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Multi-Scale Models in ImmunobiologyLearning to Guide the Behaviour of Cells
Michael P.H. Stumpf
Theoretical Systems Biology Group
04/09/2012
Multi-Scale Models in Immunobiology Michael P.H. Stumpf 1 of 26
Multi-Scale Modelling
Definitions1. Multi-scale models deal with problems which have important (and
more or less separable) features at multiple organisational, spatialand temporal scales.
2. For practical reasons we may choose to break down complexcomputational problems into different scales and couple theresulting sub-systems.
The different scales can emerge either naturally or as a consequenceof measurement, experimental or observational resolution.
Our DefinitionHere we consider problems wheremolecular processes give rise tobehaviour that are accessible at theorganismic level.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 2 of 26
Multi-Scale Modelling
Definitions1. Multi-scale models deal with problems which have important (and
more or less separable) features at multiple organisational, spatialand temporal scales.
2. For practical reasons we may choose to break down complexcomputational problems into different scales and couple theresulting sub-systems.
The different scales can emerge either naturally or as a consequenceof measurement, experimental or observational resolution.
Our DefinitionHere we consider problems wheremolecular processes give rise tobehaviour that are accessible at theorganismic level.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 2 of 26
Examples of Multi-Scale Processes
Noever et al., NASA Tech Briefs 19(4):82 (1995).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 3 of 26
Examples of Multi-Scale Processes
1950 1960 1970 1980 1990 2000 2010
38
40
42
44
46
48
Year
Tim
e/m
in
The 36 best times in the Tour de France for the Alped’Huez.
In red are shown times by cyclists whohave been found guilty of doping.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 3 of 26
Statistical Challenges of Multi-Scale Systems
Xi(t)
Yj(t)
Z (t)
When trying to write down how Z (t) depends on Y (t) = Y1(t), . . . orX (t) = X1(t), . . . a range of potential problems become apparent.• How can we write Z |Y and Y |X (or P(Z |Y ), P(Y |X ) and P(Z |X ))?• How do we relate e.g. P(Yj |Xr ,...,s) and P(Yk |Xu,...,v ) with j 6= k andr , . . . , s ∩ u, . . . , v = ∅?
• How does higher-level information flow back into dynamics at lowerlevels?
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 4 of 26
The Innate Immune Response in Zebrafish
Danio Rerio — Zebrafish• Embryos are optically transparent (as are
some mutant strain adults).• They are experimentally convenient.• Zebrafish are an outbred model organism.• Life-expectancy up to two years.
We study the innateimmune response towounding.
• How are macrophages and neutrophilsrecruited?
• Is the response different betweenaseptic and septic wounding?
• How are cellular processes insideleukocytes coupled to the tissue ororganism-wide signalling dynamics?
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 5 of 26
Leukocyte Chemotaxis in ZebrafishWe extract images,identify and trackleukocytes and thenanalyze theirtrajectories for:• random walk
behaviour• random walk
behaviour in the xdirection
• random walkbehaviour in the ydirection
• bias in thedirectionalitybetween steps.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 6 of 26
Simple Multiscale Models
Wound
Distance y
C — Cytokines
y — Leukocytes
r
R - number of receptors
αt = mean(Nc(αt−1,σ2p),Nc(0,σ2
b)||w)
varp = f1(S(C), pmax , pd) and varb = f2(S(C), bmax , bd)
S(x) =12(R+x+Kd)−
√14(R + x + Kd) − Rx
C(y , t) = unknown gradient function
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 7 of 26
Calibration of Multiscale Models
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 8 of 26
Inference for Complicated Models
We have observed data, D, that was generated by some system of ingeneral unknown structure that we seek to describe by amathematical model. In principle we can have a model-set,M = M1, . . . ,Mν, with model parameter θi .We may know the different constituent parts of the system, Xi , andhave measurements for some or all of them under some experimentaldesigns, T.
Posterior︷ ︸︸ ︷Pr(θ|T,D)=
Likelihood︷ ︸︸ ︷f (D|θ,T)
Prior︷︸︸︷π(θ)∫
Θ
f (D|θ,T)π(θ)dθ︸ ︷︷ ︸Evidence
For complicated models and/ordetailed data the likelihoodevaluation can becomeprohibitively expensive.
