Negative Feedback in Morphogen Gradients (original) (raw)

Robustness of signaling gradient in drosophila wing imaginal disc

Discrete and Continuous Dynamical Systems - Series B, 2011

Quasi-stable gradients of signaling protein molecules (known as morphogens or ligands) bound to cell receptors are known to be responsible for differential cell signaling and gene expressions. From these follow different stable cell fates and visually patterned tissues in biological development. Recent studies have shown that the relevant basic biological processes yield gradients that are sensitive to small changes in system characteristics (such as expression level of morphogens or receptors) or environmental conditions (such as temperature changes). Additional biological activities must play an important role in the high level of robustness observed in embryonic patterning for example. It is natural to attribute observed robustness to various type of feedback control mechanisms. However, our own simulation studies have shown that feedback control is neither necessary nor sufficient for robustness of the morphogen decapentaplegic (Dpp) gradient in wing imaginal disc of Drosophilas. Furthermore, robustness can be achieved by substantial binding of the signaling morphogen Dpp with nonsignaling cell surface bound molecules (such as heparan sulfate proteoglygans) and degrading the resulting complexes at a sufficiently rapid rate. The present work provides a theoretical basis for the results of our numerical simulation studies.

Regulatory feedback on receptor and non‐receptor synthesis for robust signaling

Developmental Dynamics, 2020

Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by genetic or environmental perturbations. How this might be done is still not completely understood. Previous numerical simulations on reaction‐diffusion models of Dpp gradients in Drosophila wing imaginal disc have showed that feedback (of the Hill function type) on (signaling) receptors and/or non‐(signaling) receptors are of limited effectiveness in promoting robustness. Spatial nonuniformity of the feedback processes has also been shown theoretically to lead to serious shape distortion and a principal cause for ineffectiveness. Through mathematical modeling and analysis, the present article shows that spatially uniform nonlocal feedback mechanisms typically modify gradient shape through a shape parameter (that does not change with location). This in turn enables us to uncover new multi‐feedback instrument for effective promotion of robust signaling gr...

Membrane-Associated Non-Receptors and Morphogen Gradients

Bulletin of Mathematical Biology, 2007

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.

Internalization and end flux in morphogen gradient formation

Journal of Computational and Applied Mathematics, 2006

Two simple reaction-diffusion systems of partial differential equations and auxiliary conditions governing the activities of diffusible ligands such as Dpp in anterior-posterior axis of Drosophila wing imaginal discs were previously formulated and investigated by numerical simulations in [Developmental Cell 2 (2002) 785-796]. System B focuses on diffusion, reversible binding with receptors and ligand-mediated degradation for a fixed receptor concentration uniform in time and space. System C extended this basic but meaningful model to allow for endocytosis, exocytosis and receptor synthesis and degradation. The present paper provides a mathematical underpinning for the computational studies of these two systems and some insight gained from our analysis. We will see for instance that the two boundary value problems governing the steady state for the two systems are identical in form. This result will enable us to avoid dealing with internalization explicitly when we investigate other complex morphogen activities such as the effects of (1) feedback and (2) diffusible and non-diffusible molecules competing for ligands and receptors to inhibit cell signaling and pattern formation. The principal contribution of the present work pertains to the extension of System C to allow for a ligand flux at the source end. The more general model has many significant consequences including the removal of a limitation of previous models on ligand synthesis rate for the existence of steady state behavior. Linear stability of the corresponding steady state behavior is established. While the actual decay rate of transients is less accessible in this new model, it is possible to obtain tight upper and lower bounds for the decay rate in terms of the (effective) degradation rate of the receptors and that of the ligand-receptor complexes.

The role of feedback in the formation of morphogen territories

Mathematical Biosciences and Engineering, 2008

In this paper, we consider a mathematical model for the formation of spatial morphogen territories of two key morphogens: Wingless (Wg) and Decapentaplegic (DPP), involved in leg development of Drosophila. We define a gene regulatory network (GRN) that utilizes auto-activation and cross inhibition (modeled by Hill equations) to establish and maintain stable boundaries of gene expression. By computational analysis we find that in the presence of a general activator, neither auto-activation, nor cross inhibition alone are sufficient to maintain stable sharp boundaries of morphogen production in the leg disc. The minimal requirements for a self-organizing system are a coupled system of two morphogens in-which the auto-activation and cross-inhibition have Hill coefficients strictly greater than one. In addition, the GRN modeled here describes the regenerative responses to genetic manipulations of positional identity in the leg disc.

A New Approach to Feedback for Robust Signaling Gradients

Studies in Applied Mathematics, 2014

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.

Localized Ectopic Expression of Dpp Receptors in a Drosophila Embryo

Studies in Applied Mathematics, 2009

Receptor-mediated bone morphogeneric protein (BMP) degradation has been seen to play an important role in allowing for the formation of relatively stable morphogen signaling 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 diffusing BMPs toward the ectopic, i.e., abnormally high concentration, receptor site. One possible consequence would be a depression of BMP signaling in adjacent areas because 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 BMP signaling outside the area of elevated receptor level 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.

Spatially Distributed Morphogen Production and Morphogen Gradient Formation

Mathematical Biosciences and Engineering, 2005

Partial differential equations and auxiliary conditions governing the activities of the morphogen Dpp in Drosophila wing imaginal discs were formulated and analyzed as Systems B, R, and C in [7] [9] [10]. All had morphogens produced at the border of anterior and posterior chamber of the wing disc idealized as a point, line, or plane in a one-, two-, or three-dimensional model. In reality, morphogens are synthesized in a narrow region of finite width (possibly of only a few cells) between the two chambers in which diffusion and reversible binding with degradable receptors may also take place. The present investigation revisits the extracellular System R, now allowing for a finite production region of Dpp between the two chambers. It will be shown that this more refined model of the wing disc, designated as System F, leads to some qualitatively different morphogen gradient features. One significant difference between the two models is that System F impose no restriction on the morphogen production rate for the existence of a unique stable steady state concentration of the Dpp-receptor complexes. Analytical and numerical solutions will be obtained for special cases of System F. Some applications of the results for explaining available experimental data (to appear elsewhere) are briefly indicated. It will also be shown how the effects of the distributed source of System F may be aggregated to give an approximating point source model (designated as the aggregated source model or System A for short) that includes System R as a special case. System A will be analyzed in considerable detail in [6], and the limitation of System R as an approximation of System F will also be delineated there.