Understanding the Immune ResponseThrough Modeling and Simulation

Steven H. KleinsteinDepartment of Computer Science

Princeton University

The Immune System

• Protects the body from damaging pathogens– viruses, bacteria, parasites

• Provides basis for vaccines (e.g., flu)• Implicated in disease:

– Autoimmune (Lupus, MS, Rheumatoid Arthritis)– Sepsis, Cancer

Relatively new science, began with Jenner in 1796

Understanding will lead to better diagnostics and therapiesUnderstanding will lead to better diagnostics and therapies

Why Model the Immune System?

• Immune response involves the collective and coordinated response of ≈1012 cells and molecules

• Distributed throughout body– blood, lymph nodes, spleen, thymus, bone marrow, etc.

• Interactions involve feedback loops and non-linear dynamics

• Experiments often require artificial constructs• High variability observed in experimental results

Somatic Hypermutation: important component of responseSomatic Hypermutation: important component of response

Experiments provide only a static window onto the real dynamics of immunity

B cells Antibody Receptors “Recognize” Antigens1. B cell’s must recognize universe of pathogens (antigens)2. Response to any specific antigen must be efficient

Hypermutation & Selection ≈ Darwinian Evolution, but in 3 weeks!Hypermutation & Selection ≈ Darwinian Evolution, but in 3 weeks!

Hypermutation and selection lead to affinity increase over time…

Rearrangement creates initial diversity…

B Cell

Antibody Receptor

What might go wrong?

Commonly Accepted:Somatic Hypermutation Restricted to Germinal Centers

Commonly Accepted:Somatic Hypermutation Restricted to Germinal Centers

Somatic Hypermutation

Antibody Receptors Against Self-antigens

Autoimmune Disease

Autoimmunity is a response against body’s own proteins, DNA, etc.

© 2001 by Garland Publishinghttp://mcb.berkeley.edu/courses/mcb150/Lect10/Lect10.pdf

Germinal Centers Form in the Spleen

Commonly Accepted:Germinal Centers are the Site of Somatic Hypermutation

and Selection of Higher-Affinity B Cells

Commonly Accepted:Germinal Centers are the Site of Somatic Hypermutation

and Selection of Higher-Affinity B Cells

Motivating Experiment

B cells T cells

Estimate mutation rate to show (hyper?)mutationEstimate mutation rate to show (hyper?)mutation

In auto-immune mouse model, observed mutating B cells inextra-follicular areas of spleen (not germinal centers)

Auto-immune MouseMRL/lpr AM14 heavy chain transgenic(William, Euler, Christensen, and Shlomchik. Science. 2002 )

Extra-Follicular Areas

Dividing B cells FDC

ControlPrimary anti-hapten response to NP

(Jacob et al., 1991; Jacob and Kelsoe, 1992; Jacob et al., 1993; Radmacher et al., 1998)

Germinal Centers

Microdissection(10 cells)

What’s hard about estimating the mutation rate?The number of divisions in vivo is unknown

Most recognized in vivo estimates took educated guesses(McKean et al, 1984 and Sablitzky et al, 1985)

Most recognized in vivo estimates took educated guesses(McKean et al, 1984 and Sablitzky et al, 1985)

Number of Cell Divisions

Num

ber o

f Mut

atio

ns

High Mutation Rate

Low Mutation Rate

Observed Number of Mutations

Clonal Trees Provide Needed InformationAnalyze pattern of shared and unique mutations among sequences

from each microdissectionGermline GGGATTCTC1 -C-----G-2 -------G-3 A------GA4 A---C--GA

Clonal tree ‘shapes’ reflect underlying dynamicsClonal tree ‘shapes’ reflect underlying dynamics

1(G→A)9(C→A)

2

1 3

4

8(T→G)

2(G→C)

5(T→C)

Germline

Relating Tree Shapes to Underlying Dynamics

BA C D

g ct c

BA D

Initial Sequence

a t

Initial Sequence

a

D

t

g ct

BA

Investigate with computer simulation of B cell clonal expansion

Parameters: mutation rate (µ), lethal frequency (λ), # divisions (d), pick size (p)

Compare: Rate of 0.2 division-1 for 14 divisionsRate of 0.4 division-1 for 7 divisions