Inference for Multi-Scale ModelsThe data is often collected at a level different from the one whichdetermines the dynamics. This places special demands on theinferential framework.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 9 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
θ1
θ2
Model
t
X (t)
Data, D
Simulation, Xs(θ)
d = ∆(Xs(θ),D)
Reject θ if d > ε
Accept θ if d 6 ε
Toni et al., J.Roy.Soc. Interface (2009).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Approximate Bayesian Computation
Prior, π(θ) Define set of intermediate distributions, πt , t = 1, ....,Tε1 > ε2 > ...... > εT
πt−1(θ|∆(Xs,X ) < εt−1)
πt(θ|∆(Xs,X ) < εt)
πT (θ|∆(Xs,X ) < εT )
Sequential importance sampling:Sample from proposal, ηt(θt) and weightwt(θt) = πt(θt)/ηt(θt) withηt(θt) =
∫πt−1(θt−1)Kt(θt−1, θt)dθt−1 where
Kt(θt−1, θt) is Markov perturbation kernel
Toni et al., J.Roy.Soc. Interface (2009); Toni & Stumpf, Bioinformatics (2010).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 10 of 26
Model Selection on a Joint (M, θ) Space
M1 M2
M2
M3
M3
M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1
M2
M2 M3
M3
M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1
M2
M2 M3
M3
M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2
M2 M3
M3 M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2M2 M3M3 M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)
(M3, θ5)(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2M2 M3M3 M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)
(M3, θ5)
(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2M2 M3M3 M4
(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)
(M3, θ5)
(M∗∗, θ∗∗)w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2M2 M3M3 M4(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)
(M∗∗, θ∗∗)
w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Model Selection on a Joint (M, θ) Space
M1 M2M2 M3M3 M4(M3, θ1)
(M3, θ2)
(M3, θ3)
(M3, θ4)
(M3, θ6)
(M3, θ7)
(M3, θ8)
(M3, θ9)
(M3, θ5)(M3, θ5)(M∗∗, θ∗∗)
w(M∗∗, θ∗∗)
M∗
M∗∗ ∼ KM(M |M∗)
θ∗
θ∗∗ ∼ KP(θ|θ∗)
accept / reject
calculate w
Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 11 of 26
Proof of Principle — In Vitro
Data describe the migration ofhuman neutrophils in amicrofluidic device with aknown linear IL 8 gradient.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 12 of 26
Proof of Principle — In Vitro
Models:
M1: f (y) = n0 − αy
M2: f (y) = h × en0−αy/ (1 + en0−αy)
M3: f (y) = A√4πDt
e−y2/4πt
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 12 of 26
Leukocyte Chemotaxis Models — In Vivo
Wound
Distance y
C - Cytokines
y — Leukocyte
r
R — number of receptors
M1: f (y) = n0 − αy
M2: f (y) = h × en0−αy/ (1 + en0−αy)
M3: f (y) = A√4πDt
e−y2/4πt
Model CalibrationWe use ABC to infer the shape of the gradient and how it changeswith time since wounding from the observed leucocyte trajectories.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 13 of 26
Cytokine gradient
This changing gradient explains much of the cell-to-cell variability inleukocyte chemotaxis.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 14 of 26
Robustness of Leukocyte Migration Behaviour
Liepe et al., Integrative Biology (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 15 of 26
First Lessons from this Model
Sensing Mechanisms
Σ
In our analysis the biophysicalmodel appeared to matter very little.This could be taken to mean thatthe sensing mechanism is robust tothe details of the receptor andsignalling architecture.
Probing Low Level ProcessesWe can next exploit the experimental strengths of the zebrafish modelto probe aspects of the cellular signalling machinery and its impact atmigratory patterns of leukocytes:• inhibit p38 MAPK and JNK.
and study the impact of leukocyte motility (keeping in mind thatepithelial tissue may also be affected by such inhibitors).Taylor, Liepe et al., Immun.Cell.Biol. (2012).
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Models of Immunity 16 of 26
Random walks in detailBrownian motion (BM)
∂P(x , y , t)dt
= A∇2P
biased random walk (BRW)
∂P(x , y , t)dt
= −u∇P + A∇2P
persistent random walk (PRW)
∂2Pdt2 + 2λ
∂Pdt
= A2∇2P
biased persistent random walk (BPRW)
∂2Pdt2 +(λ1+λ2)
∂Pdt
−v(λ2−λ1)∂Pdy
= A2∇2P
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 17 of 26
In Vivo Leukocyte Temporal Dynamics
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 18 of 26
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 18 of 26
In Vivo Leukocyte Spatial Dynamics
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 19 of 26
The Case for a Spatio-Temporal Perspective
If we average overspatio-temporal scaleswe miss much of theheterogeneity and caneven fail to detectsignificant functionalchanges.
