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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 & switching The dynamics of love Measuring the human genome.

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Tidbits from the Sciences: Examples for Calculus and Differential Equations

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

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

  3. Inversion of Kepler’s Equation M is easy to calculate E is easily converted to position in orbit Problem: Find E as a function of time t = time since perigee passage k = 2p/period e<1 in an elliptical orbit

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

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

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

  7. 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% the same! http://creative.gettyimages.com

  8. Gene Time Micro Array Data

  9. 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

  10. Data clusters in two dimensions y x

  11. Data clusters in two dimensions y x

  12. ABO BLOOD GROUP • Are the Slovaks and Czechs closer genealogically to the each other or to the Spanish? Use the following distance measurements: Source: http://www.bloodbook.com/world-abo.html

  13. 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!

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

  15. 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

  16. The Potato Problem (Solution)

  17. 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

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

  19. 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!!

  20. 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

  21. 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:

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

  23. Enzymatic Cascades • Traditional Enzymatic Reaction: • More common situation in nature:

  24. MAPK Cascade ModelMAPK=Mitogen Activated Protein Kinase

  25. As a cascade As chemical reactions As differential equations

  26. Differential equations

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

  28. 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

  29. 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

  30. 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)?

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

  32. Chromosomal Structure Nature, 421:396-448 (1/23/2003). http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v421/n6921/index.html

  33. Base Pairs = Barbells Thymine Adenine T A-T T-A C-G G-C A Guanine Cytosine G C A “hydrogen bonds” T C 4 flavors G

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

  35. ... and you get the “Double Helix” DNA =deoxyribonucleic acid

  36. 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 4592 miles 364,000 pages (12 point font) (100x80 char/page) GATCTACCATGAAAGACTTGTGAATCCAGGAAGAGAGACTGACTGGGCAACATGTTATTCAGGTACAAAAAGATTTGGACTGTAACTTAAAAATGATCAAATTATGTTTCCCATGCATCAGGTGCAATGGGAAGCTCTTCTGGAGAGTGAGAGAAGCTTCCAGTTAAGGTGACATTGAAGCCAAGTCCTGAAAGATGAGGAAGAGTTGTATGAGAGTGGGGAGGGAAGGGGGAGGTGGAGGGATGGGGAATGGGCCGGGATGGGATAGCGCAAACTGCCC...

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

  38. 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

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