1

Analysis of DNA Damage and Repair in Colonic Crypts

Raymond J. CarrollTexas A&M University

http://stat.tamu.edu/~carrollcarroll@stat.tamu.edu

Postdoctoral Training Program:http://stat.tamu.edu/B3NC

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Acknowledgments

• Jeffrey Morris, M.D. Anderson– Lead author

• Naisyin Wang (adducts and structure)• Marina Vannucci, Texas A&M (wavelets)• Phil Brown, University of Canterbury

(wavelets)• Joanne Lupton, Biology of Nutrition at

Texas A&M (problems and data!)

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Outline

•Introduction

•Colon Carcinogenesis Studies

•Hierarchical Functional Model

•DNA Damage: regional correlations

•Crypt Cell Architecture: modeling where the cells are located

•DNA Repair: Wavelet-based Estimation of Hierarchical Functions

•Conclusions

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Some Background

• General Goal: Study how diet affects colon carcinogenesis.

• Model: Carcinogen-induced colon cancer in rats.

• Early Carcinogenesis: DNA damage to cells, and associated repair and cell death (apoptosis)

• If not repaired or removed• Mutation• Colon cancer

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Some Background

• We are especially interested in anatomical effects

• Regions of the colon, e.g., proximal (front) and distal (back)• There are some major differences in early

carcinogenesis between these two regions

• Localized phenomena: cell locations• Apoptosis and DNA adducts differ by

location in colonic crypts

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Colon Sliced and Laid Out

Aberrant Colon Crypts

Normal Colon Crypts

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Architecture of Colon Crypts: Crosssectional View

• Stem Cells:– Mother cells

near bottom

• Depth in crypt ~ age of cells– Suggests

importance of depth

• Relative Cell Position: – 0 = bottom – 1 = top

crypts

Lumen

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Architecture of Colon Crypt: Expanded View

• The cells are more easily visible here

• Note that the cells seem smaller at the crypt bottom

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Architecture of Colon Crypt

• The general idea is to slice the colon crypt

• The cells along the left wall are assayed

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Colon Carcinogenesis Studies

• Rats are• fed different diets • exposed to carcinogen (and/or radiation)• euthanized.

• DNA adducts, DNA repair, apoptosis • measured through imaging experiments

• Hierarchical structure of data• Diet groups - rats - crypts - cells/pixels• Hierarchical longitudinal (in cell depth)

data

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Coordinated Response

• Rats were exposed to a potent carcinogen (AOM)

• At both the proximal and distal regions of the colon, ~20 crypts were assayed

• The rat-level function is gdr(t)

• For each cell within each crypt, the level of DNA damage was assessed by measuring the DNA adduct levels

• Question: how is DNA damage related in the proximal and distal regions, across rats?• We call this coordinated response

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Coordinated Response as Correlation

• We are interested in the “correlation” of the DNA damage in the proximal region with that of the distal region

• Are different regions of the colon responding (effectively) independently to carcinogen exposure?

• This sort of interrelationship of response is what is being studied in our group.• It is not cell signaling in the classic sense• We will have data on this in the near future

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Coordinated Response

• Correlation in the usual sense is not possible

• Let Y(t) = DNA adduct in a proximal cell measured by immunohistochemical staining intensity at cell depth t

• Let Z (t) = DNA adduct in a distal cell at cell depth t

• We cannot calculate correlation(Y,Z) (t) in the usual way• the same cell cannot be in both locations

• Coordinated response then has to be measured at a higher level

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Coordinated Response: Hierarchical Functional Model

• Let d = diet group• Let r = rat• Let c = crypt• Let t =tdrc= cell

position• Let Ydrc(t) = adduct

level in the proximal region

• The diet-level function is gd(t)

drc drc drct

drc drcdr

dr d dr

Y (t)=g (t)+ε

g (t)= +η (t)

=g (t)+ξ

g (t)

g (t) (t)

• Our aim: estimate the correlation between proximal and distal regions as a function of cell depth at the rat level

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Coordinated Response: Average then Smooth

• If cell depths were identical for each crypt, we could solve this by “average then smooth”

• That is, average over all crypts at any given depth, then estimate the correlation as a function of depth

• The estimated correlation would of course account for the averaging over a finite number of crypts

drc drc drct

drc drcdr

dr d dr

Y (t)=g (t)+ε

g (t)= +η (t)

=g (t)+ξ

g (t)

g (t) (t)

• Problem: data are not of this structure• Cell locations vary from crypt to crypt• Number of cells varies from crypt-to-crypt

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Coordinated Response: Smooth then Average

• Instead, we smoothed crypts via nonparametric regression

• Then average the smooth fits over the crypts (on a grid of depths)

• Then compute the correlation as before

• We actually fit REML to the fitted functions at the crypt level

dr

d

drc drct

drc

r d d

drc

d c

r

r

g (t)

g (t) g (t)

g

Y (t)=

(t)

+ε

= +η (t)

=g (t)+ξ (t)

• Problem: Is there any effect due to the initial smooth?

