patterns in nature and the mathematics behind...

SnowflakesPatterns in Nature

Turing Mechanism for Skin PatternsA Mathematical Model for Bacteria Colony Pattern

Delay Induced Oscillations in SomitogenesisA Few Comments

Patterns In Nature And The MathematicsBehind It

Peng Feng1

1Department of Physical Sciences and MathematicsFlorida Gulf Coast University

Peng Feng [email protected] Patterns In Nature




Outline

1 Snowflakes

2 Patterns in NaturePatterns in Nonbiological SystemsPatterns in Biological Systems

3 Turing Mechanism for Skin Patterns

4 A Mathematical Model for Bacteria Colony Pattern

5 Delay Induced Oscillations in Somitogenesis

6 A Few Comments





No Two Snowflakes Are the Same

Source:www.snowcrystals.com, maintained by Kenneth G. Libbrecht





1611, Kepler pondered on the question why snowflakesalways exhibit a six-fold symmetry;

1635, mathematician Descartes gave some reasonablyaccurate description;





1954, Physicist Ukichiro Nakaya performed systematicstudy on the formation of snowflakes:





Patterns in Nonbiological SystemsPatterns in Biological Systems

Outline

1 Snowflakes





6 A Few Comments






Ripples of Sand

Source:Bob Bauer, 2002






Outline

1 Snowflakes





6 A Few Comments






Shells of South West Florida

Source:http://www.meddybemps.com/shells/shells2draw.html






Animal Skin Patterns

Java fish, Zebra fish, Rabbit fish and Angel fishhttp://www.panagadivers.com/Diving/Fishes.htm






More Animal Skin Patterns

Leopard, Zebra






Patterns in Bacterial Colony

SourceCourtesy of Professor Eshel Ben-Jacob






More Patterns

Nautilus and Romanesco Broccoli:http://www.discoveringfossils.co.uk





What is Diffusion?





Modeling Diffusion

u(t , x , y):the density of a certain chemical compound attime time t and at (x , y) ∈ R2;Diffusion:The chemical will move from high density placesto low density places;Diffusion is the mechanism of many molecular and cellularmovement;It can be described by the heat equation ut = ∆u where∆u := uxx + uyy .





Simulation of Diffusion Equation

Consider the one dimensional equation: ut = kuxx where u(t , x)

represents the temperature on a beam at time t and location x ,k : heat conductivity or how fast heat will transport; Thisequation usually coupled with a boundary condition and initialcondition. For example u(0, x) = u0(x) andu(t , 0) = u(t , 1) = 0.





An Example of Reaction-diffusion System

ut = ∆u + λf (u, v), t > 0, (x , y) ∈ (0, a)× (0, b),

vt = d∆v + λg(u, v), t > 0, (x , y) ∈ (0, a)× (0, b),

ux = 0, x = 0, a, u(x , 0) = u(x , b), ux(x , 0) = ux(x , b),

vx = 0, x = 0, a, v(x , 0) = v(x , b), vx(x , 0) = vx(x , b),

u(0, x , y) = u0(x , y), v(0, x , y) = v0(x , y).

(1)Solution of the system:W(x, y, t) =

∑∞n,m=0 Cn,meλ(k2)t cos

(nπxa

)cos

(mπyb

), where

k2 = π2(

n2

a2 + 4m2

b2

).





A Few Quick Observations

When b is small, striped patterns are more likely (constantalong the y axis);When b is large, spotted patterns are more common (Notconstant along y axis);Snakes always have rings, but not spots;Animal can have spotted body and striped tailbut rarely striped body with a spotted tail.





Examples of Cos in Skin Pattern

Valais goat; Belted Galloway Cow(Oreo Cow?); Panda:

Found at: Southwestern Switzerland; Scotland; SichuanProvince, China.





Genet; Lemur; Snake:Found at: East Africa; Madagascar, South Africa; Common .





Turing

One of the greatest scientists of 20th century;Turing machine; Father of computer science;Breaking of U-boat Enigma, saving battle of Atlantic;

Source:www.turing.org.uk





Turing’s Diffusion-driven Instability

In a system of equations modeling two interactive substances,different diffusion rates could lead to nonhomogeneousdistributions of such reactants, i.e., the occurrence of patterns.





