  The fundamental biological processes of development of tissues and organs in
multicellular organisms is governed by various signaling molecules, which are
called morphogens. It is known that spatial and temporal variations in
concentration profiles of signaling molecules, which are frequently referred as
morphogen gradients, lead to cell differentiation via activating specific genes
in a concentration-dependent manner. It is widely accepted that the
establishment of the morphogen gradients involves multiple biochemical
reactions and diffusion processes. One of the critical elements in the
formation of morphogen gradients is a degradation of signaling molecules. We
develop a new theoretical approach that provides a comprehensive description of
the degradation mechanisms. It is based on the idea that the degradation works
as an effective potential that drives the signaling molecules away from the
source region. Utilizing the method of first-passage processes, the dynamics of
the formation of morphogen gradients for various degradation mechanisms is
explicitly evaluated. It is found that the linear degradation leads to a
dynamic behavior specified by times to form the morphogen gradients that depend
linearly on the distance from the source. This is because the effective
potential due to degradation is quite strong. At the same time, the nonlinear
degradation mechanisms yield a quadratic scaling in the morphogen gradients
formation times since the effective potentials are much weaker.
Physical-chemical explanations of these phenomena are presented.
