tidbits from the sciences: examples for calculus and differential equations

Tidbits from the Sciences: Tidbits from the Sciences: Examples for Calculus and Examples for Calculus and Differential Equations Differential Equations

Tidbits from the Sciences: Tidbits from the Sciences: Examples for Calculus and Examples for Calculus and Differential Equations Differential Equations

Bruce E. ShapiroBruce E. Shapiro

California State University, California State University, NorthridgeNorthridge

• Satellite navigation• Genomic variation• Cooking potatoes• Enzymatic reactions & switching• The dynamics of love• Measuring the human genome

Examples

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kt = M = E − esin E

t = time since perigee passagek = 2/periode<1 in an elliptical orbit

M is easy to calculate

E is easily converted to position in orbit

Problem: Find E as a function of time

Inversion of Kepler’s Equation

• Solve using fixed point iteration

• Since e<1 for an elliptical orbit:

• Fixed point always converges

Inversion of Kepler’s Equation

€

kt = M = E − esin E

€

g(E) = M + esin E

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| ′ g (t) |= e | cos E |≤ e <1

• Example: M= /4, e=1/4 to 3 digits:

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g(E) = 0.785 + 0.25 *sin E

E0 = 0.785

E1 = g(E0 )

= 0.785 + 0.25sin(0.785)

= 0.962

E2 = g(E1)

= 0.785 + 0.25sin(.962)

= 0.990

E3 = 0.994; E4 = 0.995


Examples

Imag

es: http://www.sciencemag.com, http://www.nature.com

Genomes are being sequenced at an exponential Rate

• Individual genomic differences occur at every 1000 “base-pairs” of our DNA: – ≈1,000,000 significant points of difference between any

two individuals• Not all the same locations in everyone

– differences in drug metabolism, disease sensitivity, eye color, ...

• Yet we are 99.9% the same!

Genomic Variation

http://creative.gettyimages.com

Micro Array Data

Time

Ge

ne

Genetic Similarity• Samples have concentration vectors

(x1,x2,….,xn), (y1,y2,….,yn) – Can be two points on a time course or

samples from two different individual!

• Come up with different measures of similarity:– Dot product/angle– Euclidean distance– Various vector norms– Projections along principal components

Data clusters in two dimensions

x

y

ABO BLOOD GROUPO A B AB

Armenians 31 50 13 6Basques 51 44 4 1Czechs 30 44 18 9Danes 41 44 11 4Dutch 45 43 9 3Hungarians 36 43 16 5Icelanders 56 32 10 3Latvians 32 37 24 7Lithuanians 40 34 20 6Slovaks 42 37 16 5Spanish 38 47 10 5Swedes 38 47 10 5Swiss 40 50 7 3

Source: http://www.bloodbook.com/world-abo.html

• Are the Slovaks and Czechs closer genealogically to the each other or to the Spanish? Use the following distance measurements:

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d1(x,y) = cos−1 x ⋅y| x || y | ⎛ ⎝ ⎜

⎞ ⎠ ⎟

d2 (x,y) = (x − y) ⋅(x − y)

d3(x,y) = | x − y |ii∑

ABO Blood Group

€

d1(Slovak,Spanish) = 69.4o

d1(Slovak,Czech) = 71.2o

d1(Czech,Spanish) = 69.9o

d2(Slovak,Spanish) = 12.32

d2(Slovak,Czech) = 14.59

d2(Czech,Spanish) = 12.37

d3(Slovak,Spanish) = 20

d3(Slovak,Czech) = 25

d3(Czech,Spanish) = 23

By all three methods the Czechs and the Slovaks are more closely related to the Spanish than they are to each other!


Examples

The Potato Problem*• The rate of change of temperature T of a

potato in a pre-heated oven is proportional to the difference between the temperature of the oven and the potato*

• Preheat the oven to 420˚• Assume room temperature is 70˚• After 3 minutes the potato is 150˚. • When will it reach 300˚?

*Newton’s law of heating as formulated by a student

The Potato Problem (Solution)

€

′ T (t) = k(420 − T ), T(0) = 70, T(3) = 150

Solve ODE :dT

T − 420∫ = − kdt∫ ⇒ T = 420 + Ce−kt

Solve for C : T(0) = 70⇒ C = -350⇒ T = 420 − 350e−kt

Solve for k :T(3) = 150⇒ 150 = 420 − 350e−3k ⇒ k = −13

ln270350

≈ 0.086

General Solution : T(t) = 420 − 350e−0.086t

T = 300 = 420 − 350e−0.086t ⇒ t = 12.44 minutes

The Potato Problem• Can be treated as either IVP or

BVP– IVP plus “fitting” data to IVP to get

second constant, or as– BVP with two boundary conditions

• Linear Separable First Order ODE• Introduces idea of Canonical forms

in nature with something other than Capacitors


Examples

• Model phenomona that appear in a wide variety of situations in nature:

• “Exponential Relaxation” of y to steady state with time constant

• One of the most common models in biology!!

