Figure 1

Unidirectional interaction: Scheme of the directed interaction in a repressor lattice.Reuse & Permissions

Figure 2

Dynamics of the system for $L=31$. While $\gamma =4.4,K=1,c=0.1$, and $h=4$ are same for both the figures above, $\alpha =3$ and 15, respectively, for the upper and the lower panels.Reuse & Permissions

Figure 3

Scheme of the directed unidirectional mixed type interaction in an activator-activator-repressor lattice. Lines with arrowheads indicate activation.Reuse & Permissions

Figure 4

Dispersion relation of the activator-activator-repressor model.Reuse & Permissions

Figure 5

Numerically validated number of different phases as a function of $L$ for the repressor-lattice (circles) and the activator-activator-repressor lattice (stars). We were unable to numerically find any limiting oscillatory state for the activator-activator-repressor lattice when $L=3$ and 4.Reuse & Permissions

Figure 6

Bidirectional interaction: Scheme of the nonperfect or asymmetrical directed interaction. The thinner repressing arrows represent the backward interactions.Reuse & Permissions

Figure 7

Inverse of the period as a function of the asymmetry $|{\alpha }_1-{\alpha }_2|$. The theoretical solid straight line is in excellent agreement with the simulation's results (blue circles). Two time series are shown as examples. ${\alpha }_1+{\alpha }_2=2.99,\gamma =4.43,K=1,c=0.1,h=4$.Reuse & Permissions

Figure 8

The black cells have higher concentration than the white cells in the high-low-low (HLL) pattern, while just the opposite happens in the low-high-high (LHH) pattern. In the homogeneous ($\mathcal{H}$) pattern, the black and the white cells have the same concentrations.Reuse & Permissions

Figure 9

Plots with tags (a), (b), and (c) are for additive bidirectional interactions while plots (d), (e), and (f) are for multiplicative bidirectional repressions. Plots (a), (b), and (c), respectively, show the maxima of the real parts of the eigenvalues ($\max \left[\mathit{\text{Re}}\right(\lambda \left)\right]$) determining stability of the fixed points $(x^{*},y^{*})$ corresponding to the LHH, the HLL, and the $\mathcal{H}$ patterns. The dashed green line in plot (b) is the boundary between the HLL and the LHH patterns in the parameter space of $c$ and $\alpha$. The HLL appearing adjacent to the green dashed line in plot (b) should not be confused with the HLL patterns appearing in plot (a) for the same $c$ and $\alpha$ values. Appearance of the two different HLL patterns for the same parameter values, as can be seen by comparing plots (a) and (b), indicates that while for some parameter values there are three positive solutions $(x^{*}>0,y^{*}>0)$ to Eq. (19), none of them corresponds to the LHH pattern. The cyan-white cross-striped regions depict the ranges of parameter space where the corresponding fixed point solutions (and hence the related pattern whether stable or unstable) are nonexistent. Plots (d), (e), and (f) should be understood along the line discussed above. For all plots, $\gamma =1$, $K=1$, and $h=3$.Reuse & Permissions

Figure 10

A typical tetrahedron in the 3D lattice formed by three cells in a particular layer and a fourth cell in the adjacent parallel layer above. One can notice, as highlighted by the dashed lines, all the triplets of adjacent cells in the tetrahedron cannot form repressilators simultaneously. Thus, consistent generalization of the unidirectional repressor lattice to three dimensions is not possible.Reuse & Permissions

Figure 11

One can see the seven-state patterns in $3\times 3\times 3$ periodic symmetrically bidirectional 3D repressor lattice with (a) additive interactions and (b) multiplicative interactions. Seven colors have been used to mark seven different equilibrium concentrations which the different cells settle for. To assist reading in black and white mode, numbers 1 to 7 have been used to tag the equilibrium concentrations. Insets have been provided to zoom into seemingly indistinguishable curves. The schematic patterns, drawn on the top of each of panels (a) and (b), have consistently numbered cells, and they show the seven-state pattern explicitly. Here we have used $c=0.1,\gamma =1,\alpha =2,K=1,h=3$ [cf. Eq. (21)].Reuse & Permissions