International Journal of Bifurcation and Chaos, Vol. 12,
No. 9 (2002) 19371938
© World Scientific Publishing Company
EDITORIAL
In the past few years, biology has changed  it requires the
attention of skilled theoreticians to understand the behavior of huge and
complex genetical and biochemical maps that summarize results obtained in
molecular and cellular biology [Focus Issue, 2001; Topical Issue, 2000].
A proper understanding can no longer be achieved by standard biological
methods only. But, most theoreticians are unaware that biology needs them,
or perhaps, they do not know where exactly their skills are required. In
addition, there is a fear of not being able to overcome the scientific and
linguistic barriers of contemporary biology. ("I don't want to learn Russian,"
joked one physicist when exposed to the biological jargon which is apparently
as difficult as Russian.) Thus, the majority of theoreticians continue to
develop details of their own science, while the Big Problems in biology
lie in wait for their tools and knowledge. However, there are some challenging
examples of high level cooperation, resulting in Nobel prize winners, for
example, theoretician A. Huxley and biologist A. Hodgkin (nerve
pulse), and theoretician F. Crick and biologist J. Watson (base
pairing in DNA).
Spatial behavior is one of the important fields of nonlinear science. This
theme section describes five biological problems related to the spatial
behavior at cellular and subcellular levels.
The first paper by Wellner et al. analyzes waves in excitable
media (for example, in cardiac muscle) that propagate at the expense of
the energy stored in the medium (in cardiac cells). The method of finite
renormalization is applied to analyze effects of wave geometry on propagation,
in particular, to obtain the correct eikonal equation without restriction
on dispersion or on the ratio of time scales.
Waves in excitable media (autowaves) display unusual properties. For example,
they annhiliate when colliding, contrary to classical waves (electromagnetic,
acoustical, etc) described in physical textbooks. These waves are strongly
nonlinear, and therefore, are not easy to analyze mathematically. The result
obtained is also of some intrinsic mathematical interest because in nonlinear
media, exact results are few and far between. The paper is written in a
didactic manner, making an easy entrance to the field.
The next paper (by Starmer) investigates initiation of spiral waves in
excitable media. The aim is to understand why the death rate was increased
by class one cardiac antiarrhythmic drugs that were widely used in clinics.
As two large CAST clinical trials discovered, about 100 additional deaths
per day in US were induced by these drugs (every year they caused six times
more deaths than the 11 September 2001 tragedy). The drugs exhibit desired
properties in a single cell, but their effect in a spatial system (tissue)
appears fatal.
Blood coagulation is analyzed experimentally and theoretically in two
subsequent papers by Ataullahanov et al. In their experiment, spatial
effects (propagation of the cloth) were found to play a crucial role, contrary
to the classical approach, which focuses attention on localized biochemistry
only. Blood coagulation was found to be an autowave, i.e. it propagates
like a wave in excitable media with the same characteristic properties (annihilation
after collisions, constant velocity, etc). But this wave stops after propagation
for a distance of several mm. A mathematical model was constructed and
analyzed, resulting in the prediction of new ways to regulate blood coagulation,
and possibly new drugs.
DNA folding dynamics is described in Noguchi's paper. In a living cell,
long chains of DNA are tightly packed. But they must be unpacked when information
reading is required. The packing is very efficient: the natural DNA chain
length may be diminished by up to 5 orders of magnitude. There is theoretical
evidence that it is a first order phase transition, while most previous
experimental studies had indicated that this transition was always a continuous
one. Noguchi showed numerically that semiflexible homopolymer chains fold
through various paths into collapsed toroidal states stochastically, passing
through kinetically trapped metastable states, like rings or rods.
Pattern formation in cultured endothelial cells is described by Takagi
et al. Cultured endothelial cells usually form a homogeneous cellular
monolayer. After several days of cultivation, the homogeneity is destroyed,
and a pattern of capillary network is created. This is an initial stage
of formation of a network of blood vessels. Pattern formation on a much
shorter time scale, of several minutes only, is described by Takagi et
al. Endothelial cells can form concentric rings, spirals or parallel
lines, when subjected to vibration applied inside a very narrow time window
of 8 min only. After this period of time, cells are anchored, and no
pattern formation can be induced. The Faraday instability leads to a method
to characterize cell attachment. It also permits the production of heterogeneous
cultures with several cell types, with a well controlled heterogeneity.
This can be used to study heterotypic cell interactions in vitro.
Maini et al. constructed and analyzed models of wound healing,
a very complex phenomenon, where many elementary processes interacted on
different scales. Wound healing is also a key process in understanding
biological pattern formation.
The mathematical meaning of the main model is clear enough. It involves
four variables: two types of cells  fibroblasts and myofibroblasts;
and two types of molecules  generic growth factor and an extracellular
matrix. These quantities obey the general conservation equation

Q
t

= f_{Q}
+ Ñ · J_{Q} , 

(1) 
where Q is the quantity in question, the first term on the righthand
side models production and degradation, and the second term models transport
(motion with flux J_{Q}).
One of the components of transport, chemotaxis, is especially interesting.
It is described by the term nÑ·a,
where n is cell density and a is the growth factor concentration.
Transport terms like this are usually not encountered in physical problems.
The presented models capture key features of wound healing. Cancerous
tumors share many common features with wound healing, and hence the results
might be interesting for cancer research as well.
These articles demonstrate that new effects arising in spatially distributed
biological systems might be extremely important (e.g. Starmer's paper),
and give evidence that concentrating on intracellular events only, which
is typical for contemporary molecular biology, is not enough. The first
paper is an example of precise results that can be offered by nonlinear
dynamics. The remaining papers are open for professional nonlinear dynamics
analysis. They may help to introduce theoreticians to exciting biological
problems.
Pierre Coullet and Valentin Krinsky
Institut NonLineaire de Nice, CNRS,
France