Modeling homeoprotein intercellular transfer unveils a parsimonious mechanism for gradient and boundary formation in early brain development

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Abstract

Morphogens are molecules inducing morphogenetic responses from cells and cell ensembles. The concept of morphogen is related to that of positional value, as the generation of morphological and physiological characteristics is function of position. Based on the observation that homeoproteins, a category of transcription factors with morphogenetic functions, traffic between abutting cells and, very often, regulate their own expression, we develop here a biophysical model of homeoprotein propagation and study the associated mathematical equations.

This mode of cell signaling can generate domains of homeoprotein expression. We study both the transient and steady-state regimes and, in this latter regime, we obtain various morphogenetic gradients, depending on the value of some parameters, such as morphogen synthesis, degradation rates and efficiency of intercellular passage. The same equations, applied to pairs of homeoproteins with auto-activation and reciprocal inhibition properties, account for border formation. They also allow us to compute how specific perturbations can either be buffered or lead to modifications in the position of borders between adjacent areas.

The model developed here, based on experimental data, and avoids theoretical obstacles associated with pluricellularity. It extends the idea that Bicoid homeoprotein is a morphogen in the fly embryo syncitium to most homeoproteins and to pluricellular systems. Because the position of borders between brain areas is of primary physiological importance, our model might lead to original views regarding epigenetic inter-individual variations and the origin of neurological and psychiatric diseases. In addition, it provides new hypotheses regarding the molecular basis of brain evolution.

Section snippets

Background

In pluricellular organisms, positional information is necessary to all developmental processes. In the neuroepithelium, positional information regulates the fate of territories, the place of borders between these territories, axonal guidance and synapse formation. It is not clear, as yet, how positional information is generated and how cells respond to this information. Within the proposed models, distinction should be made between syncitial (i.e. fly embryos or muscle fibers) and pluricellular

Mathematical models of gradient formation

We will reason on one row of cells. However, if there is a property of invariance in one direction, the conclusions can be extended to two-dimensional epithelia. The scenario is based on unidirectional morphogen fluxes across a territory not expanding trough cell division. Gradient initiation requires that an inducing step (inducer XA) triggers in a first cell the synthesis of homeoprotein A. Since A is a transcription factor with auto-catalytic activity, this synthesis labels the cell. The

Generating and maintaining gradients

We present in Fig. 2 various types of gradients generated for different values of parameter α, which measures the rate of decay from cell to cell. When α<1, gradients decay almost exponentially and after a few cells, homeoprotein concentration is already a few percent of its initial value. Interestingly, for α=1, the decay is linear, while for α>1, the gradient concentration does not decrease to zero, rather after a few cell, it converges exponentially to the asymptotic value KA(γ+γt).

In the

Messenger protein concept

The model presented here is based on the messenger protein concept (Prochiantz, 2000; Prochiantz and Joliot, 2003). This concept has been proposed following the observation that homeoprotein transcription factors signal between cells due to their unique intercellular transfer properties (Maizel et al., 2002; Joliot and Prochiantz, 2004). Taking Pax6 as an example, it was shown that blocking its extracellular activity hampers the extension of the eye territory (Lesaffre et al., 2007). In that

Conclusions

We have shown that morphogenetic gradients can be generated by local homeoprotein transfer in absence of long range diffusion. This model is parsimonious and allows us to explain the formation of boundaries and to evaluate its robustness. Although derived under very different considerations, our model lead to equations that share similarities with standard reaction–diffusion models.

Gradient and boundary formation has various consequences in the field of development and evolution. We think that

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    D. Holcman is supported by the program “Chaire d’Excellence”. D. Holcman incumbent to the Madeleine Haas Russell Career Development Chair.

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