SCEVAN Algorithm and Methodology

1.1 Quality Control

Cells are filtered based on:

• Minimum 200 expressed genes per cell
• Genes expressed in at least 1% of cells
• Minimum genes per chromosome (default: 5)

Filtering criteria:

\[\text{Keep cell } c \text{ if: } \sum_{g=1}^{G} \mathbb{1}(x_{gc} > 0) \geq 200\]

\[\text{Keep gene } g \text{ if: } \frac{\sum_{c=1}^{C} \mathbb{1}(x_{gc} > 0)}{C} \geq 0.01\]

1.2 Variance Stabilization

The Freeman-Tukey transformation is applied to stabilize variance:

\[y_{gc} = \\log\\left(\\sqrt{x_{gc}} + \\sqrt{x_{gc} + 1}\\right)\]

where \(x_{gc}\) is the raw count for gene \(g\) in cell \(c\).

This transformation:

• Approximates a variance-stabilizing transformation for Poisson data
• Handles zero counts gracefully
• Produces approximately homoscedastic data

2.1 Single-Sample Gene Set Testing

The Mann-Whitney-Wilcoxon Gene Set Test (MWW-GST) evaluates enrichment of cell-type specific signatures:

\[\\text{NES}_s = \\frac{\\bar{R}_s - \\mu_0}{\\sigma_0}\]

where:

• \(\\bar{R}_s\) is the mean rank of signature genes
• \(\\mu_0 = (n+1)/2\) is the expected mean under null
• \(\\sigma_0\) is the standard deviation under null

2.2 Cell Type Signatures

Built-in signatures include:

3.1 Greedy Optimization

The algorithm uses a priority queue to efficiently merge adjacent segments:

Algorithm: VegaMC Segmentation

Input: Expression matrix Y, parameter β
Output: Segmentation S

1. Initialize: Each probe is a segment
2. Build priority queue Q with merge costs
3. While min(Q) < β:
   a. Pop minimum cost merge (i, j)
   b. Merge segments i and j
   c. Update affected merge costs in Q
4. Return final segmentation S

3.2 Statistical Testing

For each segment, hypothesis tests determine significance:

• Loss test: \(H_0: \\mu = 0\) vs \(H_1: \\mu < 0\)
• Gain test: \(H_0: \\mu = 0\) vs \(H_1: \\mu > 0\)

P-values are computed using t-tests across cells.

4.1 Gaussian Mixture Model

Segment means are modeled as a mixture of 5 Gaussian components:

\[p(x) = \\sum_{k=0}^{4} \\pi_k \\mathcal{N}(x | \\mu_k, \\sigma_k^2)\]

4.2 EM Algorithm

E-step: Compute posterior probabilities

\[\\gamma_{ik} = \\frac{\\pi_k \\mathcal{N}(x_i | \\mu_k, \\sigma_k^2)}{\\sum_{j=0}^{4} \\pi_j \\mathcal{N}(x_i | \\mu_j, \\sigma_j^2)}\]

M-step: Update parameters

\[\\mu_k^{new} = \\frac{\\sum_i \\gamma_{ik} x_i}{\\sum_i \\gamma_{ik}}\]

\[\\sigma_k^{2,new} = \\frac{\\sum_i \\gamma_{ik} (x_i - \\mu_k^{new})^2}{\\sum_i \\gamma_{ik}}\]

5.1 Graph-Based Clustering

Subclones are identified using Louvain community detection on a shared nearest neighbor (SNN) graph:

1. Construct k-NN graph on CNA profiles
2. Build SNN graph
3. Apply Louvain algorithm
4. Filter small clusters (< 2% of cells)

5.2 Modularity Optimization

The Louvain algorithm maximizes modularity:

\[Q = \\frac{1}{2m} \\sum_{ij} \\left[ A_{ij} - \\frac{k_i k_j}{2m} \\right] \\delta(c_i, c_j)\]

where:

• \(A_{ij}\) is the adjacency matrix
• \(k_i\) is the degree of node \(i\)
• \(m\) is total edges
• \(\\delta(c_i, c_j) = 1\) if nodes are in same community

6.1 Subclone-Specific Alterations

Alterations are classified as:

• Clonal: Present in all subclones
• Shared: Present in multiple (not all) subclones
  
• Subclone-specific: Present in single subclone

6.2 Pathway Enrichment

fGSEA (fast Gene Set Enrichment Analysis) identifies enriched pathways:

\[ES = \\max_{j=1}^{n} \\left| \\sum_{i=1}^{j} \\frac{|r_i|^p}{N_R} - \\frac{j - \\text{hit}_j}{N - N_H} \\right|\]