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

Collaborators: Funding: 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 14 ligands 2 doses 4 time points x2

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

Constrain clusters’ shape. Rectangle-shaped clusters: single explanatory mechanism. Find an ODE model for each cluster.

Cell line Ligand c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2 c1 c2 l1 l2

Allowed Not 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 ∈ R36×14×2×3×2. Flattened tensor:

• Z. In this example

Z ∈ R504×12. Similarity matrix: S between the rows of

• Z. Here

S ∈ R504×504. Similarity tensor: The similarity of the data indexed by i = (i1, i2) and j = (j1, j2): si,j = sim (Z(i1, i2, :, . . . , :), Z(j1, j2, :, . . . , :)) ∈ R.

Tensor clustering Mariano Beguerisse University of Oxford

Similarity and data tensors

We summarize these relationships in the following diagram: Z S.

• Z
• S

flatten similarity of i and j similarity of rows reverse flatten

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 = (i1, i2), j = (j1, j2), where i1, j1 ∈ {1, . . . , 36} and i2, j2 ∈ {1, . . . , 14}. Partition is encoded in a (36 × 14) × (36 × 14) tensor X with entries xij =

• if i and j belong to the same cluster,

1

• therwise,

that are a coarse approximation of the “distance” between i and j. A valid assignment must fulfil Reflexivity: xii = 0, Symmetry: xij = xji, Transitivity: 0 ≤ −xik + xij + xjk ≤ 2.

Tensor clustering Mariano Beguerisse University of Oxford

Structured clustering

The (36 × 14) × m tensor Y has entries yik =

• 1

if the data indexed by i belongs to cluster k,

• therwise.

We require that

m

• k=1

yik = 1, to ensure that each data item has been assigned to exactly one cluster. The tensors X and Y are related by equation: 1 − xij =

m

• k=1

yikyjk.

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: max

X

S, (1 − X) + λ1, X, subject to bl ≤ V · vec(X) ≤ bu, where V encodes the rectangular constraints: xi1i2j1j2 = xi1j2j1i2, 0 ≤ xi1i2j1j2 − xi1i2j1i2 ≤ 1, 0 ≤ xi1i2j1j2 − xi1i2i1j2 ≤ 1. From pre-existing clustering Y: max

Y

• Y, Y,

subject to

m

• r=1

yijr = 1, − 1 ≤ yikr + yjlr − yilr ≤ 1. Both are integer programs that we optimise with a branch and cut algorithm.

Tensor clustering Mariano Beguerisse University of Oxford

Performance

Number of Cell Lines 2 4 6 8 10 12 14 16 18 20 Time (log(seconds))

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 Pre-existing clusters No prior clustering 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

E A R E A R E A R E A R E A R A Two arrow B Three arrow C Four arrow E A R E A R E A R E A R E A R E A R E A R E A R E A R E A R

A11 A12 A21 A22 A31 A32 B11 B12 B21 B22 B51 B52 B61 B62 B31 B32 B31b B32b B41 B42 B41b B42b B71 B72 B71b B72b B81 B82 B81b B82b C11 C12 C21 C22 C21b C22b C31 C32 C31b C32b C41 C42 C41b 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

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

Results

Ranking models for each cluster

Top 2nd Cluster number 1 2 3 4 5 3rd 4th

E A R E A E A E A R E A E A E A E A E A R E A E A R E A R E A E A E A R E A E A

Model rank

E A R E A R E A L L L L L L L L L L L L L L L L L L L L R R R R R R R R R R R R Activation Inhibition Ligand Receptor pAKT pERK L R E A Estimated initial average receptor (x10-1) R R R 3 - 4 < < 6 - 7 Strength of kinetics: <x10-5 x10-5 x10-4 x10-3 x10-2 x10-1 x100 x101

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