Figures and data
Pattern transformations.
The figure shows examples of pattern transformations in 1D (A-D) and in 2D (E-H). Panels represent different patterns (i.e., different spatial distributions of some gene product concentration g (t,x)). (A) and (E) are the initial patterns being transformed into the resulting patterns in (B-D) and (F-H). The x-axis is cell position along a 1D array of cells and the y-axis is the concentration of a gene product. The transformation from (A) (resp., (E)) into (B) (resp. (F)) is trivial because the resulting pattern is homogeneous. The transformation from (A) (resp., (E)) to (C) (resp., (G)) is trivial because the concentration maximum is at the same spatial position in both patterns. The transformation from (A) (resp., (E)) to (D) (resp., (H)) is non-trivial because the resulting pattern is heterogeneous and the pattern in (D) has maxima that were not present in (A). The blue, grey and pink fillings in (A-D) correspond to the sign of the derivative and are meant to highlight the maxima and minima. In (E-H) the colors represent gene product concentration.
The model transforms initial patterns (A), into resulting patterns (C), through a set of equations implementing gene networks with extracellular signaling (B).
The article considers three initial patterns (A): spike initial pattern (left), combined spike-homogeneous initial patterns (middle); and homogeneous-with-noise initial patterns, with small white noise (right). (B) Diagram of an example gene network. Black squares represent intracellular gene products, blue circles extracellular signals. Green arrows stand for positive regulations, while red arrows for negative regulations. Weights of the network are given by the J matrix; while its topology by the T matrix; D represents the diffusion coefficients and M the degradation rates (see equation (1)). (C) Example resulting patterns for each initial pattern in (A) under the gene network in (B). For each initial pattern we draw the most possible complex resulting pattern given the gene network topology in (B).
Schema of the elements and spatial relationships in the model.
Gray squares are cells. Each white square represents the extracellular space just apical of each cell. Small black squares represent intracellular gene products. Blue circles represent extracellular diffusible signals. Dashed lines represent the diffusion of the extracellular signals over extracellular space. Full green lines represent positive regulation between gene products. Red full lines represent inhibitory regulation between gene products.
Definition of chains, loops, and signal subnetworks.
(A-D) Parts of intracellular gene networks (E-F) Schema of simple extracellular regulatory loops. (G-J) Examples of gene regulatory networks with their signal subnetworks depicted. Small black squares represent intracellular gene products, and blue circles represent extracellular diffusible signals. Green arrows denote positive regulation, and red blunt arrows denote inhibition. Dots (…) indicate any positive chain of gene product interactions. (A) A pPositive regulatory chain is a sequence of interactions in which each gene product positively regulates the next and is positively regulated by the previous one in the sequence. A sequence with an even number of inhibitory interactions is also a positive regulatory chain, provided that each inhibited gene product receives a positive input from elsewhere. (B) Negative regulatory chain: as in (A) but with an odd number of inhibitory interactions. (C) A positive intracellular loop is a positive regulatory chain whose first and last gene products coincide and in which all gene products are intracellular. (D) Negative intracellular loop: as in (C) but with an odd number of inhibitory interactions. (E) A positive extracellular loop is a positive regulatory loop in which at least one gene product is an extracellular signal. (F) Negative extracellular loop: as in (E) but with an odd number of inhibitory interactions. (G) An example network in which we have surrounded its signal subnetwork with a grey rectangle. (H) Example network with two signal subnetworks. (I) Example network with three extracellular signals; each signal subnetwork is enclosed by a rectangle. Note that since signal 3 and 4 are in a loop their signal subnetworks include the same gene products. (J) Another example network with its subnetworks indicated.
Subnetwork and network classification.
(A) Schema representing each class of subnetwork. In a H signal subnetworks, the extracellular signal is not downstream of itself. In a signal L+ subnetworks, the extracellular signal is both upstream and downstream of itself, forming a positive extracellular loop (which may include any number of gene products). In a L− signal subnetwork, the extracellular signal is negatively upstream and downstream of itself. The box labeled ‘rest of subnetwork’ represents any gene network provided that the most upstream extracellular signal is not negatively upstream of itself (i.e., no negative loop leads back to it). (B) (B) Example of a H gene network containing two H subnetworks. (C) Two examples of L- gene networks, one with on L- subnetwork (left) and one with two L- subnetworks (right). (D) Example of a L+ gene network with two L+ subnetworks. (E) Two examples of L+L- gene networks. Each of them has a L+ subnetwork and a L- subnetwork. (F) Example of a H gene network. (G) Example of a gene network with a L+ subnetwork, a L- subnetwork and a H subnetwork. Colors of arrows and gene products as in figure 2 and 4.
Instances of principal branches of dispersion relations.
