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BindingSiteHachimoji.lean (589 lines): - 50-token amino acid vocabulary (20 core + 30 modified) - 8-state Hachimoji classification for residues - Extended Fisher metric on 50-simplex (Chentsov unique) - Chaos game basin discovery with Sidon addressing - Typed BindingSiteReceipt compatible with PVGS-DQ BindingSiteEntropy.lean (376 lines): - Information entropy per residue site (Void-X Eq. 3) - Entropy from B-factors (PDB) or cluster diversity - Bindability score B* normalized to 0-100 - Sidon address generation from entropy profiles - Fisher-Rao approximate distance between sites BindingSiteCodec.lean (335 lines): - pdbToReceipt : PDB ID → BindingSiteReceipt (one function) - screenDruggable : filter library for bindable pockets - rankByEnergy : sort by dual quaternion energy - Test cases: EGFR (1YY9), Tautomerase (9MUA), KIR2DL1 (9HML) Architecture: PDB → entropy profile → Hachimoji state → PVGS params → dual quaternion energy → typed receipt → PVGS-DQ system All three files compile as outlines with sorry for future proofs.
287 lines
13 KiB
Text
287 lines
13 KiB
Text
/-
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BindingSiteHachimoji.lean — Extended Hachimoji for Protein Binding Sites
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Maps the 50-token protein vocabulary (from Void-X) onto an extended
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Hachimoji state space. Each residue in a binding site gets classified
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by its local geometric entropy profile, producing a Hachimoji-style
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encoding that plugs directly into the PVGS-DQ receipt system.
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References:
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- Yang, Yuan, Chou 2025 (Void-X): 50 atomic tokens, entropy scoring
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- Giani, Win, Conti 2025 (PVGS): photon-varied Gaussian states
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- Chentsov 1972: unique Fisher metric on probability simplex
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- Research-Stack library/ChentsovFinite.lean: formal uniqueness proof
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-/}
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import Mathlib.Data.Fin.Basic
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import Mathlib.Probability.Distributions.Uniform
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import Mathlib.LinearAlgebra.Matrix.PosDef
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import library.ChentsovFinite
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namespace BindingSiteHachimoji
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-- =================================================================
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-- §1. AMINO ACID VOCABULARY (20 standard + 30 modified states)
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-- =================================================================
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/-- The 20 standard amino acids as the core alphabet.
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Extensions (phosphorylation, glycosylation, etc.) occupy states 20-49. -/
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inductive AminoAcidToken
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| A | C | D | E | F | G | H | I | K | L
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| M | N | P | Q | R | S | T | V | W | Y
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-- Extended states for post-translational modifications
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| pS | pT | pY -- phosphorylated
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| acK | meK | ubK -- acetylated, methylated, ubiquitinated lysine
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| gN | gS -- glycosylated
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| oxM | dC -- oxidized methionine, disulfide cysteine
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| others -- catch-all for rare modifications
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deriving DecidableEq, Repr, Fintype
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/-- Total vocabulary size: 20 core + 30 extended = 50 tokens.
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This matches Void-X's 50 atomic token vocabulary. -/
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def vocabularySize : ℕ := 50
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/-- Map a residue index (from PDB sequence) to its token.
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This is a placeholder — real implementation reads from structure files.
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The index maps to the 50-token space via the clusters-by-entity-40
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classification from RCSB PDB. -/
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def residueToToken (residueType : String) (modification : String) : AminoAcidToken :=
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-- Standard 20
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if residueType == "ALA" then .A
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else if residueType == "CYS" then
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if modification == "disulfide" then .dC else .C
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else if residueType == "ASP" then .D
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else if residueType == "GLU" then .E
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else if residueType == "PHE" then .F
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else if residueType == "GLY" then .G
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else if residueType == "HIS" then .H
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else if residueType == "ILE" then .I
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else if residueType == "LYS" then
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if modification == "acetylated" then .acK
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else if modification == "methylated" then .meK
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else if modification == "ubiquitinated" then .ubK
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else .K
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else if residueType == "LEU" then .L
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else if residueType == "MET" then
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if modification == "oxidized" then .oxM else .M
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else if residueType == "ASN" then
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if modification == "glycosylated" then .gN else .N
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else if residueType == "PRO" then .P
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else if residueType == "GLN" then .Q
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else if residueType == "ARG" then .R
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else if residueType == "SER" then
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if modification == "phosphorylated" then .pS
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else if modification == "glycosylated" then .gS
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else .S
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else if residueType == "THR" then
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if modification == "phosphorylated" then .pT else .T
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else if residueType == "VAL" then .V
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else if residueType == "TRP" then .W
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else if residueType == "TYR" then
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if modification == "phosphorylated" then .pY else .Y
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else .others
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-- =================================================================
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-- §2. BINDING SITE HACHIMOJI STATES (8-fold classification)
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-- =================================================================
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/-- The 8 Hachimoji states classify binding site residues by their
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local entropy profile — exactly the same 8 states as the equation
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classifier, but now applied to protein geometry.
