Estimating the Mutation RateEstimating the Mutation Ratefrom Clonal Tree Datafrom Clonal Tree Data

Steven H. Kleinstein,Yoram LouzounPrinceton University

Mark J. ShlomchikYale University

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-Dissection

B cells T cells

Micro-dissection ExperimentExtra-follicular area in autoimmune mouse

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

Clonal trees from sequence data based on pattern of shared mutations

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 artificial clonal trees with known mutation rate and number of divisions

Use distribution of shapes to estimate the mutation rateUse distribution of shapes to estimate the mutation rate

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

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

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 ideaAlso developed analytical method based on same underlying idea

ExperimentalObservations

Set of ObservedTree Shapes

Mutation Rate

Distribution ofTree Shapes

Simulation ofB cell expansion

Number of mutationsIntermediate vertices

Sequences at root

=

Experimental Data from Autoimmune Response

Data consists of 28 picks from 7 mice, 200 sequencesData consists of 28 picks from 7 mice, 200 sequences

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(Jacqueline William, Chad Euler, Sean Christensen and Mark J. Shlomchik)

B cells(Idiotype+ RF)

T cells(CD3+)

SingleMicro-dissections

Stems Pruned to Reflect Local Expansion

Closely related cells remain in close spatial proximity

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

Long stems may be artifact of local micro-dissection

Initial Sequence

Tim

e

Mutation information retained in local branching structure

Stem is removed

New tree root

Pick

Validating Methods with Limited DataUse 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)

Methods work even with limited number of clonal trees and sequences

Methods work even with limited number of clonal trees and sequences

Results for simulation method

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

Mutation Rate in an Autoimmune Response

Estimated mutation rate is 7 x 10-4 – 9 x 10-4 base-pair-1 division-1Estimated mutation rate is 7 x 10-4 – 9 x 10-4 base-pair-1 division-1

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-03

1.1E-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

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 greater than 3.7 x 10-4Estimated mutation rate is greater than 3.7 x 10-4

1.0E-04

3.0E-04

5.0E-04

7.0E-04

9.0E-04

1.1E-03

1.3E-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 EstimateData 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

Summary & Conclusions

Mutation Rate Reflected in Clonal Tree ‘Shapes’Developed Two Methods Cross-ValidatedAutoimmune Response

Data set well-suited for mutation rate estimationMutation Rate = 7 x 10-4 – 9 x 10-4 base-pair-1 division-1

Primary NP ResponseMutation Rate > 3.7 x 10-4 base-pair-1 division-1

Future Improvements with Additional Data

For more information:

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

For more information:

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

Estimating the Mutation RateEstimating the Mutation Ratefrom Clonal Tree Datafrom Clonal Tree Data

B cell somatic hypermutation of is an important component of the immune response. However, despite its significance, precise estimates of the mutation rate do not currently exist. Micro-dissection studies of mutating B cell clones provide an opportunity to measure this rate more accurately than previously possible. Each micro-dissection provides a number of clonally related sequences which, through the analysis of shared mutations, can be genealogically related to each other. The ‘shape’ of these clonal trees is influenced by many processes including the hypermutation rate. We have developed two different methods to estimate the mutation rate based on these data. These method are applied to two sets of experimental data, one from an auto-immune response and one from the anti-hapten response to NP. In addition to comparing the results of the two procedures, we have investigated the effect on our estimate of assumptions such as the fraction of lethal mutations, and experimental decisions such as the micro-dissection pick size.

Steven H. Kleinstein (stevenk@cs.princeton.edu) and Yoram LouzounPrinceton University

Mark J. ShlomchikYale University

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

Current Method Assumptions

• Common assumptions:– All trees have same underlying mutation rate– Mutation rate constant, associated with division

• Reasonable, but currently testing impact:– In each tree, all cells have same # divisions– No positive selection over time considered

Summary & Conclusions

Mutation Rate Reflected in Clonal Tree ‘Shapes’Developed Two Methods Cross-ValidatedAutoimmune Response

Data set well-suited for mutation rate estimationMutation Rate = 7 x 10-4 – 9 x 10-4 base-pair-1 division-1

Primary NP ResponseMutation Rate > 3.7 x 10-4 base-pair-1 division-1

Future Improvements with Additional Data

For more information:

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

For more information:

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