Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns In Nature And The Mathematics
Behind It
Peng Feng1
1 Department
of Physical Sciences and Mathematics
Florida Gulf Coast University
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Outline
1
Snowflakes
2
Patterns in Nature
Patterns in Nonbiological Systems
Patterns 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
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
No Two Snowflakes Are the Same
Source:
www.snowcrystals.com, maintained by Kenneth G. Libbrecht
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
1611, Kepler pondered on the question why snowflakes
always exhibit a six-fold symmetry;
1635, mathematician Descartes gave some reasonably
accurate description;
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
1954, Physicist Ukichiro Nakaya performed systematic
study on the formation of snowflakes:
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Outline
1
Snowflakes
2
Patterns in Nature
Patterns in Nonbiological Systems
Patterns 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
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Ripples of Sand
Source:
Bob Bauer, 2002
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Outline
1
Snowflakes
2
Patterns in Nature
Patterns in Nonbiological Systems
Patterns 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
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Shells of South West Florida
Source:
http://www.meddybemps.com/shells/shells2draw.html
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Animal Skin Patterns
Java fish, Zebra fish, Rabbit fish and Angel fish
http://www.panagadivers.com/Diving/Fishes.htm
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
More Animal Skin Patterns
Leopard, Zebra
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
Patterns in Bacterial Colony
Source
Courtesy of Professor Eshel Ben-Jacob
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Patterns in Nonbiological Systems
Patterns in Biological Systems
More Patterns
Nautilus and Romanesco Broccoli:
http://www.discoveringfossils.co.uk
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
What is Diffusion?
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Modeling Diffusion
u(t, x, y ):the density of a certain chemical compound at
time time t and at (x, y ) ∈ R 2 ;
Diffusion:The chemical will move from high density places
to low density places;
Diffusion is the mechanism of many molecular and cellular
movement;
It can be described by the heat equation ut = ∆u where
∆u := uxx + uyy .
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Modeling Diffusion
u(t, x, y ):the density of a certain chemical compound at
time time t and at (x, y ) ∈ R 2 ;
Diffusion:The chemical will move from high density places
to low density places;
Diffusion is the mechanism of many molecular and cellular
movement;
It can be described by the heat equation ut = ∆u where
∆u := uxx + uyy .
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Modeling Diffusion
u(t, x, y ):the density of a certain chemical compound at
time time t and at (x, y ) ∈ R 2 ;
Diffusion:The chemical will move from high density places
to low density places;
Diffusion is the mechanism of many molecular and cellular
movement;
It can be described by the heat equation ut = ∆u where
∆u := uxx + uyy .
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Modeling Diffusion
u(t, x, y ):the density of a certain chemical compound at
time time t and at (x, y ) ∈ R 2 ;
Diffusion:The chemical will move from high density places
to low density places;
Diffusion is the mechanism of many molecular and cellular
movement;
It can be described by the heat equation ut = ∆u where
∆u := uxx + uyy .
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
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; This
equation usually coupled with a boundary condition and initial
condition. For example u(0, x) = u0 (x) and
u(t, 0) = u(t, 1) = 0.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
An Example of Reaction-diffusion System


