Poisson process factorization for mutational signature analysis with genomic covariates

1. Problem Statement

Mutational signature analysis aims to decompose the complex patterns of somatic mutations in cancer genomes into latent "signatures" representing distinct mutational processes (e.g., DNA repair deficiencies, UV exposure).

• Current Limitation: The standard approach uses Non-Negative Matrix Factorization (NMF) on a matrix of aggregated mutation counts. This assumes a constant mutation rate across the entire genome.
• The Reality: Mutation rates are highly heterogeneous, varying significantly based on genomic features such as GC content, replication timing, histone modifications, nucleosome positioning, and copy number alterations (CNAs).
• Gap in Literature: Existing methods either ignore these covariates, partition the genome into discrete bins (losing resolution), or perform post-hoc associations after signature extraction. No existing method simultaneously infers mutational signatures, their activities, and the joint effect of genomic covariates on these activities in a unified probabilistic framework.

2. Methodology: Poisson Process Factorization (PPF)

The authors propose Poisson Process Factorization (PPF), a statistical framework that models mutations as realizations of an inhomogeneous Poisson point process.

Core Model Formulation

• Point Process: Mutations are treated as points in a continuous genomic space $[0, T)$. For a patient $j$ and mutation type $i$, the number of mutations in a region $A$ follows a Poisson distribution with intensity $\lambda_{ij}(t)$.
• Factorized Intensity: The intensity function is decomposed into a sum of $K$ signatures:
  $$\lambda_{ij}(t) = \sum_{k=1}^K r_{ik} \vartheta_{kj}(t)$$
  • $r_{ik}$: The probability that signature $k$ generates mutation type $i$ (constant across the genome).
  • $\vartheta_{kj}(t)$: The position-dependent activity of signature $k$ in patient $j$.
• Log-Linear Parametrization: The activity function $\vartheta_{kj}(t)$ is modeled to depend on genomic covariates and copy numbers:
  $$\vartheta_{kj}(t) = \frac{1}{2} \phi_{kj} c_j(t) \exp(\beta_k^\top x(t))$$
  • $\phi_{kj}$: Baseline activity of signature $k$ in patient $j$.
  • $c_j(t)$: Patient-specific copy number at locus $t$ (normalized by 2).
  • $x(t)$: Vector of genomic covariates (e.g., histone marks, replication timing) at locus $t$.
  • $\beta_k$: Regression coefficients quantifying the effect of covariates on signature $k$.
• Generalization: If covariates are ignored ($\beta_k=0$) and copy numbers are constant, PPF reduces to the standard Poisson NMF model.

Prior Distributions & Automatic Rank Selection

• Compressive Hyperpriors: To avoid computationally intensive rank selection, the authors employ compressive Bayesian hyperpriors (Zito and Miller, 2024).
• Mechanism: A relevance weight $\mu_k$ is assigned to each signature. The prior on $\mu_k$ is an Inverse Gamma distribution favoring small values. If a signature is redundant, its posterior activity shrinks to near zero, effectively removing it from the model. This allows the model to automatically select the number of active signatures ($\hat{K}$).

Algorithms

The paper develops two algorithms for inference:

1. Maximum A Posteriori (MAP) Estimation: Uses a Majorization-Minimization (MM) algorithm coupled with Newton's method.
   • Updates for signatures ($R$) and baseline activities ($\Phi$) are multiplicative (ensuring non-negativity).
   • Updates for regression coefficients ($B$) use Newton's method on a majorized objective function.
   • Updates for hyperparameters ($\mu, \sigma^2$) have closed-form solutions.
2. Markov Chain Monte Carlo (MCMC): A fully Bayesian sampler for posterior inference.
   • Uses data augmentation by introducing latent multinomial variables to assign each observed mutation to a specific signature.
   • Utilizes Elliptical Slice Sampling for the regression coefficients $\beta_k$ due to the non-conjugate nature of the likelihood with the Gaussian prior.

3. Key Contributions

1. Unified Framework: First method to jointly infer mutational signatures, patient-specific activities, and the multiplicative effects of continuous genomic covariates.
2. Position-Specific Attribution: Unlike NMF, which assigns a single probability distribution to all mutations of a type in a patient, PPF provides a locus-specific probability that a mutation was generated by a specific signature, based on local genomic context.
3. Automatic Rank Selection: Integration of compressive priors allows the model to determine the number of active signatures without manual tuning or cross-validation.
4. Interpretability: The regression coefficients ($\beta_k$) directly quantify how specific genomic features (e.g., H3K9me3, replication timing) enrich or deplete specific mutational signatures.

4. Results

Simulation Study

• Setup: Simulated data for 40 patients with 8 true signatures and 10 covariates (5 relevant, 5 irrelevant).
• Performance:
  • PPF with covariates significantly outperformed baseline NMF (CompNMF) in recovering true signatures ($R$), activities ($\Theta$), and regression coefficients ($B$).
  • The model correctly identified the true number of signatures ($K=8$) in almost all replicates.
  • Including irrelevant covariates had minimal impact on signature recovery but slightly reduced coefficient estimation accuracy in highly correlated scenarios.
  • MAP estimation generally provided more accurate point estimates for signatures and activities than MCMC posterior means, though MCMC provided necessary uncertainty quantification.

Application: Breast Adenocarcinoma (ICGC Cohort)

• Data: 113 patients, ~707k mutations, with 11 genomic covariates (histone modifications, replication timing, methylation, etc.) and copy number data.
• De Novo Analysis:
  • Identified 9 active signatures.
  • Recovered known signatures with high fidelity: APOBEC-related (SBS2, SBS13), Aging (SBS1, SBS5), and Homologous Recombination Deficiency (SBS3).
  • Covariate Insights:
    • Methylation: Strongly associated with SBS1 (C>T at CpG sites). Including methylation as a covariate "cleaned" SBS1, preventing it from being contaminated by other signatures (a common issue in NMF).
    • Replication Timing: Late replication timing was strongly associated with increased oxidative damage (SBS7) and APOBEC (SBS2), while early replication was linked to other processes.
    • Heterochromatin: Signatures associated with HR deficiency and oxidative stress showed increased activity in heterochromatic regions (marked by H3K9me3).
• Refitting Analysis: When fixing signatures to COSMIC references, the model successfully inferred position-specific activities.
  • Case Study: Demonstrated that the attribution of a specific mutation (A[C>A]C) could flip between signatures (e.g., SBS3 vs. SBS8) depending on the local replication timing and histone marks, a nuance lost in standard NMF.

5. Significance

• Biological Insight: PPF moves beyond "what" signatures are present to "where" and "why" they are active. It quantifies the mechanistic link between chromatin state (e.g., heterochromatin vs. euchromatin) and mutational processes.
• Clinical Relevance: By attributing mutations to specific signatures based on local genomic context, the model can better identify patients with specific deficiencies (e.g., HRD) who may benefit from targeted therapies (e.g., PARP inhibitors).
• Methodological Advancement: The framework sets a new standard for mutational signature analysis by treating the genome as a continuous, covariate-driven process rather than a bag of counts, offering a more accurate and interpretable model of cancer evolution.

In summary, this paper introduces a robust, probabilistic framework that integrates genomic context directly into the factorization of mutational signatures, offering superior accuracy, automatic model selection, and deeper biological interpretability compared to traditional NMF approaches.