Mathematical
Modeling and
Biology
Bo Deng
Mathematical Modeling and Biology
Introduction
Examples of
Models
Bo Deng
Consistency
Model Test
Mathematical
Biology
Department of Mathematics
University of Nebraska – Lincoln
Conclusion
March 10, 2016
www.math.unl.edu/∼bdeng1
1 / 24
What is modeling?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
2 / 24
Mathematical modeling is
to translate nature into mathematics
What is modeling?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Mathematical modeling is
to translate nature into mathematics
Consistency
Model Test
Mathematical
Biology
Conclusion
2 / 24
to be logically consistent
What is modeling?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Mathematical modeling is
to translate nature into mathematics
Consistency
Model Test
to be logically consistent
Mathematical
Biology
to fit the past and to predict future
Conclusion
2 / 24
What is modeling?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Mathematical modeling is
to translate nature into mathematics
Consistency
Model Test
to be logically consistent
Mathematical
Biology
to fit the past and to predict future
Conclusion
to fail against the test of time, i.e. to give way to better
models
2 / 24
Human history has two periods – before and after
calculus (1686/1687)
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
3 / 24
Issac Newton (1642-1727) is the founding father of
mathematical modeling
Human history has two periods – before and after
calculus (1686/1687)
Mathematical
Modeling and
Biology
Bo Deng
Issac Newton (1642-1727) is the founding father of
mathematical modeling
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
3 / 24
James Clerk Maxwell (1831-1879), Albert Einstein
(1879-1955), Erwin Schrödinger (1887-1961), Claude
Shannon (1916-2001) are some of the luminary disciples
Human history has two periods – before and after
calculus (1686/1687)
Mathematical
Modeling and
Biology
Bo Deng
Issac Newton (1642-1727) is the founding father of
mathematical modeling
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
3 / 24
James Clerk Maxwell (1831-1879), Albert Einstein
(1879-1955), Erwin Schrödinger (1887-1961), Claude
Shannon (1916-2001) are some of the luminary disciples
Calculus is the principle language of nature
Human history has two periods – before and after
calculus (1686/1687)
Mathematical
Modeling and
Biology
Bo Deng
Issac Newton (1642-1727) is the founding father of
mathematical modeling
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
James Clerk Maxwell (1831-1879), Albert Einstein
(1879-1955), Erwin Schrödinger (1887-1961), Claude
Shannon (1916-2001) are some of the luminary disciples
Calculus is the principle language of nature
This century is the century of mathematical biology, which
is to translate Charles Darwin’s (1809-1882) theory into
mathematics
3 / 24
Model as approximation – Newton’s planetary
motion
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
4 / 24
Planet
~r
Sun
~r2
~r1

~r1 − ~r2


m1~r¨1 = −Gm1 m2



k~r1 − ~r2 k3
~r2 − ~r1
m2~r¨2 = −Gm1 m2


k~r2 − ~r1 k3



~r = ~r1 − ~r2
Model as approximation – Newton’s planetary
motion
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Planet
~r
Sun
~r2
~r1

