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Chaotic Motifs in Gene Regulatory Networks

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Chaotic Motifs in Gene Regulatory Networks Powered By Docstoc
					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

				
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