The Art of Mathematical Modeling

The Art and Science of Mathematical Modeling Case Studies in Ecology, Biology, Medicine & Physics1Prey Predator Models2

MTBI summer 20082Observed Data3

MTBI summer 20083

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 repeat4MTBI summer 20084Why dont predators increase at the same time as the prey?5

MTBI summer 20085Simulation of Prey Predator System

67The 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.MTBI summer 20087Generic Model

f(x) prey growth term g(y) predator mortality term h(x,y) predation term e prey into predator biomass conversion coefficientMTBI summer 200889

MTBI summer 20089Lotka-Volterra Model Simulations

MTBI summer 200810

1 no species can survive2 Only A can live3 Species A out competes B4 Stable coexistence5 Species B out competes A6 Only B can liveMTBI summer 200811Hodgkin Huxley ModelHow Neurons Communicate

12Neurons 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 neuronsMost receive information from a thousand or more synapses There may be as many as 1014 synapses in the human brain.

13Neuronal CommunicationTransmission along a neuron

14Action Potential How the neuron sends a signal

15Hodgkin Huxley Model Deriving the Equations

16Hodgkin Huxley Model Deriving the Equations

17Hodgkin Huxley Model

18Hodgkin Huxley Model Deriving the Equations

19Hodgkin Huxley Model

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21HIV : Models and Treatment

22Modeling HIV InfectionUnderstand the process

Working towards a cure

Vaccination?

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The Process

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Lifespan of an HIV InfectionPoints to Note: Time in YearsT-Cell count relatively constant over a week25HIV 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

26Modeling HIV Infection

27Models 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 . 28RT Inhibitors (Reduce k!)A perfect R-T inhibitor sets k = 0:

29Protease Inhibitors

30Modeling Water Dynamics around a Protein

31Multiple Time Scaleswww.nyu.edu/pages/mathmol/quick_tour.html

32The SetupWant to study functioning of a protein given the structure

Behavior depends on the surrounding molecules

Explicit simulation is expensive due to large number of solvent molecules33

The General Program34Model IWe guess that behavior is captured by the drift and the diffusivity is the bulk diffusivity

Use the following model

Simulate using Monte Carlo methods

Calculate the bio-diffusivity and compare with MD results

35Input to the model

36Results from Model IModel does a poor job in the first hydration shell

37Model IIWe consider a more general drift diffusion model

Run Monte Carlo Simulations and compare results with Model I

38ComparisonModel II does a better job than Model I

39Moral of the StoryMathematical models have been reasonably successful

Applications across disciplines

Challenges in modeling, analysis and simulation

YES YOU CAN!!!!40Questions??

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