On the Effect of Geometry on the Topology of Data

Transcription

1 On the Effect of Geometry on the Topology of Data Monica Nicolau Stanford University Mathematics & Center for Cancer Systems Biology Topological Data Analysis IMA: October

2 big data and cancer systems biology 2

3 A little bit of history breast cancer 3

4 A little bit of history breast cancer mid 1980s distinctions among breast cancers based on single protein/gene - estrogen & progesterone receptor - her2/neu - basal cell like vs. luminal cell like - p53 mutations - BRCA genes mutations more... 4

5 A little bit of history breast cancer mid 1980s distinctions among breast cancers based on single protein/gene - estrogen & progesterone receptor - her2/neu - basal cell like vs. luminal cell like - p53 mutations - BRCA genes mutations more... breast duct 5

6 early 1990 s Large data in biology Biology changed 6

7 High throughput data several ( ) Data matrix: columns TISSUE SAMPLES 20,000 40,000 rows GENES function genes distinctions diseases 7

8 A little bit of history breast cancer mid 1980s diversity by single protein/gene early 2000s 5 molecular subtypes by microarrays basal-like ERBB2 (her2-over expressing) luminal A luminal B normal-like 8

9 early 2000s 5 molecular subtypes: basal-like ERBB2 (her2-over expressing) luminal A luminal B normal-like A little bit of history breast cancer a few genes proteins thousands of genes gene expression microarrays 9

10 A little bit of history breast cancer early 2000s 5 molecular subtypes: gene expression microarrays all genes? thousands of genes? no tumor classes 10

11 A little bit of history breast cancer early 2000s 5 molecular subtypes: gene expression microarrays thousands of genes? genes that vary a lot basal-like luminal-like CLUSTER DATA POINTS 11

12 A little bit of history breast cancer early 2000s 5 molecular subtypes: gene expression microarrays genes that vary a lot + mean center thousands of genes? many classes see distinctions better CLUSTER DATA POINTS 12

13 A little bit of history breast cancer early 2000s 5 molecular subtypes: gene expression microarrays intrinsic gene lists + mean center thousands of genes? many classes see distinctions better CLUSTER based analysis 13

14 A little bit of history breast cancer early 2000s 5 molecular subtypes: intrinsic gene lists + mean center basal-like ERBB2 (her2-over expressing) luminal A luminal B normal-like many classes see distinctions better CLUSTER based analysis 14

15 Large data - microarrays new challenges relevance significance shape 15

16 Large data - microarrays new challenges relevance significance shape geometry 16 16

17 Data is useful because it contains information 17

18 Data is useful because it contains information to extract information, we must modify profoundly the geometry of the space 18

19 example 1: 19

20 Acute Myeloid Leukemia : AML cancer affects cells in continual transition 20

21 Acute Myeloid Leukemia : AML cancer affects cells in continual transition 21

22 Acute Myeloid Leukemia : AML cancer affects cells in continual transition 22

23 Acute Myeloid Leukemia : AML cancer affects cells in continual transition HEALTHY 23

24 Acute Myeloid Leukemia : AML cancer affects cells in continual transition HEALTHY 24

25 Acute Myeloid Leukemia : AML cancer affects cells in continual transition HEMATOPOIESIS AML HSC MPP NEOPLASTIC EVENT disruption LSC LPC CMP LMPP Blasts MEP HEALTHY GMP DISEASE 25

26 NORMAL MEMORY ANALYSIS NorMA 26

27 27

28 Disease Specific Genomic Analysis DSGA T Dc.T Disease Component Nc.T Healthy State Model Normal Component DSGA Nicolau et al Bioinformatics

29 TUMORS HEALTHY TISSUE microarrays DSGA Nicolau et al Bioinformatics

30 points (HEALTHY) MATHEMATICAL MODEL FOR HEALTHY STATE DSGA Nicolau et al Bioinformatics

31 points (TUMORS) points (HEALTHY) TRANSFORMATION points (TUMORS) deviation from HEALTHY STATE MATHEMATICAL MODEL FOR HEALTHY STATE DSGA Nicolau et al Bioinformatics

