A New Approach to Feedback for Robust Signaling Gradients.

The patterning of many developing tissues is orchestrated by gradients of morphogens through a variety of elaborate regulatory interactions. Such interactions are thought to make gradients robust, that is, resistant to changes induced by genetic or environmental perturbations; but just how this might be done is a major unanswered question. Recently extensive numerical simulations suggest that robustness of signaling gradients cannot be attained by negative feedback (of the Hill's function type) on signaling receptors but can be achieved through binding with nonsignaling receptors (or nonreceptors for short) such as heparan sulfate proteoglycans with the resulting complexes degrading after endocytosis. These were followed by a number of analytical and numerical studies in support of the aforementioned observations. However, evidence of feedback regulating signaling gradients has been reported in literature. The present paper undertakes a different approach to the role of feedback in robust signaling gradients. The overall goal of the project is to investigate the effectiveness of feedback mechanisms on ligand synthesis, receptor synthesis, nonreceptor synthesis, and other regulatory processes in the morphogen gradient system. As a first step, we embark herein a proof-of-concept examination of a new spatially uniform feedback process that is distinctly different from the conventional spatially nonuniform Hill function approach.

Size-normalized Robustness of Dpp Gradient in Drosophila Wing Imaginal Disc.

Exogenous environmental changes are known to affect the intrinsic characteristics of biological organizms. For instance, the synthesis rate of the morphogen decapentaplegic (Dpp) in a Drosophila wing imaginal disc has been found to double with an increase of 5.9°C in ambient temprerature. If not compensated, such a change would alter the signaling Dpp gradient significantly and thereby the development of thewing imaginal disc. To learn how flies continue to develop "normally" under such an exogenous change, we formulate in this paper a spatially two-dimensional reaction-diffusion system of partial differential equations (PDE) that accounts for the biological processes at work in the Drosophila wing disc essential for the formation of signaling Dpp gradient. By way of this PDE model, we investigate the effect of the apical-basal thickness and antero-posterior span of the wing on the shape of signaling gradients and the robustness of wing development in an altered environment (including an enhanced morphogen synthesis rate). Our principal result is a delineation of the role of wing disc size change in maintaining the magnitude and shape of the signaling Dpp gradient. The result provides a theoretical basis for the observed robustness of wing development, preserving relative but not absolute tissue pattern, when the morphogen synthesis rate is significantly altered. A similar robustness considerqation for simultaneous changes of multiple intrinsic system characteristics is also discussed briefly.

Localized Ectopic Expression of Dpp Receptors in a Drosophila Embryo.

Receptor-mediated BMP degradation has been seen to play an important role in allowing for the formation of relatively stable P Mad patterns. To the extent that receptors act as a "sink" for BMPs, one would predict that the localized over-expression of signaling receptors would cause a net flux of freely diffused BMPs toward the ectopic, i.e., abnormally high concentration, receptor site. One possible consequence would be a depression of BMP signaling in adjacent areas since less BMPs are now available for binding with the same normal concentration of receptors at the adjacent areas. However, recent experiments designed to examine this possible effect were inconclusive. In this paper, we investigate the possibility of depression of Dpp signaling outside the area of elevated tkv in a Drosophila embryo by modeling mathematically the basic biological processes at work in terms of a system of nonlinear reaction diffusion equations with spatially varying (and possibly discontinuous) system properties. The steady state signaling morphogen gradient is investigated by the method of matched asymptotic expansions and by numerical simulations.

Membrane-associated non-receptors and morphogen gradients.

A previously investigated basic model (System B) for the study of signaling morphogen gradient formation that allows for reversible binding of morphogens (aka ligands) with signaling receptors, degradation of bound morphogens and diffusion of unbound morphogens is extended to include the effects of membrane-bound non-signaling molecules (or non-receptors for short) such as proteoglycans that bind reversibly with the same morphogens and degrade them. Our main goal is to delineate the effects of the presence of non-receptors on the existence and properties of the steady-state concentration gradient of signaling ligand-receptor complexes. Stability of the steady-state morphogen gradients is established and the time to reach steady-state behavior after the onset of morphogen production will be analyzed. The theoretical findings offer explanations for observations reported in several previous experiments on Drosophila wing imaginal discs.

Aggregation of a Distributed Source in Morphogen Gradient Formation.

In the development of a biological entity, ligands (such as Decapentaplegic (Dpp) along the anterior-posterior axis of the Drosophila wing imaginal disc) are synthesized at a localized source and transported away from the source for binding with cell surface receptors to form concentration gradients of ligand-receptor complexes for cell signaling. Generally speaking, activities such as diffusion and reversible binding with degradable receptors also take place in the region of ligand production. The effects of such morphogen activities in the region of localized distributed ligand source on the ligand-receptor concentration gradient in the entire biological entity have been modeled and analyzed as System F in [1]. In this paper, we deduce from System F, a related end source model (System A) in which the effects of the distributed ligand source is replaced by an idealized point stimulus at the border between the (posterior) chamber and the ligand production region that simulates the average effects of the ligand activities in the production zone. This aggregated end source model is shown to adequately reproduce the significant implications of System F and to contain the corresponding ad hoc point source model, System R of [2], as a special case. Because of its simpler mathematical structure and the absence of any limitation on the ligand synthesis rate for the existence of steady-state gradients, System A type models are expected to be used widely. An example of such application is the recent study of the inhibiting effects of the formation of nonsignaling ligand-nonreceptor complexes [3].