Estimating Mutation Rates from Clonal Tree Data (using modeling to understand the immune system) Steven H. Kleinstein Department of Computer Science Princeton.

Estimating Mutation RatesEstimating Mutation Ratesfrom Clonal Tree Datafrom Clonal Tree Data

(using modeling to understand the immune system)(using modeling to understand the immune system)

Steven H. KleinsteinDepartment of Computer Science

Princeton University

The Immune System

• Protects the body from dangerous 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

Talk Outline

• Background Immunology

1. What are clonal trees?2. What can we learn from them?3. Method for estimating the hypermutation rate

Simulation Method Analytical Method

4. Results on simulation and experimental data

• Summary, Conclusions and Open Problems

GerminalCenter

EffectorCells

Overview of Humoral Immune Response

AgB

BB

B

B

Ag

Germinal Centers are the Site of Affinity Maturation(selection for cells with affinity-increasing mutations)

Germinal Centers are the Site of Affinity Maturation(selection for cells with affinity-increasing mutations)

B

B

Foreign Pathogen(Antigen)

B

B

B

More about Germinal Centers

• Hundreds of germinal centers in spleen

• Composed of many cell types (B, T, FDC)

• Spatial structure that changes over time

• Driven by stochastic mutation process

• Observation involves sacrifice of animal

While experiments have elucidated basic mechanisms,

how these fit together is not well understood

While experiments have elucidated basic mechanisms,

how these fit together is not well understood

How do we learn about somatic hypermutation?

1. Inject mouse with antigen to provoke an immune response

2. Harvest spleen, cross-section and stain to identify proliferating cells

B cells T cells

3. Microdissect groups of cells 4. Sequence DNA coding for B cell receptor

Experimental Microdissection Data

How can we estimate the rate of mutation?How can we estimate the rate of mutation?

Estimating the B Cell Mutation RateCounting mutations is not sufficient,

number of cell divisions at time of measurement is unknown

Also, problems of mutation schedule and positive/negative selection Also, problems of mutation schedule and positive/negative selection

Number of Cell Divisions

Num

ber

of M

utat

ions

High Mutation Rate

Low Mutation Rate

Observed Number of Mutations

Clonal Tree Data from Micro-DissectionClonal trees from sequence data based on pattern of shared mutations

Germline GGGATTCTC1 -C-----G-2 -------G-3 A------GA4 A---C--GA

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

1(GA)9(CA)

3

1 3

4

8(TG)

2(GC)

5(TC)

Germline

Clonal Tree ‘Shape’ Reflects 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

Simulate clonal trees with known mutation rate and number of divisions

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

Use simulation to identify relevant shape measuresUse simulation to identify relevant shape measures

0

1

2

3

4

5

6

7

8

9

10

11

EDGES

VERTICIE

S

NODES

LEAVES

LEAF C

ELLS

INTERM

IDIA

TE VERTIC

ES

INTRM

EDIATE C

ELLS

REPEAT NODES

REPEAT CELL

S

INTERM

EDIATE N

ODES

INTERM

EDIATE N

ODE CELL

S

GERMLI

NE

UNIQUE M

UTATIO

NS

TOTAL M

UTATIO

NS

ROOT CELL

S

ANY ROOT

Ave

rag

e N

um

ber

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 2.5

Total 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 RateFind 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(Kleinstein, Louzoun and Shlomchik, The Journal of Immunology, In Press)

Also developed analytical method based on same underlying idea(Kleinstein, Louzoun and Shlomchik, The Journal of Immunology, In Press)

ExperimentalObservations

Set of ObservedTree Shapes

Mutation Rate

Distribution ofTree Shapes

Simulation ofB cell expansion

Number of mutationsIntermediate vertices

Sequences at root

=

Simulation Method

1 2 … d

Tree 1

Tree 2

…

Tree n

# divisions (d)

1 2 … d

Tree 1

Tree 2

…

Tree n

# divisions (d)

Equivalent, E(t,d)

Observable, O(t,d)

Used version of Golden Section Search to optimize mutation rate ()Used version of Golden Section Search to optimize mutation rate ()

( , )( | , )

( , )

d

d

E t dL t

O t d

( ) ( | , )t

L L t

2. Likelihood of experimentally observed tree t:

3. Likelihood of experimental dataset:

1. Run simulation many times to fill in equivalent and observable matrices

Reasonable clone size?

