Download ORDINARy DIFFERENTIAL EqUATIONS AND CELLULAR

Theoretical models in immunology
Ordinary Differential
Equations and cellular
automata.
Nicolas Rapin
Algorithms in Bioinformatics
The Bioinformatics Centre, Copenhagen University
Biotech Research and Innovation Center , Copenhagen University
Sunday, October 24, 2010
Why models?
Mathematical modeling allows to:
•Understand better the phenomena
•distinguish what is relevant/redundant
•Make hypothesis, test different scenarios
•analyse dynamics
•Estimate parameters.
Sunday, October 24, 2010
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.
Sunday, October 24, 2010
Bioinformatics centre
a PE model
108
107
ρ
N
r
PE
M
δM = 0
α
106
Cell number
THE ROLE
OF MODELS
IN
UNDERSTAN
-DING CD8+
T-CELL
MEMORY
Apoptosis
105
104
ρ = 2.9 day –1
Ton = 1.3 days
Toff = 5.9 days
r = 0.009 day –1
α = 0.57 day –1
103
dPE
= ƒ(t)ρPE – [1–ƒ(t)]( α + r)PE
dt
dM
= r[1–ƒ(t)]PE – δ MM
dt
102
101
0
10
Where ƒ(t) =1, if Ton ≤ t < Toff ; and ƒ(t) = 0, otherwise
108
E
δM = 0
α
Apoptosis
BEST FIT
A procedure that estimates the
parameters in a model by
minimizing the differences
between the predictions of the
14-06-10
model and experimental
data.
Sunday,
October 24, 2010
1 0 4 | FEBRUARY
2005
105
104
102
dE
dt
101
= ƒ(t)rPM – α [1 – ƒ(t)]E
Unconstrained
ρ = 172 day –1
Ton = 2.54 days
Toff = 5.98 days
r = 169.3 day –1
α = 0.58 day –1
103
dPM
= ƒ(t)( ρ – r)PM – δ MPM
dt
Where ƒ(t) =1, if Ton ≤ t < Toff ; and ƒ(t) = 0, otherwise
| VOLUME 5
Constrained ρ = 5 day –1
106
r
Cell number
PM
N
40
Unconstrained ρ =172 day –1
107
ρ
30
Time (days)
Antia et al.
NRI05
b PM model
20
0
10
20
30
Constrained
ρ = 5 day –1
Ton = 1.25 days
Toff = 5.04 days
r = 2.30 day –1
α = 0.11 day –1
40
Time (days)
www.nature.com/reviews/immunol
Two kinds of mathematical models
Descriptive models, where the aim is to fit biological data
to achieve prediction through a statistical model
calibration.
‣Machine based learning. NetMHC.
Analytical 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.
Sunday, October 24, 2010
Modeling choices
mathematical space
continuous vs. discrete time and/or space
•Continuous time, continuous space (PDE)
•Continuous time, discrete space
•Discrete time, continuous space (coupled
maps)
•Discrete time, discrete space (cellular
automata)
Presence of non-determinism
•Deterministic models
•Stochastic models
Sunday, October 24, 2010
Aims
Mathematical models in immunology:
• Rely on concepts
• Theoretical
• Stirred models (averaged populations)
Bioinformatic models in immunology
• Rely on collected data
• Data driven
Sunday, October 24, 2010
Continuous Time Models
Sunday, October 24, 2010
Continuous Time Models
Differential equation for the growth of a population of
size x as a function of continuous time t :
f
Where is a function that relates the populationʼs rate of
change
dx(t)/dt to its size x(t) at the time t.
After integration, we get x as a function of time.
Sunday, October 24, 2010
Continuous Time Models
Differential equation for the growth of a population of
size x as a function of continuous time t :
f
Where is a function that relates the populationʼs rate of
change
dx(t)/dt to its size x(t) at the time t.
After integration, we get x as a function of time.
Sunday, October 24, 2010
Continuous Time Models
Differential equation for the growth of a population of
size x as a function of continuous time t :
f
Where is a function that relates the populationʼs rate of
change
dx(t)/dt to its size x(t) at the time t.
After integration, we get x as a function of time.
Sunday, October 24, 2010
Per capita rates
We define
C(x)=f(x)/x
as the per capita growth
rate, where :
f(x)
x
is the function describing the growth/death rate.
is the number of individuals in the population.
Sunday, October 24, 2010
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.
Note that f is not a function of P.
example: bank account.
k
P
Sunday, October 24, 2010
constant growth rate : dP/dt = k
15
k
10
dy/dt = k
5
0
Sunday, October 24, 2010
solution: y(t) = k*t
5
10
15
20
25
constant growth rate : dP/dt = k
15
k
10
dy/dt = k
5
solution: y(t) = k*t
dP/dt= k - s = k’
0
Sunday, October 24, 2010
5
10
15
20
25
constant growth rate : dP/dt = k
15
k
10
k’
dy/dt = k
5
solution: y(t) = k*t
dP/dt= k - s = k’
0
Sunday, October 24, 2010
5
10
15
20
25
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: decay of radioactive particles.