~r1 − ~r2


m1~r¨1 = −Gm1 m2



k~r1 − ~r2 k3
~r2 − ~r1
m2~r¨2 = −Gm1 m2


k~r2 − ~r1 k3



~r = ~r1 − ~r2
A few calculus maneuvers lead to
r(θ) =
ρ
1 + cos θ
with the eccentricity 0 ≤ < 1 for elliptic orbits
4 / 24
Special Relativity – Einstein’s model of space and
time
Mathematical
Modeling and
Biology
Bo Deng
Introduction
One Assumption:
The speed of light is constant for every stationary observer
Examples of
Models
ȳ
y
Consistency
Model Test
v
K
K̄
Mathematical
Biology
Conclusion
0
5 / 24
0
x
x̄
Special Relativity – Einstein’s model of space and
time
Mathematical
Modeling and
Biology
Bo Deng
Introduction
One Assumption:
The speed of light is constant for every stationary observer
Examples of
Models
ȳ
y
Consistency
Model Test
v
K
K̄
Mathematical
Biology
Conclusion
0
0
x
x̄
A few calculus maneuvers lead to E = mc2 , and more
5 / 24
Special Relativity — Einstein’s model of space and
time
Mathematical
Modeling and
Biology
Bo Deng
One Assumption:
The speed of light is constant for every stationary observer
Examples of
Models
Consistency
ȳ
y
Introduction
K
ct
Model Test
L
K̄
√
2
2
[ c − v ]t = ct̄
Mathematical
Biology
Conclusion
6 / 24
0
vt 0
x
x̄
Special Relativity — Einstein’s model of space and
time
Mathematical
Modeling and
Biology
Bo Deng
One Assumption:
The speed of light is constant for every stationary observer
Examples of
Models
Consistency
ȳ
y
Introduction
K
ct
Model Test
L
K̄
√
2
2
[ c − v ]t = ct̄
Mathematical
Biology
Conclusion
0
vt 0
x
x̄
Prediction: Time dilation for K-frame observer
L
L
t= p
> = t̄
2
c
c 1 − (v/c)
6 / 24
General Relativity — Model of space and time in
acceleration
Mathematical
Modeling and
Biology
ȳ
y
Bo Deng
v1 = a∆t + v0
Introduction
c∆t
Examples of
Models
Consistency
Model Test
Mathematical
Biology
c∆t
v1 ∆t
Conclusion
v0 ∆t
7 / 24
x
x̄
General Relativity — Model of space and time in
acceleration
Mathematical
Modeling and
Biology
ȳ
y
Bo Deng
v1 = a∆t + v0
Introduction
c∆t
Examples of
Models
Consistency
Model Test
Mathematical
Biology
c∆t
v1 ∆t
Conclusion
v0 ∆t
x
x̄
Prediction: Light beam bends
under acceleration or near massive
bodies
7 / 24
Mathematical model need not be mathematical
Mathematical
Modeling and
Biology
Bo Deng
Gregor Johann Mendel (1822-1884) found the first
mathematical model in biology, leading to the discovery of
gene
Introduction
Parent Genotype
mrr × frr
mrD × frD
mDD × fDD
mrr × frD or
mrD × frr
mrr × fDD or
mDD × frr
mrD × fDD or
mDD × frD
Examples of
Models
1
0
0
1/4
1/2
1/4
0
0
1
1/2
1/2
0
0
1
0
0
1/2
1/2
Consistency
Model Test
Mathematical
Biology
Conclusion
Offspring
Genotype
0
zrr
0
zrD
0
zDD
8 / 24
One More Example: Structure of DNA by modeling
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Rosalind Franklin and Maurice Wilkins had the data, but
James D. Watson and Francis Crick had the frame of
mind to model the data (1953)
9 / 24
Another More – Predation in Ecology
Mathematical
Modeling and
Biology
The mathematical model was discovered by Crawford Stanley
(Buzz) Holling (1930- ) in 1959
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
10 / 24
Td — average time a predator takes to discover a prey
Another More – Predation in Ecology
Mathematical
Modeling and
Biology
The mathematical model was discovered by Crawford Stanley
(Buzz) Holling (1930- ) in 1959
Bo Deng
Introduction
Td — average time a predator takes to discover a prey
Examples of
Models
Tk — average time a predator takes to kill a prey
Consistency
Model Test
Mathematical
Biology
Conclusion
10 / 24
Another More – Predation in Ecology
Mathematical
Modeling and
Biology
The mathematical model was discovered by Crawford Stanley
(Buzz) Holling (1930- ) in 1959
Bo Deng
Introduction
Td — average time a predator takes to discover a prey
Examples of
Models
Tk — average time a predator takes to kill a prey
Consistency
Model Test
Mathematical
Biology
Conclusion
10 / 24
Td,k = Td + Tk — average time a predator takes to
discovery and kill a prey
Another More – Predation in Ecology
Mathematical
Modeling and
Biology
The mathematical model was discovered by Crawford Stanley
(Buzz) Holling (1930- ) in 1959
Bo Deng
Introduction
Td — average time a predator takes to discover a prey
Examples of
Models
Tk — average time a predator takes to kill a prey
Consistency
Model Test
Mathematical
Biology
Conclusion
10 / 24
Td,k = Td + Tk — average time a predator takes to
discovery and kill a prey
1
— rate of discovery, i.e. number of preys a
Rd =
Td
