The Art and Science of Mathematical Modeling

Case Studies in Ecology, Biology, Medicine &

Physics

Prey Predator Models

2

3

Observed Data

A verbal model of predator-prey cycles:

1. Predators eat prey and reduce their numbers2. Predators go hungry and decline in number3. With fewer predators, prey survive better and

increase4. 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: Assumptions1. 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

x -y

0 0 ,3 0 ,6 0 ,9 1 ,2 1 ,5

x

0

1 ,6

3 ,2

4 ,8

6 ,4

8

y

x -y

0 0 ,3 0 ,6 0 ,9 1 ,2 1 ,5

x

0

4

8

1 2

1 6

2 0

y

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 ModelHow 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 1011 neurons

– Most receive information from a thousand or more synapses

– There may be as many as 1014 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

Think of ion channels as variable conductances

membrane capacitance

membrane potential

external current

net ionic current

m ion ext

m

ext

ion

ion K K KK K

ion Na Na K K L L

dVC I I

dtC

V

I

I

I I g V V

I g V V g V V g V V

Hodgkin Huxley Model –Deriving the Equations

What is ?

- Each channel has several gates.

- Need all gates to be open.

- Probability of one gate open

- Probability of channel being open

and 1

Hodgkin Huxley modeled Na

i

i

iK K i i i i i

i

g

p

dpg g p V p V p

dt

3

4

channel with 3

gates of 'm' and one 'h':

Similarly, they modeled K channel with 4 gates

of type 'n':

Na Na

K K

g g m h

g g n

Hodgkin Huxley Model

3 4

Some Notations:

Potential Difference , Na Activation

Na Inactivation , K Activation

The Equations:

m Na Na K K L L

m

h

n

V m

h n

dVC g m h V V g n V V g V V

dtdm

V m V mdtdh

V h V hdtdn

V n V ndt

Hodgkin Huxley Model –Deriving the Equations

3 4

Hence our equations are:

1

1

1

What are 's and 's?

1

m Na Na K K L L

m m

h h

n n

i i

i ii i

dVC g m h V V g n V V g V V

dtdm

V m V mdtdh

V h V hdtdn

V n V ndt

i V i VV V

V V

Hodgkin Huxley Model

3 4

So finally our equations become:

m Na Na K K L L

m

h

n

C V g m h V V g n V V g V V

V m m V m

V h h V h

V n n V n

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 YearsT-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

Produced by max DeathLymphaticSystem Clonal Selection

So the equation is: 1dT T

s rT Tdt T

Modeling HIV Infection

.tocomparedsmallis point, At this :Note

1

:Equations

Population Virus

Cells-T Infected

Cells-T Normal

*

*

**

max

*

TT

kVTcVTNdt

dV

TkVTdt

dT

kVTTT

TrTs

dt

dT

V

T

T

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:

cttct

t

eec

TNeVV

eTT

cVTNV

TT

TT

TrTsT

*0

0

*0

*

*

**

max

Zero toDecays

Zerolly toExponentia Decays

Reserves Population Cell1

:Become EquationsOur

Protease Inhibitors

max

* *

*

0

100% perfect Protease Inhibitor:

1

Infected virions

Non-infected virions

Before therapy, 0 0, 0 0and .

T-Cell population goes

I

I

I I I

NI NI NI

ctNI I I

TT s rT T kV T

T

T kV T T

V cV V

V N T cV V

V V V V e

0

* * 00 0

to zero in absence of virus.

Assuming within a short time after therapy, constant:

Then, (0) and t ct

t t ct ctNI

T T

e e cV cT T e kV T V e e te

c c c

Modeling Water Dynamics around a Protein

Multiple Time Scales

www.nyu.edu/pages/mathmol/quick_tour.html

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??