Reaction-Diffusion System

Consider the hypothetical chemical reactions:

U + 2V → 3V (2)V → P (3)

Assume: U is fed into the reaction at a constant rate F andprecipitate P is removed;Rate of change of U: remove rate uv2; feed rate F (1 − u);Thus

ut = Du∆u − uv2 + F (1 − u) (4)vt = Dv∆v + uv2 − (F + k)v (5)





Consider the equilibrium solution (u, v) = (u∗, v∗) of the system

ut = Du∆u − uv2 + F (1 − u)

vt = Dv∆v + uv2 − (F + k)v ⇔0 = −uv2 + F (1 − u)

0 = uv2 − (F + k)v⇒ u∗ = 1

2

(1±

√1 − 4δ2F

), v∗ = 1

2δ

(1∓

√1 − 4δ2F

).

Recall : For y ′(t) = 1 − y2, the equilibrium solution y = 1 isstable⇔ limt→∞ y(t)→ 1 for y(t) with suitable initial condition.An idea : Instability of the constant solutions implies possiblecomplicated patterns. Even though the type of patterns willrequire more detailed analysis.





Pattern Developing Process 1:

k=0.062; F=0.05






k=0.057; F=0.018






k=0.001; F=0.005





Phase Diagram for Bacillus

Source:K. Kawasaki et. al. Journal of Theoretical Biology, 1997.





A Simple Looking Mathematical Model for BacteriaColony Pattern

bt = Db 5 · nb5 b + nb, in Ω

nt = Dn∆n − nb, in Ω

n(x , y , 0) = n0, b(x , y , 0) = b0(x , y)∂n∂ν = ∂b

∂ν = 0, ∂Ω

(6)





Shouldn’t gene be playing ahuge role here???





Mutation in Fruit Fly

Figure: Bithorax mutant Figure: Antennapediamutant

Figure: Small wingmutant

Caused by a single homeodomain gene.





Gene Expression

Figure: Fushi Tarazu (ftz) mRNA inDrosophila embryo, by in situhybridization;(Hafen, Kuroiwa andGehring, Cell, 1984)

Figure: Another example, courtesy ofSean Carroll, HHMI





Formation of Somites

What is Somite:Paired blocks of mesoderm cells along the vertebrate body axisthat form during early vertebrate development and differentiateinto dermal skin, bone and muscle.Mouse: 60 somites; Chick: 28 somites; (Every 90 minutes)Snake: can reach up to 500; Gene: Chick: C-hairy-1 Zebrafish:her-1





Biology 101





A Mathematical Model for Gene Regulation in A Cell

dpdt

= am(t − τp) − bp(t), (7)

dmdt

= f (p(t − τm)) − cm(t), (8)

τp: time lag between the initiation of the translation and theappearance of a mature proteinτm: time lag between transcription and the appearance of amature mRNA molecule.b and c: degradation rates of proteins and mRNAs,respectively.a is the translational constant.f (p) = kθn

θn+pn : Switch-like phenomena.





Parameter Valuea 4.5 molecules/minb 0.23 molecules/minc 0.23 molecules/mink 33 molecules/cell· minn 5θ 40 moleculesτp 2.8 minτm 10.2 min∼ 31.5 min

Table: Estimated parameters for her1 gene.





Let r = τm/τp and we can reduce the system to the followingone with dimensionless time t/τp, which is again denoted by t :

dpdt

= τp[am(t − 1) − bp(t)], (9)

dmdt

= τp[f (p(t − r)) − cm(t)]. (10)

The steady states E∗ = (p∗, m∗) satisfy

am − bp = 0, (11)

kθn

θn + pn − cm = 0. (12)





Theoretical Result

Theorem

When c > kabθ

n−1n

n√

n − 1, E∗ = (p∗, m∗) is asymptotically stablefor all r > 0. When c is small, there exists a critical value r0such that the steady state E∗ is asymptotically stable forr ∈ [0, r0) and unstable for r > r0, where

r0 =1

ξ+arccos

ξ2

+ − bcτ2p

bc1τ2p

− 1, (13)

and

ξ2+ = −

12(b2 + c2)τ2

p +12τ2

p[(b2 − c2)2 + 4b2c21 ]1/2 (14)





Future Direction

This model only deals with a single cell;It does give us some insight into the mechanism;Modeling the cell network communications can be verycomplex;One mechanism for cell communication is Delta-NotchPathway.





What does it take to model biological problem?

A sound understanding and appreciation of the biologicalproblem;A realistic mathematical representation of the importantbiological phenomena;A biological interpretation of the mathematical analysis andresults in terms of insights and predictions.


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