predator would find in a unit time
1
Rk =
— rate of killing, i.e. number of preys a predator
Tk
would kill in a unit time
1
1
Rd,k =
=
— rate of discovery and killing
Td,k
Td + Tk
Model of Predation in Ecology
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
11 / 24
And Holling’s predation function form:
Rd,k =
1/Td
Rd
1
=
=
Td + Tk
1 + Tk (1/Td )
1 + Tk Rd
Model of Predation in Ecology
Mathematical
Modeling and
Biology
And Holling’s predation function form:
Bo Deng
Rd,k =
Introduction
Examples of
Models
Consistency
Model Test
1/Td
Rd
1
=
=
Td + Tk
1 + Tk (1/Td )
1 + Tk Rd
Prediction: Assume the discovery rate is proportional to
the prey population X, Rd = aX. Then the Holling Type
II predation rate must saturate as X → ∞
Mathematical
Biology
aX
1
=
X→∞ 1 + Tk aX
Tk
lim Rd,k = lim
X→∞
Conclusion
Rd,k
1
Tk
0
11 / 24
X
Consistency
Mathematical
Modeling and
Biology
Bo Deng
Not every piece of mathematics can be a physical law or
model. Logical consistency is the first and necessary
constraint
Introduction
Examples of
Models
Time Invariance Principle (TIP)
Consistency
A model must has the same functional form for every time
independent observation
Model Test
Mathematical
Biology
Conclusion
12 / 24
Consistency
Mathematical
Modeling and
Biology
Bo Deng
Not every piece of mathematics can be a physical law or
model. Logical consistency is the first and necessary
constraint
Introduction
Examples of
Models
Time Invariance Principle (TIP)
Consistency
A model must has the same functional form for every time
independent observation
Model Test
Mathematical
Biology
Conclusion
Newtonian mechanics is TIP-consistent:
s
t
x(s, x0 )
x0
12 / 24
x(t + s, x0 ) = x(t, x(s, x0 ))
Special Relativity is self-consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
13 / 24
Let P be a point, having K = (x, y, z, t) coordinate in the
K-frame and K̄ = (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.
Then they are exchangeable via a linear transformation
depending the speed v:
K̄ = KL(v)
Special Relativity is self-consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Let P be a point, having K = (x, y, z, t) coordinate in the
K-frame and K̄ = (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.
Then they are exchangeable via a linear transformation
depending the speed v:
K̄ = KL(v)
Consistency
Model Test
Mathematical
Biology
Conclusion
Let K̃ = (x̃, ỹ, z̃, t̃) be the coordinate of the same point in
a K̃-frame moving at speed u with respect to the
K̄-frame. Then we have
K̃ = K̄L(u) = KL(v)L(u) = KL(w) with w =
13 / 24
u+v
1 + uv
c2
Special Relativity is self-consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Let P be a point, having K = (x, y, z, t) coordinate in the
K-frame and K̄ = (x̄, ȳ, z̄, t̄) coordinate in the K̄-frame.
Then they are exchangeable via a linear transformation
depending the speed v:
Examples of
Models
K̄ = KL(v)
Consistency
Model Test
Mathematical
Biology
Conclusion
Let K̃ = (x̃, ỹ, z̃, t̃) be the coordinate of the same point in
a K̃-frame moving at speed u with respect to the
K̄-frame. Then we have
K̃ = K̄L(u) = KL(v)L(u) = KL(w) with w =
u+v
1 + uv
c2
u+v
for elements u, v ∈ (−c, c)
1 + uv
c2
defines a commutative group
The operation u ⊕ v =
13 / 24
Holling’s predation model is consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
14 / 24
Tc — average time to consume a prey
Holling’s predation model is consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
14 / 24
Tc — average time to consume a prey
Td,k,c = Td + Tk + Tc — average time to discover, kill,
and consume a prey
Holling’s predation model is consistent
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Tc — average time to consume a prey
Td,k,c = Td + Tk + Tc — average time to discover, kill,
and consume a prey
Then the rate of predation is self-consistent:
Rd,k,c =
Conclusion
=
14 / 24
1
Td,k,c
=
1
Td + Tk + Tc
Rd,k
Rd
=
1 + Tc Rd,k
1 + (Tk + Tc )Rd
Pay the TIP, or else
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
15 / 24
All differential equation models are TIP-consistent
Pay the TIP, or else
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
15 / 24
All differential equation models are TIP-consistent
Most mapping models in ecology are TIP-inconsistent
Pay the TIP, or else
Mathematical
Modeling and
Biology
Bo Deng
All differential equation models are TIP-consistent
Most mapping models in ecology are TIP-inconsistent
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