Data Treatment• Finding the right level of
averaging andhomogenizing data is pivotalfor meaningful analysis andmodelling.
• In the absence of physicalarguments we can employinformation theoreticapproaches to determinerelevant scales.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 20 of 26
Summarizing Multi-Scale Systems Statistics
Xi(t)
Yj(t)
Z (t)
We assume that dynamics are governed by processes at the lowestlevel. That means the system is completely specified by the Xi .
Statistical Inference Based on Summary StatisticsWe can often interpret Y /Z as summaries of X , e.g.Yj = g(Xr , . . . ,Xs). Then we have to ensure that
P(θ|Z ) = P(θ|Y ) = P(θ|X )
which is not automatically the case.For parameter estimation and especially for model selection we haveto account for relationships between data at different levels.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 21 of 26
Summarizing Multi-Scale Systems Statistics
Xi(t)
Yj(t)
Z (t)
We assume that dynamics are governed by processes at the lowestlevel. That means the system is completely specified by the Xi .
Information Theoretical PerspectiveIn many circumstances we can interpret a higher levels as aninformation compression device. Then we should ensure that themutual information
I(Θ;X ) =
∫Ω
∫X
p(θ, x) logp(θ, x)
p(θ)p(x)dθdx = I(θ,Y )
In this case Y = g(X ) is a sufficient statistic of the lower-level data X .Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 21 of 26
Haematopoietic Stem Cells
Similar problems ofbiological processes notbeing observable at allrelevant scales abound.
Bone Marrow in Mouse, Cristina Lo Celso.
Stem Cell Niche DynamicsStem cell niche
HSC A D
LSC T
Blood stream
Sub niche
Bone marrow
D
T
lineage
determination
differentiation migration
differentiation migration
Here, too, we observe at thetissue/organismic level but are reallytrying to resolve processes at themolecular level. Doing bothsimultaneously is not yet possible.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Haematopoiesis 22 of 26
Stem Cell Niche Dynamics with Leukaemia from aBayesian Perspective
Here we have mapped out the parameter regions that would allowHSCs to win over leukaemic rivals.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Haematopoiesis 23 of 26
Bayesian Design or “Robustness of Behaviour”
1A Design objectives
Modelinput output
input I
t
output O
t
1B Model definitions
x ∼ fM1(θ)
θ ∼ π(θ|M1)
x ∼ fM2(θ)
θ ∼ π(θ|M2)
x ∼ fM3(θ)
θ ∼ π(θ|M3)
x ∼ fM4(θ)
θ ∼ π(θ|M4)
∆(x, O)
1C System evolution
1 2 3 4 5
population
1
2
3
4
5
1D Posterior distribution
p(M |D)
M
p(θi|M1,D)
p(θj |M1,D)
p(θi|M2,D)
p(θj |M2,D)
Barnes et al., PNAS (2011); Silk et al.Nature Communication (2011).
A
B C
1
I
O
A
B C
2
I
O
A
B C
3
I
O
A
B C
4
I
O
A
B C
5
I
O
A
B C
6
I
O
A
B C
7
I
O
A
B C
8
I
O
A
B C
9
I
O
A
B C
10
I
O
A
B C
11
I
O
1 2 3 4 5 6 7 8 9 10 11
model
post
erio
r
0.0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 10 11
model
post
erio
r
0.0
0.1
0.2
0.3
0.4
A
B
C
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Haematopoiesis 24 of 26
Conclusion and Outlook
Uses of Multi-Scale ModelsThey can be used as a computational device or a convenientdescription of natural processes. Here we used them in the lattersense.Progress will require careful selection of experimental methodologiesand integration of different (though often collinear) data sources.
Some Caveats• Often, especially in medical applications, data cannot be obtained
at lower levels. This can have far-reaching consequences forstatistical models.
• Generally, likelihoods are difficult to assess unless suitableapproximations are available.
• “Simple models can pretty much fit anything (up to a point).”
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Conclusions 25 of 26
Acknowledgements
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Conclusions 26 of 26