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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CEN
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BiC BioCentrum-DTUTechnical University of Denmark
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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CEN
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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!