15 / 24
Example: Logistic map
xn+1 = Qλ (xn ) = λxn (1 − xn )
cannot be a model for which n represents time
Pay the TIP, or else
Mathematical
Modeling and
Biology
Bo Deng
All differential equation models are TIP-consistent
Most mapping models in ecology are TIP-inconsistent
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Example: Logistic map
xn+1 = Qλ (xn ) = λxn (1 − xn )
cannot be a model for which n represents time
Conclusion
The time n + 2 observation yields a different functional
form:
xn+2 = Qλ (xn+1 ) = Qλ (Qλ (xn )) 6= Qµ (xn )
for any value µ. Strike one on the logistic map
15 / 24
Model Test – Finding the Best Fit
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
16 / 24
x̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for a
natural process which are modeled by competing models
y(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,
and initial state y0 , z0
Model Test – Finding the Best Fit
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
x̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for a
natural process which are modeled by competing models
y(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,
and initial state y0 , z0
Model selection criterion: All else being equal whichever
has a smaller error is the benchmark model by default:
Model Test
Mathematical
Biology
Conclusion
Ey = min
(y0 ,p)
Ez = min
(z0 ,q)
16 / 24
n
X
[y(ti ; y0 , p) − x̄i ]2
i=1
n
X
i=1
[z(ti ; z0 , q) − x̄i ]2
Model Test – Finding the Best Fit
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
x̄1 , . . . , x̄n — Observed states at time t1 , . . . , tn for a
natural process which are modeled by competing models
y(t; y0 , p) and z(t; z0 , q), respectively, with parameter p, q,
and initial state y0 , z0
Model selection criterion: All else being equal whichever
has a smaller error is the benchmark model by default:
Model Test
Mathematical
Biology
Conclusion
Ey = min
(y0 ,p)
Ez = min
(z0 ,q)
16 / 24
n
X
[y(ti ; y0 , p) − x̄i ]2
i=1
n
X
[z(ti ; z0 , q) − x̄i ]2
i=1
A parameter value is only meaningful to its model, and it
can only be derived by best-fitting the observed data to
the model
Model Test – Fit the past, predict the future
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
17 / 24
Edmond Halley (1656-1742) used Newtonian mechanics to
predict the 1758 return of Halley’s Comet, giving the
comet its name
Model Test – Fit the past, predict the future
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
17 / 24
Edmond Halley (1656-1742) used Newtonian mechanics to
predict the 1758 return of Halley’s Comet, giving the
comet its name
Arthur Eddington (1882-1944) used the total solar eclipse
of May 29, 1919 to confirm general relativity’s prediction
for the bending of starlight by the Sun, making Einstein an
instant world celebrity
Model Test – Fit the past, predict the future
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
17 / 24
Edmond Halley (1656-1742) used Newtonian mechanics to
predict the 1758 return of Halley’s Comet, giving the
comet its name
Arthur Eddington (1882-1944) used the total solar eclipse
of May 29, 1919 to confirm general relativity’s prediction
for the bending of starlight by the Sun, making Einstein an
instant world celebrity
Gregor Mendel’s Laws of Inheritance (1866) was
rediscovered in 1900, ushering in the science of modern
genetics
Model Test – Fit the past, predict the future
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Edmond Halley (1656-1742) used Newtonian mechanics to
predict the 1758 return of Halley’s Comet, giving the
comet its name
Arthur Eddington (1882-1944) used the total solar eclipse
of May 29, 1919 to confirm general relativity’s prediction
for the bending of starlight by the Sun, making Einstein an
instant world celebrity
Gregor Mendel’s Laws of Inheritance (1866) was
rediscovered in 1900, ushering in the science of modern
genetics
Holling’s model of predation is ubiquitous in theoretical
ecology
17 / 24
Mathematical Biology — To Translate Evolution to
Mathematics
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
18 / 24
Example: One Life Rule
Every organism lives only once and must die in any finite time
in the presence of infinite population density
Mathematical Biology — To Translate Evolution to
Mathematics
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
18 / 24
Example: One Life Rule
Every organism lives only once and must die in any finite time
in the presence of infinite population density