Sunday, October 24, 2010
P
k
Integration of exp. decay
Sunday, October 24, 2010
Integration of exp. decay
Shuffle variables,
so that Ps are on
one side and other
terms on the other
Then, integrate
with respect to P
and time
Sunday, October 24, 2010
Exponential growth
The variable P change at a rate proportional to the current value of P (times a factor
µ).
The solution is well known as:
P
µ
Sunday, October 24, 2010
Fitting bacterial growth:
2400
2000
1600
1200
800
400
0
Sunday, October 24, 2010
2.5
5
7.5
Fitting bacterial growth:
2400
2000
1600
1200
800
400
0
Sunday, October 24, 2010
2.5
5
7.5
Fitting bacterial growth:
µ= 0.69 = ln(2)/1
growth rate is 69% per hour.
2400
2000
1600
1200
800
400
0
Sunday, October 24, 2010
2.5
5
7.5
Logistic growth
Exponential growth doesn’t always reflect reality completely.
P
d
n
Sunday, October 24, 2010
Logistic growth
Exponential growth doesn’t always reflect reality completely.
P
d
n
Sunday, October 24, 2010
Logistic growth
Exponential growth doesn’t always reflect reality completely.
Two solutions:
‣Density dependent death rate and fixed growth
rate.
‣Density dependent growth rate and fixed death
rate.
Sunday, October 24, 2010
Density-dependent per capita rates
B ir
th
rat
e
Death rate
individual birth/death rate
individual birth/death rate
Maximal rate
D
te
a
r
h
eat
Birth rate
Minimal rate
Population size
Sunday, October 24, 2010
Population size
Density-dependent per capita rates
equilibrium point
B ir
th
rat
e
Death rate
individual birth/death rate
individual birth/death rate
Maximal rate
D
te
a
r
h
eat
Birth rate
Minimal rate
Population size
Population size
carrying capacity
Sunday, October 24, 2010
Density-dependent rates
Density independent death rate:
rate is constant
Density-dependent death rate:
rate is a (linear) function of the population
Sunday, October 24, 2010
Density-dependent rates
Density independent death rate:
rate is constant
Density-dependent death rate:
rate is a (linear) function of the population
Sunday, October 24, 2010
Logistic growth
The model becomes:
Sunday, October 24, 2010
Logistic growth
The model becomes:
Sunday, October 24, 2010
Logistic growth
The model becomes:
Sunday, October 24, 2010
Logistic growth
The model becomes:
1500
Exponential growth
dP/dt = rN
1250 K = 1250
1000
Logistic growth
dP/dt = rN (1- P/K )
750
500
250
0
Sunday, October 24, 2010
2.5
5
7.5
10
12.5
15
Steady states
Sometimes, analytic solutions are hard to find.
Sunday, October 24, 2010
Steady states
Sometimes, analytic solutions are hard to find.
Sunday, October 24, 2010
Steady states
Sometimes, analytic solutions are hard to find.
It is possible to find the equilibrium point, or Steady
Sate of this equation, by assuming that at steady
state, the change is zero. i.e. dP/dt = 0
Sunday, October 24, 2010
Steady States
in mathematical terms, we assume the following:
This is for a stable steady state.
t
Sunday, October 24, 2010
Steady States
in mathematical terms, we assume the following:
This is for a stable steady state.
t
Sunday, October 24, 2010
Mass action law, interacting populations
The rate of a chemical reaction is directly
proportional to the product of the effective
concentrations of each participating molecule.
Sunday, October 24, 2010
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.
a
P
R
µ
Sunday, October 24, 2010
d
Lotka-Volterra
Sunday, October 24, 2010
Interacting populations
240
200
160
Pumas
numbers 120
80
40
0
2500
5000
7500
Rabbits numbers
Sunday, October 24, 2010
1!104
1.25!104
Interacting populations
240
Stable limit cycle
200
160
Pumas
numbers 120
80
40
0
2500
5000
7500
Rabbits numbers
Sunday, October 24, 2010
1!104
1.25!104
Interacting populations
Pumas and rabbits can be replaced by immunologicaly
relevant entities, such as target cells and infected cells in the
case of viral infection.
Target cells
Infected cells
T
I
β
δT
I
T
σ
Sunday, October 24, 2010
δI
Steady states, Exercise, 10 minutes.
Find the steady states of the
model.
Sunday, October 24, 2010
Steady states, Exercise, 10 minutes.
Find the steady states of the
model.
Sunday, October 24, 2010
Simulations with Discrete Models
A step towards artificial life
Nicolas Rapin
Algorithms in Bioinformatics
Center for Biological Sequence analysis.
Sunday, October 24, 2010
Cellular automata
‣An
Automaton is an entity which can exist
in one of a set of states and make
transition between these states based on a
set of rules
‣A Cellular Automaton (CA) consists of a set
of objects (usually) arranged in a regular
grid (Lattice).
