Input: Detection Techniques

• Cytogenetic techniques, e.g. Array CGH
• Virtual karyotyping with SNP array
• Welcome to the digital world: Next-gen sequencing

ArrayCGH

SNP array

Next-gen sequencing

The problem

• Read vector: $R = \{r_1, r_2, \ldots, r_N\}$, $N \sim 10^9$ # of reads in the library
• Copy vector: $C = \{c_1, c_2, \ldots, c_K\}$, $K \sim 3\times 10^9$ length of the genome
• Model prior: $M_0$, prior belief of the “world”, imposes smoothness of $C$ and includes the knowledge of the sequencing process
• Solve the MAP problem: $$\arg\max_{M} \sum_C P(C, R \mid M_0, M)$$
• Denoising problem, $R$ is a noisy measure of $C$, otherwise $C$ is simply normalized coverage
• Actual output: break point

Noisy Measure

Intuition: Balance

• Use anchor reads to place ambiguous reads
• Could potentially fix sequencing errors
• Final placement of an ambiguous read is a balance of local and contextual “happiness”

Ambiguous reads

Formally...

• Solve the MAP problem, the likelihood: $$ \begin{aligned} P(R \mid \theta, M_0) &= \sum_{C} P(R, C \mid \theta, M_0) \\ &= \sum_{C} P(R \mid C, \theta, M_0) P(C \mid \theta, M_0) \end{aligned} $$
• Impose sensible smoothness (regularization on model complexity) $$ P(C \mid \theta, M_0) = P(c_1) \prod_i P(c_i \mid c_{i-1}) $$
• Introduce new latent variables - true mapping for each reads: $Z=\{z_1, z_2, \ldots, z_N\}$
  • $z_j$ indicates the locations to which $r_j$ could be mapped.

• Inovation: $$ \begin{aligned} P(R \mid C, \theta, M_0) &= \prod_i P(r_i \mid C, \theta, M_0) \\ &= \prod_i \sum_{z_i} P(r_i, z_i \mid C, \theta, M_0) \\ &= \prod_i \sum_{z_i} P(r_i \mid z_i, C, \theta, M_0) P(z_i \mid C, \theta, M_0) \\ &= \prod_i \sum_{z_i} P(r_i \mid z_i, M_0) P(z_i \mid C, M_0) \end{aligned} $$
• The simple alignment model $P(r_i \mid z_i, M_0)$: match contributes $1 - \epsilon$ and dismatch contributes $\epsilon \mathbin{/} 3$
• Based on the knowledge of sequencing experiments: $$P(z_i \mid C, M_0) = P(z_i \mid c_i, M_0) \propto c_i$$
• Learning algorithm is trivial if we look at the structure of the MRF in which $P$ factorized...

Result

• Simulation
• Real data: basically it's good

Simulation Result

Real Data Result

Intuition: Orthogonal Information

• A waste if simply compressing all read info into coverage
• The reads directly carries the information of break point
• However the read info has even more noise (from HTS and alignment), e.g. $175k$ break points in chr1
• Must find a way to combine coverage and read info

Reads & Break Point

Insertion Size Distribution

Pipeline

Donor Graph

Events

Min-cost Flow Problem Formulation

• Why min-cost flow?
• $k_e$ observed coverage, $f_e$ latent coverage (flow). $$ P(k_e \mid f_e) = \frac{e^{-\lambda f_e l_e}(\lambda f_e l_e)^{k_e}}{k_e !} $$
• The problem: $$ \begin{align} \arg\max_{f} \prod_e P(k_e \mid f_e) &= \arg\min_{f} \sum_e \lambda_e f_e l_e - k_e \ln(\lambda_e f_e l_e) \end{align} $$ Subject to a set of linear constraint defined by the graph and the property of “flow”
• Convex optimization problem and thus in P

Why min-cost flow

• The sample genome is a walk in the graph
• However that's the same as genome assembly
• Only need the traversal counts
• Classical problem could be solved in P, e.g. linear programming

Result

• Good sensitivity, but worse specificity than Bentley et al.
• The resolution is within tens of bases.

Accuracy comparison