Two models of self-maintaining systems are presented. The equation of motion for the surface of the system in the first chapter is a function only of the surface itself. Moreover, for the sake of simplicity, I assume that this is a...
moreTwo models of self-maintaining systems are presented. The equation of motion for
the surface of the system in the first chapter is a function only of the surface itself.
Moreover, for the sake of simplicity, I assume that this is a function of the local
mean curvature only. This equation of motion leads to a stationary shape of the
system (circle, sphere) with a determined size, but it also destabilizes the shape.
To counteract this fact, a term which behave like a surface tension is added.
The model of a self-maintaining system in the second chapter is based on reaction-diffusion
equations. Consider a system surrounded by a nutrient solution. This
system essentially consists of a building-material A, which I assume to be homogeneous
and incrompressible. The nutrient N is diffussing in- and outside the
system and has a constant concentration at infinity. Inside the system there are
two chemical reactions. One from the nutrient to the building-material and the
other from the building-material to a decay-material Z. The decay-material do I
not consider, cause it should leave the system quickly enough. This process leads
also to a stationary shape (circle, sphere) of the system with a determined size.
Again the surface tension stabilizes the shape.
Although the two models of self-maintaining systems look very different, they
have some common properties. If the surface tension is strong enough, the stationary
shape is stable. Decreasing the surface tension leads to an instability of
the surface. The dominant mode is ℓ=2. This means in two dimensions the trigonometrical
functions sin(2 φ) and cos(2 φ) and in three dimensions the spherical
harmonics Y₂,ₘ.
But what happens in the unstable region? The tools to answer this question are
the so-called bifurcation analysis (up to second and third order) and numerical
calculations of the model equations.
For both models there exists - in three dimensions - a certain parameter region
in which the instability drives the system to a division (like a cell division). But
for the model in the first chapter there exist other parameter regions, too, where
the instability drives the system to new stable shapes which look like oblate or
prolate spheres (like the shape of erythrocytes and rod-shaped bacteria).
It should be noted, that such models of self-maintainig systems, although seeming
oversimplified compared with real living cells, share some essential properties with
them.
