Crn Topology

CRN Topology Skill: Reaction Network Generation

Status: ✅ Production Ready Trit: +1 (PLUS - generator) Color: #D82626 (Red) Principle: Network structure → Dynamical behavior Frame: Hypergraph topology of chemical reactions

Overview

CRN Topology generates and analyzes the graph structure of chemical reaction networks. The topology determines qualitative dynamics—multistability, oscillations, and computational capacity.

1. Species-reaction graph: Bipartite hypergraph
2. Stoichiometric matrix: Linear algebra of reactions
3. Deficiency: Gap between complexes and rank
4. Persistence: Network admits no extinctions

Core Formula

Deficiency δ = n - ℓ - s
  n = number of complexes
  ℓ = number of linkage classes  
  s = rank of stoichiometric matrix

Zero deficiency theorem:
  δ = 0 and weakly reversible ⟹ unique stable equilibrium

def crn_deficiency(network: CRN) -> int:
    n = len(network.complexes)
    l = network.linkage_classes()
    s = np.linalg.matrix_rank(network.stoichiometry)
    return n - l - s

Key Concepts

1. Stoichiometric Matrix Generation

class CRNGenerator:
    def __init__(self, species: list[str]):
        self.species = species
    
    def random_reaction(self) -> Reaction:
        """Generate topology-valid reaction."""
        reactants = self.sample_complex()
        products = self.sample_complex()
        return Reaction(reactants, products)
    
    def stoichiometry_matrix(self, reactions) -> np.ndarray:
        """S[i,j] = net change in species i from reaction j."""
        S = np.zeros((len(self.species), len(reactions)))
        for j, rxn in enumerate(reactions):
            S[:, j] = rxn.products - rxn.reactants
        return S

2. Network Motif Generation

def generate_oscillator_topology() -> CRN:
    """Generate Brusselator-like topology."""
    return CRN([
        "A → X",
        "2X + Y → 3X",
        "B + X → Y + D", 
        "X → E"
    ])

def generate_bistable_topology() -> CRN:
    """Generate Schlögl-like bistability."""
    return CRN([
        "A + 2X ⇌ 3X",
        "X ⇌ B"
    ])

3. Deficiency Analysis

def analyze_topology(crn: CRN) -> dict:
    """Determine dynamical properties from topology."""
    delta = crn_deficiency(crn)
    wr = is_weakly_reversible(crn)
    return {
        "deficiency": delta,
        "weakly_reversible": wr,
        "unique_equilibrium": delta == 0 and wr,
        "multistability_possible": delta > 0,
        "complex_balanced": check_complex_balance(crn)
    }

Commands

# Generate CRN with target properties
just crn-generate --oscillator --species 3

# Compute deficiency
just crn-deficiency network.crn

# Visualize reaction hypergraph
just crn-topology network.crn

Integration with GF(3) Triads

assembly-index (-1) ⊗ turing-chemputer (0) ⊗ crn-topology (+1) = 0 ✓  [Molecular Complexity]
persistent-homology (-1) ⊗ turing-chemputer (0) ⊗ crn-topology (+1) = 0 ✓  [Topological CRN]

Related Skills

• turing-chemputer (0): Execute reactions in CRN
• assembly-index (-1): Validate molecular complexity
• acsets (0): Algebraic representation of CRN hypergraph

Skill Name: crn-topology Type: Reaction Network Generator Trit: +1 (PLUS) Color: #D82626 (Red)

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