[5, 6]. The lowest common denominator of all of these systems is
that they all belong to the realm of non-equilibrium thermody-
namical processes [7] subject to complex bifurcations characteriz-
ing their dissipative dynamics. During several decades, many
scientists have looked for the best physical and mathematical tools
to adopt to describe these dynamics appropriately. All of these
systems have, depending on the spatiotemporal scale involved, a
discrete structure starting from atoms up to molecules, cells, cells
aggregates, and even complex living beings populations. In this
zooming-out tale, a quantum description rapidly seems to fade
out for a classical and sometimes coherent [8, 9] one, possibly
complicated by a stochastic flavor which however can still be seen
as an echo of the underlying smaller scales quantum probabilistic
dynamics. Such a large-scale description can be quite successfully
obtained by using ordinary or partial differential equations (the
latter seen as a continuum limit for discrete systems). Variations
on the theme as stochastic or delay differential equations, as well as
integrodifferential mathematical descriptions or maps and cellular
automata, are possible (although less popular) options that can be
successfully used too (for a complete review on some of these
mathematical tools, we refer to refs.10, 11). Partial differential
equations of Reaction-Diffusion (RD) class [12] have a long-lasting
and fruitful history in treating non-equilibrium phenomena in
chemistry and biology.
The pioneer work for using RD equations in Biology is
described in the 1952 classic article “The Chemical Basis of Mor-
phogenesis” by Alan Turing [13]. Here, by using as actors some
chemicals, known as morphogens (equivalently described as activa-
tors vs. inhibitors), a possible “toy model” explanation for animal
coat patterns appearance as well as developmental processes as
Hydra tentacles, for instance, is discussed. Turing article has been
widely recognized as the source for the interpretative paradigm
which has driven plenty of quantitative biology research in the
past 65 years.
However, another 1952 article would have shared with Tur-
ing’s the title of one of the most influential articles in quantitative
biology, i.e., the classical Hodgkin and Huxley study (Nobel Prize
awarded) on action potential propagation in nerves [14]. This work
combined both experiments and equations (of reaction-diffusion
class) to give quantitative predictions on the bioelectrical behavior
of a squid giant axon.
The wise reader would be at this stage tempted to ask at which
level and to which extent such a quantitative description of living
beings meets all the decades (in some cases centuries) of studies in
Biology and Physiology, Chemistry, and Biochemistry as well as in
Medicine. Stated—say—40 years ago, such a question would have
received an “almost none” answer. The investigation tools and the
methodologies of all of these branches of Science were at that time204 Christian Cherubini et al.