Scale invariance of morphogen-mediated patterning by flux optimization

Abstract

Abstract-The development of an organism from a single cell into an adult requires exquisite control of spatial patterning so that the correct tissues are formed at the correct place and at the correct time. A number of molecular factors that regulate spatial patterning of tissues during development have been identified, however, relatively little is known about how they form consistent spatial patterns of cell signaling or gene expression that scales in proportion to the size of the tissue being patterned. In the model organism Drosophila melanogaster, embryonic morphogen gradients establish the anterior/posterior (AP) and dorsal/ventral (DV) axes to establish the boundaries of gene expression that are further subdivided into distinct cell types. Both AP and DV patterning systems exhibit a high degree of scaling, however, they have evolved distinct mechanisms to achieve scaling. Our analysis herein shows that recently discovered data for Bicoid (the principle AP morphogen) embryonic patterning is consistent for a mechanism that optimizes the flux of Bcd molecules into the anterior end. This mechanism is distinct from mechanisms suggested in other systems that target biophysical properties that regulate morphogen range, thereby stretching or shrinking the morphogen distribution accordingly. Instead, flux optimization changes the morphogen peak levels to provide "approximate" scaling over a large region of space and perfect scaling at a single spatial location. While remarkably simple, this mechanism of scale-invariance is sufficient for increases and decreases in tissue size by 50% and confers less than 10% error in spatial patterning.

Growth control in the Drosophila wing disk

“…In effect, Pent fine-tunes the balance between local and long-range Dpp signaling, and both dally and pent expression is repressed by Dpp. It is known how spatial scaling of solutions of reaction-transport models such as Equation ( 1) can be obtained by modulation of transport or kinetic parameters (Othmer & Pate, 1980;Umulis & Othmer, 2013b), and a model for active modulation of the Dpp gradient based on Dpp, Tkv, Pent, and Dally has been proposed to explain the observed scaling of the Dpp distribution (Ben-Zvi et al, 2011;Umulis & Othmer, 2013a). Further computational studies could serve to predict the profile of other genes under different conditions and compare them with the experimental observations (Bollenbach et al, 2008).…”

Mechanisms of scaling in pattern formation

“…An alternative interpretation of the volume-dependent production of Bicoid scaling is that the input flux is optimized to the levels needed for adequate scaling (Umulis and Othmer, 2013). Solution of Eqn 7 with input flux optimization produces profiles that provide adequate, but not perfect, scaling of Bicoid (Fig.…”