multi-scale models in immunobiology

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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

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Page 1: Multi-Scale Models in Immunobiology

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

Page 2: Multi-Scale Models in Immunobiology

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

Page 3: Multi-Scale Models in Immunobiology

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

Page 4: Multi-Scale Models in Immunobiology

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

Page 5: Multi-Scale Models in Immunobiology

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

Page 6: Multi-Scale Models in Immunobiology

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

Page 7: Multi-Scale Models in Immunobiology

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

Page 8: Multi-Scale Models in Immunobiology

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

Page 9: Multi-Scale Models in Immunobiology

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

Page 10: Multi-Scale Models in Immunobiology

Calibration of Multiscale Models

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Spatio-Temporal Immune Response in Zebrafish 8 of 26

Page 11: Multi-Scale Models in Immunobiology

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

Page 12: Multi-Scale Models in Immunobiology

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

Page 13: Multi-Scale Models in Immunobiology

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

Page 14: Multi-Scale Models in Immunobiology

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

Page 15: Multi-Scale Models in Immunobiology

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

Page 16: Multi-Scale Models in Immunobiology

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

Page 17: Multi-Scale Models in Immunobiology

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

Page 18: Multi-Scale Models in Immunobiology

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

Page 19: Multi-Scale Models in Immunobiology

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

Page 20: Multi-Scale Models in Immunobiology

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

Page 21: Multi-Scale Models in Immunobiology

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

Page 22: Multi-Scale Models in Immunobiology

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

Page 23: Multi-Scale Models in Immunobiology

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

Page 24: Multi-Scale Models in Immunobiology

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

Page 25: Multi-Scale Models in Immunobiology

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

Page 26: Multi-Scale Models in Immunobiology

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

Page 27: Multi-Scale Models in Immunobiology

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

Page 28: Multi-Scale Models in Immunobiology

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

Page 29: Multi-Scale Models in Immunobiology

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

Page 30: Multi-Scale Models in Immunobiology

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

Page 31: Multi-Scale Models in Immunobiology

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

Page 32: Multi-Scale Models in Immunobiology

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

Page 33: Multi-Scale Models in Immunobiology

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

Page 34: Multi-Scale Models in Immunobiology

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

Page 35: Multi-Scale Models in Immunobiology

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

Page 36: Multi-Scale Models in Immunobiology

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

Page 37: Multi-Scale Models in Immunobiology

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

Page 38: Multi-Scale Models in Immunobiology

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

Page 39: Multi-Scale Models in Immunobiology

In Vivo Leukocyte Temporal Dynamics

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 18 of 26

Page 40: Multi-Scale Models in Immunobiology

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 18 of 26

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In Vivo Leukocyte Spatial Dynamics

Multi-Scale Models in Immunobiology Michael P.H. Stumpf Probing Cellular Processes — Observing Tissues 19 of 26

Page 42: Multi-Scale Models in Immunobiology

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

Page 43: Multi-Scale Models in Immunobiology

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

Page 44: Multi-Scale Models in Immunobiology

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

Page 45: Multi-Scale Models in Immunobiology

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

Page 46: Multi-Scale Models in Immunobiology

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

Page 47: Multi-Scale Models in Immunobiology

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

Page 48: Multi-Scale Models in Immunobiology

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

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Acknowledgements

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