PHASE: encoding global protein ensembles with local Hamiltonians and all-atom backmapping

Technical Summary of PHASE: Encoding Global Protein Ensembles with Local Hamiltonians and All-Atom Backmapping

Problem Statement
Protein function is governed by conformational ensembles—high-dimensional probability distributions over molecular structures—rather than single static structures. While Molecular Dynamics (MD) simulations provide raw trajectories of these ensembles, the output is a finite, temporally correlated collection of frames rather than a compact, interpretable probabilistic model. Existing statistical models like Markov State Models often describe global conformations, which become intractable as system size increases. Conversely, deep generative models (e.g., Boltzmann Generators, BioEmu) can sample ensembles but often distribute statistical information across opaque neural parameters, making mechanistic analysis difficult. There is a need for a framework that converts atomistic ensembles into an explicit, interpretable statistical model capable of sampling, interrogation, and reconstruction at all-atom resolution, while preserving the statistical dependencies of the original data.

Methodology
The authors introduce PHASE (Protein Hamiltonians for Sampling of Ensembles), a system-specific framework that bridges atomistic simulation and explicit statistical modeling through three main stages:

1. Shared Residue-State Representation:

   • Atomistic trajectories are projected onto a discrete space where each residue $r$ occupies one of $K_r$ microstates.
   • Microstates are defined by clustering local backbone ($\phi, \psi, \omega$) and side-chain ($\chi$) torsional angles.
   • Crucially, a shared microstate basis is learned from a pooled set of reference trajectories (covering various conditions like ligand binding and activation states). This ensures that different ensembles (e.g., inactive vs. active) are compared within the same discrete configuration space $X = \prod_{r=1}^N \{1, \dots, K_r\}$.

2. Sparse Potts Hamiltonian Fitting:

   • A pairwise Potts Hamiltonian $E_\theta(x)$ is fitted to the discrete configurations using Pseudolikelihood Maximization (PLM).
   • The model includes single-site fields $h_r(x_r)$ and pairwise couplings $J_{rs}(x_r, x_s)$.
   • Locality Constraint: Couplings are restricted to a sparse contact graph $G=(V, E)$, where an edge exists only if residues $r$ and $s$ come within a spatial cutoff (e.g., 6 Å) in the atomistic ensemble. This encodes 3D locality directly into the interaction graph.
   • Parameters are expressed in a zero-sum gauge to ensure consistent, interpretable comparisons between independently fitted Hamiltonians.

3. Sampling and All-Atom Backmapping:

   • Sampling: New configurations are generated via Gibbs sampling and Replica-Exchange Monte Carlo (REMC) from the fitted Hamiltonian.
   • Backmapping: A cluster-conditioned structure generator, based on a fine-tuned SimpleFold model (a flow-matching transformer), reconstructs all-atom coordinates from the sampled discrete microstate vectors. The model is conditioned on the specific microstate cluster identity for each residue.
   • Optimization: The discrete Hamiltonian admits a QUBO (Quadratic Unconstrained Binary Optimization) encoding, enabling exploration via classical and potential future quantum annealing.

Key Results
The framework was applied to ten conformational ensembles of the adenosine A2A receptor (A2A), derived from ~37 µs of atomistic MD simulations covering inactive, active, and intermediate states with various ligands and effectors (mini-Gs).

• Statistical Fidelity with Local Interactions:

  • A contact cutoff of 6 Å was found to be sufficient. Hamiltonians with interactions restricted to this local neighborhood reproduced both residue-wise microstate populations and pairwise joint statistics (including correlations between residues not directly coupled in the model) with low Jensen-Shannon (JS) divergence.
  • This demonstrates that non-local correlations emerge naturally from the network of local statistical dependencies.

• Preservation of Activation Landscape:

  • Independently fitted Hamiltonians for "inactive" (INzma) and "active" (FANG) endpoints were used to define an endpoint preference score ($\Delta E$).
  • Sampling from state-specific Hamiltonians (e.g., agonist-bound, apo, mini-Gs coupled) produced configurations that ordered systematically along this score, correctly reflecting the known pharmacological hierarchy (e.g., inverse agonists favoring inactive, agonists favoring active) without these labels being provided as model inputs.
  • The model successfully distinguished conformational regimes within the same trajectory (e.g., early vs. late segments of an apo trajectory), capturing spontaneous relaxation.

• Regional Decomposition:

  • Decomposing the energy score into extracellular (ligand-facing) and intracellular (transducer-facing) regions revealed distinct statistical signatures.
  • While ligand effects were strongly encoded in the extracellular region, effector coupling (mini-Gs) produced specific shifts in the intracellular region. This allowed the model to detect ensembles with similar extracellular states but different intracellular conformations (e.g., pAs vs. FApoG).

• Generative Cycle Closure:

  • The backmapping model achieved sub-angstrom accuracy (0.95 Å RMSD on structured C$\alpha$) and recovered the input microstate cluster identities with ~78% accuracy on structured regions.
  • The cycle successfully generated all-atom structures from Potts-sampled configurations that were consistent with the prescribed microstates.

• QUBO and Annealing:

  • Classical simulated annealing on the QUBO representation successfully explored the energy landscape, accessing valid configurations distinct from the initialization, though with varying validity rates depending on penalty scales.

Significance and Claims
The authors claim that PHASE provides a protein-general procedure for constructing compact, interpretable, and atomistically realizable statistical models of protein conformational ensembles.

• Interpretability: Unlike deep generative models, PHASE exposes an explicit energy model where contributions can be associated with individual residues and pairs, allowing for mechanistic analysis of conformational preferences.
• Generative Capability: It enables the direct sampling of new, statistically weighted residue-state combinations and their reconstruction into all-atom structures, closing the coarse-graining–sampling–backmapping cycle.
• Comparative Analysis: By projecting different conditions into a shared residue-state space, PHASE allows for the direct comparison of ensembles (e.g., different ligands or mutations) via their Hamiltonian parameters and endpoint preference scores.
• Optimization Pathway: The QUBO formulation offers a route for constrained optimization and energy landscape exploration using annealing algorithms, potentially on quantum hardware.

The paper explicitly notes limitations: the model cannot recover basins absent from the training data (inheriting MD sampling biases), and the endpoint preference score is a relative statistical measure, not a physical free-energy difference. However, it establishes a robust framework for transforming complex simulation data into manipulable statistical models.