Table 1.
Attractor distributions in random GRNs.
Table 2.
Attractor distributions in oscillatory networks in Table 1.
Figure 1.
A, D: Two 3-node chaotic GRNs. Solid (dashed) lines denote active (repressive) regulations. B, E: Bifurcation diagrams of A and D, respectively (peak values of plotted as functions of
with
). C, F: Chaotic attractors (C for A with
, F for D with
). From A to D, we discard a single interaction (
, blue line in A). The removal does not essentially affect the chaotic motion. The bifurcation diagrams and the chaotic attractors of the two GRNs in A and D are similar. On the other hand, all the cross interactions in D (cross refers to interaction between different nodes) are irreducible, and removal of any of them can surely suppress chaos no matter how to adjust the parameters, initial conditions and self-regulations of different nodes. The cross interaction structure of D is identified as a 3-node chaotic motif.
Figure 2.
The same as Fig.1 with three 4-node GRNs considered. A-C: The GRNs under investigations. D−F: Their corresponding bifurcation diagrams (D for A with ; E for B with
; F for C with
). G-I: The corresponding chaotic attractors (G for A with
; H for B with
; I for C with
). From A to B the interactions
and
are removed, and from B to C the interaction
is deleted. These deletions do not essentially affect the chaotic motions. All bifurcation diagrams to chaos (D−F) and chaotic attractors of the three GRNs (G−I) are similar. Network C is irreducible for chaos in the sense that removal of any cross interaction of it can absolutely terminate chaos. The cross interaction structure of C is thus considered as a 4-node chaotic motif.
Table 3.
Attractor distributions in chaotic motifs.
Figure 3.
By chaotic motifs, we mean that these GRNs can produce chaos with certain parameters, initial conditions and suitable self-regulations, and removal of any cross interaction in these GRNs can definitely terminate chaos. Chaotic motifs are the minimal and irreducible building blocks for chaotic motions in GRNs. Some significant motifs are shown with amount of chaotic realizations more than 100 in Table 3. There are four conditions (i)−(iv) for chaotic motifs. All these motifs possess two feedback loops (i), and at least one of them is NFL (ii). In all the motifs which contain one PFL and one NFL, the PFL must have at least one node not included in the NFL (iii). In most of the chaotic motifs, there is a node regulated by both repressive and active regulations (iv). If not, the motifs more often contain two NFLs.
Figure 4.
All 3-node and 4-node TLSs, which fulfill conditions (i) and (ii) while violate condition (iii), can never produce chaos. The competition between the two feedback loops in each GRN does not work for chaos due to the fact that the PFLs are completely controlled by the corresponding NFLs. Therefore, all these TLSs can never produce chaos. For each of these TLSs, more than random tests are made by varying self-regulations, parameter sets and initial conditions, and none of these tests yields chaos. These structures may bring some advantages on rhythmic functions, avoiding the chaotic disturbances definitely.
Figure 5.
distributions of the chaotic GRNs of Figs.1 and 2. A for Fig.1A; B for Fig.1D; C for Fig.2A; D for Fig.2B; E for Fig.2C. The numbers associated to all cross interactions indicate the
of Eq.(7). The total period of measurement is about
cycles of chaotic orbits. It is shown that most of the interactions reducible for chaos have almost zero
, while all the interactions irreducible for chaos in the chaotic motifs in Fig.1D and Fig.2C have sufficiently large
. Note that, the two interactions
and
in D have comparable
. Discarding different one of them can construct different chaotic motifs (motifs (22) and (67) in Fig. S1 by discarding
and
, respectively). On the other hand, both the interactions of
and
in D are important for the competition between the two loops and thus essential for chaos. There is only one loop in the GRN after removal of
(breaking condition (i)); and the PFL is included in the NFL after deletion of
(breaking condition (iii)), and both of the two operations can securely suppress chaotic motions.
Figure 6.
Different distributions in chaos and limit cycles.
A ,B: Two periodically oscillatory TLSs. C ,D: Two 4-node chaotic motifs. E, F: limit cycle solutions of A and B. E with for A; F with
for B. G, H: Chaotic solutions of C and D. G with
for C; H with
for D.
of the corresponding states are labeled on all the cross interactions of the GRNs A-D. It is demonstrated that, there are single effective loops (
in A and
in B) which dominate the oscillation of limit cycles while the two feedback loops possess comparable
to common nodes (node 4 in C and node 2 in D) in chaotic states.
Figure 7.
Statistic analysis of chaotic GRNs.
A, B: Relative frequency distributions (and
) of the 105 TLSs embedded in all chaotic GRNs observed in Table 2 (A for 4-node chaotic GRNs and B for 5-node ones). Black squares represent
and red cycles denote
, which are computed by Eq.(1) in original topological chaotic GRNs (
) and in dynamically reduced chaotic GRNs by deleting all interactions with
(
), respectively. The significant chaotic motifs in Fig.3 are labeled out with corresponding indexes. Most of them have atypically high
and
, and they more often contain two NFLs. Existence of two independent and competitive oscillatory modes guarantees strong competitions in these subnetworks, leading to chaotic motions.
Figure 8.
A detailed demonstration of statistical method counting the frequencies of TLSs.
A randomly constructed GRN A is considered as an example to show how to compute all the TLSs contained in a GRN. Obviously, there are two 4-node TLSs (B and C) and three 3-node TLSs (D−F) embedded in A. All the 105 TLSs in Fig. S1 are taken into account in our counting. Similar analysis can be applied to all chaotic samples in Table 2 to obtain the results of Figs.7A, 7B.