ut = ∆u + λf (u, v ),





vt = d∆v + λg(u, v ),
ux = 0, x = 0, a, u(x, 0) = u(x, b),



vx = 0, x = 0, a, v (x, 0) = v (x, b),




u(0, x, y ) = u0 (x, y ),
Solution of the
P system: λ(k 2 )t
W(x, y, t) = ∞
C e
cos
2 n,m=0
n,m
2
n
4m
k 2 = π2 a 2 + b 2 .
Peng Feng pfeng@fgcu.edu
t > 0, (x, y ) ∈ (0, a) × (0, b),
t > 0, (x, y ) ∈ (0, a) × (0, b),
ux (x, 0) = ux (x, b),
vx (x, 0) = vx (x, b),
v (0, x, y ) = v0 (x, y ).
(1)
nπx
a
cos
Patterns In Nature
mπy ,
b
where
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few Quick Observations
When b is small, striped patterns are more likely (constant
along the y axis);
When b is large, spotted patterns are more common (Not
constant along y axis);
Snakes always have rings, but not spots;
Animal can have spotted body and striped tail
but rarely striped body with a spotted tail.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few Quick Observations
When b is small, striped patterns are more likely (constant
along the y axis);
When b is large, spotted patterns are more common (Not
constant along y axis);
Snakes always have rings, but not spots;
Animal can have spotted body and striped tail
but rarely striped body with a spotted tail.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few Quick Observations
When b is small, striped patterns are more likely (constant
along the y axis);
When b is large, spotted patterns are more common (Not
constant along y axis);
Snakes always have rings, but not spots;
Animal can have spotted body and striped tail
but rarely striped body with a spotted tail.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few Quick Observations
When b is small, striped patterns are more likely (constant
along the y axis);
When b is large, spotted patterns are more common (Not
constant along y axis);
Snakes always have rings, but not spots;
Animal can have spotted body and striped tail
but rarely striped body with a spotted tail.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few Quick Observations
When b is small, striped patterns are more likely (constant
along the y axis);
When b is large, spotted patterns are more common (Not
constant along y axis);
Snakes always have rings, but not spots;
Animal can have spotted body and striped tail
but rarely striped body with a spotted tail.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Examples of Cos in Skin Pattern
Valais goat; Belted Galloway Cow(Oreo Cow?); Panda:
Found at: Southwestern Switzerland; Scotland; Sichuan
Province, China.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Genet; Lemur; Snake:
Found at: East Africa; Madagascar, South Africa; Common .
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Turing
One of the greatest scientists of 20th century;
Turing machine; Father of computer science;
Breaking of U-boat Enigma, saving battle of Atla
Source:
www.turing.org.uk
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Turing’s Diffusion-driven Instability
In a system of equations modeling two interactive substances,
different diffusion rates could lead to nonhomogeneous
distributions of such reactants, i.e., the occurrence of patterns.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
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 and
precipitate P is removed;
Rate of change of U: remove rate uv 2 ; feed rate F (1 − u);
Thus
ut
vt
= Du ∆u − uv 2 + F (1 − u)
2
= Dv ∆v + uv − (F + k )v
Peng Feng pfeng@fgcu.edu
Patterns In Nature
(4)
(5)
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Consider the equilibrium solution (u, v ) = (u∗ , v∗ ) of the system
ut
vt
= Du ∆u − uv 2 + F (1 − u)
⇔
= Dv ∆v + uv 2 − (F + k)v
√
⇒ u∗ = 12 1 ± 1 − 4δ2 F , v∗ =
0 = −uv 2 + F (1 − u)
uv 2 − (F + k )v
0 =
√
1
2
2δ 1 ∓ 1 − 4δ F .
Recall : For y 0 (t) = 1 − y 2 , the equilibrium solution y = 1 is
stable⇔ limt→∞ y (t) → 1 for y (t) with suitable initial condition.
An idea : Instability of the constant solutions implies possible
complicated patterns. Even though the type of patterns will
require more detailed analysis.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Pattern Developing Process 1:
k=0.062; F=0.05
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Pattern Developing Process 2:
k=0.057; F=0.018
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Pattern Developing Process 3:
k=0.001; F=0.005
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Phase Diagram for Bacillus
Source:
K. Kawasaki et. al. Journal of Theoretical Biology, 1997.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Simple Looking Mathematical Model for Bacteria
Colony Pattern


bt = Db 5 · {nb 5 b} + nb,



n = D ∆n − nb,
n
t
n(x, y , 0) = n0 , b(x, y , 0) = b0 (x, y )



