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. 2015 May 15;142(10):1860-8.
doi: 10.1242/dev.113688.

Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues

Cristóbal Quiñinao  1 , Alain Prochiantz  2 , Jonathan Touboul  3
Affiliations

Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues

Cristóbal Quiñinao et al. Development. .

Abstract

Boundary formation in the developing neuroepithelium decides on the position and size of compartments in the adult nervous system. In this study, we start from the French Flag model proposed by Lewis Wolpert, in which boundaries are formed through the combination of morphogen diffusion and of thresholds in cell responses. In contemporary terms, a response is characterized by the expression of cell-autonomous transcription factors, very often of the homeoprotein family. Theoretical studies suggest that this sole mechanism results in the formation of boundaries of imprecise shapes and positions. Alan Turing, on the other hand, proposed a model whereby two morphogens that exhibit self-activation and reciprocal inhibition, and are uniformly distributed and diffuse at different rates lead to the formation of territories of unpredictable shapes and positions but with sharp boundaries (the 'leopard spots'). Here, we have combined the two models and compared the stability of boundaries when the hypothesis of local homeoprotein intercellular diffusion is, or is not, introduced in the equations. We find that the addition of homeoprotein local diffusion leads to a dramatic stabilization of the positioning of the boundary, even when other parameters are significantly modified. This novel Turing/Wolpert combined model has thus important theoretical consequences for our understanding of the role of the intercellular diffusion of homeoproteins in the developmental robustness of and the changes that take place in the course of evolution.

Keywords: Boundary formation; Homeoprotein diffusion; Morphogenesis; Stability.

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Conflict of interest statement

The authors declare no competing or financial interests.

Figures

Fig. 1
Fig. 1. Different models of cell differentiation.
Pure gene competition with small diffusion in the absence of spatial cues classically yields Turing patterns composed of unpredictable abutting territories, while the PI model by Wolpert shows a fixed global patterning driven by the morphogen gradients but with imprecise salt-and-pepper boundaries. The combination of the two phenomena yields precise and predictable patterning.
Fig. 2
Fig. 2. Ambiguous boundary in the absence of non cell-autonomous processes.
Simulations of the system with distinct initial conditions (top row) in the absence of HP diffusion σA=σB=0. For each point, the combination of levels of morphogen gradients either corresponds to an univocal or to an ambiguous region (see supplementary material Model and Fig. S1). We chose a simple two-dimensional square topology to illustrate the phenomenon with gA/gB=1, SA/SB=1 unitary parameters and linear morphogen gradients. (Top) From left to right: random initial values; structured initial values with a small predominance of TA in a centered square; and a large predominance of TA (close to the steady state) in a rectangle that exceeds the ambiguous region. (Bottom) End state of the differentiation process: two differentiated regions emerge with a fuzzy interface. When the initial condition shows a small predominance of TA, a clear bias in this region to A type is found and the salt-and-pepper interface persists. Important predominance of TA leads to a differentiation of all cells in the region into A cells within the ambiguous region. A salt-and-pepper boundary persists away from the region of high initial TA.
Fig. 3
Fig. 3. Precise patterning for competitive systems with spatial cues and HP diffusion.
(Left) Simulation of a ‘neural tube’ with Shh (floor plate) and BMP (roof plate) morphogen sources. Gradients are formed through morphogen diffusion; symmetry is broken by considering a BMP gradient larger than Shh (ratio between BMP and Shh 3:2). The absence of HP diffusion leads to a salt-and-pepper boundary, whereas the presence of HP diffusion (σA=σB=10−2) makes the boundary sharp, precise and smooth (top center, bottom center). Phenomena ensuring that this stabilization and regularization rely only on HP diffusion, even limited, and are heuristically depicted on the right: misplaced cells or irregular boundaries will evolve according to the influence of their neighboring cells to yield a unique possible outcome of the differentiation process.
Fig. 4
Fig. 4. Stability of the boundary to heterogeneous variations of the parameters.
One-dimensional field made of 100 cells, diffusion constants σA=σB=10−4 and linear gradients. (Left) Stationary solution of the neural differentiation process with constant unit values of gA and gB, or heterogeneous values centered at 1 with a variance of 0.2 (20%). (Right) Histograms of boundary positions for 500 realizations of the process, for heterogeneity levels of 100% (variance 1, center) or 20% (right).
Fig. 5
Fig. 5. Schematic description of the model of neural differentiation.
Transcription factor synthesis is driven by external morphogens organized along gradients (which form through diffusion from different morphogen sources) and by the dynamic competition of gene expression. Diffusion of HPs to the nearest neighboring cells takes into account the non cell-autonomous transfer properties.
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