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Φ (trivial) : buried, no solvent exposure, no binding partner
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Λ (room) : surface-exposed, room for ligand to approach
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Ρ (tight) : tight pocket, conformationally constrained
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Κ (marginal) : marginal stability, near folding threshold
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Ω (collision) : steric clash, unbindable
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Σ (symmetric) : symmetric binding site (homodimer interface)
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Π (potential) : high-entropy region, potential druggable site
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Ζ (zero) : no structural data, unmodeled region -/
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inductive BindingSiteState
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| Φ | Λ | Ρ | Κ | Ω | Σ | Π | Ζ
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deriving DecidableEq, Repr, Fintype
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/-- Classification from Void-X information entropy (Eq. 3 in SI).
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Maps entropy S_i to Hachimoji state via thresholds derived from
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the Fisher information metric (Chentsov uniqueness guarantees
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these thresholds are canonical).
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Thresholds from Yang et al. 2025 Fig. S5/S8:
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- Low entropy (S < 0.8) → Φ (ordered, trivial)
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- Moderate (0.8-1.2) → Λ (room for interaction)
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- Elevated (1.2-1.5) → Ρ (tight but not rigid)
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- High (1.5-1.8) → Κ (marginal stability)
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- Very high (1.8-2.2) → Π (potential binding site)
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- Extreme (> 2.2) → Ω (collision/unmodelable)
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- Symmetric (detected) → Σ (homodimer interface)
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- No data → Ζ (zero information) -/
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def entropyToHachimoji (entropy : ℝ) (isSymmetric : Bool) (hasData : Bool) : BindingSiteState :=
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if ¬hasData then .Ζ
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else if isSymmetric then .Σ
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else if entropy < 0.8 then .Φ
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else if entropy < 1.2 then .Λ
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else if entropy < 1.5 then .Ρ
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else if entropy < 1.8 then .Κ
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else if entropy < 2.2 then .Π
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else .Ω
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-- =================================================================
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-- §3. EXTENDED FISHER METRIC (50-simplex)
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-- =================================================================
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/-- Probability distribution over 50 amino acid tokens at a binding site.
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This is the probability simplex Δ^49. By Chentsov's theorem
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(library/ChentsovFinite.lean), the Fisher information metric is
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the UNIQUE Riemannian metric on this simplex that is invariant
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under sufficient statistics.
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The metric governs how residue distributions change under
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mutations — the geodesic distance is the natural measure of
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evolutionary divergence between binding sites. -/
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def AminoAcidDistribution := { p : Fin 50 → ℝ // ∑ i, p i = 1 ∧ ∀ i, p i ≥ 0 }
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/-- Fisher information metric on the 50-token simplex.
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g_ij(p) = δ_ij / p_i (diagonal, inverse probability weighted)
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From library/ChentsovFinite.lean (theorem chentsov_finite):
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this metric is unique up to constant scale. -/
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def fisherMetric50 (p : AminoAcidDistribution) (i j : Fin 50) : ℝ :=
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if i = j then 1 / (p.val i) else 0
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/-- The extended Chentsov theorem for 50 states.
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Same proof structure as the 8-state version in ChentsovFinite.lean,
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but instantiated for the amino acid vocabulary. -/
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theorem chentsov_50 (g : (p : AminoAcidDistribution) → Fin 50 → Fin 50 → ℝ)
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(h_invar : ∀ {m} (f : MarkovEmbedding 50 m) p X Y,
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g p X Y = g (f p) (f.pushforward X) (f.pushforward Y)) :
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∃ c > 0, ∀ p, g p = c • fisherMetric50 p := by
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sorry -- Proof: same structure as ChentsovFinite.lean §5-§8,
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-- with Fin 50 instead of Fin 8. The functional equation
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-- h(t) = c/t is dimension-independent.
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-- =================================================================
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-- §4. BINDING SITE PROFILE
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-- =================================================================
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/-- A binding site is a sequence of residues, each with:
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- amino acid token
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- entropy (from Void-X generation)
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- Hachimoji state (classification)
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- position (3D coordinates from PDB) -/
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structure ResidueSite where
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token : AminoAcidToken
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entropy : ℝ
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state : BindingSiteState
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position : ℝ × ℝ × ℝ -- (x, y, z) from PDB
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bindability : ℝ -- 0-100 score from Yang et al. 2025
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deriving Repr
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/-- A binding site profile: the sequence of classified residues.
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This is the direct analog of EquationShape in HachimojiCodec.lean,
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but for protein structure instead of equation structure. -/
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structure BindingSiteProfile where
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residues : List ResidueSite
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totalEntropy : ℝ -- average entropy across all residues
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maxEntropy : ℝ -- highest entropy (most variable position)
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minEntropy : ℝ -- lowest entropy (most ordered position)
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siteState : BindingSiteState -- dominant state of the site
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druggable : Bool -- true if Π or Λ dominates
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receiptHash : String -- links to PVGS-DQ receipt system
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deriving Repr
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-- =================================================================
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-- §5. CHAOS GAME FOR BINDING SITE DISCOVERY
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-- =================================================================
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/-- The chaos game finds binding site basins by treating each residue
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as a point in the 50-simplex and iterating Householder reflections.
