(iii) Prior p(Z) on (S × C) binary matrix Z,
prior p(w) on mixture weights wtc for composition
(i).
Slide 1
Modeling Tumor Heterogeneity
Slide 4
Peter Müller, UT Austin
0 0.2
2
2
4
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
⇒
⇒
Goal: Reconstruct cell subpopulations = estimate Z and C.
Problem: Deconvolution of pst as a mixture of binary indicators Zsc
pst =
P
c
wtc Zsc
+wt0 ps0
5
4
3
⇒
2
3
2
1
4
3
1
2
4
1
3
3
SUBCLONES
binary matrix
Z = [Zsc ]
SNV s, clone c
1
SNV
4
SAMPLES
1
subclones
subclones
0.6
Value
5
0
Column Z−Score
21
1
44
58
59
68
69
70
89
90
91
94
92
95
115
32
93
110
31
47
43
41
39
8
7
9
10
65
57
40
21
62
1 77
52
44
96
58
100
59
12
68
6
69
26
70
89
87
9073
88
91
97
94
42
92
11
95
79
115
76
32
50
93
37
110
51
31
66
47
116
43
117
41
67
39
8
5
7 4
2
9
3
10
35
65
27
57
113
40
114
77
62
29
5249
28
96
34
100
45
12
46
6
33
26
73
36
8753
56
88
60
97
61
42
72
11
75
79
78
76
50
13
3714
111
51
112
66
16
116
15
117
17
67
4
19
5 18
20
2
25
3
81
35
82
27
83
113
84
114
49
86
2985
98
28
64
34
99
45
24
46
23
33
101
53
36
102
5674
104
60
105
61
106
72
107
75
108
78
14
103
13109
118
111
30
112
22
16
38
15
54
17
55
18
19
71
2063
48
25
80
81
82
83
84
85
86
98
64
99
24
23
101
74
102
104
105
106
107
108
109
103
118
30
22
38
54
55
63
71
48
80
Inference
0
samples
SNV
0 0.5 1
Color Key
and Histogram
150
Count
1.5
50
4
Count
0
150
Count
50
0
0.5
SNV
Column Z−Score
−1
−0.5
Row Z−Score
Color Key
and
−1 Histogram
0 0.5 1
50
150
−1.5
0
Count
Color Key
and Histogram
8
12
Juhee Lee, UCSC, Yuan Ji, U Chicago & NorthShore, K.
Gulukota, NorthShore Health System
Color Key
and Histogram
SUBCLONES
composition
weights
wt = (wtc )
sample t
SAMPLES
data: nts
mutations
Slide 2
Tumor Heterogeneity
plus “background noise”
Real problem: Z is latent, need to infer Z from the data.
Identifiability: In principle even feasible with one sample.
Weights are identified across mutations s.
Slide 5
• Mutations acquired over a tumor’s life history
Prior
• Every new mutation gives rise to a new subpopulation of
cells (“subclone”)
Latent cell types: p(Z) on (S × C) binary matrix, w. random C.
• −→ heterogeneous population of cells, composed of subpopulations with varying numbers of mutations.
Feature allocation: Think of SNV s selecting cell types c
Features (dishes) = c; experimental units (customers)
• Tumor history imprinted in each sample as the mosaicism
=s
of mutations.
Random feature allocation: define p(Z) as
• p(Zsc = 1 | πc ) = πc , c = 1, . . . , C
α
• πc ∼ Be C
,1
Slide 3
• Drop unselected features
Data
SNV: point mutations, s = 1, . . . , S
IBP as C → ∞.
Composition of sample t as mix of cell types:
(wtc , c = 1, . . . , C) ∼ Dir(·).
Data: Nst = # reads mapped to locus of SNV s in sample t.
nst = # of these with SNV.
Sampling model: nst ∼ Bin(Nst , pst )
Prior: in words,
(i) pst arises as a composition of sample t as a mixture
of C latent cell subclones.
(ii) Mutation s in subclone c is either present (Zsc = 1)
or not (Zsc = 0).
Z c = (Zsc , s = 1, . . . , S) defines subclone, c.
1
This is for normal sampling, asymptotically for small
variance and shrinking total mass.