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Coordinated Response: Asymptotics

• General theory available: kernel regression• Allows explicit calculations• Can we estimate the correlation function just as

well as if the crypt-level functions were known?

• Complex higher order expansions necessary• The asymptotic theory is for large numbers of

• Rats• Crypts• Cells

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Coordinated Response: Asymptotics

• Possibility #1: Use standard methods at the crypt level • Optimal at the crypt level• Double-smoothing phenomenon (at crypt then

across crypts)• Effect of smoothing does not disappear

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Coordinated Response: Asymptotics

• Possibility #2: Under-smoothing at crypt level • Known to work for other double-smoothing

problems• Is optimal for this problem

• Explicit simple adjustments for under-smoothing derived• Divide optimal bandwidth by the 1/5th power of

the number of crypts

• Result: no asymptotic effect due to the initial smoothing

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Coordinated Response: Results

• Simulations: we found that this simple bit of under-smoothing works well.

• Data: extraordinary lack of sensitivity to the smoothing parameter• other smoothers give the same basic

answers• In principle:

• Regular Smooth then Average: sub-optimal

• Undersmooth then Average: better

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Coordinated Response: Asymptotics

• Alternatives:• Random coefficient polynomial models:

REML/Bayes• Hierarchical regression splines

• Major Point:• The method should not matter too much• Estimation of Crypt level functions has no

asymptotic effect

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Results: Correlation Functions for Proximal and Distal Regions

The negative correlation in the corn oil diet is unexpected

May suggest localization of damage: consistent with damage in the proximal or distal regions, but not both

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Results: Correlation Functions for Proximal and Distal Regions

For basic reasons, as well as robustness reasons, we were led to study whether this was an artifact of the use of relative as opposed to actual cell depth

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Modeling Cell Crypt Architecture• Most analyses of cell

depth measure cells on a relative basis

• Thus, if there are 11 cells, the depths are listed as 0/10, 1/10, …, 10/10

• This is not the same as actual depth

• Indeed, it effectively suggests that cells are uniformly spaced along the crypt wall

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Cell Crypt Architecture: Two Questions

• We are interested in the first place in the architecture:• Are the cells uniformly distributed

within a crypt?• It is also extremely tedious to measure actual

cell depth• Almost any statistical analysis extant uses

nominal cell depth: i.e., cell i of n has nominal depth (i-1)/(n-1) • Are downstream analyses affected by

the use of nominal instead of actual cell depth?

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Cell Crypt Architecture: Two Questions

• Downstream analyses: affected by the use of nominal instead of actual cell depth?

• Let X = true cell depth = Beta(0.5,1.0) with n = 30

• Let W = nominal cell depth

• Let E(Y|X) = X• What is E(Y|W)? • Plot order statistics of X

versus W

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Cell Crypt Architecture

• We have data on 30 rats• ~20 colonic crypts per rat• ~45 cells per crypt• For each rat, 3 crypts were analyzed to

measure their actual cell positions• Thus, we have incomplete data: true cell

positions are missing on ~ 17 crypts per rat• Question: is the negative proximal-distal

correlation in the corn-oil group a consequence of measuring only nominal cell position?

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Cell Crypt Architecture: Order Statistics

• The actual cell positions are on [0,1]• We model the true cell positions for each

crypt as the order statistics from Beta(a,b)

• We fit the crypt level functions via parametric cubic random effects models

• General problem: data missing as a group but subject to ordering constraints

• The order statistic model greatly speeds up computation

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Cell Crypt Architecture

• MCMC approach: various tricks to speed up especially the generation of the missing cell positions (~600 per animal)

• Missing cell positions can be generated simultaneously at the crypt level• Simpler than cell-by-cell generation• Faster than cell-by-cell generation• If generation were cell-by-cell, the order

constraints would have to be accounted for

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Cell Crypt Architecture: Results

• Proximal architecture is almost exactly U[0,1]

• Distal architecture is clearly not uniform: Beta(a = 0.8,b = 1.0)

• Here is the posterior mean density

• The correlation analysis was virtually unchanged– Appears that measuring

exact cell positions is not necessary

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Cell DNA Damage and Repair

• The same data structure occurs for DNA repair enzyme data as it does for DNA damage (adduct) data

• It is clearly of great interest to understand the relationship between the two• also as a function of cell depth

• Repair is measured on a pixel-by-pixel basis averaging across the crypt

• A problem arises: the DNA repair data are not nearly so smooth as the adduct data

