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

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

t = time since perigee passage k = 2  /period e<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 Example: M=  /4, e=1/4 to 3 digits:

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

Images: 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

Micro Array Data Time Gene

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

x y

ABO BLOOD GROUP Source: Are the Slovaks and Czechs closer genealogically to the each other or to the Spanish? Use the following distance measurements:

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!

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

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

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

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 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 Normalize variables (N=1) Solution:

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

MAPK Cascade Model MAPK=Mitogen Activated Protein Kinase

As chemical reactions As a cascade As differential equations

Differential equations

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

Romeo and Juliet General Model: –Can a cautious lover(a 0) find true love with an eager beaver (c>0,d>0)? –Can two equally cautious lovers get together? (a=d 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)?

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

Chromosomal Structure Nature, 421: (1/23/2003).

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

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

... and you get the “Double Helix” 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 GATCTACCATGAAAGACTTGTGAATCCAGGAAGAGAGACTGACTGGGCAACATGTTAT TCAGGTACAAAAAGATTTGGACTGTAACTTAAAAATGATCAAATTATGTTTCCCATGC ATCAGGTGCAATGGGAAGCTCTTCTGGAGAGTGAGAGAAGCTTCCAGTTAAGGTGACA TTGAAGCCAAGTCCTGAAAGATGAGGAAGAGTTGTATGAGAGTGGGGAGGGAAGGGGG AGGTGGAGGGATGGGGAATGGGCCGGGATGGGATAGCGCAAACTGCCC miles 364,000 pages (12 point font) (100x80 char/page)

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

Research examples can … Awaken Motivate Consolidate –Relate math to other disciplines Contact for more information: shapiro.com/presentations.html