Sametree shape?

For each mutation rate () and lethal fraction ()…

t

t

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

y = 1.2073x

R2 = 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

Results for simulation method

Estimate of method precision(SD = 0.035 division-1)

Analytical MethodFormulas to approximate tree shapes…

Minimize error X() 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 P

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 d

d

t

eSR

)1( 1

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

Validating the Analytical Method

y = 1.075x

R2 = 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)

Pre

dict

ed 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

Experimental Data from Autoimmune Response

Data set consists of 31 trees from 7 mice, average 6 sequences / treeData set consists of 31 trees from 7 mice, average 6 sequences / tree

Data set particularly well-suited for estimating mutation rate

• Defined pick sizes provide upper-bound on clone size• Small picks (< 50 cells) minimize positive selection

Micro-dissection from extra-follicular areas in MRL/lpr AM14 heavy chain transgenic mice(William, Euler, Christensen, and Shlomchik. Science. 2002 )

B cells(Idiotype+ RF)

T cells(CD3+)

SingleMicro-dissections

Stems Pruned to Reflect Local Expansion

Closely related cells remain in close spatial proximity

Mutation information retained in local branching structureMutation information retained in local branching structure

Long stems may be artifact of local micro-dissection

Initial Sequence

Tim

e

Stem is removed

New tree root

Pick

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

0 0.1 0.2 0.3 0.4 0.5 0.6

Fraction of FWR Replacement Mutations Lethal ()

Est

imat

ed M

utat

ion

Rat

e

Simulation Estimate

Analytical Estimate

Analytical Estimate (10,000)

Mutation Rate in an Autoimmune Response

Estimated mutation rate is 0.7 x 10-3 – 0.9 x 10-3 base-pair-1 division-1Estimated mutation rate is 0.7 x 10-3 – 0.9 x 10-3 base-pair-1 division-1

Consider range of values for lethal mutation frequency ()

Likely value based on FWR R:S Ratios

Mutation Rate in the Primary NP ResponseData set consists of 23 trees, average 7 sequences / tree

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

Estimated mutation rate is 0.9 x 10-3 – 1.1 x 10-3 base-pair-1 division-1Estimated mutation rate is 0.9 x 10-3 – 1.1 x 10-3 base-pair-1 division-1

Data based on many large picks:

Clone sizes may be very largeIn estimation, limited to:103 in simulation estimate104 in analytical estimate

Positive selection clearly factor

Still evaluating validity of methods for this NP data set

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

0 0.1 0.2 0.3 0.4 0.5 0.6

Fraction of FWR Replacement Mutations Lethal (l)

Est

imat

ed M

utat

ion

Rat

e

Simulation Estimate

Analytical Estimate (10,000)

Analytical Estimate (15,000)

Summary for Estimating Mutation Rate

Mutation rate reflected in clonal tree ‘shapes’Developed simulation and analytical methodsSynthetic datasets to estimate precisionApplied to autoimmune response & NP responseMost accurate mutation rate estimate to dateFuture improvements with additional data

Noisy Optimization

2.50E-27

3.00E-27

3.50E-27

4.00E-27

4.50E-27

5.00E-27

0 64000 128000 192000 256000 320000

Simulation Runs per Likelihood Evaluation

Lik

elih

oo

d

Can we adapt the number of runs at each parameter setting to the minimum required to determine which direction to look?

Can we adapt the number of runs at each parameter setting to the minimum required to determine which direction to look?

How many runs should be used for each likelihood evaluation?

Bias in Simulation Method

0.25

0.27

0.29

0.31

0.33

0.35

0.37

0.39

0.41

0.43

0.45

0 10 20 30 40 50 60

Number of Sequences per Dataset

Est

imat

ed M

utat

ion

Rat

e

Can we remove the bias from the simulation method?Can we remove the bias from the simulation method?

