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Exploring Real-World Applications of Calculus and Differential Equations

This comprehensive text delves into various applications of calculus and differential equations in real-world scenarios. Through engaging examples, it illustrates topics such as satellite navigation, genomic variation, and enzymatic reactions. The book explains complex concepts like Kepler's Equation and Newton's law of heating using relatable problems like cooking potatoes. By connecting mathematical theories to everyday phenomena—including the dynamics of love and genetic similarities—it provides an intuitive understanding of advanced mathematical principles.

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Exploring Real-World Applications of 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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