Contrary to popular opinion, the School of Mathematics is home to some of the
University's most innovative programs, colorful personalities, & perplexing
research problems
For centuries, scientists have been stumped by the hydra's ability
to regenerate its severed head. The phenomenon, first observed in
1744, is the oldest known documented case of morphogenesis.
But thanks to Professor Wei-Ming Ni—whose recent work includes
a mathematical verification of the theory explaining the hydra's
regenerative powers—the biological and mathematical mystery seems
to have been solved.
One of the central issues in developmental biology is understanding
how structure and form emerge from stable, homogeneous environments,
explains Ni. Although biologists know that genes play a role, they
are still searching for the mechanism that propels them.
Experiments prove that, under certain conditions, a piece of cell
tissue taken from an area near the hydra's head will develop into
a new head when placed in a lower position on the body. Researchers
theorize that the diffusion of certain chemicals within the hydra's
body stimulates that regenerative response.
Although diffusion seems to be a smoothing process and thus shouldn't
generate patterns, University of Manchester professor Alan Turing
predicted in 1952 that two interacting substances with different
rates of diffusion could result in a nonhomogeneous distribution.
According to his hypothesis, known today as “diffusion-driven
instability,” small perturbations under different diffusion
rates can result in concentrations of reactants. These concentrations
literally can be conceptualized as stripes or spots, similar to
the markings in the coats of animals. In the case of the hydra,
the concentrations grow into heads.
These concentrations, or “Turing patterns,” can be used
to explain the existence of many patterns in nature, from morphogenesis
to population dynamics.
However, Turing and his contemporaries lacked the mathematical
tools to solve this puzzle. During the 1960s and 1970s, when nonlinear
equations were first widely used, scientists and mathematicians
around the world eagerly explored the mathematics of Turing patterns
in search of the mechanism that creates them.
The Gierer-Meinhardt model, developed in 1972, was one of the most
successful to emerge from Turing's hypothesis. This model described
a state in which two reactants—a slowly diffusing activator and
a rapidly diffusing inhibitor—are in a state of equilibrium.
Following small perturbations, the activator begins to diffuse.
The inhibitor is then activated and quickly spreads out evenly over
the spatial domain, confining the activator to an area near its
original location and creating “peaks” of the activator
—the concentration that Turing predicted more than 40 years ago.
In a different direction, Shigesada, Kawasaki, and Teramoto developed
the theory of cross-diffusion, which takes into account the pressures
created by two competing species. Their objective was to bring the
models closer to reality by incorporating nonlinearities and cross-diffusion.
By further developing modern theories and methods in nonlinear
partial differential equations, Ni and his collaborators mathematically
proved the existence of patterns of concentration, or “spike-layers,”
in mathematical terms.
In particular, Ni has proved that spike-layers exist within the
Gierer-Meinhardt model and that they are stable under certain conditions.
His work established a mathematical precedent for the proof and
mapping of spike-layers. By unlocking the mathematical key to this
model, he has opened the door to infinitely more complex problems
involving pattern formation.
Ni's pioneering work in spatial pattern formation can be applied
to a range of disciplines, including mathematics, biology, chemistry,
and physics. For instance, he is currently working to understand
pattern formation in cross-diffusion, a complex but more realistic
process that takes into account the interspecific pressures created
by competing species in population dynamics.
Still, he cautions that solutions to some complex problems are
currently beyond the realm of possibility; however, he acknowledges
that each step forward translates into great strides in the mathematical
world.
"We try harder and [develop] more sophisticated models as
we gain experience with each earlier model,” says Ni.