In math translation: Let xt be the population at time t.
Then the per-capita change must satisfy
xt − x0
xt
=
− 1 ≥ −1
x0
x0
Mathematical Biology — To Translate Evolution to
Mathematics
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Example: One Life Rule
Every organism lives only once and must die in any finite time
in the presence of infinite population density
In math translation: Let xt be the population at time t.
Then the per-capita change must satisfy
xt − x0
xt
=
− 1 ≥ −1
x0
x0
Lead to
xt − x0
= −1
One Life Rule ⇐⇒ lim
x0 →∞
x0
and to the logistic equation
ẋ(t) = rx(t)[1 − x(t)/K]
18 / 24
with x(t) = xt , r the max per-capita growth rate, and K
the carrying capacity
Footnote: model or no model, generalization or
relativism is often the problem
Mathematical
Modeling and
Biology
Bo Deng
Strike two on the logistic map x1 = λx0 (1 − x0 ):
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
19 / 24
lim
x0 →∞
x1 − x0
= lim [λ(1 − x0 ) − 1] = −∞ =
6 −1
x0 →∞
x0
Footnote: model or no model, generalization or
relativism is often the problem
Mathematical
Modeling and
Biology
Bo Deng
Strike two on the logistic map x1 = λx0 (1 − x0 ):
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
19 / 24
lim
x0 →∞
x1 − x0
= lim [λ(1 − x0 ) − 1] = −∞ =
6 −1
x0 →∞
x0
While the logistic equation, x0 (t) = rx(t)(1 − x(t)/K),
dogged another consistency bullet
x(t; x0 ) − x0
K
lim
= lim
− 1 = −1
x0 →∞ x0 + (K − x0 )e−rt
x0 →∞
x0
Footnote: model or no model, generalization or
relativism is often the problem
Mathematical
Modeling and
Biology
Bo Deng
Strike two on the logistic map x1 = λx0 (1 − x0 ):
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
lim
x0 →∞
x1 − x0
= lim [λ(1 − x0 ) − 1] = −∞ =
6 −1
x0 →∞
x0
While the logistic equation, x0 (t) = rx(t)(1 − x(t)/K),
dogged another consistency bullet
x(t; x0 ) − x0
K
lim
= lim
− 1 = −1
x0 →∞ x0 + (K − x0 )e−rt
x0 →∞
x0
There should be no different versions of the same reality,
but refined approximations
19 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
20 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
The AT pair has one weak O-H bond but the GC pair has
two O-H bonds. Hence, the GC pair takes longer to
complete binding than the AT pair does
20 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
21 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Start with a conceptual model: DNA replication is a
communication channel
21 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Start with a conceptual model: DNA replication is a
communication channel
Every communication is characterized by the transmission
data rate in bits per second, i.e. the information entropy
per second
21 / 24
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
22 / 24
For 2n paired bases, the replication rate is
R2n =
log2 (2n)
τ12 +τ34 +···+τ(2n−1)(2n)
n
in bits per time
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
For 2n paired bases, the replication rate is
Bo Deng
R2n =
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
22 / 24
If
5
3
≤
τGC
τAT
log2 (2n)
τ12 +τ34 +···+τ(2n−1)(2n)
n
≤ 2.7, then:
in bits per time
max R2n = R4
n
One More Example: Why DNA is coded in 4 bases?
Mathematical
Modeling and
Biology
For 2n paired bases, the replication rate is
Bo Deng
R2n =
Introduction
Examples of
Models
Consistency
If
5
3
≤
τGC
τAT
log2 (2n)
τ12 +τ34 +···+τ(2n−1)(2n)
n
≤ 2.7, then:
in bits per time
max R2n = R4
n
Model Test
Mathematical
Biology
Conclusion
Punch Line: Life is a reality show on your DNA channel
22 / 24
Closing Comments
Mathematical
Modeling and
Biology
Bo Deng
Mathematics is driven by open problems, but science is
driven by existing solutions
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
Mathematical modeling is to find the equation to which
nature fits as a solution
Mathematics is to create more hays but modeling is to
find the needle in haystack
Mathematical biology is not to solve mathematical
problems of models but to find mathematical models for
biological problems
Training to be a mathematical modeler does need to solve
mathematical problems of reasonable models.
23 / 24
Mathematical
Modeling and
Biology
Bo Deng
Introduction
Examples of
Models
Consistency
Model Test
Mathematical
Biology
Conclusion
24 / 24
Mathematical modeling is to construct the picture so
that the consequence of the picture is the picture of
the consequence.
– Anonymous or by Heinrich Hertz (1857-1894)