Scaling of morphogen gradients by an expansion-repression integral feedback control

Abstract

Despite substantial size variations, proportions of the developing body plan are maintained with a remarkable precision. Little is known about the mechanisms that ensure this adaptation (scaling) of pattern with size. Most models of patterning by morphogen gradients do not support scaling. In contrast, we show that scaling arises naturally in a general feedback topology, in which the range of the morphogen gradient increases with the abundance of some diffusible molecule, whose production, in turn, is repressed by morphogen signaling. We term this mechanism "expansionrepression" and show that it can function within a wide range of biological scenarios. The expansion-repression scaling mechanism is analogous to an integral-feedback controller, a key concept in engineering that is likely to be instrumental also in maintaining biological homeostasis.development | patterning | theoretical biology M ulticellular organisms develop through a sequence of patterning events in which uniform fields of cells differentiate into patterned tissues and organs. Positional information is commonly encoded by morphogen gradients, whereby a signaling pathway is activated in a spatially graded manner over a field of cells and induces distinct gene expression domains in a concentration-dependent manner. In the standard paradigm, a signaling molecule-the morphogen-is secreted from a local source and diffuses through the tissue, establishing a gradient that peaks at the source. The gradient is shaped further by factors that impact on morphogen diffusion or degradation. The activity and abundance of these regulators is often influenced by the morphogen signaling itself through a variety of feedbacks (1-6).Developing individuals of the same species vary in size. However, the proportions of their body plans are kept remarkably constant. To achieve this proportionate patterning, morphogen gradients ought to scale with the size of the tissue. Despite intense research, little is known about the means by which field size is measured and how this information feeds back to shape the morphogen gradient (7)(8)(9)(10)(11)(12)(13)(14)(15).Theoretical studies have shown that scaling is not a general property of morphogen models but requires specialized mechanisms. Proposed mechanisms include (i) two diffusible molecules that emanate from opposing poles and define the activation profile through their ratio; (ii) a mechanism that maintains a constant, size-independent receptor number; and (iii) a topology that restricts morphogen degradation to only the distal edge of the field (a "perfect sink") (9-11, 13, 15). Whereas those mechanisms likely apply in some cases, they invoke fine-tuned interactions, posing significant constraints on the design of the patterning network (SI Text).Here we show that scaling emerges as a natural consequence of a feedback topology, which we term "expansion repression". In this case, patterning is defined by a single morphogen, whose profile is shaped by a diffusible molecule, the "expander". The expander functions, directl...

Mechanisms of scaling in pattern formation

“…Examples of active modulator mechanisms include the expander repressor feedback motif (Ben-Zvi and Barkai, 2010;Ben-Zvi et al, 2011) and other examples in which feedback between multiple modulator species or multiple morphogens modulate the reaction and transport rates to produce proper scaling. The desired outcome of active modulation is the scaling of diffusion and/or reactions as before, but in active modulation, feedback on the modulator dynamics provides a dynamic regulation of the biophysical parameters to achieve the desired scaling of components.…”

Positional information and patterning revisited

“…There are several proposed mechanisms such as two diffusible molecules that emanate from opposing poles and define the activation profile through their ratio. A model based on expansionrepression by Ben-Zvi and Barkai (2010) suggests that scaling emerges as a natural consequence of a feedback topology. Patterning is defined by a single morphogen, whose profile is shaped by a diffusible molecule, the "expander".…”

Dynamic scaling of morphogen gradients on growing domains

“…1b). Different models have been proposed to explain the scaling of morphogen gradients in the various systems where scaling has been analysed [9][10][11][12][13][14][15][16][17] . However, these models have remained controversial because of conflicting experimental observations 16,[18][19][20][21] .…”

“…These adjustments could be achieved either by passive or active modulators 16 . An active modulation mechanism for the Dpp gradient in the wing imaginal disc has recently been proposed 17 as an extension of a previously proposed expander-repressor scheme 13 . However, as recently noted, for this expander-repressor mechanism, the amplitude of the morphogen-receptor complex decreases more than twofold for e d…”