Slide 6
c2 c1 SNP 1 c2 c3 c1 SNP 1 c2 c3 c4 c1 SNP 1 c2 c3 c4 c5 c1 SNP 1 SNP 1 SNP 2 SNP 2 SNP 2 SNP 2 SNP 2 SNP 3 SNP 3 SNP 3 SNP 3 SNP 3 SNP 4 SNP 4 SNP 4 SNP 4 SNP 4 SNP 5 SNP 5 SNP 5 SNP 5 SNP 5 SNP 6 SNP 6 SNP 6 SNP 6 SNP 6 SNP 7 SNP 7 SNP 7 SNP 7 SNP 7 SNP 8 SNP 8 SNP 8 SNP 8 SNP 8 SNP 9 SNP 9 SNP 9 SNP 9 SNP 9 SNP 10 SNP 10 SNP 10 SNP 10 SNP 10 SNP 11 SNP 11 SNP 11 SNP 11 SNP 11 SNP 12 SNP 12 SNP 12 SNP 12 SNP 12 SNP 13 SNP 13 SNP 13 SNP 13 SNP 13 SNP 14 SNP 14 SNP 14 SNP 14 SNP 14 SNP 15 SNP 15 SNP 15 SNP 15 SNP 16 SNP 16 SNP 16 SNP 16 SNP 17 SNP 17 SNP 17 SNP 17 SNP 18 SNP 18 SNP 18 SNP 19 SNP 19 SNP 19 SNP 19 SNP 20 SNP 20 SNP 20 SNP 20 SNP 21 SNP 21 SNP 21 SNP 22 SNP 22 SNP 22 SNP 22 SNP 23 SNP 23 SNP 23 SNP 23 SNP 24 SNP 24 SNP 24 SNP 25 SNP 25 SNP 25 SNP 25 SNP 26 SNP 26 SNP 26 SNP 26 SNP 27 SNP 27 SNP 27 SNP 28 SNP 28 SNP 28 SNP 28 SNP 29 SNP 29 SNP 29 SNP 29 SNP 30 SNP 30 SNP 30 SNP 30 SNP 30 SNP 31 SNP 31 SNP 31 SNP 31 SNP 31 SNP 32 SNP 32 SNP 32 SNP 32 SNP 33 SNP 33 SNP 33 SNP 33 SNP 33 SNP 34 SNP 34 SNP 34 SNP 34 SNP 34 SNP 35 SNP 35 SNP 35 SNP 35 SNP 35 SNP 36 SNP 36 SNP 36 SNP 36 SNP 36 SNP 37 SNP 37 SNP 37 SNP 37 SNP 37 SNP 38 SNP 38 SNP 38 SNP 38 SNP 39 SNP 39 SNP 39 SNP 39 SNP 40 SNP 40 SNP 40 SNP 40 SNP 41 SNP 41 SNP 41 SNP 41 SNP 42 SNP 42 SNP 42 SNP 42 SNP 42 SNP 43 SNP 43 SNP 43 SNP 43 SNP 43 SNP 44 SNP 44 SNP 44 SNP 44 SNP 44 SNP 45 SNP 45 SNP 45 SNP 45 SNP 45 SNP 46 SNP 46 SNP 46 SNP 46 SNP 46 SNP 47 SNP 47 SNP 47 SNP 47 SNP 47 SNP 48 SNP 48 SNP 48 SNP 48 SNP 48 SNP 49 SNP 49 SNP 49 SNP 49 SNP 49 SNP 50 SNP 50 SNP 50 SNP 50 SNP 50 SNP 51 SNP 51 SNP 51 SNP 51 SNP 51 SNP 52 SNP 52 SNP 52 SNP 52 SNP 52 SNP 53 SNP 53 SNP 53 SNP 53 SNP 53 SNP 54 SNP 54 SNP 54 SNP 54 SNP 54 SNP 55 SNP 55 SNP 55 SNP 55 SNP 55 SNP 56 SNP 56 SNP 56 SNP 56 SNP 56 SNP 57 SNP 57 SNP 57 SNP 57 SNP 57 SNP 58 SNP 58 SNP 58 SNP 58 SNP 58 SNP 59 SNP 59 SNP 59 SNP 59 SNP 59 SNP 60 SNP 60 SNP 60 SNP 60 SNP 60 SNP 61 SNP 61 SNP 61 SNP 61 SNP 62 SNP 62 SNP 62 SNP 62 SNP 63 SNP 63 SNP 63 SNP 63 SNP 64 SNP 64 SNP 64 SNP 64 SNP 65 SNP 65 SNP 65 SNP 65 SNP 65 SNP 66 SNP 66 SNP 66 SNP 66 SNP 66 SNP 67 SNP 67 SNP 67 SNP 67 SNP 67 SNP 68 SNP 68 SNP 68 SNP 68 SNP 68 SNP 69 SNP 69 SNP 69 SNP 69 SNP 69 SNP 70 SNP 70 SNP 70 SNP 70 SNP 70 SNP 71 SNP 71 SNP 71 SNP 71 SNP 71 SNP 72 SNP 72 SNP 72 SNP 72 SNP 72 SNP 73 SNP 73 SNP 73 SNP 73 SNP 73 SNP 74 SNP 74 SNP 74 SNP 74 SNP 74 SNP 75 SNP 75 SNP 75 SNP 75 SNP 75 SNP 76 SNP 76 SNP 76 SNP 76 SNP 76 SNP 77 SNP 77 SNP 77 SNP 77 SNP 77 SNP 78 SNP 78 SNP 78 SNP 78 SNP 78 SNP 79 SNP 79 SNP 79 SNP 79 SNP 79 SNP 80 SNP 80 SNP 80 SNP 80 SNP 80 SNP 81 SNP 81 SNP 81 SNP 81 SNP 81 SNP 82 SNP 82 SNP 82 SNP 82 SNP 82 SNP 83 SNP 83 SNP 83 SNP 83 SNP 83 SNP 84 SNP 84 SNP 84 SNP 84 SNP 84 SNP 85 SNP 85 SNP 85 SNP 85 SNP 85 SNP 86 SNP 86 SNP 86 SNP 