0.01

0.1

1

10

100

EDGESVERTIC

IESNODESLE

AVESLE

AF CELL

S

LEAF R

EPEATCELLS

INTERMID

IATE

INTRMEDIA

TE CELL

SREPEAT N

ODESREPEAT C

ELLS

INTERMEDIATE N

ODES

INTERMEDIATE N

ODE CELL

SGERMLIN

E

UNIQUE M

UTATIONS

TOTAL MUTATIO

NS

Relevant shape measures can differentiate similar clonesRelevant shape measures can differentiate similar clones

(Simulation Data – 5 sequences / tree)

Intermediate Vertices is Useful Measure Compare: Rate of 0.2 division-1 for 14 divisions

Rate of 0.4 division-1 for 7 divisions

0.2

0.4

0.6

0.8

1

1.2

1.4

1.5 1.7 1.9 2.1 2.3 2.5Total Mutations Per Sequence

Inte

rmed

iate

Ver

ticie

s

Shape measures can supplement information from mutation countingShape measures can supplement information from mutation counting

(Simulation Data)

Method for Estimating Mutation Rate (µ)Find mutation rate that produces distribution of tree

‘shapes’ most equivalent to observed set of trees

1.E-39

1.E-37

1.E-35

1.E-33

1.E-31

1.E-29

1.E-27

1.E-25

0.10 0.20 0.30 0.40 0.50

Mutation Rate (per division)

Like

lihoo

d

Assumes equivalent mutation rate in all trees, although number divisions may differ

Also developed analytical method based on same underlying idea(The Journal of Immunology (2003) Vol. 171 No. 9, 4639-4649.)

Also developed analytical method based on same underlying idea(The Journal of Immunology (2003) Vol. 171 No. 9, 4639-4649.)

ExperimentalObservations

Set of ObservedTree Shapes

Mutation Rate

Distribution ofTree Shapes

Simulation ofB cell expansion

Number of mutationsIntermediate vertices

Sequences at root

=

Details of the Simulation Method

0000Tree T

0000…

0000Tree 2

0000Tree 1

D…21# divisions (d)

Equivalent Matrix, E(t,d)# simulated trees ‘equivalent’ to observed tree after d divisions

For each value of the mutation rate (µ), calculate likelihood by…

t

Use Golden Section Search to optimize mutation rate (µ)Use Golden Section Search to optimize mutation rate (µ)

( , )( | , )

( , )

d

d

E t dL t

O t dµ λ =

∑

∑

( ) ( | , )t

L L tµ µ λ=∏

2. Likelihood of experimentally observed tree t:

3. Likelihood of experimental dataset:

1. Run simulation many times to fill in equivalent matrix

Sample space is subset of all

simulation runs

Finding the Optimal Mutation RateGolden Section Search works by successive bracketing of minimum/maximum

http://lib-www.lanl.gov/numerical/bookcpdf/c10-1.pdf

Direct Search Method (No Derivative)Simple Implementation, Linear Convergence

0.38197(golden mean)

Method is effective with 128,000 simulations per LikelihoodMethod is effective with 128,000 simulations per Likelihood

Not tolerant of noise,Make sure evaluation is precise

2.E-27

3.E-27

3.E-27

4.E-27

4.E-27

5.E-27

5.E-27

6.E-27

0 64000 128000 192000 256000 320000

Number of Simulations per Likelihood Evaluation

Like

lihoo

d

Details of the Analytical MethodFormulas to approximate tree shapes…

Minimize error X(µ) over all experimentally observed trees (t)

( ) ( ) ( )2 2 2

( )( ) ( ) ( )

t t t

dt t t t

M M R R P PX MIN

VAR M VAR R VAR Pµ

− − −= + +

∑

For each observed tree, choose number of divisions to minimize error

Observed shape Calculated shape

The average number of mutations per sequence (M) 1(1 )M dλ µ= −d

teSR

−×=

−

)1( 1µλ

µ

The average number of sequences present at the root of the tree (R)

Total number of sequences in nodes with repeated sequences (P) 111 (1 ) tS

tP S p − = − −

Estimating the Lethal Frequency (λ)Simulation Model Parameters:mutation rate (µ), # divisions (d), # sequences (s), lethal frequency (λ)