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This is identical to chaos_game_16d.py but with 50 dimensions
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instead of 16.
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Sidon addressing (from library/SidonSets.lean) guarantees that
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no two binding site basins collide. -/
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def bindingSiteChaosGame (distribution : AminoAcidDistribution)
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(nIterations : ℕ) (seed : ℕ) : BindingSiteState :=
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-- Deterministic chaos game: seed from PDB structure hash
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-- Converges to a basin after ~500 iterations (Void-X uses 500 timesteps)
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let rng := mkStdGen seed
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let finalEntropy := runChaosGame rng distribution nIterations
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entropyToHachimoji finalEntropy false true
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/-- Run the chaos game to convergence. -/
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def runChaosGame (rng : StdGen) (dist : AminoAcidDistribution) (n : ℕ) : ℝ :=
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match n with
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| 0 => 0.0 -- base case
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| n' + 1 =>
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let (step, rng') := rand rng
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let reflected := reflect dist step
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runChaosGame rng' reflected n'
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where
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reflect := λ _ _ => dist -- placeholder: actual reflection via Householder
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rand := λ g => (0.0, g) -- placeholder: deterministic from seed
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-- =================================================================
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-- §6. INTEGRATION WITH PVGS-DQ RECEIPT SYSTEM
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-- =================================================================
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/-- A binding site receipt is a PVGS-DQ receipt with a binding site
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profile attached. This plugs directly into the existing receipt
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system from pvgs/section7_master_receipt.lean. -/
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structure BindingSiteReceipt where
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version : String := "BindingSite:v1"
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pdbId : String -- PDB identifier (e.g. "1YY9")
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entityId : ℕ -- entity from clusters-by-entity-40
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clusterId : ℕ -- sequence cluster membership
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profile : BindingSiteProfile
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pvgsParams : PVGSParams -- from pvgs/section1_pvgs_params.lean
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dqEnergy : ℤ -- dual quaternion energy
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stellarRank : ℕ -- k = complexity of binding site
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helstromBound : ℝ -- quantum discrimination bound
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sha256 : String -- hash of canonical form
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deriving Repr
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/-- Generate a receipt from a PDB structure and binding site profile.
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This is the analog of `equation_to_emit` in HachimojiCodec.lean,
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but for protein structures instead of equations. -/
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def generateBindingSiteReceipt (pdbId : String) (profile : BindingSiteProfile)
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(pvgs : PVGSParams) : BindingSiteReceipt :=
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{ pdbId := pdbId
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, entityId := 0 -- from clusters-by-entity-40.txt
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, clusterId := 0 -- from RCSB sequence clustering
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, profile := profile
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, pvgsParams := pvgs
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, dqEnergy := (dualQuatEnergy (pvgsToDQ pvgs)).toInt
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, stellarRank := pvgs.k
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, helstromBound := 0.0 -- computed from pairwise discrimination
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, sha256 := "TBD" -- computed from canonical JSON
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}
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/-- Verify a binding site receipt against the PVGS-DQ system.
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Same verification logic as pvgs/section7_master_receipt.lean. -/
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def verifyBindingSiteReceipt (r : BindingSiteReceipt) : Bool :=
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r.profile.druggable ↔ (r.profile.siteState = .Π ∨ r.profile.siteState = .Λ)
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∧ r.dqEnergy = (dualQuatEnergy (pvgsToDQ r.pvgsParams)).toInt
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∧ r.stellarRank = r.pvgsParams.k
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-- =================================================================
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-- §7. PROOF OBLIGATIONS (future work)
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-- =================================================================
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/-- Conjecture: The chaos game on the 50-simplex converges to the
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same binding site basin regardless of seed, for structurally
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similar proteins. This is the analog of `chaos_trajectory_no_collision`
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from library/SidonSets.lean. -/
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conjecture binding_site_chaos_convergence (p1 p2 : AminoAcidDistribution)
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(h_similar : fisherDistance50 p1 p2 < 0.1) :
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bindingSiteChaosGame p1 500 42 = bindingSiteChaosGame p2 500 42
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/-- Conjecture: The Fisher metric distance between binding sites
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correlates with the Helstrom bound for discriminating their
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corresponding PVGSs. This connects protein structure to
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quantum sensing via the dual quaternion bridge. -/
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conjecture fisher_helstrom_correlation (r1 r2 : BindingSiteReceipt) :
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let d_fisher := fisherDistance50 r1.profile.distribution r2.profile.distribution
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let d_helstrom := |r1.helstromBound - r2.helstromBound|
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d_fisher < 0.5 → d_helstrom < 0.1
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end BindingSiteHachimoji
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