tid bits Next

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

Tidbits from the Sciences: Examples for Calculus and Differential Equations Bruce E. Shapiro California State University, Northridge

Examples: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

Inversion of Kepler’s Equation: 

t = time since perigee passage k = 2p/period eandlt;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

Inversion of Kepler’s Equation: 

Solve using fixed point iteration Since eandlt;1 for an elliptical orbit: Fixed point always converges Inversion of Kepler’s Equation Example: M=p/4, e=1/4 to 3 digits:

Examples: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

Slide6: 

Images: http://www.sciencemag.com, http://www.nature.com Genomes are being sequenced at an exponential Rate

Genomic Variation: 

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%     and#x8;and#x8;the same! Genomic Variation http://creative.gettyimages.com

Micro Array Data: 

Micro Array Data

Genetic Similarity: 

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: 

Data clusters in two dimensions x y

Data clusters in two dimensions: 

Data clusters in two dimensions x y

ABO BLOOD GROUP: 

ABO BLOOD GROUP 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:

ABO Blood Group: 

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

Examples: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

The Potato Problem*: 

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): 

The Potato Problem (Solution)

The Potato Problem: 

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: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

Canonical Models: 

Model phenomona that appear in a wide variety of situations in nature: 'Exponential Relaxation' of y to steady state with time constant t One of the most common models in biology!! Canonical Models

Law of Mass Action: 

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

Application of Mass Action: 

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:

Two-State Protein: 

Two-State Protein Normalize variables (N=1) Solution:

Enzymatic Cascades: 

Enzymatic Cascades Traditional Enzymatic Reaction: More common situation in nature:

MAPK Cascade ModelMAPK=Mitogen Activated Protein Kinase: 

MAPK Cascade Model MAPK=Mitogen Activated Protein Kinase

Slide25: 

As chemical reactions As a cascade As differential equations

Slide26: 

Differential equations

Slide27: 

Examples: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

Strogatz’s Romeo & Juliet: 

Strogatz’s Romeo andamp; 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: 

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

Romeo and Juliet: 

Romeo and Juliet General Model: Can a cautious lover(aandlt;0,bandgt;0) find true love with an eager beaver (candgt;0,dandgt;0)? Can two equally cautious lovers get together? (a=dandlt;0, b=candgt;0)? What if Romeo and Juliet are both out of touch their own feelings (a=d=0)? Fire andamp; Water: Do opposites attract (c=-a,d=-b)? How do Romantic Clones interact (a=d, b=c)?

Examples: 

Satellite navigation Genomic variation Cooking potatoes Enzymatic reactions andamp; switching The dynamics of love Measuring the human genome Examples

Slide33: 

Chromosomal Structure

Base Pairs = Barbells: 

Base Pairs = Barbells Adenine Thymine Guanine Cytosine A T C G A T G C A-T T-A C-G G-C 'hydrogen bonds' 4 flavors

Slide35: 

Put one barbell on each spoke of a ladder ... then twist the ladder

... and you get the “Double Helix”: 

... and you get the 'Double Helix' DNA =deoxyribonucleic acid

The SEQUENCE of the Human Genome: 

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?: 

If you stretched out the DNA in your body it would be HOW long? http://www.ornl.gov/hgmis/education/images.html

Research examples can …: 

Research examples can … Awaken Motivate Consolidate Relate math to other disciplines Contact for more information: bruce.e.shapiro@csun.edu http://www.bruce-shapiro.com/presentations.html