生物数学中的斑图生成Pattern Formations in Math Biology

Junping Shi 史峻平Department of MathematicsCollege of William and MaryWilliamsburg, 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

The biggest question remains:

What is life? Where does it come from?

Can we create life? What is the secret of life? ?

(Mr. P. Fermat and Mr. A. Wiles)

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

• 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 non-constant equilibrium solutions which have more complicated spatial structure.

Reaction-Diffusion Equation (1) Reaction Eqution (2)

An example of Turing patterns by James Murray (author of books: Mathematical Biology)

Emeritus Professor University of Washington, SeattleOxford University, Oxfordhttp://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 reaction-diffusion 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: rectangleBoundary 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 andstriped 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

stripes

Fingerprint Zebra stripes

CIMA reaction spots

Fish skin Leopard body

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.

Junping Shi, Transactions of AMS, Vol 354, 3117-3154 (2002).

Bifurcation diagrams of tree or mushroom structures

Conclusions

• Pattern formation could answer the basic question in biology: What is life? Where does it come from?

• Pattern formation is a multi-disciplinary research area between biology, chemistry, physics and mathematics.

• Mathematical understanding of pattern formation is still lacking, but they are important in the progress of mathematical biology.

Lord Robert May

President of Royal Society

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 inwhich 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