Canonical Models

€

dydt

=y∞ − y

τ

€

y∞

Law of Mass Action• The rate of a reaction is

proportional to the concentrations of the reactants

• Single Reactant:• Multiple Reactants:

•Multiple Reactions: add terms from each reaction

€

x → y ⇒ ′ x = −kx, ′ y = kx

€

w + x + y → z ⇒ ′ z = kxyz = − ′ x = − ′ y = − ′ z

Application of Mass Action• Protein in Two States

– x=amount in “on” state – y=amount in “off” state

• Conservation of mass x+y=N=constant

• Chemical Equation:

€

xα

⏐ → ⏐ y

yβ

⏐ → ⏐ x

Two-State Protein• Normalize variables (N=1)

• Solution:

€

xα

⏐ → ⏐ y

yβ

⏐ → ⏐ x

€

′ x = −αx + βy = −αx + β (1− x)

= β − (α + β )x

= (α + β )β

α + β− x

⎛ ⎝ ⎜

⎞ ⎠ ⎟

=x∞ − x

τ, x∞ =

βα + β

, τ =1

α + β

€

x + y = 1

Enzymatic Cascades• Traditional Enzymatic Reaction:

• More common situation in nature:

€

Source + Enzyme → Product + Enzyme

€

Source + Enzyme → Complex

Complex → Source + Enzyme

Complex → Product + Enzyme

written as

Source ⇒ ProductEnzyme

or :

Source + EnzymeF Complex → Product + Enzyme

MAPK Cascade ModelMAPK=Mitogen Activated Protein Kinase

10

100

1000

10000

100000

2 3 4 5 6# Slots

# ReactionsSingle PhosphorylationDouble Phosphorylation

€

KKK ⇒ KKKpsignal

KK ⇒KKKp

KKp ⇒KKKp

KKpp

K ⇒KKKpp

Kp ⇒KKKpp

Kpp

€

KKK ⇒ KKKpsignal

KK ⇒KKKp

KKp ⇒KKKp

KKpp

K ⇒KKKpp

Kp ⇒KKKpp

KppAs chemical

reactions

As a cascade

As differential equations

Differential equations


Examples

Strogatz’s Romeo & Juliet• Juliet is strangely attracted to Romeo:

– The more Romeo loves Juliet, the more she wants to run away

– When Romeo gets discouraged, she finds him strangely attractive

• Romeo echo’s Juliet’s love:– he warms up when she loves him– he loses interest when she hates him

Romeo and Juliet• R(t) = Romeo’s Love/Hate for Juliet• J(t)=Juliet’s Love/Hate for Romeo• Postive Values signify love, negative

values hate• Dynamical Model: • Outcome: a never-ending cycle of love

and hate with a center at (R,J)=(0,0); they manage to simultaneously love one another 25% of the time€

′ R = aJ , ′ J = −bR

Romeo and Juliet• General Model:

– Can a cautious lover(a<0,b>0) find true love with an eager beaver (c>0,d>0)?

– Can two equally cautious lovers get together? (a=d<0, b=c>0)?

– What if Romeo and Juliet are both out of touch their own feelings (a=d=0)?

– Fire & Water: Do opposites attract (c=-a,d=-b)?

– How do Romantic Clones interact (a=d, b=c)?

€

′ R = aR + bJ , ′ J = cR + dJ


Examples

Chromosomal Structure

Nat

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(1/

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Base Pairs = BarbellsAdenine

Thymine

GuanineCytosineA

T

CG

AT

GC

A-T T-AC-G G-C

“hydrogen bonds”

4 flavors

Put one barbell on each spoke of a ladder

... then twist the ladder

... and you get the “Double Helix”

Sugar-Phosphate "backbone"deoxyribose

phosphodiester bond

base

hydrogen bondsAdenine

Thymine

Guanine

Cytosine

BASES

DNA =deoxyribonucleic acid

The SEQUENCE of the Human Genome

• 23 chromosome pairs• 2.91 Giga Base Pairs• 691 MB (at 2 bits/Base Pair)• 39,114 genes: functional units• 26,383 “known” function• Average gene ≈27 kBP• Genes ≈ 1/3 of genome

GATCTACCATGAAAGACTTGTGAATCCAGGAAGAGAGACTGACTGGGCAACATGTTATTCAGGTACAAAAAGATTTGGACTGTAACTTAAAAATGATCAAATTATGTTTCCCATGCATCAGGTGCAATGGGAAGCTCTTCTGGAGAGTGAGAGAAGCTTCCAGTTAAGGTGACATTGAAGCCAAGTCCTGAAAGATGAGGAAGAGTTGTATGAGAGTGGGGAGGGAAGGGGGAGGTGGAGGGATGGGGAATGGGCCGGGATGGGATAGCGCAAACTGCCC...

4592 miles 364,000 pages(12 point font)

(100x80 char/page)

If you stretched out the DNA in your body it would be HOW long?

€

2.9 ×109 bases in the genome

× 1013 cells in the human body

× 2 copies in each cell

× 0.34 ×10-9 meters/base (0.34 nM)

× 1 kilometer /1000 meters

× 1 A.U./1.5 ×108 kilometers

= 131 A.U.

http://www.ornl.gov/hgmis/education/images.html

Research examples can …• Awaken • Motivate• Consolidate

– Relate math to other disciplinesContact for more information:[email protected]

http://www.bruce-shapiro.com/presentations.html

tidbits from the sciences: examples for calculus and differential equations

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