 ∂n = ∂b = 0,
∂ν
∂ν
Peng Feng pfeng@fgcu.edu
Patterns In Nature
in Ω
in Ω
∂Ω
(6)
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Shouldn’t gene be playing a
huge role here???
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Mutation in Fruit Fly
Figure: Bithorax mutant
Figure: Antennapedia
mutant
Caused by a single homeodomain gene.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Figure: Small wing
mutant
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Gene Expression
Figure: Fushi Tarazu (ftz) mRNA in
Drosophila embryo, by in situ
hybridization;(Hafen, Kuroiwa and
Gehring, Cell, 1984)
Peng Feng pfeng@fgcu.edu
Figure: Another example, courtesy of
Sean Carroll, HHMI
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Formation of Somites
What is Somite:
Paired blocks of mesoderm cells along the vertebrate body axis
that form during early vertebrate development and differentiate
into 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
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Biology 101
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Mathematical Model for Gene Regulation in A Cell
dp
= am(t − τp ) − bp(t),
(7)
dt
dm
= f (p(t − τm )) − cm(t),
(8)
dt
τp : time lag between the initiation of the translation and the
appearance of a mature protein
τm : time lag between transcription and the appearance of a
mature mRNA molecule.
b and c: degradation rates of proteins and mRNAs,
respectively.
a is the translational constant.
θn
f (p) = θkn +p
n : Switch-like phenomena.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Parameter
a
b
c
k
n
θ
τp
τm
Value
4.5 molecules/min
0.23 molecules/min
0.23 molecules/min
33 molecules/cell· min
5
40 molecules
2.8 min
10.2 min∼ 31.5 min
Table: Estimated parameters for her1 gene.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Let r = τm /τp and we can reduce the system to the following
one with dimensionless time t/τp , which is again denoted by t:
dp
= τp [am(t − 1) − bp(t)],
dt
dm
= τp [f (p(t − r )) − cm(t)].
dt
The steady states E ∗ = (p∗ , m∗ ) satisfy
(9)
(10)
am − bp = 0,
(11)
k θn
− cm = 0.
θn + pn
(12)
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Theoretical Result
Theorem
√
ka n−1 n
When c > bθ
n − 1, E ∗ = (p∗ , m∗ ) is asymptotically stable
n
for all r > 0. When c is small, there exists a critical value r0
such that the steady state E ∗ is asymptotically stable for
r ∈ [0, r0 ) and unstable for r > r0 , where
ξ2+ − bcτ2p
1
arccos
r0 =
− 1,
(13)
ξ+
bc1 τ2p
and
1
1
ξ2+ = − (b2 + c 2 )τ2p + τ2p [(b2 − c 2 )2 + 4b2 c12 ]1/2
2
2
Peng Feng pfeng@fgcu.edu
Patterns In Nature
(14)
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
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 very
complex;
One mechanism for cell communication is Delta-Notch
Pathway.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
What does it take to model biological problem?
A sound understanding and appreciation of the biological
problem;
A realistic mathematical representation of the important
biological phenomena;
A biological interpretation of the mathematical analysis and
results in terms of insights and predictions.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
What does it take to model biological problem?
A sound understanding and appreciation of the biological
problem;
A realistic mathematical representation of the important
biological phenomena;
A biological interpretation of the mathematical analysis and
results in terms of insights and predictions.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
What does it take to model biological problem?
A sound understanding and appreciation of the biological
problem;
A realistic mathematical representation of the important
biological phenomena;
A biological interpretation of the mathematical analysis and
results in terms of insights and predictions.
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Further Readings
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
A Few People
James Murray, Emeritus Professor of Mathematics,
University of Washington
Philip Maini, Center of Mathematical Biology, University of
Oxford
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
E. Ben-Jacob, Tel Aviv University, Former President of
Israel Physical Society
Hans Meinhardt, Max-Planck Institute
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Are we abusing mathematics?
Or are we ignoring it?
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Are we abusing mathematics?
Or are we ignoring it?
Peng Feng pfeng@fgcu.edu
Patterns In Nature
Snowflakes
Patterns in Nature
Turing Mechanism for Skin Patterns
A Mathematical Model for Bacteria Colony Pattern
Delay Induced Oscillations in Somitogenesis
A Few Comments
Thank you!
Peng Feng pfeng@fgcu.edu
Patterns In Nature