Utilizing spectral interpolation methods from multi-manifold learning to interpolate between interaction matrices in protein-ligand docking will enhance the prediction of binding modes in structurally diverse drug candidates.

Applying spectral interpolation techniques derived from multi-manifold learning (specifically, interpolating between interaction matrices in a shared spectral embedding space) to protein-ligand docking will yield statistically significant improvements in binding mode prediction accuracy—measured by RMSD to crystallographic poses—compared to standard docking pipelines (AutoDock Vina, Glide, or equivalent) on a benchmark set of ≥200 structurally diverse drug-like ligands across ≥20 distinct protein families, with mean RMSD reduction of ≥0.5 Å and success rate (RMSD < 2.0 Å) improvement of ≥10 percentage points.

Phase 1 — Baseline Establishment (Days 1–15): Reproduce standard docking results on PDBbind v2020 refined set (≥200 diverse complexes). Record top-1 RMSD, success rate, and runtime per target. Phase 2 — Manifold Construction (Days 16–35): Extract interaction matrices from all training-set co-crystal structures. Compute graph Laplacians, perform spectral embedding, and validate manifold geometry (eigenvalue gap analysis, reconstruction error). Phase 3 — Interpolation Integration (Days 36–60): Implement spectral interpolation module; integrate with AutoDock Vina scoring pipeline as a post-processing re-ranking step. Run on validation set. Phase 4 — Benchmark Evaluation (Days 61–80): Blind evaluation on held-out test set (n=200 complexes). Statistical comparison vs. baseline. Ablation studies (linear interpolation, no interpolation, spectral only). Phase 5 — Analysis and Reporting (Days 81–90): Subgroup analysis by protein family, ligand diversity, and data availability. Failure mode characterization.

# ============================================================
# Spectral Interpolation Docking Re-ranker — Architecture Sketch
# ============================================================

import numpy as np
from scipy.sparse.linalg import eigsh
from scipy.spatial.distance import cdist
from sklearn.preprocessing import normalize

# --- STEP 1: Interaction Matrix Construction ---
def build_interaction_matrix(protein_coords, ligand_coords, sigma=3.5, cutoff=10.0):
    """
    protein_coords: (Np, 3) — Cα coordinates of residues within cutoff
    ligand_coords:  (Nl, 3) — heavy atom coordinates of ligand
    Returns M: (Np, Nl) normalized interaction matrix
    """
    D = cdist(protein_coords, ligand_coords)  # (Np, Nl)
    M = np.exp(-D / sigma) * (D < cutoff)
    M = normalize(M, norm='l2', axis=1)       # row-normalize
    return M  # shape: (Np, Nl)

# --- STEP 2: Graph Laplacian for Protein Family Cluster ---
def compute_spectral_embedding(matrices_list, k_neighbors=5, n_components=10):
    """
    matrices_list: list of (Np, Nl) matrices for a protein family
    Returns: embedding (N, n_components), eigenvectors for Nystrom
    """
    N = len(matrices_list)
    # Flatten matrices to vectors for distance computation
    vecs = np.array([M.flatten() for M in matrices_list])  # (N, Np*Nl)
    
    # Pairwise Frobenius distances
    dist_matrix = cdist(vecs, vecs, metric='euclidean')     # (N, N)
    
    # k-NN affinity graph
    W = np.zeros((N, N))
    for i in range(N):
        nn_idx = np.argsort(dist_matrix[i])[1:k_neighbors+1]
        W[i, nn_idx] = np.exp(-dist_matrix[i, nn_idx]**2 / (2 * np.median(dist_matrix)**2))
    W = (W + W.T) / 2  # symmetrize
    
    # Normalized graph Laplacian
    D_deg = np.diag(W.sum(axis=1))
    D_inv_sqrt = np.diag(1.0 / np.sqrt(W.sum(axis=1) + 1e-10))
    L = np.eye(N) - D_inv_sqrt @ W @ D_inv_sqrt
    
    # Spectral decomposition (bottom n_components eigenvectors)
    eigenvalues, eigenvectors = eigsh(L, k=n_components, which='SM')
    
    return eigenvectors, eigenvalues, vecs  # embedding: (N, n_components)