‣ Each of these objects is a cell.
‣Discrete
dynamics: at each discrete unit
time, each cell updates its current state
according to a transition rule that
Sunday, October 24, 2010
Modeling Choices
Geometry: dimension and topology.
Boundary conditions: mirrored, periodic.
Neighborhood: von Neumann, Moore.
Cell: defined through the set Σ.
Transition rule(s).
Sunday, October 24, 2010
Topology
3 neighbors
Sunday, October 24, 2010
4 neighbors
6 neighbors
Topology
3 neighbors
Sunday, October 24, 2010
4 neighbors
6 neighbors
Topology
3 neighbors
Sunday, October 24, 2010
4 neighbors
6 neighbors
Topology
3 neighbors
Sunday, October 24, 2010
4 neighbors
6 neighbors
Boundaries
Sunday, October 24, 2010
Boundaries
Sunday, October 24, 2010
Boundaries
Sunday, October 24, 2010
Neighborhoods in 2D CA
von Neumann: four cells, the cell
above and below, right and left
from each cell. The radius r of this
definition is 1.
Moore: enlargement of the von
Neumann neighbourhood
containing the diagonal cells. In
this case, the radius r = 1.
Extended Moore neighbourhood
equivalent to description of Moore
neighbourhood, but neighbourhood
reaches over the distance of the
next adjacent cells. r = 2 (or
Sunday, October 24, 2010
von
Neumann
Moor
e
Moore
(r=2)
Neighborhoods in 2D CA
von Neumann: four cells, the cell
above and below, right and left
from each cell. The radius r of this
definition is 1.
Moore: enlargement of the von
Neumann neighbourhood
containing the diagonal cells. In
this case, the radius r = 1.
Extended Moore neighbourhood
equivalent to description of Moore
neighbourhood, but neighbourhood
reaches over the distance of the
next adjacent cells. r = 2 (or
Sunday, October 24, 2010
von
Neumann
Moor
e
Moore
(r=2)
Neighborhoods in 2D CA
von Neumann: four cells, the cell
above and below, right and left
from each cell. The radius r of this
definition is 1.
Moore: enlargement of the von
Neumann neighbourhood
containing the diagonal cells. In
this case, the radius r = 1.
Extended Moore neighbourhood
equivalent to description of Moore
neighbourhood, but neighbourhood
reaches over the distance of the
next adjacent cells. r = 2 (or
Sunday, October 24, 2010
von
Neumann
Moor
e
Moore
(r=2)
Neighborhoods in 2D CA
von Neumann: four cells, the cell
above and below, right and left
from each cell. The radius r of this
definition is 1.
Moore: enlargement of the von
Neumann neighbourhood
containing the diagonal cells. In
this case, the radius r = 1.
Extended Moore neighbourhood
equivalent to description of Moore
neighbourhood, but neighbourhood
reaches over the distance of the
next adjacent cells. r = 2 (or
Sunday, October 24, 2010
von
Neumann
Moor
e
Moore
(r=2)
Conway’s game of life
‣Moore’s neighborhood, square lattice.
‣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).
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
Small example
Who in the class says the state changes?
Sunday, October 24, 2010
The game of life
Sunday, October 24, 2010
The future: Agent based models (ABMs)
Agent-Based Modeling (ABM), a relatively new computational
modeling paradigm, is the modeling of phenomena as
dynamical systems of interacting agents. Other names for
ABM is individual-based, micro-simulation and multi-agents
modeling.
Sunday, October 24, 2010
ABMs
A promising technique made feasible with the advent of greater
computer power.
Each agent consists of state variables and a set of rules that
governs its behaviour.
Agents can interact either directly with each other or indirectly
through the environment.
Because all individuals in a population are explicitly
represented, they can have unique histories and behaviours.
Sunday, October 24, 2010
ABMs
One important characteristic of ABMs, which
distinguishes them from Cellular
Automata, is the potential asynchrony of
the interactions among agents and between
agents and their environments. In ABM
agents typically do not simultaneously
perform actions at constant time-steps, as
in CAs.
Their actions follow discrete-event cues or a
sequential schedule of interactions. The
discrete-event setup allows for the
cohabitation of agents with different
environmental experiences. Also ABMs are
not necessarily grid-based nor do agents
"tile" the environment.
Sunday, October 24, 2010
The emergence of complex behavior
ABM represents dynamic systems in a
manner permitting the systems to
evolve over time through agent
interactions, with a minimum of apriori assumptions. Macroscopic
system behaviors are then observed
as emergent properties (emergent
behavior).
In particular, the richness of detail one
can take into account in ABM makes
this methodology very appealing for
the simulation of biological and social
systems, where the behavior and the
Sunday, October 24, 2010
Definition and properties of ABMs
When building an ABM to simulate a certain phenomenon, we
need:
• to identify the actors first (the agents).
• We then need to consider the processes (rules) governing
the interactions among the agents.
