Download Models for cell and organism development

Models for cell and organism
development
ODE, PDE, and Boolean models
Jan-Åke Larsson
Informationskodning, ISY
11th May 2012
Proteins, proteins, proteins
Essential Cell Biology, 3rd ed, Alberts et al, 2009
Examples of modeling tasks
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Metabolism
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Signal transduction
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Gene regulation
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...
(As on Wikipedia)
Examples of modeling tasks
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Metabolism
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Signal transduction
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Gene regulation
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...
Nyman et al, Trends Endocr. Metabol. 23:107, 2012
Examples of modeling tasks
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Metabolism
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Signal transduction
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Gene regulation
I
...
Maduro et al, Biochim Biophys Acta, 2009
Examples of modeling tasks
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Metabolism
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Signal transduction
I
Gene regulation
I
...
Essential Cell Biology, 3rd ed, Alberts et al, 2009
Model types
pnas.org
I
Graph models
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Dynamic models
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Stochastic models
Graph models
pnas.org
Graph models
cycles point at
feedback relations
connected
subgraphs point at
functional modules
pnas.org
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Indicates dependence
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Can handle complex networks
I
Dynamics not included
I
Statistical (Bayesian-network)
methods common
may reveal missing
regulatory
interactions
provides clues about
redundancy
comparing different
organisms can show
evolutionary
relations
Model types
pnas.org
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Graph models
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Dynamic models
I
Stochastic models
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
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Graph models
I
Dynamic models
I
Stochastic models
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
I
Graph models
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Dynamic models
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Stochastic models
Boolean models
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Represented using binary vectors
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Usually synchronous updating, discretized time
x(t + 1) = F x(t)
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Finite state space
T
x(t) = xhb (t), xKr (t), xpdm (t), xcas (t)

0
 +1


x(t+1) = F 
−5
−5
0
0
0 −5
+1 0
−5 +1


0

0 
 x(t)


−5
0
Boolean models
I
Represented using binary vectors
I
Usually synchronous updating, discretized time
x(t + 1) = F x(t)
I
Finite state space
9
hb
C
D
Kr
6
T
x(t) = xhb (t), xKr (t), xpdm (t), xcas (t)
pdm
cas
3
5

7
A
B
E
F
0
0
 +1 0

x(t+1) = F 
 −5 +1
−5 −5
0
−5
0
+1


0

0 
 x(t)

