Multi-Scale Models in Immunobiology

Multi-Scale Models in Immunobiology
Learning 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

Deﬁnitions
1. Multi-scale models deal with problems which have important (and
more or less separable) features at multiple organisational, spatial
and temporal scales.
2. For practical reasons we may choose to break down complex
computational problems into different scales and couple the
resulting sub-systems.

The different scales can emerge either naturally or as a consequence
of measurement, experimental or observational resolution.

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Multi-Scale Systems 2 of 26

Multi-Scale Modelling

Deﬁnitions
1. Multi-scale models deal with problems which have important (and
more or less separable) features at multiple organisational, spatial
and temporal scales.
2. For practical reasons we may choose to break down complex
computational problems into different scales and couple the
resulting sub-systems.

The different scales can emerge either naturally or as a consequence
of measurement, experimental or observational resolution.
Our Deﬁnition
Here we consider problems where
molecular processes give rise to
behaviour that are accessible at the
organismic level.


Examples of Multi-Scale Processes

Noever et al., NASA Tech Briefs 19(4):82 (1995).


Examples of Multi-Scale Processes

48 In red are shown times by cyclists who
have been found guilty of doping.

46

44
Time/min

42

40

38

1950 1960 1970 1980 1990 2000 2010
Year
The 36 best times in the Tour de France for the Alpe
d’Huez.

Statistical Challenges of Multi-Scale Systems

Z (t )

Yj ( t )

Xi ( t )

When trying to write down how Z (t ) depends on Y (t ) = {Y1 (t ), . . .} or
X (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 = k and
{r , . . . , s } ∩ {u , . . . , v } = ∅ ?
• How does higher-level information ﬂow back into dynamics at lower
levels?

The Innate Immune Response in Zebrafish

Danio Rerio — Zebrafish
• Embryos are optically transparent (as are
some mutant strain adults).
• They are experimentally convenient.
We study the innate
• Zebrafish are an outbred model organism.
immune response to
• Life-expectancy up to two years. wounding.

• How are macrophages and neutrophils
recruited?
• Is the response different between
aseptic and septic wounding?
• How are cellular processes inside
leukocytes coupled to the tissue or
organism-wide signalling dynamics?
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 5 of 26

Leukocyte Chemotaxis in Zebraﬁsh
We extract images,
identify and track
leukocytes and then
analyze their
trajectories for:
• random walk
behaviour
• random walk
behaviour in the x
direction
• random walk
behaviour in the y
direction
• bias in the
directionality
between steps.

Simple Multiscale Models

αt = mean(Nc (αt −1 , σ2 ), Nc (0, σ2 )||w )
p b

Wound varp = f1 (S (C ), pmax , pd ) and varb = f2 (S (C ), bmax , bd )

1 1
S (x ) = (R +x +Kd )− (R + x + Kd ) − Rx
2 4

C (y , t ) = unknown gradient function

y — Leukocytes

r

C — Cytokines R - number of receptors

Distance y

Calibration of Multiscale Models


Inference for Complicated Models
We have observed data, D, that was generated by some system of in
general unknown structure that we seek to describe by a
mathematical 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 , and
have measurements for some or all of them under some experimental
designs, T .
Likelihood Prior
Posterior
f (D|θ, T)π(θ) For complicated models and/or
Pr(θ|T, D)= detailed data the likelihood
f (D|θ, T)π(θ)d θ evaluation can become
Θ
prohibitively expensive.
Evidence

Inference for Multi-Scale Models
The data is often collected at a level different from the one which
determines the dynamics. This places special demands on the
inferential framework.
Multi-Scale Models in Immunobiology Michael P.H. Stumpf Approximate Bayesian Computation 9 of 26

Approximate Bayesian Computation

Model Data, D

θ2 X (t )

t

θ1

Toni et al., J.Roy.Soc. Interface (2009).



