Design principles of stripe-forming motifs: the role of positive feedback

Abstract

Interpreting a morphogen gradient into a single stripe of gene-expression is a fundamental unit of patterning in early embryogenesis. From both experimental data and computational studies the feed-forward motifs stand out as minimal networks capable of this patterning function. Positive feedback within gene networks has been hypothesised to enhance the sharpness and precision of gene-expression borders, however a systematic analysis has not yet been reported. Here we set out to assess this hypothesis, and find an unexpected result. The addition of positive-feedback can have different effects on two different designs of feed-forward motif- it increases the parametric robustness of one design, while being neutral or detrimental to the other. These results shed light on the abundance of the former motif and especially of mutual-inhibition positive feedback in developmental networks.

Figure 1. Examples of functional motifs in biological networks.

After identifying the biological and dynamical functions of individual network motifs, either genetic or biochemical, a major question still remains: will a given motif have consistent effects when added to different networks?

Figure 2. The framework of the current study.

An example of single-stripe formation in the positional-information scenario: the concentration of a morphogen M is monotonically increasing along a one-dimensional “tissue” of N isogenic cells. The underlying genetic network transforms the increasing input into a single stripe in the expression of one of the genes. Each transcription factor X acting at the promoter region of gene Y has an effect on gene expression characterised by the interaction strength ω_XY (Interaction matrix). The activating interactions have ω_XY > 0 and inhibiting interactions, ω_XY < 0. For each gene, the contributions from multiple transcription factors affecting it are subsequently summed. The resultant transcription rate from the promoter is proportional to the sigmoidal-filtering of this total contribution. The parameters a and b of the sigmoidal function f(x) control the steepness and location of the threshold value of the regulation function. The morphogen M is considered to be received only by gene A (see also Methods section).

Figure 3. The formation of the single-stripe pattern by 3-gene networks based on the four Incoherent Feed-Forward Motifs.

The upper part of the figure shows the stripe-forming IFFMs reinforced with positive feedbacks, having from 4 to 6 interactions. Only interactions that generate positive feedbacks have been included. Within the I₁ and I₃ branches of stripe-forming networks, thick solid-line arrows imply adding C self-activation, the dashed-line arrows, B self-activation, and the dotted-line arrows, mutual-interaction (i.e. mutual-inhibition or mutual-activation). The lower rectangle in the figure includes the four IFFMs and the underlying process of stripe formation. It shows the generic stripes of the I₁ and I₃ networks, the lack of stripe for I₄ and the barely-detectable one for I₂. Within the rectangle, the upper graphs single out the individual contributions into the stripe-gene promoter for an increasing morphogen, M: from gene A in red, and from gene C in blue. For simplicity, we chose to use the concentration of A for the horizontal axis, instead the morphogen concentration. The lower graphs show the resultant expression from gene B. The Gray areas represent non-biological negative values for the concentration A and conceptually illustrate that I₄ would require a negative concentration of A to show a stripe. The range of (non-biological) negative concentration for protein A is also included for illustrative purposes.

Figure 4. Changes in the functional parameter space produced by adding positive feedbacks to the IFFMs.

The positive feedbacks added to the IFFMS are: two self-activations (of gene B and C), and the B-C interaction. We show thus three graphs associated to these three added positive feedbacks. The graphs illustrate the change in the extent of the functional parameter space as the positive feedback increases. More precisely, they include the fraction of the functional parameter space of the RIFFMs, each scaled to the extension of the functional parameter space of a lower level RIFFM. This comparison methodology is illustrated in the lower panels for the mutual-inhibition and the mutual-activation of the 4-link networks. In these two examples, the parameter space to be compared is (ω_AB, ω_AC, ω_CB) with (absolute) values ω_XY ∈ [0, 10] (Gray cube). The functional region of parameter space is represented by the red shape. The behaviour illustrated in these three graphs is generic for the addition of each of the three types of positive feedback, independent of the existing positive feedbacks in the network. We wish to point out the parametric boost brought by the mutual-inhibition in the I₁ branch, and the complete neutrality introduced by the mutual-activation and stripe-gene self-activation in the I₃ branch.

Figure 5. Parametric boost versus neutrality: a mechanistic view.

The figure illustrates the generic consequences of introducing B-C interaction leading to indirect positive feedback. Middle panels show the stripe (or lack of it) in the pure IFFMs and in the corresponding IFFM reinforced with the mutual-interaction. The dotted lines in the stripes represent attractor-switching, while the solid lines follow the attractor's movement. The Gray vertical bands qualitatively indicate the three regions, Low-High-Low, constituting the stripe in gene B. The yellow-background square panels are (C, B) phase plots generically corresponding to the three regions. There, the steady states (stable, S or unstable, U) are the intersections of nullcline curves (where one variable does not change in time – notation in the upper left corner). The black star indicates the initial condition close to the origin. The red arrows in these phase plots show that the nullclines move only horizontally or vertically in response to the morphogen gradient. When creating mutual inhibition, even a weak B-C inhibition allows single-stripe formation through attractor-switching process. When creating mutual activation, any value of B-C activation sharpens the stripe that already exists, and allows attractor-switching in the posterior border. The thin line passing through the unstable state shows the separatrix, delimiting the basins of attraction for the two stable states.