Analysis of DNA Damage and Repair
in Colonic Crypts
Raymond J. Carroll
Texas A&M University
http://stat.tamu.edu/~carroll
carroll@stat.tamu.edu
Postdoctoral Training Program:
http://stat.tamu.edu/B3NC
1
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!)
2
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
3
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
4
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
5
Colon Sliced and Laid Out
Normal Colon Crypts
Aberrant Colon Crypts
6
Architecture of Colon Crypts:
Crosssectional View
• Stem Cells:
Lumen
– Mother cells
near bottom
• Depth in crypt
~ age of cells
– Suggests
importance of
depth
• Relative Cell
Position:
crypts
– 0 = bottom
– 1 = top
7
Architecture of Colon Crypt: Expanded
View
• The cells are
more easily
visible here
• Note that the
cells seem
smaller at the
crypt bottom
8
Architecture of Colon Crypt
• The general idea
is to slice the
colon crypt
• The cells along
the left wall are
assayed
9
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
10
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
11
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
12
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
13
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)
Ydrc (t)=g drc (t)+ε drct
g drc (t)=g dr (t)+ηdrc (t)
g dr (t)=g d (t)+ξ dr (t)
• Our aim: estimate the
correlation between
proximal and distal regions
as a function of cell depth
at the rat level
14
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
Ydrc (t)=gdrc (t)+ε drct
gdrc (t)=gdr (t)+ηdrc (t)
gdr (t)=gd (t)+ξ dr (t)
• Problem: data are not
of this structure
• Cell locations vary
from crypt to crypt
• Number of cells varies
from crypt-to-crypt
15
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
Ydrc (t)=g drc (t)+ε drct
g drc (t)=g dr (t)+ηdrc (t)
g dr (t)=g d (t)+ξ dr (t)
• Problem: Is there any
effect due to the initial
smooth?
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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?
25
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
26
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?
27
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
28
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
29
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
30
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
31
DNA Adduct (Damage) Data: 4 crypts
with Regression Spline Fits
32
DNA Repair Data Plots
DNA Repair Enzyme for Selected Crypts
F
,
T
is
h
3
C
,
,T
o
R
r
9
n
1
M
G
T
0 20 40 60 80
h
M
G
T
0 20 40 60 80
M
G
T
0 20 40 60 80
F is
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
t
h
F
,
T
is
M
G
T
0 20 40 60 80
M
G
T
0 20 40 60 80
F is
t
h
3
C
,
,T
o
R
r
9
n
1
M
G
T
0 20 40 60 80
t
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
0
0
.0
0
.0
2
.0
4
.1
6
. 8
. 0
t
t
t
33
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
34
Hierarchical Functional Model
• 2-level HF model:
Ydrc =g drc (t)+ε drct ,
g drc (t)=g dr (t)+ηdrc (t)
g dr (t)=g d (t)+ξ dr (t)
where
ε drc ~MVN(0,σe2 I),
ηdrc (•) and ξ dr (•) : mean 0 with covariance
matrices Σ1 (t1 ,t 2 ) and Σ 2 (t1 ,t 2 ).
35
Wavelets & Wavelet Regression
• Data space model: y = f(t) + e
– t = equally spaced grid, length n=2J, on (0,1)
– Here e = MVN(0,s2)
• In wavelet space: d = Wy =  + e*
– d = ‘empirical’ wavelet coefficients
–  = ‘true’ wavelet coefficients
• By orthogonality, e* ~ MVN(0,s2)
36
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
37
Wavelet Space Model
• Wavelets: families of orthonormal basis functions
• ddrc = { d
j,k
drc}
= W ydrc
Discrete Wavelet Transform
d ~N(θ ,σ )
j,k
drc
j,k
drc
2
e
θ ~N(θ ,σ )
j,k
drc
j,k
dr
2
1,j
θ ~N(θ ,σ )
j,k
dr
j,k
d
2
2,j
-1.0 0. 1.0
Daubechies Basis Function
-1.
-0.
0.
0
0.
5
0
1.
5
1.
0
2.
5
0
38
Shrinkage Prior
• Prior on θdj,k
is a 0-normal mixture
θ ~N(0,γ τ )
j,k
d
j,k 2
d
j
γ ~Bernoulli(p j )
j,k
d
• Nonlinear shrinkage -- denoises data
2
• p j and τ j
regularization parameters
• Hierarchical model fit using MCMC
39
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
40
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!
41
Results: DNA Repair
Estimates & 90% posterior bounds by diet/time
9h
0 5 10 MGTlev 15 20 25 30
6h
0 5 10 MGTlev 15 20 25 30
3h
0 5 10 MGTlev 15 20 25 30
Fish Oil
0h
O l
i
F
s
i
h
0
O
h
r
l
i
F
s
i
h
3
O
h
r
l
i
F
s
i
h
6
O
h
F
r
l
i
s
i
h
9
O
h
0 5 10 MGTlev 15 20 25 30
0 5 10 MGTlev 15 20 25 30
F
s
i
h
12 h
00
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
80
R
C
e e
a l
l
t
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
0 5 10 MGTlev 15 20 25 30
0 5 10 MGTlev 15 20 25 30
0 5 10 MGTlev 15 20 25 30
Corn Oil
e
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
P
v o
i
e
O
C
l
i
o
r n
0
O
h
C
r
l
i
o
r n
3
O
h
C
r
l
i
o
r n
6
O
h
C
o
r
l
i
r n
9
O
0 5 10 MGTlev 15 20 25 30
i
v
C
o
r n
0 5 10 MGTlev 15 20 25 30
R
e l
a t
00
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
8
0
0
0
.
0
.
0
0
.
2
0
.
4
1
.
6
.
80
R
e l
a t
i
v
e
R
C
e e
a l
l
t
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
P
v o
i
e s
R
t
i
C
e i
o
e
a n
l
l
t
P
v o
i
e
42
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
43