生物数学中的斑图生成
Pattern Formations in Math Biology
Junping Shi 史峻平
Department of Mathematics
College of William and Mary
Williamsburg, VA 23187,USA
中国哈尔滨师范大学数学学院
shij@math.wm.edu
中国上海华东师范大学, 2004年五月二十六日
What is life?
Where did it come from?
问生命究竟为何物 ?
The progress of sciences is based on the curiosity of the
human beings:
•Why do we have day and night?
•Why are there four seasons, rain and clouds?
•Can we fly like birds? (Wright Brothers)
•Can we solve
(Mr. P. Fermat and Mr. A. Wiles)
The biggest question remains:
What is life? Where does it come from?
Can we create life? What is the secret of life?
?
Charles Darwin (1809-1882)
• Origin of Species,
published in 1859
• Theory of Natural
Selection
• Question Darwin
can’t answer:
How could complex
patterns of life be
produced by natural
selection?
Discovery of DNA structure
• Francis Crick
(Cambridge University)
• James Watson
(Harvard University)
1962 Nobel Prize
(1953 “Nature” paper)
http://www.nature.com/nature/dna50/watsoncrick.pdf
Is DNA the secret of life?
• Inside every living creature on earth, there is a
complex molecular DNA code (gene), which
prescribes the creature’s form, growth,
development, and behavior
• Genes are not engineering blueprints, they are
recipes in a cookbook. Recipes are different
from meals
• The mathematical laws of physical and
chemistry control the growing organism's
response to its genetic instructions.
Ian Stewart “Life’s other secret”
Example
淮南桔，而淮北枳
(California’s orange cannot grow
in Florida)
Although with same genes, the
different chemical environment
may produce different creatures.
Diffusion
• u(t,x) : density function of a chemical
• The chemical will move from high density
places to lower density places, this is called
diffusion
• Diffusion is the mechanism of many molecular
or cellular motions
• Diffusion can be described by a heat equation
Reaction-diffusion equations
• Let U(x,t) and V(x,t) be the density functions of
two chemicals or species which interact or react
Alan Turing (1952 Phil. Trans. Roy. Soc. )
“The Chemical Basis of Morphogenesis”
Morphogenesis (from the Greek morphê shape and genesis creation) is
one of three fundamental aspects of developmental biology along with
the control of cell growth and cellular differentiation. Morphogenesis is
concerned with the shapes of tissues, organs and entire organisms and
the positions of the various specialized cell types.
Alan Turing (1912-1954)
One of greatest
scientists in 20th
century
Designer of Turing
machine (a theoretical
computer) in 1930’s
Breaking of U-boat
Enigma, saving battle
of the Atlantic
Initiate nonlinear
theory of biological
growth
http://www.turing.org.uk/
Turing’s idea
Reaction-Diffusion Equation (1)
Reaction Eqution (2)
• A constant solution u(t,x)=u0, v(t,x)=v0 can be a
stable solution of (2), but an unstable solution of (2).
Thus the instability is induced by diffusion.
• On the other hand, there must be stable nonconstant equilibrium solutions which have more
complicated spatial structure.
An example of Turing patterns by James Murray
(author of books: Mathematical Biology)
Emeritus Professor
University of Washington, Seattle
Oxford University, Oxford
http://www.amath.washington.edu/people/faculty/murray/
Why do animals’ coats have
patterns like spots, or stripes?
Murray’s theory
Murray suggests that a single mechanism could be
responsible for generating all of the common patterns
observed. This mechanism is based on a reactiondiffusion system of the morphogen prepatterns, and the
subsequent differentiation of the cells to produce
melanin simply reflects the spatial patterns of
morphogen concentration.
Melanin: pigment that affects skin, eye, and hair color in humans and
other mammals.
Morphogen: Any of various chemicals in embryonic tissue that
influence the movement and organization of cells during
morphogenesis by forming a concentration gradient.
Reaction-diffusion systems
Domain: rectangle
Boundary conditions:
head and tail (no flux),
body side (periodic)
The full reaction-diffusion system:
Solution of the system:
“Theorem 1”: Snakes always have striped (ring)
patterns, but not spotted patterns.
Turing-Murray Theory: snake is the example of b/a is
large.
Snake pictures (stripe patterns)
“Theorem 2”: There is no animal with striped body and
spotted tail, but there is animal with spotted body and
striped tail.
Turing-Murray theory: The body is always wider than the
tail. The same reaction-diffusion mechanism should be
responsible for the patterns on both body and tail. Then if
the body is striped, and the parameters are similar for tail
and body, then the tail must also be striped since the
narrower geometry is easier to produce strips.
Examples: zebra, tiger (striped body and tail), leopard (spotted body and
tail), genet, cheetah (spotted body and striped tail)
Spotted body and striped tail or legs
Cheetah (upper), Okapi (lower)
Tiger (upper), Leopard (lower)
Spotted body and striped tail
Genet (left), Giraffe (right)
Natural Patterns of cos(kx)
cos(x): Valais goat
(single color: f(x)=1, a lot of examples)
Cos(2x): Galloway belted Cow
cos(2x): Giant Panda
Other related researches
Patterns of sea shells
Patterns of tropical fishes
Waves in the Belousov-Zhabotinsky reaction
Boris P. Belousov
(Soviet Union, 1951,left)
Anatol M. Zhabotinsky
(Soviet Union, 1961,right)
Chemical reactions can be oscillatory (periodic)!