32 Tumor data T Nc.T Dc.T transformed tumor data vector of residuals Healthy State Model [Null Hypothesis Space] Normal tissue data DSGA Nicolau et al Bioinformatics

33 NorMA STEP 1: Data deconvolution T Dc.T Disease Component Nc.T Healthy State Model Normal Component 33

34 Why should HSM be subspace rather than manifold? T TUMOR DATA VECTOR DISEASE COMPONENT Dc.T Nc.T NORMAL COMPONENT LINEAR SPACE GENERATED FROM NORMAL TISSUE GENE EXPRESSION ARRAYS: N 1, N 2,..., N k healthy Figure 1 DSGA Nicolau et al Bioinformatics

35 Why should HSM be subspace rather than manifold? T TUMOR DATA VECTOR DISEASE COMPONENT Dc.T Nc.T NORMAL COMPONENT LINEAR SPACE GENERATED FROM NORMAL TISSUE GENE EXPRESSION ARRAYS: healthy Figure 1 N 1, N 2,..., N k cell type 1 cell type 2 DSGA Nicolau et al Bioinformatics

36 T Dc.T NorMA STEP 2: Signatures in Normal Component Nc.T Healthy State Model T = normal tissue data Data 42 normal tissue samples: HSC, MPP, LMPP, CMP, GMP, MEP SAM analysis: 901 genes with 90% FDR = 0 make centroids 2 step data centering 36

37 NorMA STEP 3: Assign normal memory stage to tumors: compare to signatures 37

38 Results CD34+/CD38+ CD34+/CD38- CD34-38

39 KM survival Probability of survival Probability of survival Probability of survival Time months 39

40 example 2: Breast Cancer different mathematical method 40

41 DECOMPOSITION of data into NORMAL memory ABERRANT part hypothesis definition & testing DSGA SHAPE OF DATA simplify data network by collapse to small graph applied topology and persistence robustness Mapper Topology based data analysis identifies subgroup of breast cancers with unique mutational profile and excellent survival Nicolau M, Levine AJ, Carlsson G Proc Nat Acad Sci

42 DECOMPOSITION ABERRANT DISEASE DATA HEALTHY - LIKE DSGA Nicolau et al Bioinformatics

43 network collapse Mapper f Singh, Memoli, Carlsson Point Based Graphics

44 DSGA transformed data from tumors & normals: disease component tumors Mapper filter: (L p -vector magnitude) q R normal 0 Health State Model Nicolau et al PNAS

45 Progression Analysis of Disease: PAD running Mapper on DSGA-transformed data ER-- sequence 9.9E+5 3.2E+6 4.7E+6 7.0E+6 9.4E+6 1.2E+7 1.5E+7 FILTER COLOR SCALE Basal-like sparse data sparse data sparse data sparse data detached tumor bins very sparse data c-myb+ tumors ER+ sequence Normal-Like & Normal Nicolau et al PNAS

46 survival analysis K-M survival cmyb+group Nicolau et al PNAS

47 survival analysis K-M survival cmyb+group Group is homogeneous & distinct mathematically biologically predictor variables few (1 or 2) significant variables biologically meaningful Group is new: doesn t follow old classification Luminal A Luminal B unclassified Nicolau et al PNAS

48 Thanks Computational: Gunnar Carlsson (Stanford Mathematics) Sylvia Plevritis (Stanford Cancer Center for Systems Biology) Rob Tibshirani (Stanford Statistics) Andrew Gentles (Stanford Cancer Center for Systems Biology) Biology: Arnold Levine (Princeton IAS, School of Natural Studies) Anne-Lise Børresen-Dale (Genetics, University of Oslo, Norway) Stefanie Jeffrey (Surgery, Stanford) Janine Erler (Cell and Molecular Biology, Institute of Cancer Research, London, UK) Ravindra Majeti (Hematology, Stanford) Garry Nolan (Microbiology and Immunology, Stanford) Denise Monack (Microbiology and Immunology,Stanford) 48