86 SNP 86 SNP 87 SNP 87 SNP 87 SNP 87 SNP 87 SNP 88 SNP 88 SNP 88 SNP 88 SNP 88 SNP 89 SNP 89 SNP 89 SNP 89 SNP 89 SNP 90 SNP 90 SNP 90 SNP 90 SNP 91 SNP 91 SNP 91 SNP 91 SNP 92 SNP 92 SNP 92 SNP 92 SNP 93 SNP 93 SNP 93 SNP 93 SNP 94 SNP 94 SNP 94 SNP 94 SNP 95 SNP 95 SNP 95 SNP 96 SNP 96 SNP 96 SNP 97 SNP 97 SNP 97 SNP 98 SNP 98 SNP 90 SNP 91 SNP 92 SNP 93 SNP 94 SNP 95 SNP 96 SNP 97 SNP 98 SNP 95 SNP 96 SNP 97 SNP 98 SNP 98 SNP 99 SNP 99 SNP 99 SNP 99 SNP 99 SNP 100 SNP 100 SNP 100 SNP 100 SNP 100 C=2
C=3
C=4
= truth
Approx posterior: use k-means with different starting values to characterize posterior.
SNP 61 SNP 62 SNP 63 SNP 64 IBP: Broderick et al. (2013) extend a similar argument to
the IBP, with normal sampling and small variance and
shrinking rate of new features,
IBP with binomial sampling: same argument can be
made :-)
using increasing scaling of Bin with β and shrinking IBP
par γ, using γ = exp(−βλ2 )
SNP 38 SNP 39 SNP 40 SNP 41 c6 SNP 27 SNP 28 SNP 29 SNP 32 c5 SNP 24 SNP 25 SNP 26 SNP 27 c4 SNP 21 SNP 22 SNP 23 SNP 24 c3 SNP 18 SNP 19 SNP 20 SNP 21 MAP Z
for fixed C
SNP 15 SNP 16 SNP 17 SNP 18 c2 SNV
c1 Simulation
C=5
Slide 10
C=6
Simulation
Slide 7
True and estimated Z
1"
12
8
0
4
Count
0.5
1.5
Row
Z−Score
SNV
2
2"
3"
4"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
1"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
11"
12"
13"
14"
15"
16"
17"
18"
19"
20"
21"
22"
23"
24"
25"
26"
27"
28"
29"
30"
31"
32"
33"
34"
35"
36"
37"
38"
39"
40"
41"
42"
43"
44"
45"
46"
47"
48"
49"
50"
51"
52"
53"
54"
55"
56"
57"
58"
59"
60"
61"
62"
63"
64"
65"
66"
67"
68"
69"
70"
71"
72"
73"
74"
75"
76"
77"
78"
79"
80"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
2"
3"
4"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
1"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
11"
12"
13"
14"
15"
16"
17"
18"
19"
20"
21"
22"
23"
24"
25"
26"
27"
28"
29"
30"
31"
32"
33"
34"
35"
36"
37"
38"
39"
40"
41"
42"
43"
44"
45"
46"
47"
48"
49"
50"
51"
52"
53"
54"
55"
56"
57"
58"
59"
60"
61"
62"
63"
64"
65"
66"
67"
68"
69"
70"
71"
72"
73"
74"
75"
76"
77"
78"
79"
80"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
λ2 = 8, 10
Truth
""
2"
""
3"
""
4"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
5"
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
λ2 = 6
3
3
2
subclones
4
2
4
p(zsc = 1 | data)
1
3
SUBCLONE
1
1
2
4
1
subclones
3
Column Z−Score
5
4
0 0.5 1
21
1
44
58
59
68
69
70
89
90
91
94
92
95
115
32
93
110
31
47
43
41
39
8
7
9
10
65
57
40
21
62
1 77
52
44
96
58
100
59
68
6
6912
26
70
73
89
90
88
9187
97
94
42
92
11
95
115
76
3279
50
93
37
110
31
66
4751
116
43
117
41
67
39
8
5
7 4
2
9
3
10
35
65
57
113
4027
114
77
49
62
52
28
9629
34
100
45
12
46
6
26
53
7333
36
87
56
88
97
61
4260
72
11
75
79
78
76
50
13
3714
111
51
112
66
16
116
15
117
17
67
18
4
19