No bias is present in the analytical method

y = 1.2073x

R2 = 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)

Acknowledgements

For more information:[email protected], www.cs.princeton.edu/~stevenk

Yoram Louzoun (Bar-Ilan University)

Mark Shlomchik (Yale University)

(Kleinstein, Louzoun and Shlomchik, The Journal of Immunology, In Press)

PICASsoPICASsoProgram in Integrative Information, Computer and Application Sciences

• Interdisciplinary Computational Seminars– Graduate student-oriented– Wednesdays at noon

• PICSciE Computational Colloquium– Tuesdays at 4pm

• Information and mailing list (see website)

www.cs.princeton.edu/picassowww.cs.princeton.edu/picasso

Summary of Experimental DataAn Total number of cells used to generate the sequences

Un Number of unique sequences used to create the tree

Tn The average number of mutations per sequence

Mn The number of unique mutations in the tree

IN The number of intermediate nodes (with sampled sequences)

RN The number of sampled sequences at the root of the tree

Autoimmune Response Primary NP ResponseMouse Pick/Tree An Un Tn Mn In Rn 2205 5a2,3 30 7 1.38 11 0 2 2205 5a5 20 6 1.00 7 0 2 2205 5f1,2.K 100 2 0.17 1 0 5 2205 5f1,2.L 100 1 0.00 0 0 5 2205 5f1,2.M 100 1 0.00 0 0 2 2540 11f,g 85 6 2.17 7 2 0 2540 11g1 30 2 1.20 3 0 0 4270 10j1 10 1 0.00 0 0 6 4270 10j2 20 3 1.83 6 1 0 5281 14c1 30 3 6.00 3 1 0 5281 14a1,3 20 3 0.18 2 0 9 5281 14c2 10 3 2.00 4 1 2 7976 16b3.A 15 3 2.17 6 1 0 7976 16b3.B 15 2 1 0 1 7976 16c1 50 1 0.00 0 0 5 7976 16c2.A 20 3 0.29 2 0 5 7976 16c2.B 20 1 0.00 0 0 7 7976 16d1 20 3 0.25 2 0 6 7976 16d2 20 2 0.14 1 0 6 7976 16c3.A 10 2 0.75 1 0 1 7976 16c3.B 10 1 0.00 0 0 1 7983 17a4.A 10 2 0.33 2 0 3 7983 17a4.B 10 2 0.25 1 0 2 7983 17a4.C 10 1 0.00 0 0 1 7983 17a5.A 50 6 3.25 15 1 1 7983 17a5.B 50 1 0.00 0 0 1 4641 12a2,b1 17 3 2 1 2 4641 12c1 10 1 0.00 0 0 6 4641 12c2 6 2 0.14 1 0 6 4641 12d2 50 4 2.80 9 1 1 4641 12e1,2 20 2 0.09 1 0 10

Germinal Center Day An Un Mn In Rn 61AM40 8 ? 7 11 1 1 61AM41.A 8 ? 5 7 1 0 61AM41.B 8 ? 3 3 1 0 61AM14 8 ? 6 10 0 7 61AM16 8 ? 11 20 1 2 61AB08 10 ? 2 1 0 1 B12 10 ? 3 3 0 1 B17.A 10 ? 2 1 0 2 B17.B 10 ? 3 2 0 1 B17.C 10 ? 2 4 0 0 L1AB01 10 ? 2 1 0 8 L1AB02 10 ? 2 1 0 9 L1AB03 10 ? 2 1 0 1 L1AD01.A 14 ? 5 11 1 4 L1AD01.B 14 ? 2 1 0 1 L1AD02 14 ? 7 9 3 3 L1AD03 14 ? 4 9 2 0 L1AD05 14 ? 7 10 4 0 61AD01 14 ? 7 9 3 1 61AD02 14 ? 10 51 4 0 61AA02 16 ? 5 7 1 0 GC8 16 ? 10 23 6 0 GC24 16 ? 14 38 6 0

Acknowledgements

For more information:[email protected], www.cs.princeton.edu/~stevenk

Yoram Louzoun (Bar-Ilan University)

Mark Shlomchik (Yale University)

(Kleinstein, Louzoun and Shlomchik, The Journal of Immunology, In Press)