Model Data, D

θ2 X (t ) Simulation, Xs (θ)

t

θ1




Model Data, D


t
d = ∆(Xs (θ), D)
θ1




Model Data, D


t
d = ∆(Xs (θ), D)
θ1 Reject θ if d >
Accept θ if d




Prior, π(θ) Deﬁne set of intermediate distributions, πt , t = 1, ...., T
1 > 2 > ...... > T

πt −1 (θ|∆(Xs , X ) < t −1 )

πt (θ|∆(Xs , X ) < t)

πT (θ|∆(Xs , X ) < T)

Sequential importance sampling:
Sample from proposal, ηt (θt ) and weight
wt (θt ) = πt (θt )/ηt (θt ) with
ηt (θt ) = πt −1 (θt −1 )Kt (θt −1 , θt )d θt −1 where
Kt (θt −1 , θt ) is Markov perturbation kernel

Toni et al., J.Roy.Soc. Interface (2009); Toni & Stumpf, Bioinformatics (2010).


Model Selection on a Joint (M , θ) Space

M1 M2 M3 M4

Toni & Stumpf, Bioinformatics (2010); Barnes et al., PNAS (2011); Barnes et al., Stat.Comp. (2012).



M∗

M1 M2 M3 M4




M∗

M ∗∗ ∼ KM (M |M ∗ )
M1 M2 M3 M4




(M3 , θ3 )
(M3 , θ7 )
M∗
(M3 , θ6 )
(M3 , θ2 )
M ∗∗ ∼ KM (M |M ∗ )
(M3 , θ8 ) (M3 , θ5 )
(M3 , θ1 )
(M3 , θ4 ) θ∗
(M3 , θ9 )




(M3 , θ3 )
(M3 , θ7 )
M∗
(M3 , θ6 )
(M3 , θ2 )
M ∗∗ ∼ KM (M |M ∗ )
(M3 , θ8 ) (M3, θ5)(M , θ ) 3 1
(M3 , θ4 ) θ∗
(M3 , θ9 )




(M3 , θ3 )
(M3 , θ7 )
M∗
(M3 , θ6 )
(M3 , θ2 )
M ∗∗ ∼ KM (M |M ∗ )
(M3 , θ8 ) (M3, θ5)(M , θ ) 3 1
(M3 , θ4 ) θ∗
(M3 , θ9 )
θ∗∗ ∼ KP (θ|θ∗ )




M∗

∗∗ ∗∗ M ∗∗ ∼ KM (M |M ∗ )
(M , θ )
θ∗

θ∗∗ ∼ KP (θ|θ∗ )

accept / reject




M∗

∗∗ ∗∗ M ∗∗ ∼ KM (M |M ∗ )
w (M , θ )
θ∗

θ∗∗ ∼ KP (θ|θ∗ )

accept / reject

calculate w


Proof of Principle — In Vitro

Data describe the migration of
human neutrophils in a
microﬂuidic device with a
known 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 )
2
M3 : f ( y ) = √ A e−y /4πt
4πDt


Leukocyte Chemotaxis Models — In Vivo
Wound M1 : f (y ) = n0 − αy
M2 : f (y ) = h × en0 −αy / (1 + en0 −αy )
2
M3 : f (y ) = √4A Dt e−y /4πt
π

y — Leukocyte

r

C - Cytokines R — number of receptors

Distance y
Model Calibration
We use ABC to infer the shape of the gradient and how it changes
with time since wounding from the observed leucocyte trajectories.

Cytokine gradient

This changing gradient explains much of the cell-to-cell variability in
leukocyte chemotaxis.

Robustness of Leukocyte Migration Behaviour

Liepe et al., Integrative Biology (2012).


First Lessons from this Model
Sensing Mechanisms
In our analysis the biophysical
model appeared to matter very little.
This could be taken to mean that
Σ the sensing mechanism is robust to
the details of the receptor and
signalling architecture.

Probing Low Level Processes
We can next exploit the experimental strengths of the zebraﬁsh model
to probe aspects of the cellular signalling machinery and its impact at
migratory patterns of leukocytes:
• inhibit p38 MAPK and JNK.
and study the impact of leukocyte motility (keeping in mind that
epithelial tissue may also be affected by such inhibitors).
Taylor, Liepe et al., Immun.Cell.Biol. (2012).