# --- STEP 3: Nyström Extension for Query Projection ---
def nystrom_project(query_matrix, training_vecs, eigenvectors, eigenvalues, sigma_nys=1.0):
    """
    Project a new interaction matrix into the spectral embedding space.
    """
    q_vec = query_matrix.flatten()
    dists = np.linalg.norm(training_vecs - q_vec, axis=1)
    k_vec = np.exp(-dists**2 / (2 * sigma_nys**2))
    # Nyström approximation
    embedding = k_vec @ eigenvectors / (eigenvalues + 1e-10)
    return embedding  # shape: (n_components,)

# --- STEP 4: Geodesic-Weighted Interpolation ---
def spectral_interpolate(query_embedding, training_embeddings, training_matrices,
                          k_neighbors=3, lambda_geo=1.0):
    """
    Interpolate interaction matrix from k nearest neighbors in spectral space.
    """
    dists = np.linalg.norm(training_embeddings - query_embedding, axis=1)
    nn_idx = np.argsort(dists)[:k_neighbors]
    nn_dists = dists[nn_idx]
    
    # Geodesic-distance weights
    weights = np.exp(-lambda_geo * nn_dists)
    weights /= weights.sum()
    
    # Weighted combination of interaction matrices
    M_interp = sum(w * training_matrices[i] for w, i in zip(weights, nn_idx))
    return M_interp

# --- STEP 5: Spectral Consistency Scoring ---
def spectral_consistency_score(M_query, M_interp):
    """
    Lower score = better consistency with manifold.
    """
    diff = M_query - M_interp
    score = np.linalg.norm(diff, 'fro') / (np.linalg.norm(M_query, 'fro') + 1e-10)
    return score  # in [0, 1], lower is better

# --- STEP 6: Combined Re-ranking ---
def rerank_poses(vina_poses, vina_scores, protein_coords, 
                 training_vecs, training_matrices, training_embeddings,
                 eigenvectors, eigenvalues, alpha=0.5):
    """
    vina_poses: list of (Nl, 3) ligand coordinate arrays
    vina_scores: list of Vina docking scores (kcal/mol, negative = better)
    Returns: ranked list of (pose_idx, final_score)
    """
    final_scores = []
    for i, (pose, vina_score) in enumerate(zip(vina_poses, vina_scores)):
        M_query = build_interaction_matrix(protein_coords, pose)
        q_embed = nystrom_project(M_query, training_vecs, eigenvectors, eigenvalues)
        M_interp = spectral_interpolate(q_embed, training_embeddings, training_matrices)
        s_score = spectral_consistency_score(M_query, M_interp)
        
        # Normalize Vina score to [0,1] range (lower = better for both)
        vina_norm = -vina_score / 15.0  # rough normalization
        final_score = alpha * vina_norm + (1 - alpha) * (-s_score)
        final_scores.append((i, final_score))
    
    return sorted(final_scores, key=lambda x: -x[1])  # descending

# --- STEP 7: Validation Loop ---
def evaluate_benchmark(test_complexes, model_params):
    results = []
    for complex_id, protein, ligand_true, ligand_poses, vina_scores in test_complexes:
        ranked = rerank_poses(ligand_poses, vina_scores, protein, **model_params)
        top1_pose = ligand_poses[ranked[0][0]]
        rmsd = compute_rmsd(top1_pose, ligand_true)
        results.append({'id': complex_id, 'rmsd': rmsd, 'success': rmsd < 2.0})
    
    mean_rmsd = np.mean([r['rmsd'] for r in results])
    success_rate = np.mean([r['success'] for r in results])
    return mean_rmsd, success_rate, results

# --- ARCHITECTURE SUMMARY ---
# Input:  Protein PDB + Ligand SMILES + N Vina poses
# Stage 1: build_interaction_matrix() for each pose
# Stage 2: nystrom_project() into pre-built spectral space
# Stage 3: spectral_interpolate() to get manifold-consistent matrix
# Stage 4: spectral_consistency_score() as auxiliary scoring term
# Stage 5: rerank_poses() with tuned alpha
# Output: Re-ranked pose list; top-1 pose selected