Deep Mechanistic Models reveal pathway-extrinsic drivers of mammary MAPK signalling heterogeneity

Technical Summary: Deep Mechanistic Models Reveal Pathway-Extrinsic Drivers of Mammary MAPK Signalling Heterogeneity

Problem Statement
Cellular responses to environmental cues are governed by signalling pathways, such as the EGFR/MAPK cascade, where dynamics shape cell fate even within genetically identical populations. However, existing computational approaches to model this heterogeneity are bifurcated. Mechanistic models (typically ordinary differential equations, ODEs) encode pathway topology and kinetics but struggle to represent variation arising from factors outside the explicitly modelled pathway. Conversely, representation-learning methods (e.g., PCA, autoencoders) map high-dimensional genome-wide data to low-dimensional manifolds but lack mechanistic interpretability, failing to explain how resulting cell states execute biological functions. Reconciling these views to explain signalling heterogeneity in a manner that is both data-driven and mechanistically interpretable has remained a significant challenge.

Methodology: Deep Mechanistic Models (DMMs)
The authors introduce Deep Mechanistic Models (DMMs), a framework that couples semi-supervised representation learning with a fixed mechanistic ODE model of EGFR/MAPK signalling. The architecture operates as follows:

1. Encoder: A neural network (autoencoder setup) encodes high-dimensional molecular measurements (transcriptomic, proteomic, or phosphoproteomic) into a low-dimensional latent embedding.
2. Mechanistic Head: This latent representation parameterizes deviations in the kinetic parameters of a fixed ODE model. The ODE model simulates the dynamics of receptor tyrosine kinases (EGFR, ERBB2, ERBB3) and downstream phosphorylation events (MEK, ERK, p90rsk) under various perturbations (EGF stimulation, MEK inhibitors, RTK inhibitors).
3. Decoder: A parallel decoder reconstructs baseline molecular features from the latent embedding.
4. Training: The system is trained end-to-end using gradient-based optimization (implemented in JAX and AMICI). The loss function combines the mechanistic fit to dynamic signalling data (supervised) with feature reconstruction (unsupervised), ensuring the latent representation captures information relevant to both pathway dynamics and molecular state.

Unlike Neural ODEs or Universal Differential Equations, the ODEs in DMMs are fixed and mechanistically interpretable, operating in the observed space rather than a latent space.

Key Results
The framework was applied to multiplexed CyTOF signalling data from 63 breast cancer cell lines (training set: 58 lines; test set: 5 held-out lines).

• Generalization and Performance: DMMs demonstrated strong generalization to held-out cell lines, achieving low root mean squared error (RMSE) on test data comparable to training data. Models trained on multimodal inputs (combining transcriptomics, proteomics, and CyTOF) outperformed those trained on single modalities and surpassed baseline linear regression models.
• Identification of Heterogeneity Drivers: Sensitivity analysis of the inferred parameter deviations revealed that most signalling heterogeneity is driven by pathway-extrinsic factors rather than variation in core MAPK components. Two major axes were identified:
  1. Baseline ERBB2 Activation: A parameter governing ERBB2 auto-phosphorylation, strongly correlated with baseline p-ERBB2 levels and sustained ERK activation.
  2. Ca²⁺/p38 Signalling Axis: A parameter governing signal transmission from ERK to p90rsk. This axis correlated with markers of calcium signalling, PLCG, and p38, suggesting crosstalk between the MAPK pathway and calcium-mediated signalling regulates signal amplification to p90rsk.
• Mechanistic Integration vs. Feature Augmentation: Integrating EGFR transcript levels directly into the mechanistic model (rescaling synthesis rates) reshaped the latent representation. While this ablated the separation of luminal and basal subtypes in the latent space (as EGFR levels were now explicitly accounted for by the mechanism), it maintained predictive power. This demonstrated that DMMs can switch between a "molecular lens" (where EGFR drives phenotypes via mechanism) and a "systems lens" (where EGFR acts as a subtype marker) with equivalent predictive performance.
• Model Failure as Discovery: Discrepancies between model predictions and experimental data (model failure) were systematically analyzed to identify biological insights:
  • Rare Mutations: High error in specific cell lines (e.g., HCC2218, DU4475) was linked to rare mutations (MET exon-14 deletion, BRAF V600E) that alter signalling responses, which standard regression models failed to capture without mechanistic priors.
  • Recurrent Programmes: Systematic errors in groups of cell lines pointed to recurrent programmes, including a putative AMPK–BRAF MEK-inhibitor resistance axis (associated with PRKAA2 downregulation) and a cytoskeletal/secretory programme (associated with Rho GTPase mutations and genes like TAGLN2 and GSN) driving heterogeneity in ERK-to-p90rsk signalling.
• Limitations and Bottlenecks: Analysis of training set size indicated diminishing returns beyond 30 cell lines, suggesting that the current predictive ceiling is limited by the mechanistic model architecture rather than data quantity. Feature augmentation improved performance slightly, but mechanistic integration of specific biological hypotheses (e.g., ligand-driven ERBB2 phosphorylation) did not always yield gains, acting as a filter for hypotheses.

Significance and Claims
The paper claims that DMMs offer a general framework for fusing mechanism with learning, enabling:

1. Interpretability: Providing a mechanistic account of how data-driven cell states execute functions, bridging the gap between statistical correlation and biological causality.
2. Discovery of Extrinsic Regulation: Successfully attributing heterogeneity to pathway-extrinsic factors (e.g., Ca²⁺/p38 crosstalk) that fixed mechanistic models alone could not capture.
3. Systematic Failure Analysis: Turning model failure into a systematic route to discover candidate biology, such as rare mutations and recurrent signalling programmes, by flagging discrepancies that standard models miss.
4. Modality Agnosticism: The framework is naturally extendable to other modalities (e.g., imaging) and can reconcile molecular and systems-level descriptions of the same data without forcing a choice between them.

The authors conclude that while DMMs currently face limitations in model architecture (specifically the coarse-grained nature of the ODEs), they represent a significant step toward unifying data-driven representation learning with mechanistic biological modelling.