically Fig. 5 Nonchaotic MTLSs with cindition (iii) violated IV. Analysis by applying concept and method of dominant phase-advanced drivings (DPAD) relations between nodes through effective (1) Driving interactions l Chaotic motifs characterize the conditions of topology of GRNs for chaos l For actually realizing chaos, neccesary dynamic driving relations between the nodes of motifs are required l Study of chaotic motifs: Find relations between GRN topology and chaotic dynamics Driving relations between nodes through effective interactions How nodes drive each other in complex networks? (2) Concept of dominant phase-advanced drivings (DPAD) (a) DPAD and DPAD time fraction (DTF) Any single node of a GRN can not oscillate individually. It can oscillate only through the cross interactions Defining include two types of interactions: l One favorable to generate oscillation of node i , called phase-advanced driving (PAD) for the Δt time segment; l Others not , called phase-delayed interactions NOTEs l A node may be PAD driven by multiple neighbors; l Among which the single driving providing the largest contribute most to the oscillation of node i; l Called dominant phase-advanced driving (DPAD) (b) DTF The above DPAD is defined for time Δt Weighted time interval : Total weighted time : Total weighted time when node j serves as the DPAD of node i DPAD time fraction of node j driving node i ( ) is the quantitative measurement of the function of node j Small (large) indicates weak (strong) driving effect of node j on node i NOTEs l Large (small) DTF implies that in large (small) fraction of the total time period, j playes the most important role in driving node i to oscillation l DTF distributions can present the dynamic driving relations in networks producing oscillations and even chaos quantitatively l DTF can also manifest competitions between different dynamic loops leading to chaos (3) Distributions of DTF in oscillatory and chaotic GRNs (a) Example of periodically oscillatory GRNs (i) Peridic oscillations A single NFL completely control the oscillations. Loop 2→1→4→2 and 4→1→2→4 (ii) Chaotic motions l Two loops with a comparable importance l Strong competition between the two loops Through the jointly regulated nodes (center node, node 4 and node 2) (iii) Network reduction by reading DTFs A→B Removing 3→1 C, D Competitons through two nodes, node 1 and node 3 C→D Removing 1→2, 3→2 D→E Removing 1→3, motif 67 Removing 2→3, motif 22 Competition through node 1 is irreducible for chaos Removing 3→1 breaks condition (i) Removing 2→1 breaks condition (iii) (4) Statistics of motifs in chaotic GRNs Frequency study on chaotic motifs Compute the frequencies of all the 105 MTLSs appearing in random GRNs and scale their frequencies. They are identical (1.0) for all the 105 MTLSs Take all available chaotic samples in Table 2 Ø Compute DTFs of all interactions of chaotic samples Ø Discarding all links with DTF ≤ 0.10 Call the reduced subnetworks as dynamic netowrks Ø Compute the frequencies of all the 105 MTLSs in these dynamically effective chaotic GRNs, called We obtain 4-node 5-node Significant chaotic motifs with high frequency in networks l 12 most significant chaotic motifs in Table 2 l 12 most significant chaotic motifs measured by 8 overlaps: 11,12,16,23,48,58,63,92 IV. Conclusions (1) Extreme rareness of chaotic networks (a) Strict requirements on network structures (b) Strict requirements on dynamic oscillatory competitons (2) Chaotic motifs (a) Minimal and irreducible subnetworks where all cross interactions are essential for chaos Elementary building blocks of chaotic GRNs (b) Some of the above subnetworks appearing atypically frequent in chaotic GRNs (3) Chaotic motifs — interaction topology (a) Necessary and sufficient conditions of chaotic GRNs on topology structures (b) Numerical verifications of these conditions (4) Chaotic motifs — dynamic oscillation competiton Dominant phase-advanced drivings (DPADs) Time fraction of DPAD, quantitive mesurement of driving relations and competition intensities