Morphogen gradient formation

Abstract

How morphogen gradients are formed in target tissues is a key question for understanding the mechanisms of morphological patterning. Here, we review different mechanisms of morphogen gradient formation from theoretical and experimental points of view. First, a simple, comprehensive overview of the underlying biophysical principles of several mechanisms of gradient formation is provided. We then discuss the advantages and limitations of different experimental approaches to gradient formation analysis.

Figure 1.

Tissue geometry and simplifications. (A) Gradients in epithelia (left) and mesenchymal tissues (right). Because of symmetry considerations, one row of cells (red outline) is representative for the whole gradient. (B) Magnified view of the red row of cells shown in A. Cells with differently colored nuclei (brown, orange, and blue) express different target genes. (C) A continuum model in which individual cells are ignored and the concentration is a function of the positions x. The morphogen activates different target genes above different concentration thresholds (brown and orange).

Figure 2.

Mechanisms of gradient formation. (A–D) Steady state (solid lines) and pre-steady-state gradients (dashed lines) in the target tissue (x > 0). (Insets) Steady state solutions. The picture below each graph is a schematic view of the respective gradient formation mechanism (the respective differential equation at the bottom). (Blue circles) Nuclei, (concentric circles) degradation (darker color corresponds to higher % degradation). (A) Gradients formed by diffusion do not have a steady state: If there is a constant flux of molecules coming from the source, concentration in the target continuously increases. (B) Steady-state gradients formed by diffusion and linear degradation have exponential shape. (C) Nonlinear degradation leads to the formation of power-law gradients. (D) Power-law gradients formed by cell-lineage transport.

Figure 3.

Range of exponential and power-law gradients. “Range” is defined as the width of a target gene domain x*, responding to a concentration threshold c*. (A) Exponential gradients. (Left panel) A change in c₀ (blue profile) or λ (green profile), with respect to a reference profile (black), affects the shape of the gradient; (right panel) changes in kinetic parameters D (green), k (red), and j₀ (blue) affect the range of the gradient in different ways. Note that changing D can lead to increase or decrease in x*, depending on the concentration threshold c*. (B) Power-law gradients. (Left) The shape depends on: A (blue), x_b (green), and m (magenta). Changes in either affect the range of the gradient. (Right) Changes in the kinetic parameters D, k, j₀, and n qualitatively have similar effects on x* as corresponding effects with exponential gradients (A).

Figure 4.

FRAP in epithelia. (A) (top) A vertical stripe next to the source of a two-dimensional epithelial sheet is bleached. Recovery of fluorescence is measured in the bleached stripe (red rectangle). In the adjacent region (dashed rectangle), there is loss of fluorescence (FLIP). (bottom) Theoretical recovery curves for the red and dashed regions in the top panel, depending on the spreading mechanism (diffusion and degradation [middle] vs. anomalous diffusion [bottom]). The double logarithmic plot (A, bottom) allows to distinguish the time-scales of recovery for different mechanisms. Recovery by anomalous diffusion (dashed red line) has a bigger initial slope of recovery than regular diffusion with linear or nonlinear degradation (solid red lines), respectively, in which the initial slope of recovery is close to 0.5 (dotted black line). (B) (top) A stripe perpendicular to the source is bleached. Recovery is measured in four regions at different distances to the source. Qualitative features of the recovery curves allow distinguishing between underlying mechanisms (B, center and bottom). For instance, in diffusion with linear degradation (B, center), the difference between FI recovery in two different regions continuously increases (arrows), whereas in diffusion with nonlinear degradation (B, bottom), it increases and then decreases (arrows). Depending on endogenous parameter values and experimental noise, this effect may be hard to distinguish in an experiment.