Sunday, October 24, 2010
Definition and properties of ABMs
The range of agent interactions:
• global interaction (every agent interacts with every other agent);
• local interaction (every agent only interacts with a local neighborhood of
other agents);
• local interaction with some degree of global reach (e.g., small-world
networks).
Agents' behaviors are determined by rules. These rules range from simple
first order predicate logic to algorithms comprising thousands of lines of
code. (this is called a video game in that case)
In many ABM applications, modelers introduce some form of spatial landscape
(e.g., lattice) that constrains potential agent interactions. In some cases
this spatial landscape is represented as a purely passive platform upon
which agents interact. In other cases, however, the spatial landscape is
itself represented as an agent with its own internal states and behavioral
rules (e.g., a region of land with naturally growing food sources).
Sunday, October 24, 2010
Definition and properties of ABMs
The system evolves over time.
The interactions of the agents take place in a
certain order. This order should, in principle,
not be sequential since agents behave
individually in parallel with each other.
In practice however, since computers are
sequential in nature,the order needs to be
serialized though randomized. Simulation of
agents can also be done on parallel
machines, in this case the asynchrony is
easily and better represented.
Sunday, October 24, 2010
Predator / prey system
Rules:
•When
a rabbit meets another
rabbit, they mate and produce one
offspring if there is space for it.
They can only die when eaten.
They move around.
•When
a puma is near a rabbit,
the rabbit gets eaten. A puma can
eat more than one rabbit at a time.
there a probability that puma
produce an offspring if it has eaten
during a time step and if there is
space for it. Pumas move around
and die if they don’t eat for 10
time steps.
Sunday, October 24, 2010
Craig Reynolds Boids
Cohesion: steer to
move toward the
average position of
local flockmates
Separation:
steer to avoid
crowding local
flockmates
Alignment:
steer towards
the average
heading of
local
flockmates
Sunday, October 24, 2010
a boid's neighborhood
Sunday, October 24, 2010
ABMs in immunology
more states, movements, diffusion even
complete organs and much more!!
Sunday, October 24, 2010
ABMs in immunology
LEGEND
Input
IL12
Naive
Th0
MF
(1)
becomes
(2)
(1)
secretes
(2)
(1)
affects (+/-)
(2)
(1)
contact
(2)
TNF_
Input
TGF`
(+)
(+)
Th1
(-)
(+)
(-)
(-)
(-)
Activated
Th0
(+)
Th2
Naive
B
IL4
INFa
(class switch
of B cell)
(class switch
of B cell)
Plasmocyte
IgG2
IgM
helper
T cell
(+)
IgG1
Sunday, October 24, 2010
Activated
B
Parameter estimation
Sunday, October 24, 2010
Parameter estimation
Parameters value need to:
Sunday, October 24, 2010
Parameter estimation
Parameters value need to:
‣Be estimated from experiments.
Sunday, October 24, 2010
Parameter estimation
Parameters value need to:
‣Be estimated from experiments.
‣Guessed.
Sunday, October 24, 2010
Parameter estimation
Parameters value need to:
‣Be estimated from experiments.
‣Guessed.
Mathematical model allow to explore the parameter space or
test some assumptions.
Sunday, October 24, 2010
Parameter estimation
Parameters value need to:
‣Be estimated from experiments.
‣Guessed.
Mathematical model allow to explore the parameter space or
test some assumptions.
It is best to have few parameters & processes than many!
Sunday, October 24, 2010
Merging it all: System Biology paradigm
Population plots
Parameters
file
Graphical spatial
output
Input
data
HLA database
C-ImmSim
Output
data
antigen's peptide
database
Distributions
>vif
MGARASVLSGGKLDK
WEKIRLRPGGKKKYR
LKHIVWASRELERYAL
NPGLLETSEGCKQIIG
QLQPAIQTGTEELRS
LYNTVATLYCVHKGID
VKDTKEALEKMEEEQ
NKSKKKAQQAAADT
GNNSQVSQNYPIVQ
NLQGQMVHQAIS
BLOSUM matrices
Sunday, October 24, 2010
bits
Last position-specific scoring matrix computed
A
R
N
D
C
Q
E
G
H
I
L
K
M
F
P
S
T
W
Y
V
1 A -0.758 -0.822 0.373 -0.982 2.230 0.082 -1.238 -0.140 0.355 0.050 0.474 -0.347 -0.157 0.696 -0.529 -0.026 -0.355 -0.187
2 A 0.624 -1.175 -1.497 -2.260 -2.194 -0.141 -1.838 -2.304 -2.619 -0.730 -0.492 -1.734 0.259 -2.789 -2.358 3.532 3.744 -3.511
3 A 0.093 -2.321 0.208 4.649 0.179 -0.638 2.466 -1.503 -1.906 -2.336 -2.903 -1.261 -0.994 -1.097 -0.905 0.652 -1.337 -3.538