Real Turing patterns?
• Turing’s patterns are from a theory. Many
phenomena may be explained by Turing’s
theory, but it does not mean the real biological
mechanism is governed by these equations. (we
can only say maybe)
• Are there such patterns existing in real chemical
reactions? (Biology is more complicated than
chemistry)------It was not known for many years,
so Turing’s theory was only a theory, after all.
But……
Chlorite-Iodide-Malonic Acid (CIMA) reaction
CIMA reaction
spots
CIMA reaction
stripes
Fish skin
Leopard body
Fingerprint
Zebra stripes
Work on CIMA Chemical Reactions
•
•
•
Experimental evidence of a sustained standing Turing-type nonequilibrium chemical
pattern (PRL 1990) V. Castets, E. Dulos, J. Boissonade, and P. De Kepper
(Université Bordeaux I, France)
Quasi-two-dimensional Turing patterns in an imposed gradient (PRL1992) István
Lengyel, Sándor Kádár, and Irving R. Epstein (Brandeis University,USA)
Transition from a uniform state of hexagonal and striped Turing patterns (Nature
1992) Q. Ouyang and H. L. Swinney (University of Texas, USA)
欧阳颀，(Qi Ouyang)
北京大学物理学院长江学者特聘教授，
北京大学理论生物学中心副主任
“反应扩散方程的斑图理论”
上海科技出版社
Theoretical Models of Turing Patterns
• Gierer-Meinhardt model (1972)
Mathematical works: W.-M. Ni, I. Takagi, J. Wei, M. Winters, C. Gui,
X. Chen, M. Kowalczyk, M. Ward, and many others
Gray-Scott Model
• Chemical reaction: U+2V  3V, V P
Turing and I
• From Mathematical point of view,
Turing bifurcation is just primary
ones (bifurcations from constants).
Real Turing patterns may be
generated through secondary or
even more levels of bifurcations.
• Mathematically the secondary
bifurcations are much more difficult
to study.
• For scalar equation (instead of
system in Turing pattern formation),
secondary bifurcations are studied,
and hopefully they can be useful in a
more comprehensive mathematical
theory of Turing patterns.
Bifurcation diagrams of tree or
mushroom structures
Junping Shi, Transactions of AMS, Vol 354, 3117-3154 (2002).
Conclusions
• Pattern formation could answer the
basic question in biology: What is
life? Where does it come from?
• Pattern formation is a multidisciplinary research area between
biology, chemistry, physics and
mathematics.
• Mathematical understanding of
Lord Robert May
pattern formation is still lacking, but
they are important in the progress President of Royal Society
of mathematical biology.
One of pioneers in math biology
Chaos in
An article by Robert May (Feb 2004, Science)
……A paradigmatic account of the uses of mathematics in the natural
sciences comes, in deliberately oversimplified fashion, from the classic
sequence of Brahe, Kepler, Newton: observed facts, patterns that give
coherence to the observations, fundamental laws that explain the patterns.
Tycho Brahe (1546-1601) made the most precise instruments
available before the invention of the telescope. The instruments
of Brahe allowed him to determine the detailed motions of the
planets. In particular, Brahe compiled extensive data on the
planet Mars.
Johannes Kepler (1571-1630), Brahe's assistant. By using Brahe’s
data, he formulated the correct theory of the Solar System: Three
Laws of Planetary Motion. But he was not able to prove them
mathematically.
Isaac Newton (1642-1727) is the greatest scientist ever. He
demonstrated that the motion of objects on the Earth could be
described by three new Laws of motion, and then he went on to show
that Kepler's three Laws of Planetary Motion were special cases of
Newton's three Laws. In fact, Newton went even further: he showed that
Kepler's Laws of planetary motion were only approximately correct, and
supplied the quantitative corrections that with careful observations
proved to be valid.
……Consider the role played by applications of mathematics in sequencing
the human and other genomes. This adventure began with the recognition
of the doubly helical structure of DNA and its implications, an oft-told tale in
which classical mathematical physics played a central role. Brilliant
biochemical advances, allowing the 3 billion-letter-long human sequence to
be cut up into manageable fragments, were a crucial next step. The actual
reassembling of the sequence fragments, to obtain a final human genome
sequence, drew on both huge computational power and complex software,
itself involving new mathematics. The sequence information, however,
represents only the Tycho Brahe stage. Current work on various genomes
uses pattern-seeking programs to sort out coding sequences
corresponding to individual genes from among the background that is
thought to be noncoding. Again, elegant and sometimes novel mathematics
is involved in this Keplerian stage of the “work in progress.” We are only
just beginning, if that, the Newtonian stage of addressing the deeper
evolutionary questions posed by these patterns.
In this Newtonian quest, mathematical models will help in a different way
than in earlier stages. Various conjectures about underlying mechanisms
can be made explicit in mathematical terms, and the consequences can be
explored and tested against the observed patterns.
21st century belongs to biology.
Who is the next Newton in biology?
• It could be YOU!
•
I know not what I appear to the world, but to
myself I seem to have been only like a boy
playing on the sea-shore, and diverting
myself in now and then finding a smoother
pebble or a prettier shell, whilest the great
ocean of truth lay all undiscovered before
me.
•
我不知道世人對我的看法如何，我只覺得自己
好像是個在海濱遊戲的男孩，有時為了找到一
塊光滑的石子或比較美麗的貝殼而高興，而真
理的海洋仍然在我的前面而未被發現。
----Isaac Newton