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 "expansion-repression" 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.

Fig. 1.

The expansion-repression mechanism. (A) Scheme of the expansion-repression mechanism: Morphogen signaling represses the production of the expander. The expander in turn expands the morphogen gradient. (B) Dynamics of the expansion-repression mechanism. Initially (time = t₁), the morphogen gradient is sharp, enabling expression of the expander in a wide domain (orange bar). At later times (time = t₂), the expander accumulates, the gradient expands, and consequently the expander production domain shrinks. At steady state, the expander has accumulated and the gradient is wide enough to repress expander production everywhere. Specifically, the profile at x = L is slightly higher than T_rep, which corresponds to a complete repression of expander production. (C) Screen for scaling in the expansion-repression topology: Shown is the distribution of parameters corresponding to systems that scaled (total of 2,372 sets). Such systems had two main characteristics: (i) flat expander profile (λ_E > L, with λ_E the expander decay length) and (ii) the initial profile is sharp enough to induce the expander. L_min is the length for which M(L) = T_rep when [E] = 0. (D and E) Examples of morphogen gradients defined by the expansion-repression topology: A gradient that does not scale is shown in D whereas a gradient that does scale is shown in E. The profiles are shown for fields of length L (black curve) and 2L (gray curve). Distance from the source (x axis) is shown as a function of the relative position x/L. Profiles in nonscaled coordinates are shown in the Inset. Red bars mark the change in the positions of three thresholds used to measure scaling. (F and G) Scaling in a growing field: Simulation of the expansion-repression system in a field that grew from 100 to 200 μm during patterning is shown in F for four subsequent time points. The same simulations are shown in G but the gradients are scaled with the size of the field. Parameters for numerical solutions are given in the SI Text.

Fig. 2.

Scaling in a morphogen-inhibitor system. (A) The morphogen-inhibitor system: A morphogen M binds an inhibitor I forming an inert complex MI, which is not capable of binding the receptor R. The morphogen is degraded through receptor-mediated endocytosis. Morphogen–receptor signaling represses inhibitor production. (B) Screen for systems that scale: Shown is the distribution of parameter sets corresponding to systems that scaled (total of 309 sets). Such systems were characterized by (i) flat profile of the inhibitor λ_I > L, where λ_I denotes the inhibitor decay length, and (ii) the system length being large enough to allow induction of the inhibitor. (C and D) Morphogen gradients established by a morphogen-inhibitor system: (C) a gradient that does not scale and (D) a gradient that scales. Notations are as in Fig. 1 D and E. (E) Schematic representation of morphogen and a competitive extracellular inhibitor. The morphogen M competes with the inhibitor I for binding the receptor R. The morphogen–receptor complex is able to signal and represses production of the inhibitor. The morphogen is degraded through receptor-mediated endocytosis. (F) Scaling in a competitive-inhibitor system: Shown are solutions for fields of length L (black curve) and 2L (gray curve). Distance from the source (x axis) is a function of the relative position x/L. Nonscaled coordinates are shown in the Inset. Parameters for numerical solutions are given in the SI Text.

Fig. 3.

The expansion-repression mechanism as an integral feedback controller. Morphogen gradient (system output) is measured by production of the expander in responding cells (sensor). The region where the expander is repressed (measured output) is compared with L, the region where the expander should be expressed when the gradient is scaled (reference signal). The expander is produced in the region corresponding to the difference between the measured error and the reference (current error). Its accumulation is the time integrator (controller). Following production, the elevated expander level (system input) increases the length scale of the gradient (system), which expands the gradient (system output). This process ends when the error is zero; i.e., the expander is not produced by any cell.