The Art and Science of Mathematical Modeling Case Studies in Ecology, Biology, Medicine & Physics

Prey Predator Models 2

Observed Data 3

A verbal model of predator-prey cycles: 1.Predators eat prey and reduce their numbers 2.Predators go hungry and decline in number 3.With fewer predators, prey survive better and increase 4.Increasing prey populations allow predators to increase And repeat… 4

Why don’t predators increase at the same time as the prey? 5

Simulation of Prey Predator System

7 The Lotka-Volterra Model: Assumptions 1.Prey grow exponentially in the absence of predators. 2.Predation is directly proportional to the product of prey and predator abundances (random encounters). 3.Predator populations grow based on the number of prey. Death rates are independent of prey abundance.

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

Lotka-Volterra Model Simulations

1 – no species can survive 2 – Only A can live 3 – Species A out competes B 4 – Stable coexistence 5 – Species B out competes A 6 – Only B can live

Hodgkin Huxley Model How Neurons Communicate

Neurons generate and propagate electrical signals, called action potentials Neurons pass information at synapses: The presynaptic neuron sends the message. The postsynaptic neuron receives the message. Human brain contains an estimated neurons – Most receive information from a thousand or more synapses – There may be as many as synapses in the human brain.

Neuronal Communication Transmission along a neuron

Action Potential How the neuron ‘sends’ a signal

Hodgkin Huxley Model –Deriving the Equations

Hodgkin Huxley Model

Hodgkin Huxley Model –Deriving the Equations

Hodgkin Huxley Model

HIV : Models and Treatment

Modeling HIV Infection Understand the process Working towards a cure Vaccination?

The Process

Lifespan of an HIV Infection Points to Note: Time in Years T-Cell count relatively constant over a week

HIV Infection Model (Perelson- Kinchner) Modeling T-Cell Production: – Assumptions: Some T-Cells are produced by the lymphatic system Over short time the production rate is constant At longer times the rate adjusts to maintain a constant concentration T-Cells are produced by clonal selection if an antigen is present but the total number is bounded T-Cells die after a certain time

Modeling HIV Infection

Models of Drug Therapy – Line of Attack R-T Inhibitors: HIV virus enters cell but can not infect it. Protease Inhibitors: The viral particle made RT, protease and integrase that lack functioning.

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

Protease Inhibitors

Modeling Water Dynamics around a Protein

Multiple Time Scales

The Setup Want 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 molecules

The General Program

Model I We 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

Input to the model

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

Model II We consider a more general drift diffusion model Run Monte Carlo Simulations and compare results with Model I

Comparison Model II does a better job than Model I

Moral of the Story Mathematical models have been reasonably successful Applications across disciplines Challenges in modeling, analysis and simulation YES YOU CAN!!!!

Questions??