4 A -1.325 -1.641 -0.347 2.046 -0.505 0.850 0.133 0.359 0.311 -0.520 -0.420 -0.244 0.103 -0.251 1.513 -0.946 0.385 -3.002
5 A -0.690 -0.011 -0.206 -1.269 -1.482 0.584 0.164 -0.393 -0.881 -0.530 -0.307 0.087 -0.282 1.174 0.488 0.625 -0.142 -0.126
6 A -1.104 -0.176 0.989 0.422 -1.513 -0.300 -1.516 -0.696 0.878 -0.242 0.046 -1.193 0.233 1.762 0.832 0.027 0.420 -0.180
7 A -0.108 -0.059 1.286 0.777 -1.478 0.581 -0.527 -0.891 1.288 -1.094 0.466 -0.990 0.639 -0.196 0.062 0.710 0.011 1.815
8 A -0.337 -0.825 0.394 0.155 0.373 -1.192 0.077 -1.057 -0.369 -0.147 1.063 -0.335 0.820 1.111 -1.980 -0.109 -0.046 -0.310
9 A -2.783 -2.685 -3.101 -4.039 -3.936 -2.294 -2.882 -4.172 0.893 -2.354 -1.911 -1.371 -1.940 1.723 -3.701 -2.263 -2.765 0.891
1
1.999 -0.607
-2.723 0.203
-0.461 -2.935
0.687 -0.419
2.253 -0.150
1.607 -0.970
0.013 -1.692
0.514 -0.135
7.946 -2.035
|
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# gibbss_mc -onlymat -swt 1 -wlc 200 -m matrices/A0101.mat --
LOGO files
Mathematical / Bioinformatic model of HIV infection
Sunday, October 24, 2010
Bioinformatics centre
Modelling HIV infection
with a twist of bioinformatics
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Mutations
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV TIME COURSE
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Large scale simulation of HIV
infection:
- The idea
Model with immune memory cells.
•One single non-mutating HIV should
be cleared.
Several strains may co-exist. (ABM-like
model)
•Virus sequences are allowed to
mutate.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Large scale simulation of HIV
infection
Bioinformatics methods evaluate:
•fitness (infectivity) as a function of viral
genome.
•epitopes (given a set of MHCs, haplotype).
Immune system pressure as a function of
epitope/MHC predicted binding affinity.
No super infection of cells.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
flowchart of HIv infection model
k
Infection rate
p
Virions per day
I
Infected Cells
b
Death
V
Virus
T
Target Cells
c
Clearance
Perelson A. nri:02
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Model of HIV infection
dI
kV
T
I
dt
dV
pI cV
dt
dT
kV
T
dt
Ordinary
Differential
Equations
(ODE)
14-06-10
Sunday, October 24, 2010
k
Infection rate
p
Virions per day
I
Infected Cells
b
Death
V
Virus
T
Target Cells
c
Clearance
Perelson A. nri:02
Bioinformatics centre
Model of HIV infection
dI
kV
T
I
dt
dV
pI cV
dt
dT
kV
T
dt
Ordinary
Differential
Equations
(ODE)
14-06-10
Sunday, October 24, 2010
k
Infection rate
p
Virions per day
I
Infected Cells
b
Death
V
Virus
T
Target Cells
c
Clearance
Perelson A. nri:02
Bioinformatics centre
Role of memory cells in primary and
persistent infection
Memory is associated with resistance to reinfection.
Viral infection models predict that diseases are
chronic if R0 (per capita growth) is larger
than one.
R0 is > 1 in HIV models
chronic behaviour.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Cell number
A
R0 > 1
Chronic infection
Time
Cell number
B
R0 < 1
Transient infection
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
The problem
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV Models predict
persistence
Cell number
The problem
Time (days)
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV Models predict
persistence
HIV should be cleared like
flu
Cell number
The problem
Time (days)
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV should be cleared like
flu
New model where
memory CTLs help clear
infection
Time (days)
Cell number
HIV Models predict
persistence
Cell number
The problem
Time (days)
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV should be cleared like
flu
New model where
memory CTLs help clear
infection
Model no longer represents
reality
Time (days)
Cell number
HIV Models predict
persistence
Cell number
The problem
Time (days)
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV should be cleared like
flu
New model where
memory CTLs help clear
infection
Time (days)
Cell number
HIV Models predict
persistence
Cell number
The problem
Model no longer represents
reality
There must be other mechanisms
that need to be considered to
restore chronic behavior (escape
Mutations)
14-06-10
Sunday, October 24, 2010
Time (days)
Bioinformatics centre
Conclusion
The new mathematical model predicts
clearance of the virus.
We can apply this to a more descriptive
model.
Can genetic resolution, and mutation
give the chronic behavior back?
