Non-trivial pattern transformations.

The figures shows four patterns (i.e., four spatial distributions of a gene product concentration) an initial one (A) and three resulting ones (B-D). The transformation from (A) to (B) is trivial because the resulting pattern is homogeneous. From (A) to (C), the pattern transformation is trivial because the sign of the spatial derivative of the gene product concentration (shown in blue when positive and in red when negative) is the same as in the initial pattern (A). The transformation from (A) to (D) is non-trivial because the resulting pattern is heterogeneous and the sign of the spatial derivative ofg (x) in (A) and (D) is different (see section S1 in SI for details).

The model transforms initial patterns (A), into resulting patterns (C), through a set of equations implementing gene networks (B).

The article considers three initial patterns (A): spike initial pattern (left), combined spike-homogeneous initial patterns (middle); and homogeneous initial patterns, with small white noise (right). (B) Diagram of example gene network. Black squares represent intracellular gene products, blue circles extracellular signals. Green arrows stand for activatory regulations, while red arrows for inhibitory regulations. Weights of the network are given by theJ matrix; while its topology by theT matrix;Drepresents the diffusivities andM the degradation rates (see equation 1). (C) The resulting patterns from each initial pattern in (A) under the gene network in (B).

Instances of gene regulatory networks.

The one-signal subnetworks corresponding to each extracellular signal, or pair of signals, are surrounded by a square with a different color. Some subnetworks are part of bigger subnetworks. Network colors and shapes as in Figure 2.

Gene networks capable of pattern formation and their resulting patterns.

The first column depicts simple examples of each type 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 topology. Note that pattern transformations in (B1), (C1) and (A3) are trivial. Network colors and shapes as in Figure 2.Pstands for the gene product plotted as resulting pattern whileI stands for the gene product in the initial pattern. Simulations were run using a Forward-Euler algorithm on the Maini-Miura model (see S8 in SI for parameter values).

Variational properties of the diamond H network.

(A) Diamond H network (see S5.5 in SI). Network colors and shapes as in Figure 2. (B) Initial pattern in 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. Simulations as in Figure 4 (see S8 in SI for parameter values).

Some variational properties of the fundamental gene network topologies capable of pattern transformation.

Network colors and shapes as in Figure 2.