Algebra and tensors give interpretable groups for crosstalk - PowerPoint PPT Presentation

Algebra and tensors give interpretable groups for crosstalk mechanisms in breast cancer Mariano Beguerisse D´ ıaz Mathematical Institute University of Oxford June 12, 2018 Tensor clustering Mariano Beguerisse University of Oxford

Acknowledgements Funding: Collaborators: Anna Seigal (UC Berkeley) Heather Harrington (Oxford) Mario Niepel (Harvard) Birgit Schoeberl (Merrimack) Pre-print: arXiv:1612.08116 Tensor clustering Mariano Beguerisse University of Oxford

Biological motivation Chemotherapy is a blunt tool that kills indiscriminately all rapidly dividing cells. Cancer physiology is complex. Need for focused therapies to target cellular decision making of cancer cells. Tensor clustering Mariano Beguerisse University of Oxford

Tensor data 36 cell lines 4 time points x2 2 doses 14 ligands Five dimensional tensor containing results of 36 × 14 experiments. The challenge is to determine the signalling mechanisms at play in these data. Tensor clustering Mariano Beguerisse University of Oxford

Clustering experiments Cluster experiments with similar responses. Can be difficult to interpret mechanistically. Need to impose constraints to facilitate interpretation. Tensor clustering Mariano Beguerisse University of Oxford

Rectangular clusters Ligand l 1 l 2 l 1 l 2 l 1 l 2 Constrain clusters’ shape. c 1 c 1 c 1 c 2 c 2 c 2 Rectangle-shaped clusters : single explanatory mechanism. l 1 l 2 l 1 l 2 l 1 l 2 Cell line c 1 c 1 c 1 Find an ODE model for each cluster. c 2 c 2 c 2 l 1 l 2 l 1 l 2 l 1 l 2 c 1 c 1 c 1 c 2 c 2 c 2 Not Allowed allowed Tensor clustering Mariano Beguerisse University of Oxford

Rectangular clusters Tensor clustering Mariano Beguerisse University of Oxford

Overview of method Tensor clustering Mariano Beguerisse University of Oxford

Similarity and data tensors Notation Multi-indexed data Z : In this example Z ∈ R 36 × 14 × 2 × 3 × 2 . Flattened tensor: � Z . In this example � Z ∈ R 504 × 12 . Similarity matrix: � S between the rows of � Z . Here � S ∈ R 504 × 504 . Similarity tensor: The similarity of the data indexed by i = ( i 1 , i 2 ) and j = ( j 1 , j 2 ): s i , j = sim ( Z ( i 1 , i 2 , : , . . . , :) , Z ( j 1 , j 2 , : , . . . , :)) ∈ R . Tensor clustering Mariano Beguerisse University of Oxford

Similarity and data tensors We summarize these relationships in the following diagram: similarity of i and j Z S . flatten reverse flatten similarity of rows � � Z S Where i and j are the multi-indices of experiments (i.e., cell-type/ligand combinations). Tensor clustering Mariano Beguerisse University of Oxford

Structured clustering Given S we cluster the experiments indexed by i = ( i 1 , i 2 ), j = ( j 1 , j 2 ), where i 1 , j 1 ∈ { 1 , . . . , 36 } and i 2 , j 2 ∈ { 1 , . . . , 14 } . Partition is encoded in a (36 × 14) × (36 × 14) tensor X with entries � 0 if i and j belong to the same cluster, x ij = 1 otherwise, that are a coarse approximation of the “distance” between i and j . A valid assignment must fulfil Reflexivity: x ii = 0 , Symmetry: x ij = x ji , Transitivity: 0 ≤ − x ik + x ij + x jk ≤ 2 . Tensor clustering Mariano Beguerisse University of Oxford