(A-B) Dispersion relations of RD-unstable gene networks of the first kind (i.e. there is only a number of wavenumbers, x-axis, for the which there are eigenvalues with a positive real part). The real part of the principal branch of a dispersion relation can diverge to −∞ for large wavenumbers (A); it can converge to a negative finite value (B). (C-D) Dispersion relation of RD-unstable gene networks of the second kind (i.e. there is an infinite number of wavenumbers with eigenvalues with a positive real part). The real part of the principal branch of a dispersion relation cannot diverge to +∞ (Klika et al., 2012).
Diagram showing the topological requirements for pattern transformation in hierarchical gene networks
using example gene networks. (A) Requirement (RH2a): the positive regulation of 3 by A (discontinuous green arrow) has a different non-linearity than that of 4 by A. (B) Requirement (RH2b): the negative regulation of 5 by 3 (discontinuous red arrow) has a different non-linearity than that of 5 by 4. (C) Requirement (RH2c): two extracellular signals downstream of 1 and upstream of 4. The signals have different concentration profiles over space because either they have different diffusion coefficient or different intrinsic degradation rates. Each of these requirements lead to the formation of two new concentration peaks in the most downstream gene product. Network colors and shapes as in Figure 2. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model (see S6 above for parameter values). Notice that in (A) gene product 3 is an example of whatcall gene product j in the main text, while in (B) gene product j corresponds to gene product 5.
Classes of gene networks capable of pattern formation and their resulting patterns in 1D.
The first column depicts simple examples of each class of gene network topology capable of non-trivial pattern transformation. The upper row shows the three initial patterns. Intermediate panels show each type of possible resulting pattern arising from each combination of initial pattern and gene network topologies. The network topologies in the left column are only simple ones, for a more detailed description of the possible ones check the main text. For the over-Turing topology we chose to represent a intracellular loop with two gene products (but we could have chosen one). Note that some of the pattern transformations are trivial (over-Turing and Turing on spike initial pattern). H network can lead to non-trivial pattern transformations from homogeneous-with-noise initial patterns. Network colors and shapes as in Figure 2, except that P stands for the gene product plotted as resulting pattern while I stands for a gene product present in the initial pattern. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model for f (see S6 in SI for parameter values).
Classes of gene networks capable of pattern formation and their resulting patterns in 2D.
The first column depicts simple examples of each class of gene network topology capable of non-trivial pattern transformation. The upper row shows the three initial patterns. Intermediate panels show each type of possible resulting pattern arising from each combination of initial pattern and gene network topologies. In the case of spike and spike-homogeneous initial patterns, the resulting patterns correspond to the revolution of the patterns in Figure 8 around the center of the spike given the radial symmetry of equation (1) and the spike. In homogeneous-with-noise initial patterns, the white noise breaks such symmetry. In the case of Turing networks acting on homogeneous-with-noise initial pattern other types of periodical resulting patterns (e.g., dots or stripes) have been reported in the literature (Murray, 2002). Network colors and shapes as in Figure 8, Simulations were run using a Forward-Euler algorithm on the Maini-Miura model for f (see S6 in SI for parameter values).
Peak addition and subtraction in the resulting pattern of downstream gene products from hierarchic gene networks.
The most upper row shows an idealized pattern of a gene product i with several concentration peaks. The middle row that of another gene product, j, with several concentration peaks. In (A) both gene products positively regulate the same gene product, k. As a result gene product k has concentration peaks either i or j have them. In (B), instead, j inhibits k while i activates it. (C) shows an example network in which the pattern of two gene products, 4 and 5, are effectively added into that of gene product 6. (D) shows and example gene network in which the patterns of gene products 4 is effectively subtracted from that of gene product 5 to give rise to the pattern of gene product 6. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model for f (see S6 in SI for parameter values).
Variational properties of the diamond H network.
(A) Diamond H network (see S4 in SI). Network colors and shapes as in Figures 2 and 4. (B) Initial pattern in gene product 1. (C-D) The resulting patterns consist in two symmetric peaks around the initial spike. The height (C) and position (D) of such peaks can be independently modified by tuning the model parameters. Peak height can be varied by changing a single parameter, that is the activation of gene product 5 by gene product 4 (i.e. J54). To change the position of the peak while peaking its height constant (D) two parameters need to be changed at the same time, the diffusion coefficient of extracellular signal and the activation of gene product 5 by gene product 4. Just changing the diffusion coefficient the position of the peaks will vary but so will their height. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model for f (see S6 in SI for parameter values).
The combinations of hierarchical, over-Turing and Turing networks are and their resulting patterns.
The first row depicts simple examples of each class of gene network topology acting as the upstream subnetwork of the combination in series. The first column shows simple examples of each the same gene network topologies acting as the downstream subnetwork of an in series combination. Intermediate panels show each type of possible resulting pattern arising from a spike-homogeneous initial pattern from each combination. Note that all pattern transformations are non-trivial. Network colors and shapes as in Figure 2. Dashed arrows represent the different, and strictly non-linear, regulations in the H0 networks. P stands for the gene product plotted as resulting pattern while I stands for the gene product in the initial pattern. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model for f (see S6 in SI for parameter values).
