Case Studies in Ecology Biology Medicine amp Physics Prey Predator Models 2 Observed Data 3 A verbal model of predatorprey cycles Predators eat prey and reduce their numbers Predators go hungry and decline in number ID: 1036966
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1. The Art and Science of Mathematical Modeling Case Studies in Ecology, Biology, Medicine & Physics
2. Prey Predator Models2
3. Observed Data3
4. A verbal model of predator-prey cycles:Predators eat prey and reduce their numbersPredators go hungry and decline in numberWith fewer predators, prey survive better and increaseIncreasing prey populations allow predators to increase ...........................And repeat…4
5. Why don’t predators increase at the same time as the prey?5
6. Simulation of Prey Predator System
7. 7The Lotka-Volterra Model: AssumptionsPrey grow exponentially in the absence of predators.Predation is directly proportional to the product of prey and predator abundances (random encounters).Predator populations grow based on the number of prey. Death rates are independent of prey abundance.
8. Generic Model f(x) prey growth term g(y) predator mortality term h(x,y) predation term e prey into predator biomass conversion coefficient
9. 9
10. Lotka-Volterra Model Simulations
11. 1 – no species can survive2 – Only A can live3 – Species A out competes B4 – Stable coexistence5 – Species B out competes A6 – Only B can live
12. Hodgkin Huxley ModelHow Neurons Communicate
13. Neurons generate and propagate electrical signals, called action potentialsNeurons pass information at synapses:The presynaptic neuron sends the message.The postsynaptic neuron receives the message.Human brain contains an estimated 1011 neurons Most receive information from a thousand or more synapses There may be as many as 1014 synapses in the human brain.
14. Neuronal CommunicationTransmission along a neuron
15. Action Potential How the neuron ‘sends’ a signal
16. Hodgkin Huxley Model –Deriving the Equations
17. Hodgkin Huxley Model –Deriving the Equations
18. Hodgkin Huxley Model
19. Hodgkin Huxley Model –Deriving the Equations
20. Hodgkin Huxley Model
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22. HIV : Models and Treatment
23. Modeling HIV InfectionUnderstand the processWorking towards a cureVaccination?
24. The Process
25. Lifespan of an HIV InfectionPoints to Note: Time in YearsT-Cell count relatively constant over a week
26. HIV Infection Model (Perelson- Kinchner)Modeling T-Cell Production:Assumptions:Some T-Cells are produced by the lymphatic systemOver short time the production rate is constantAt longer times the rate adjusts to maintain a constant concentrationT-Cells are produced by clonal selection if an antigen is present but the total number is boundedT-Cells die after a certain time
27. Modeling HIV Infection
28. Models of Drug Therapy – Line of AttackR-T Inhibitors: HIV virus enters cell but can not infect it.Protease Inhibitors: The viral particle made RT, protease and integrase that lack functioning .
29. RT Inhibitors (Reduce k!)A perfect R-T inhibitor sets k = 0:
30. Protease Inhibitors
31. Modeling Water Dynamics around a Protein
32. Multiple Time Scaleswww.nyu.edu/pages/mathmol/quick_tour.html
33. The SetupWant to study functioning of a protein given the structureBehavior depends on the surrounding moleculesExplicit simulation is expensive due to large number of solvent molecules
34. The General Program
35. Model IWe guess that behavior is captured by the drift and the diffusivity is the bulk diffusivityUse the following modelSimulate using Monte Carlo methodsCalculate the ‘bio-diffusivity’ and compare with MD results
36. Input to the model
37. Results from Model IModel does a poor job in the first hydration shell
38. Model IIWe consider a more general drift diffusion modelRun Monte Carlo Simulations and compare results with Model I
39. ComparisonModel II does a better job than Model I
40. Moral of the StoryMathematical models have been reasonably successful Applications across disciplinesChallenges in modeling, analysis and simulationYES YOU CAN!!!!
41. Questions??