theoretical models in immunology - cbs · theoretical models in immunology ... predator/preys...

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Theoretical models in immunology

Nicolas RapinImmunological bioinformatics

Center for Biological Sequence analysis.

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Why models?

Models make simplifications of the reality, mainly an unknown reality.

A model can describe different parts of the reality, its function or its appearance.

Models have different levels of abstraction.

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Two kinds of mathematical models

Descriptive models, where the aim is to fit biological data to achieve prediction through a statistical model calibration. ‣development of experimental design techniques.

Analytic models, based on systems of ordinary differential equations (ODEs), take into consideration the mechanisms involved in the studied system.‣a famous example is the Lotka-Volterra model for predator/preys interactions.

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A very simple model

The variable P increases at rate k per unit of time.

The ODE describes the change in P.k is a free parameter.

example: bank account.P

k

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constant growth rate : dP/dt = k

0 5 10 15 20 25

5

10

15

dy/dt = k

solution: y(t) = k*t

dP/dt= k - s = k’

k

k’

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Exponential decay

The variable P decreases at a rate proportional to the current value of P (times a factor k).

The solution is well known as:

examples are decay of radioactive particles.

P k

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Exponential decay: dy/dt = -k*y

0 5 10 15 20 25

5

10

15

dy/dt = -d*y

solution: y(t) = Y0*exp(-k*t)

Half life is calculated as follow

Here, all the curves have the same k, but different starting points

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Half life and life span

Half life is the time it takes for a given population to be halved.

Life span is the inverse, 1/half life.

so if a cell in a culture live on average for 2 days, the half life of the cell population is:

1/2 = 0,5 days.

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Exponential growth

The variable P change at a rate proportional to the current value of P (times a factor n).

The solution is well known as:

P

n

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Fitting bacterial growth:

‣The experiment:

‣The assumptions derived from experiments:‣ The population doubles every hour. ‣ On average, each Bacteria enter mitosis every hour.

‣Can one predict the population size at a given time point?

Time(h) 0 1 2 3 4

# bacteria 10 20 40 80 160

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Fitting bacterial growth:

0 2.5 5 7.5

400

800

1200

1600

2000

2400

0 2.5 5 7.5

400

800

1200

1600

2000

2400

n= 0.69 = ln(2)/1growth rate is 69% per hour.

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A first realistic model

We change make the assumption that the more money you have in the bank, the more you spend, whatever you get from your earning.

Will you ever become rich?

k

dP

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constant growth - exp. decay

0 5 10 15 20 25

5

10

15

dy/dt = k - d*y

solution: y(t) = k/d*(1-exp(-dt))+Y0*exp(-dt)

dy/dt = k-dY_= 0 => Y_ = k/d

this is the steady state of the system

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Mass action law

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Interacting populations

The predator/prey system:Rabbit population grows exponentially, and is

a source of food for pumas.Puma population increases by eating rabbits,

and decrease following exp. decay.

R

n

d

a

P

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Interacting populations

0 2500 5000 7500 1!104

1.25!104

40

80

120

160

200

240

Rabbits numbers

Pumasnumbers

Stable limit cycle

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Interacting populations

Target cells TInfected cells I

Pumas and rabbits can be replaced by immunologicaly relevant entities, such as target cells and infected cells in the case of viral infection.

T

v

b

I dIdT

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Logistic growth

Exponential growth doesn’t always reflect reality completely.

n

dP

Two solutions:‣Density dependent death rate and fixed growth

rate.‣Density dependent growth rate and fixed death

rate.

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Density-dependent rates

Maximal rate

indi

vidu

al b

irth

/dea

th ra

te

Population size

Death rate

Birth rate

Minimal rate

indi

vidu

al b

irth

/dea

th ra

tePopulation size

Death rate

Birth rate

equilibrium point

carrying capacity

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Density-dependent rates

Density independent death rate:rate is always

Density-dependent death rate:rate is a (linear) function of the population

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Logistic growth

The model becomes:

0 2.5 5 7.5 10 12.5 15

250

500

750

1000

1250

1500

K = 1250

Exponential growthdP/dt = rN

Logistic growthdP/dt = rN (1- P/K )

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Cellular automata

‣The environment is a matrix like a chess

board. (Lattice)

‣Each cell has a “state”

‣On, Off, Alive, Infected, Dead...

‣Update of the state of a cell defined by rules

according to neighborhood.

‣Not continuous, works step by step.

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The game of life

‣Each cell has 8 "neighbors", which are the cells adjacent to it.

‣Each cell can be either alive or dead.‣A dead cell with exactly 3 live neighbors becomes

alive (or is born). ‣A live cell with 2 or 3 live neighbors stays alive;

otherwise it dies (from loneliness or overcrowding).

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Small example

Who in the class says the state changes?

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The game of life

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The future, agent based models

NaiveTh0

ActivatedTh0

Th1

helperT cell

Th2

IL12

IL4

TNFαTGFβ

NaiveB

ActivatedB

Plasmocyte

INFγ

MF

IgM

IgG1

IgG2

(class switchof B cell)

(class switchof B cell)

becomes(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

secretes

affects (+/-)

contact

LEGEND

(+)

(+)

(+)(+)

(+) (-)(-)

(-) (-)

Input

Input

more states, movements, diffusion even complete organs and much more!!

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Parameter estimation

Parameters value need to:

‣Be estimated from experiments.‣Guessed.

Mathematical model allow to explore the parameter space or test some assumptions.

It si best to have few parameters & processes than many!