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Full HIV model
Immune
Response
Immune
Response
Immune
Response
Virus
Virus
Target
cell
Infected
cell
Immune
Response
Infected
cell
Immune
Response
Immune
Response
Immune
Response
Immune
Response
Immune
Response
Virus
Immune
Response
Infected
cell
Immune
Response
Immune
Response
Immune
Response
14-06-10
Sunday, October 24, 2010
Immune
Response
Immune
Response
Sunday, October 24, 2010
Target cells
infected cells
Productively
infected cells (i.e.
those the immune
system sees)
Naive CTL cells
Effector CTL cells
Memory CTL cells
Sunday, October 24, 2010
fi
fi
Sunday, October 24, 2010
Need a way to
put numbers for
f (fitness) and
kappa (CTL
killing
efficiency)
Bioinformatics centre
How to assign fitness to a strain in the
model?
Survival is a balance between loss of
fitness and immune escape.
The bad: mutations may lead to
impaired protein functions.
The good: at the same time, mutations
can be beneficial to a virus, as
epitopes may be lost.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Predicting fitness from sequence.
sequence
structure
function.
Mutation in protein can lead to
impaired function.
Impaired functions reduce fitness.
fitness = f
(ATCGCGTGAGCG...GTGAGCGCGA)
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
How different is my query
sequence from the consensus?
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
BLOSUM Matrix
14-06-10
A R N D C Q E G H I L K M F P S T WY V
A 4 -1 -2 -2 0 -1 -1 0 -2 -1 -1 -1 -1 -2 -1 1 0 -3 -2 0
R -1 5 0 -2 -3 1 0 -2 0 -3 -2 2 -1 -3 -2 -1 -1 -3 -2 -3
N -2 0 6 1 -3 0 0 0 1 -3 -3 0 -2 -3 -2 1 0 -4 -2 -3
D -2 -2 1 6 -3 0 2 -1 -1 -3 -4 -1 -3 -3 -1 0 -1 -4 -3 -3
C 0 -3 -3 -3 9 -3 -4 -3 -3 -1 -1 -3 -1 -2 -3 -1 -1 -2 -2 -1
Q -1 1 0 0 -3 5 2 -2 0 -3 -2 1 0 -3 -1 0 -1 -2 -1 -2
E -1 0 0 2 -4 2 5 -2 0 -3 -3 1 -2 -3 -1 0 -1 -3 -2 -2
G 0 -2 0 -1 -3 -2 -2 6 -2 -4 -4 -2 -3 -3 -2 0 -2 -2 -3 -3
H -2 0 1 -1 -3 0 0 -2 8 -3 -3 -1 -2 -1 -2 -1 -2 -2 2 -3
I -1 -3 -3 -3 -1 -3 -3 -4 -3 4 2 -3 1 0 -3 -2 -1 -3 -1 3
L -1 -2 -3 -4 -1 -2 -3 -4 -3 2 4 -2 2 0 -3 -2 -1 -2 -1 1
K -1 2 0 -1 -3 1 1 -2 -1 -3 -2 5 -1 -3 -1 0 -1 -3 -2 -2
M -1 -1 -2 -3 -1 0 -2 -3 -2 1 2 -1 5 0 -2 -1 -1 -1 -1 1
F -2 -3 -3 -3 -2 -3 -3 -3 -1 0 0 -3 0 6 -4 -2 -2 1 3 -1
P -1 -2 -2 -1 -3 -1 -1 -2 -2 -3 -3 -1 -2 -4 7 -1 -1 -4 -3 -2
S 1 -1 1 0 -1 0 0 0 -1 -2 -2 0 -1 -2 -1 4 1 -3 -2 -2
T 0 -1 0 -1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -2 -1 1 5 -2 -2 0
W -3 -3 -4 -4 -2 -2 -3 -2 -2 -3 -2 -3 -1 1 -4 -3 -2 112 -3
Y -2 -2 -2 -3 -2 -1 -2 -3 2 -1 -1 -2 -1 3 -3 -2 -2 2 7 -1
V 0 -3 -3 -3 -1 -2 -2 -3 -3 3 1 -2 1 -1 -2 -2 0 -3 -1 4 Henikoff et al.
(1992)
Sunday, October 24, 2010
Bioinformatics centre
Position specific scoring matrix for fitness evaluation
A
R
N
D
C
Q
E
G
H
I
L
K
M
F
P
S
T
W
Y
V
S L Y N T V A T L ..
.