Structured clustering The (36 × 14) × m tensor Y has entries � 1 if the data indexed by i belongs to cluster k , y i k = 0 otherwise. We require that � m y i k = 1 , k =1 to ensure that each data item has been assigned to exactly one cluster. The tensors X and Y are related by equation: m � 1 − x ij = y i k y j k . k =1 Tensor clustering Mariano Beguerisse University of Oxford

Two implementations Need to classify experiments i into rectangular clusters. Two ways to do this: Starting from scratch (i.e., no previous clustering information). Starting from a pre-existing, non-rectangular clustering of experiments. Tensor clustering Mariano Beguerisse University of Oxford

Two implementations Starting from scratch: From pre-existing clustering � Y : max � S , ( 1 − X ) � + λ � 1 , X � , X � � max Y , Y � , subject to b l ≤ V · vec ( X ) ≤ b u , Y where V encodes the rectangular subject to constraints: m � y ijr = 1 , x i 1 i 2 j 1 j 2 = x i 1 j 2 j 1 i 2 , r =1 0 ≤ x i 1 i 2 j 1 j 2 − x i 1 i 2 j 1 i 2 ≤ 1 , − 1 ≤ y ikr + y jlr − y ilr ≤ 1 . 0 ≤ x i 1 i 2 j 1 j 2 − x i 1 i 2 i 1 j 2 ≤ 1 . Both are integer programs that we optimise with a branch and cut algorithm. Tensor clustering Mariano Beguerisse University of Oxford

Performance 5 Pre-existing clusters No prior clustering 4 3 2 Time (log(seconds)) 1 0 -1 -2 -3 -4 -5 2 4 6 8 10 12 14 16 18 20 Number of Cell Lines Tensor clustering Mariano Beguerisse University of Oxford

Results No prior clustering Test on HR + cells and Triple Negative Breast Cancer (TNBC) only. Tensor clustering Mariano Beguerisse University of Oxford

Results Prior clustering Test on all cells based starting on initial non-rectangular partitions into 3 and 5 clusters. Begin from 3 clusters Begin from 5 clusters Tensor clustering Mariano Beguerisse University of Oxford

Results Systematic search for models A Two arrow B Three arrow C Four arrow R R R R A E A E A E A E A11 B11 B31 B31b C11 A12 B12 B32 B32b C12 R R R R A E A E A E A E A21 B21 B41 B41b C21 C21b A22 B22 C22 B42 B42b C22b R R R R A E A E A E A E A31 C31 B51 B71 B71b C31b A32 B52 B72 C32 C32b B72b R R R A E A E A E B61 B81 B81b C41 C41b B62 B82 B82b C42 C42b Tensor clustering Mariano Beguerisse University of Oxford

Results Systematic search for models Tensor clustering Mariano Beguerisse University of Oxford

Results Ranking models for each cluster 45 Cluster 1 Cluster 2 Cluster 3 40 Cluster 4 Cluster 5 35 Model's AIC 30 25 20 15 1 2 3 4 5 6 7 8 9 10 Model rank in cluster Tensor clustering Mariano Beguerisse University of Oxford

Results Ranking models for each cluster Cluster number Model 1 2 3 5 4 rank L L L L L R R R R R Top A E A E A E A E A E L L L L L R R R R R 2 nd A E A E A E A E A E L L L L L R R R R R 3 rd A E A E A E A E A E L L L L L R R R R R 4 th A E E A E A E A A E L Ligand R Receptor E pERK A pAKT Activation Inhibition Estimated initial average receptor (x10-1) Strength of kinetics: 6 - 7 3 - 4 < < <x10-5 x10-5 x10-4 x10-3 x10-2 x10-1 x100 x101 R R R Tensor clustering Mariano Beguerisse University of Oxford

Recap Method for clustering multi-indexed data. Encode interpretatibility constraints as algebraic constraints in integer program. Clustering from scratch or find nearest compliant clustering to initial guess. 36 cell lines with 14 ligands into 5 clusters with ranking of mechanistic hypotheses. arXiv:1612.08116 Thank you! Tensor clustering Mariano Beguerisse University of Oxford