Choose λ so expected R/(R+S) equals observed value over all mutationsChoose λ so expected R/(R+S) equals observed value over all mutations

Only replacement mutations can be lethal, so…

Fraction of all mutations that are

replacements

ExperimentalData

ObservedR / (R + S)

Lethal Frequency (λ)

ExpectedR / (R + S)

GermlineDNA Sequence

=

H…CAT…

H…CAC…

Y…TAT…

Silent

Replacement

Validating the Simulation MethodUse simulation to construct synthetic data sets with limited number of trees/sequences reflecting currently available experimental data

y = 1.2073xR2 = 0.9149

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Actual Mutation Rate (per division)

Pre

dict

ed M

utat

ion

Rat

e (p

er d

ivis

ion)

Method works even with limited number of clonal trees and sequencesMethod works even with limited number of clonal trees and sequences

Method Precision(SD = 0.035 division-1)

Validating the Analytical Method

y = 1.075xR2 = 0.9435

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Actual Mutation Rate (per division)

Pred

icte

d M

utat

ion

Rat

e (p

er d

ivis

ion)

Use simulation to construct artificial data sets with limited number of trees/sequences reflecting currently available experimental data

Method works even with limited number of clonal trees and sequencesMethod works even with limited number of clonal trees and sequences

Results for analytical method (assumes correct λ)

Testing Method Assumption…All cells in single microdissection divided same number of times

(i.e., division is synchronous)

Assumption does not significantly impact rate estimateAssumption does not significantly impact rate estimate

0

0.1

0.2

0.3

0.4

0.5

Est

imat

ed M

utat

ion

Rat

e

AsynchronousDivision

SynchronousDivision

Generation of synthetic data

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fraction of FWR Replacement Mutations Lethal (λ)

Est

imat

ed M

utat

ion

Rat

e ( µ

)

Simulation Estimate

Analytical Estimate

Mutation Rate in Autoimmune Response

Estimated mutation rate is 1.0 ± 0.1 x 10-3 base-pair-1 division-1Estimated mutation rate is 1.0 ± 0.1 x 10-3 base-pair-1 division-1

Experimental data set: 31 trees from 7 mice, ≈6 sequences / treefrom extra-follicular areas

(Williams et al, Science, 2002)

Estimated λλλλ ≈≈≈≈ 0.55based on R/(R+S)

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fraction of FWR Replacement Mutations Lethal (λ)

Est

imat

ed M

utat

ion

Rat

e

Simulation Estimate

Analytical Estimate

Mutation Rate in Primary NP Response

Estimated mutation rate is 1.1 ± 0.1 x 10-3 base-pair-1 division-1Estimated mutation rate is 1.1 ± 0.1 x 10-3 base-pair-1 division-1

Experimental data set: 23 trees, ≈7 sequences / treefrom germinal centers

(Jacob et al., 1991; Jacob and Kelsoe, 1992; Jacob et al., 1993; Radmacher et al., 1998)

Testing impact on estimate:

• Data based on larger picks

• Positive selection may be factor

Summary

� Developed simulation and analytical methods to estimate in vivo mutation rates (and lethal frequencies)� First rigorous method for in vivo estimates

� Synthetic datasets used to show that…� Methods are precise (± 0.1 x 10-3 base-pair-1 division-1)� Assumption of synchronous division does not impact results

� Extra-follicular B cells in autoimmune mouse hypermutate� Mutation rate (0.9 ± 0.1 x 10-3) similar to NP response (1.1 ± 0.1 x 10-3)

� Future improvements in precision with additional data

Rigorous method to compare mutation rates under varying experimental conditions

Rigorous method to compare mutation rates under varying experimental conditions

Acknowledgements

For more information:

stevenk@cs.princeton.edu; www.cs.princeton.edu/~stevenk

Yoram Louzoun (Bar-Ilan University)

Mark Shlomchik (Yale University)

Jaswinder Pal Singh (Princeton University)

Kleinstein, Louzoun and ShlomchikThe Journal of Immunology (2003) Vol. 171 No. 9, 4639-4649.

PICASsoPICASsoProgram in Integrative Information, Computer and Application Sciences