1 -1 -2 -2 0 0 4 0 -1
-1 -2 -2 0 -1 -3 -1 -1 -2
1 -3 -2 6 0 -3 -2 0 -3
0 -4 -3 1 -1 -3 -2 -1 -4
-1 -1 -2 -3 -1 -1 0 -1 -1
0 -2 -1 0 -1 -2 -1 -1 -2
0 -3 -2 0 -1 -2 -1 -1 -3
0 -4 -3 0 -2 -3 0 -2 -4
-1 -3 2 1 -2 -3 -2 -2 -3
-2 2 -1 -3 -1 3 -1 -1 2
-2 4 -1 -3 -1 1 -1 -1 4
0 -2 -2 0 -1 -2 -1 -1 -2
-1 2 -1 -2 -1 1 -1 -1 2
-2 0 3 -3 -2 -1 -2 -2 0
-1 -3 -3 -2 -1 -2 -1 -1 -3
4 -2 -2 1 1 -2 1 1 -2
1 -1 -2 0 5 0 0 5 -1
-3 -2 2 -4 -2 -3 -3 -2 -2
-2 -1 7 -2 -2 -1 -2 -2 -1
-2 1 -1 -3 0 4 0 0 1
14-06-10
Sunday, October 24, 2010
V
0
-3
-3
-3
-1
-2
-2
-3
-3
3
1
-2
1
-1
-2
-2
0
-3
-1
4
Virus
SLYNTVATL
FQYMFHGAI
GLNYPLMPH
TKYRLIALQ
APKNVVIRR
CSLPNIPTK
MIGEKSTLL
DIEINDRTR
FRPLAVYTA
HLLQVELKY
SAFGNMKKL
DAYPNISCG
SDHSMILDR
Fitness
1.0
0.242
0.246
0.418
0.365
0.265
0.262
0.234
0.309
0.248
0.374
0.291
0.280
Sequence used to
build the PSSM is
the consensus
sequence
Bioinformatics centre
Kappa value.
The CTLs killing efficiency.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Infected
cell
A*0201
14-06-10
Sunday, October 24, 2010
GCTCTTCAGACAGGATCAGAAGAACTTAGATCA
TTATATAATACAGTAGCAACCCTCTATTGTGTA
CATCAAAGGATAGAGGTAAAAGACACCAAGGAA
Virus
Bioinformatics centre
Infected
cell
A*0201
14-06-10
Sunday, October 24, 2010
GCTCTTCAGACAGGATCAGAAGAACTTAGATCA
TTATATAATACAGTAGCAACCCTCTATTGTGTA
CATCAAAGGATAGAGGTAAAAGACACCAAGGAA
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
Virus
Bioinformatics centre
GCTCTTCAGACAGGATCAGAAGAACTTAGATCA
TTATATAATACAGTAGCAACCCTCTATTGTGTA
CATCAAAGGATAGAGGTAAAAGACACCAAGGAA
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
ALQTGSEEL
SLYNTVATL
NTVATLYCV
TLYCVHQRI
Infected
cell
Virus
A*0201
CTL
Response
CTL
Response
CTL
Response
CTL
Response
ALQTGSEEL SLYNTVATL NTVATLYCV TLYCVHQRI
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
Infected
cell
CTL
Response
CTL
Response
CTL
Response
CTL
Response
ALQTGSEEL SLYNTVATL NTVATLYCV TLYCVHQRI
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
Infected
cell
CTL
Response
CTL
Response
CTL
Response
CTL
Response
ALQTGSEEL SLYNTVATL NTVATLYCV TLYCVHQRI
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
Infected
cell
CTL
Response
CTL
Response
CTL
Response
CTL
Response
ALQTGSEEL SLYNTVATL NTVATLYCV TLYCVHQRI
Infected
cell
ALQTGSEELRSLINTVATLYCVHQRIEVKDTKE
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
ALQTGSEELRSLYNTVATLYCVHQRIEVKDTKE
Infected
cell
CTL
Response
CTL
Response
CTL
Response
CTL
Response
ALQTGSEEL SLYNTVATL NTVATLYCV TLYCVHQRI
CTL
Response
SLINTVATL
Infected
cell
No cross
reactivity!!
ALQTGSEELRSLINTVATLYCVHQRIEVKDTKE
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
HIV total population analysis
T=2e11 sigma=2e9 tmax=500 disp=1 mutarate=1e-4 seqshor.txt
1e+12
T
I
V
E
P+I+T
Numbers #/mm
1e+10
1e+12
1e+10
1e+08
1e+08
1e+06
1e+06
10000
10000
100
1
100
0
14-06-10
Sunday, October 24, 2010
100
200
300
Time (days)
400
1
500
Bioinformatics centre
IS CTL KILLING OF HIV INFECTED
CELLS CONSTANT?
debate: once activated, CTLs kill
infected cells with the same
efficiency.
•some claim it’s true.
•other claim it is not.
is killing or proliferation correlated to
MHC binding?
14-06-10
Sunday, October 24, 2010
D (CTL)
Bioinformatics centre
One would
expect
that D(CTL) is
close to one
because of
dominance.
14-06-10
Sunday, October 24, 2010
D (HIV)
A
B
Bioinformatics centre
conclusion
Developed a model of HIV where
bioinformatics and mathematical
modelling are used at the same time
Mutation is necessary for the virus to
remain.
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Key points
System level models enable:
Simulations of millions of experiments
Years of experimentation can be
simulated in hours
help select “best” or “most informative”
experiments for actual screening
14-06-10
Sunday, October 24, 2010
Bioinformatics centre
Conclusion
The Immune System is complex
Models increase our understanding of the
immune system
•If you can make a simple model of a
system, you capture its main components
Genomic resolution is important to understand
host/pathogen interaction
System level models address the biological
complexity
14-06-10
Sunday, October 24, 2010
CA of Predator / prey system
Sunday, October 24, 2010
Definition and properties of ABMs
When building an ABM to simulate a certain phenomenon, we
need:
• to identify the actors first (the agents).