Random walks in detail
Brownian motion (BM)

∂P (x , y , t ) 2
=A P
dt

biased random walk (BRW)

∂P (x , y , t ) 2
= −u P + A P
dt

persistent random walk (PRW)

∂2 P ∂P
+ 2λ = A2 2
P
dt 2 dt

biased persistent random walk (BPRW)

∂2 P ∂P ∂P
+(λ1 +λ2 ) −v (λ2 −λ1 ) = A2 2
P
dt 2 dt dy

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 17 of 26

In Vivo Leukocyte Temporal Dynamics


In Vivo Leukocyte Spatial Dynamics


The Case for a Spatio-Temporal Perspective

Data Treatment
• Finding the right level of
averaging and
homogenizing data is pivotal
for meaningful analysis and
If we average over modelling.
spatio-temporal scales • In the absence of physical
we miss much of the arguments we can employ
heterogeneity and can information theoretic
even fail to detect approaches to determine
signiﬁcant functional relevant scales.
changes.


Summarizing Multi-Scale Systems Statistics

Z (t )

Yj ( t )

Xi ( t )

We assume that dynamics are governed by processes at the lowest
level. That means the system is completely speciﬁed by the Xi .
Statistical Inference Based on Summary Statistics
We 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 have
to account for relationships between data at different levels.

Summarizing Multi-Scale Systems Statistics

Z (t )

Yj ( t )

Xi ( t )

We assume that dynamics are governed by processes at the lowest
level. That means the system is completely speciﬁed by the Xi .
Information Theoretical Perspective
In many circumstances we can interpret a higher levels as an
information compression device. Then we should ensure that the
mutual information
p(θ, x )
I (Θ; X ) = p(θ, x ) log d θdx = I (θ, Y )
Ω X p(θ)p(x )
In this case Y = g (X ) is a sufﬁcient statistic of the lower-level data X .

Haematopoietic Stem Cells

Similar problems of
biological processes not
Stem Cell Niche Dynamics
being observable at all
relevant scales abound. Stem cell niche

lineage differentiation migration
HSC A D D
determination

Bone marrow Blood stream

differentiation migration
Sub niche LSC T T

Here, too, we observe at the
tissue/organismic level but are really
trying to resolve processes at the
molecular level. Doing both
simultaneously is not yet possible.
Bone Marrow in Mouse, Cristina Lo Celso.

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Haematopoiesis 22 of 26

Stem Cell Niche Dynamics with Leukaemia from a
Bayesian Perspective

Here we have mapped out the parameter regions that would allow
HSCs to win over leukaemic rivals.

Bayesian Design or “Robustness of Behaviour”
I I I
A
1 A 2 A 3 A
1A Design objectives 1B Model deﬁnitions
B C B C B C

x ∼ fM1 (θ) x ∼ fM2 (θ) O O O

input output θ ∼ π(θ|M1 ) θ ∼ π(θ|M2 ) I I I
Model
4 A 5 A 6 A

B C B C B C
∆(x, O)
O O O
input I output O
I I I

7 A 8 A 9 A
x ∼ fM3 (θ) x ∼ fM4 (θ)
θ ∼ π(θ|M3 ) θ ∼ π(θ|M4 ) B C B C B C

O O O
t t
I I
1C System evolution 1D Posterior distribution 10 A 11 A

p(M |D)
1 2 3 4 5 B C B C

O O
population

0.4
B

0.3
posterior
1

0.2
2 M

0.1
p(θi |M1 ,D) p(θi |M2 ,D)

0.0
1 2 3 4 5 6 7 8 9 10 11
model
3

0.4
4 C

0.3
posterior
5
0.2
p(θj |M1 ,D) p(θj |M2 ,D) 0.1
0.0

Barnes et al., PNAS (2011); Silk et al.Nature Communication (2011). 1 2 3 4 5 6
model
7 8 9 10 11


Conclusion and Outlook

Uses of Multi-Scale Models
They can be used as a computational device or a convenient
description of natural processes. Here we used them in the latter
sense.
Progress will require careful selection of experimental methodologies
and 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 for
statistical models.
• Generally, likelihoods are difﬁcult to assess unless suitable
approximations 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

Multi-Scale Models in Immunobiology

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