5
2
25
3 20
81
35
82
27
83
113
114
85
4984
86
29
98
28
34
99
4564
24
46
23
33
101
53
36
102
5674
104
60
105
61
72
107
75106
108
78
109
14
103
13
118
111
30
112
22
16
38
15
54
17
18
63
1955
71
20
48
25
80
81
82
83
84
85
86
98
64
99
24
23
101
74
102
104
105
106
107
108
109
103
118
30
22
38
54
55
63
71
48
80
samples
0
SUBCLONE
Slide 11
E(wtc | data)
Results – Pancreatic Cancer
n = 5 samples of pancreatic cancer (PDAC, pancreatic ductal adenocarcinoma). Estimated wtc :
Color Key
and Histogram
Count
Slide 8
0 2 4 6 8
−1
0
1
0.01%
5
0.28%
0.40%
0.31%
0.00%
0.00%
1
0.00%
0.60%
0.00%
0.00%
0.39%
3
0.01%
0.00%
0.00%
0.59%
0.40%
0.00%
0.00%
0.57%
0.00%
0.42%
Samples%
SAMPLES
0.00%
1
0.12%
3
Slide 9
0.41%
4
. . . is a pain.
0.45%
5
Computation
0.05 0.00 0.00 0.00 0.04 0.01 0.00 0.00 0.45 0.04 0.02 2
0.43 0.00 4
0.00 0.41 0.00 0.58 0.28 DP mixture: Kulis & Jordan (2012) recognize log posterior
≈ criterion function in k-means – voila!
2
Results – Breast Cancer
0.00 0.38 0.00 0.55 0.40 0.41 0.12 0.00 0.01 0.00 0.36 0.00 0.40 0.31 0.00 0.00 0.00 0.60 0.00 0.00 0.39 0.42 0.00 0.00 0.14 0.00 0.01 SUBLONES
Cell%types%
MAD Bayes for TH
with Yanxun XU, UT Austin; Yuan YUAN, Baylor C.of
S = 118 SNV’s in KEGG pathway
Med.;
Yuan JI and Kamalakar Gulukota, NorthShore HospiSlide 12
tal.
0.56 0.60 0.00 0.00 0.00 0.00 0.59 0.40 0.00 0.01 0.00 0.00 0.57 0.00 0.42 Cell types SUBLONES
S = 7000 SNV’s
SAMPLES
Row Z−Score
2
−1
−0.5
SAMPLE
−1.5
Column Z−Score
SNV
150
50
Color Key
and
−1 Histogram
0 0.5 1
50
150
Color Key
and Histogram
Results – Pancreatic Cancer
n = 5 samples of pancreatic cancer (PDAC, pancreatic ductal adenocarcinoma).
0
Count
0
Count
Color Key
and Histogram
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
11"
12"
13"
14"
15"
16"
17"
18"
19"
20"
21"
22"
23"
24"
25"
26"
27"
28"
29"
30"
31"
32"
33"
34"
35"
36"
37"
38"
39"
40"
41"
42"
43"
44"
45"
46"
47"
48"
49"
50"
51"
52"
53"
54"
55"
56"
57"
58"
59"
60"
61"
62"
63"
64"
65"
66"
67"
68"
69"
70"
71"
72"
73"
74"
75"
76"
77"
78"
79"
80"
Samples Horvath et al. (2013): n = 17 BC patients, S = 329 SNV’s.
4
3
2
1
TNBC1
TNBC2
TNBC3
TNBC4
TNBC5
TNBC6
non−TNBC1
non−TNBC2
non−TNBC3
non−TNBC4
non−TNBC5
non−TNBC6
HER2−1
HER2−2
HER2−3
HER2−4
HER2−5
Proportion
Estimated
wtc with C = 4.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Slide 13
Summary
TH: Model-based estimation of cell subpopulations is possible – and seems to work.
Big data: MCMC is not feasible anymore – alternative approaches remain feasible.
Limitations: and extensions
Tumor phylogenetics: Without condition on phylogenetic tree of subclones
A priori independent cell types: indpendent z c =
(Z1...S,c ), with p(z c = z c0 ) > 0, a priori (i know
— arrgh!)
Alternative dependent prior using DPP or others.
CNV: we conditioned on Nst .
Could use Nst to learn about CNV.
3