• We then need to consider the processes (rules) governing
the interactions among the agents.
Sunday, October 24, 2010
CA of Predator / prey system
Rules:
•When a rabbit meets another rabbit,
they mate and produce one offspring
if there is space for it. They can only
die when eaten. They move around.
•When
a puma is near a rabbit, the
rabbit gets eaten. A puma can eat
more than one rabbit at a time. there
a probability that puma produce an
offspring if it has eaten during a time
step and if there is space for it.
Pumas move around and die if they
don’t eat for 10 time steps.
Sunday, October 24, 2010
step by step creation of a CA
Information of the environment stored in a matrix.
Cells can be either:
• empty (0)
• rabbit
(1)
• puma (2); pumas need another way to remember their age. use another
matrix.
The result of every time step is stored in a temporary matrix.
Sunday, October 24, 2010
Initialisation:
• Create environment matrix. (made of zeros)
• randomly put preys (1) and predators (2) inside.
Loop:
• for each cell in the matrix , apply evolution rules.
Sunday, October 24, 2010
Empty
Do nothing
Sunday, October 24, 2010
Rabbit
count rabbits
around
Do nothing
no
n>1
yes
generate random
position
Do nothing
no
cell is empty
yes
generate random
position
put a new rabbit
there
cell is empty
yes
move rabbit there
Sunday, October 24, 2010
no
Do nothing
Puma
decrease counter
yes
counter = 0
Puma is terminated
no
count rabbits
around
Do nothing
no
n>0
yes
Generate random
position.
Remove ALL
rabbits around
cell is empty
yes
no
Do nothing
generate random
position
put a new puma
there
cell is empty
yes
move puma there
Sunday, October 24, 2010
no
Do nothing
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
1
0
0
Sunday, October 24, 2010
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
Sunday, October 24, 2010
0
0
1
0
0
0
1
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
2
0
0
0
0
0
2
0
0
0
1
0
0
0
2
0
0
0
Sunday, October 24, 2010
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
9
0
0
0
0
0
8
0
0
0
0
0
0
0 10 0
0
0
Sunday, October 24, 2010
The problem
Heavy computations require:
• fast compiled languages. (C, fortran ... ).
• fast computers.
• or many computers in parallel.
• IF execution speed is an issue:
• Use an already written C / fortran(number crushing)
library
• Write your own library
• IF not (think development time):
• Use Maltab or R or python.
Sunday, October 24, 2010
GNU Scientific library / SUNDIALS
‣GNU Scientific libraries provide a complete
set of scientific functions.
‣Can
replace Matlab but is not necessary
faster (as Matlab uses extremely well
optimised functions).
‣Documentation is human readable.
Sunday, October 24, 2010
SUNDIALS
SUNDIALS (SUite of Nonlinear and DIfferential/
ALgebraic equation Solvers) consists of the following
four solvers:
CVODE solves initial value problems for ordinary differential
equation (ODE) systems.
CVODES solves ODE systems and includes sensitivity
analysis capabilities (forward and adjoint).
IDA solves initial value problems for differential-algebraic
equation (DAE) systems.
KINSOL solves nonlinear algebraic systems.
In addition, SUNDIALS provides a Matlab interface to
CVODES and KINSOL, sundialsTB.
Sunday, October 24, 2010
Open MP
We all have now 2+ processors in our computer.
www.openmp.org
The idea is to annotate C source code to make it parallel, with
close to no knowledge of traditional parallel framework like
MPI.
Sunday, October 24, 2010
vfgen
A Vector Field File Generator:
• Generate code for ordinary and
delay differential equations.
• Specify your equations once, and
export to multiple file formats.
• Generate analytical expressions
for Jacobians and higher derivatives
automatically.
• Available export formats include
MATLAB, Scilab, AUTO, GNU Scientific
Library (GSL), SciPy, PyDSTool, XPP
and more.
Sunday, October 24, 2010
Processing
Easy visualisation....
and programing.
Now it’s up to you!
Sunday, October 24, 2010
Python
Numerical libraries
• SciPy
• Numpy
• Rpy2
Machine learning libraries
• mlpy
• mdp
• hcluster
Visualization:
• pylab (matplotlib)
• mayavi
Bioinformatics libraries
• BioPython
Takes 1 week to be up an running using the free book “Dive
Into Python”.
Sunday, October 24, 2010
Bioinformatics centre
Acknowledgements
Ole Lund, Morten Nielsen, Claus
Lundegaard
CBS
The vaccine group
Binf @ KU.
BRIC @ KU.
14-06-10
Sunday, October 24, 2010
Boids - prettier.
Sunday, October 24, 2010