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DNA Adduct (Damage) Data: 4 crypts with Regression Spline Fits

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DNA Repair Data Plots

DNA Repair Enzyme for Selected CryptsFish , T 3 , R 1 , C 6

t

MG

MT

0.0 0.2 0.4 0.6 0.8 1.0

02

04

06

08

0

Fish , T 9 , R 3 , C 8

t

MG

MT

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

Corn , T 9 , R 3 , C 19

t

MG

MT

0.0 0.2 0.4 0.6 0.8 1.0

02

04

06

08

0

Fish , T 3 , R 1 , C 23

t

MG

MT

0.0 0.2 0.4 0.6 0.8 1.0

02

04

06

08

0

Fish , T 9 , R 3 , C 19

t

MG

MT

0.0 0.2 0.4 0.6 0.8 1.0

02

04

06

08

0

Corn , T 9 , R 3 , C 20

tM

GM

T

0.0 0.2 0.4 0.6 0.8 1.0

02

04

06

08

0

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Cell DNA Repair

• The irregularity of the DNA repair data suggests that new techniques are necessary

• We are going to use wavelet methods around an MCMC calculator

• The multi-level hierarchical data structure makes this a new problem

• The images are pixel-by-pixel:• We “connected the dots”• Split into 256 (2**8) “observations”• Forces regularly spaced data

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Hierarchical Functional Model

• 2-level HF model:

drc drc drct

drc dr drc

dr d dr

Y =g (t)+ε ,

g (t)=g (t)+η (t)

g (t)=g (t)+ξ (t)

2drc ewhere ε ~MVN(0,σ I),

drc dr

1 1 2 2 1 2

η (•) and ξ (•) : mean 0 with covariance

matrices Σ (t ,t ) and Σ (t ,t ).

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Wavelets & Wavelet Regression

• Data space model: y = f(t) + – t = equally spaced grid, length n=2J, on (0,1)– Here = MVN(0,2)

• In wavelet space: d = Wy = + *

– d = ‘empirical’ wavelet coefficients

– = ‘true’ wavelet coefficients

• By orthogonality, * ~ MVN(0,2)

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Overview of Wavelet Method

• Convert data Yabc to wavelet space dabc

• Involves 1 DWT for each crypt• Fit hierarchical model in wavelet space to

obtain• Posterior distribution of ‘true’ wavelet

coefficients d corresponding to gd(t)

• Variance component estimates to assess relative variability

• Use IDWT to obtain posterior distribution of gd(t) for estimation and inference

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Wavelet Space Model• Wavelets: families of orthonormal basis functions• ddrc = { } = W ydrc

j,kdrcd

j,k j,k 2drc drc e

j,k j,k 2drc dr 1,j

j,k j,k 2dr d 2,j

d ~N(θ ,σ )

θ ~N(θ ,σ )

θ ~N(θ ,σ )

Discrete Wavelet Transform

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-1.0

0.01.0

Daubechies Basis Function

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Shrinkage Prior

• Prior on is a 0-normal mixture

• Nonlinear shrinkage -- denoises data• regularization parameters• Hierarchical model fit using MCMC

j,kdθ

j,k j,k 2d d jθ ~N(0,γ τ )

j,kd jγ ~Bernoulli(p )

2j jp and τ

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Some General Comments

• We focused on marginal (diet level) analyses• The marginalization allowed for efficient

MCMC• Some fairly difficult calculations are required• Much more efficient than brute-force

• Enables analysis of subsampling units, e.g., individual rats• This we have not yet done in our data

• Enables assessment of variance components

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Summary

• Method to fit hierarchical longitudinal data• Nonparametrically estimate mean profiles for:

• Treatments• Individuals• Subsampling units

• Estimates of relative variability at hierarchical levels• We find that 90% of the variability is from

crypt-to-crypt• Do lots of crypts!

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Results: DNA Repair Estimates & 90% posterior bounds by

diet/time

Fish Oil

Corn Oil

0 h 3 h 6 h 9 h 12 h

Fish Oil 0 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Fish Oil 3 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Fish Oil 6 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Fish Oil 9 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Fish Oil 12 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Corn Oil 0 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Corn Oil 3 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Corn Oil 6 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Corn Oil 9 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

Corn Oil 12 hr

Relative Cell Position

MG

MT

le

ve

l

0.0 0.2 0.4 0.6 0.8 1.0

05

10

15

20

25

30

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Conclusion

• Cell-based colon carcinogenesis studies• Hierarchical Longitudinal/Functional data • Rich in information -- challenging to extract

• Methods developed• Kernel methods for longitudinal correlations• Method for missing data with order

constraints• Wavelet regression methods for longitudinal

hierarchical data