−5 
0
Boolean models
I
Represented using binary vectors
Usually synchronous updating, discretized time
x(t + 1) = F x(t)
I
Finite state space
I
Wuensche, Artificial Life Models in Software, 2nd Ed, eds. Komosinski et al, 2009
Boolean models
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Can be used to study large-scale behaviour of random
networks
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For example, the number of attractors seem to scale as
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One can also distinguish chaotic from ordered structures
Wuensche, Kybernetes 32:77, 2003
√
n
Generalized Boolean models
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Asynchronous update
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Multi-valued variables
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Random update rules
A
Nakajima et al, PLoS Comput
Biol 6:e1000760, 2010
B
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
I
Graph models
I
Dynamic models
I
Stochastic models
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
I
Graph models
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Dynamic models
I
Stochastic models
Continuous (ODE-) models
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Time-dependent vectors in Rn
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Usually continuous time (but numerics use discrete time)
x0 (t) = F x(t)
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The function F should be nonlinear
h+ (xj , θj , m) =
xjm
xjm
+ θjm
, or
σ(xj , θj , m) =
1
1 + e m(θj −xj )
de Jong, J Comput Biol 9:67, 2002
Continuous (ODE-) models
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Stability is more difficult to analyze
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The number of attractors is hard to find
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In some cases, proving that there is one is difficult
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Parameter values are unknown
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Usable in less complicated networks
A common form is
xi0 (t) = σ
X
ωij xj (t) − θi
j
This is linear close to the “thresholds” θi
Continuous (ODE-) models
One approach is piecewise linearization
de Jong, J Comput Biol 9:67, 2002
This will simplify the analysis greatly, but also introduce spurious
steady states on boundaries between the linearized regions
Continuous (ODE-) models
Example:
dMi (t)
= γM [Fi (P(t)) − Mi (t)] + ξi (t)
dt
dPi (t)
= γP [Mi (t)) − Pi (t)]
dt
P
[g (Si + j Jij Pj (t))]m
P
Fi (P(t)) = m
KM + [g (Si + j Jij Pj (t))]m
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
Continuous (ODE-) models
Unfortunately, there are more complications. The reactions take
time, resulting in a delay from presence of a transcription factor
to production of the next protein in the chain
x0 (t) = F (x1 (t − τ1 ), x2 (t − τ2 ), . . . , xN (t − τN ))T
Ruan, in Delay Differential Equations and Applications, ed Arino et al, Springer 2006
Continuous (PDE-) models
In embryonal development, position of cells (or as in Drosophila,
to the right, the nucleii) also plays
a role
Long range signals are often mediated via diffusion of a transcription
factor, leading to a PDE like
∂2x
∂x
= F (x) + c · 2
∂t
∂s
This goes back to Turing (1951)
Essential Cell Biology, 3rd ed, Alberts et al, 2009
Continuous (PDE-) models
In embryonal development, position of cells (or as in Drosophila,
to the right, the nucleii) also plays
a role
Long range signals are often mediated via diffusion of a transcription
factor, leading to a PDE like
∂2x
∂x
= F (x) + c · 2
∂t
∂s
Essential Cell Biology, 3rd ed, Alberts et al, 2009
This goes back to Turing (1951)
Continuous (ODE-) models
Example:
dMi (t)
= γM [Fi (P(t)) − Mi (t)] + ξi (t)
dt
dPi (t)
= γP [Mi (t)) − Pi (t)]
dt
P
[g (Si + j Jij Pj (t))]m
P
Fi (P(t)) = m
KM + [g (Si + j Jij Pj (t))]m
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
Continuous (ODE-) models
↓
Example:
dMi (t)
= γM [Fi (P(t)) − Mi (t)] + ξi (t)
dt
dPi (t)
= γP [Mi (t)) − Pi (t)]
dt
P
[g (Si + j Jij Pj (t))]m
P
Fi (P(t)) = m
KM + [g (Si + j Jij Pj (t))]m
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
I
Graph models
I
Dynamic models
I
Stochastic models
Model types
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
I
Graph models
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Dynamic models
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Stochastic models
Stochastic models
Concentrations of substances do
not vary continuously and
deterministically
There is gene expression noise,
arising mainly in transcription
Signals on the order of tens of
molecules can make a difference
Essential Cell Biology, 3rd ed, Alberts et al, 2009
Model: a stochastic master equation, or alternatively a stochastic
differential equation
Sometimes, but not always, this can be written as an ODE with a
noise term: “Langevin equations”
Stochastic models
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good description of
molecular gene regulation
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requires detailed knowledge
of the reaction mechanisms
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. . . and reaction rates
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stochastic simulation is
costly
On a larger time-scale, stochastic
effects may level out, so that
continuous and deterministic
models form a good
approximation
Essential Cell Biology, 3rd ed, Alberts et al, 2009
Model types, briefly
Graph models: Highly complex
networks, no dynamics, general
features, functional modules
Boolean models: Moderately complex networks, coarse-grained dynamics, synergy effects can be
handled
Continuous models: Less complex
networks, general dynamics, nonlinearity and synergy more difficult
to handle
Stochastic models: Used on single
pathways, requires detailed knowledge, gives detailed answers
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010
Example: Drosophila melanogaster
Example: Drosophila
Example: Drosophila

P

1, Pj Jij xj (t) > 0
xi (t + 1) = θi ,
Jij xj (t) = 0

Pj

0,
j Jij xj (t) < 0
Example: Drosophila
Example: Drosophila
Example: Drosophila
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Boolean modeling can
give hints on missing
parts of the network
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Here, the factor X
would give both
wild-type and mutant
behaviour
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Stochastic modeling
can give more insight
into the stability
Example: Drosophila
Example: Drosophila
Example: Cænorhabditis elegans
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A well-studied organism
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Cell lineage tree completely mapped (959 cells in adult })
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Genome completely mapped (∼ 20 000 genes)
Part of lineage tree (Sulston et al, 83)
Part of lineage tree (Sulston et al, 83)
Boolean model for regulation
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Notation: “promotor functions” g(A|B)
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Synergy as logical AND in g(A|B,C)
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Cell division is controlled by special promotors g(ab|Z)
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Can generate all trees
Part of lineage tree (Sulston et al, 83)
Part of lineage tree (Sulston et al, 83)
Part of lineage tree (Sulston et al, 83)
Extension to discretized levels
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C5 : The factor C has concentration 5
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g(h5 |C5 ): Concentration 5 of the factor h is produced, if
the factor C has concentration 5
Part of lineage tree (Sulston et al, 83)
Part of lineage tree (Sulston et al, 83)
Another extension: external signals,
for example, at cell-cell contact
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Based on physical cell proximity
(morphology) — not lineage tree
Two main mechanisms
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Distinguish two initially identical
daughters
Polarize dividing cell to generate two
distinct daughters
Signaling is part of the known network
Maduro et al, Biochim Biophys Acta, 2009
Part of lineage tree (Sulston et al, 83)
Part of lineage tree (Sulston et al, 83)
Model types, briefly
Graph models: Highly complex
networks, no dynamics, general
features, functional modules
Boolean models: Moderately complex networks, coarse-grained dynamics, synergy effects can be
handled
Continuous models: Less complex
networks, general dynamics, nonlinearity and synergy more difficult
to handle
Stochastic models: Used on single
pathways, requires detailed knowledge, gives detailed answers
Nakajima et al, PLoS Comput Biol 6:e1000760, 2010