Toward Quantitative Simulation Toward Quantitative Simulation of Germinal Center Dynamicsof Germinal Center Dynamics

Biological and Modeling Insights from Experimental Validation

Department of Computer Science, Princeton University

Steven KleinsteinSteven Kleinsteinstevenk@cs.princeton.edu

J.P. SinghJ.P. Singhjps@cs.princeton.edu

DUICMI 2000

Talk Outline

Germinal center model during typical immune responseGerminal center model during typical immune response

How to simulate the specific response to oxazoloneHow to simulate the specific response to oxazolone

Experimental validation: average dynamics - individual dynamicsExperimental validation: average dynamics - individual dynamics

Why the model fails and how to fix itWhy the model fails and how to fix it

Preliminary results using extended modelPreliminary results using extended model

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The Oprea-Perelson Model(Liu et al. Immunity 1996 4: 241) Includes mechanism underlying affinity maturation

Oprea, M., and A. Perelson. 1997. J. Immunol. 158:5155.

Affinity-DependentSelection

Proliferate & Diversify

Dark-Zone Centroblasts Light-Zone Centrocytes

Memory

Death

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Oprea-Perelson Model Equations

CBmC

BBmi

iiLCbB

iMiRRi

iRiRioffi

ioffiion

i

iioffiionim

i

iBimiRRiCbiiiCbidi

dB

ii

td

LLfdt

dL

LLfBpdt

dL

MdRmpdt

dM

RdRmXkdt

dR

XkSCkdt

Xd

CXkSCkBfdt

dC

BdBfRmpBpBiiBiiBiipBtkdt

dB

BtkM

BBp

dt

dB

XeStS S

3 Flow

3 FloweProliferat

Exit

ExitUnbind

UnbindBind

UnbindBind2 Flow

2 FlowExiteProliferat

11

1 Flow

0,

1 Flow

0

2

1

)1(

),1(),1(),(2)(

)(1

)(

CBmC

BBmi

iiLCbB

iMiRRi

iRiRioffi

ioffiion

i

iioffiionim

i

iBimiRRiCbiiiCbidi

dB

ii

td

LLfdt

dL

LLfBpdt

dL

MdRmpdt

dM

RdRmXkdt

dR

XkSCkdt

Xd

CXkSCkBfdt

dC

BdBfRmpBpBiiBiiBiipBtkdt

dB

BtkM

BBp

dt

dB

XeStS S

3 Flow

3 FloweProliferat

Exit

ExitUnbind

UnbindBind

UnbindBind2 Flow

2 FlowExiteProliferat

11

1 Flow

0,

1 Flow

0

2

1

)1(

),1(),1(),(2)(

)(1

)(

Oprea, M., and A. Perelson. 1997. J. Immunol. 158:5155.

A complex model that includes many details

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Simulated Germinal Center Dynamics

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0 5 10 15 20 25Day

Num

ber

of B

cel

ls

3210 (Germline)-1-2

Seed Grow Mature

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Does model apply to specific system?

Compare dynamics with data from oxazolone response

GeneralGeneral

ParametersParameters

Response SpecificResponse Specific

Affinity & Mutation:Germline kon & koff

Transition ProbabilitiesAffinity Factor

Half-lifeMigration Rates

Physical Capacity

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Experimental Validation Step #1

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Days post-immunization

Fra

ctio

n of

cel

ls th

at a

re h

igh-

affin

ity (Berek, Berger and Apel, 1991)

The dynamics of splenic germinal center B cellsThe dynamics of splenic germinal center B cells

Among cells with canonical receptor,Key Mutation is highly selected

22 )( ESR 22 )( ESR

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Searching Parameter Space

1.E-02

1.E+011.E+04

1.E-061.E-03

1.E+001.E+03

0.2

0.25

0.3

0.35

R2

koff (day-1)

kon (FDC sites/GC)-1day-1

Two selection pressures:Rescue from apoptosis

Competition for antigen

Two selection pressures:Rescue from apoptosis

Competition for antigen

NOTE: Lower R2 is better fit

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Agreement (under realistic assumptions)

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Days post-immunization

Fra

ctio

n o

f ce

lls th

at a

re h

igh

-a

ffin

ity

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Days post-immunization

Fra

ctio

n o

f ce

lls th

at a

re h

igh

-a

ffin

ity

But, this is not the whole story...

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Experimental Validation Step #2The dynamics within individual germinal centersThe dynamics within individual germinal centers

(Ziegner, Steinhauser and Berek, 1994)

SingleFounder

SingleFounder

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Experimental Validation Step #2

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Fraction of B cells that are high-affinity at day 14

Fra

ctio

n o

f germ

inal c

ente

rs

Statistical model for NP responseshowing all-or-none property

(Radmacher, Kelsoe and Kepler, 1998)

The dynamics within individual germinal centersThe dynamics within individual germinal centers

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A New Implementation is Needed

Differential equations implicitly model average-case dynamics and have no notion of individual cells

Differential equations implicitly model average-case dynamics and have no notion of individual cells

Create new discrete/stochastic simulation of the Oprea-Perelson model

Create new discrete/stochastic simulation of the Oprea-Perelson model

Follows individual cells

Predicts distribution of behaviors

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Model Differs From Experiment

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0.005

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0.015

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0.025

0.03

0.035

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Fraction of B cells that are high-affinity at day 14

Fra

ctio

n o

f ge

rmin

al c

en

ters

Model predicts6 founding cells

Model predicts6 founding cells

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The Root of the ProblemToo many high-affinity clones, too soonToo many high-affinity clones, too soon

Additional mechanisms for “overlooking” potential founders have been proposed(Radmacher, Kelsoe and Kepler, 1998)

Additional mechanisms for “overlooking” potential founders have been proposed(Radmacher, Kelsoe and Kepler, 1998)

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Days post-immunization

Num

ber

High-affinity clones

High-affinity founders

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Implementing Rare SelectionDecrease the probability of recyclingDecrease the probability of recycling

Affinity-DependentSelection

Proliferate & Diversify

Dark-Zone Centroblasts Light-Zone Centrocytes

Memory

Death

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The Effect of Rare Selection

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Probability of Recycling

Num

ber

of F

ound

ers

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0.15

0.2

0.25

0.3

0.35

0.4

R2

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A Fundamental ProblemSelection is on the critical path to clonal dominanceSelection is on the critical path to clonal dominance

Affinity-DependentSelection

Affinity-IndependentProliferation

Dark-Zone Centroblasts Light-Zone Centrocytes

Memory

Death

DUICMI 2000

Fixing the Oprea-Perelson ModelSelected cells have a faster effective division rateSelected cells have a faster effective division rate

Affinity-DependentSelection

Selection-DependentProliferation

Dark-Zone Centroblasts Light-Zone Centrocytes

Memory

Death

DUICMI 2000

Preliminary Agreement with Experiment

Model predicts1 founding cell

Model predicts1 founding cell

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0.14

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Fraction of B cells that are high-affinity at day 14

Fra

ctio

n o

f ge

rmin

al c

en

ters

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Days post-immunization

F(d)

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Oprea-Perelson model (applied to oxazolone)• Predicts average GC dynamics• Fails to predict individual GC behavior

Different mechanisms are required for:• Selection of high-affinity founder• Clonal dominance

Extended Oprea-Perelson model works - so far

Summary & Conclusions

DUICMI 2000

Quantitative analysis of extended model• Develop optimization algorithms

Incorporate additional validation constraints• Size, Mutation, Clonal Trees, etc.

Incorporate additional biological constraints• Spatial aspects, biased mutation, etc.

Apply model to other experimental systems

Future Work