CatColab Regulatory Networks: Molecular Biology Modeling

Trit: -1 (MINUS - validator/inhibitor) Color: Red (#DC143C)

Overview

Regulatory Networks in CatColab model molecular interactions that control gene expression:

• Nodes: Genes, proteins, RNA, metabolites
• Positive edges: Activation/promotion (+)
• Negative edges: Inhibition/repression (-)

These signed graphs capture the control logic of biological systems.

Mathematical Foundation

A regulatory network is a signed graph or signed category:

┌─────────────────────────────────────────────────────┐
│              REGULATORY NETWORK                      │
├─────────────────────────────────────────────────────┤
│  Nodes (Genes/Proteins):                             │
│    GeneA, GeneB, GeneC, ProteinX                     │
│                                                      │
│  Positive Edges (Activation):                        │
│    GeneA ──(+)──► GeneB                              │
│    ProteinX ──(+)──► GeneC                           │
│                                                      │
│  Negative Edges (Inhibition):                        │
│    GeneB ──(-)──► GeneC                              │
│    GeneC ──(-)──► GeneA  (negative feedback)         │
│                                                      │
│  Motifs:                                             │
│    Feedforward loop: A→B→C, A→C                      │
│    Negative feedback: A→B→C⊣A                        │
└─────────────────────────────────────────────────────┘

Double Theory

// Signed category double theory
pub fn th_signed_category() -> DiscreteDblTheory {
    let mut cat = FpCategory::new();

    // Object type
    cat.add_ob_generator(name("Node"));

    // Morphism types (signed edges)
    cat.add_mor_generator(name("Positive"), name("Node"), name("Node"));
    cat.add_mor_generator(name("Negative"), name("Node"), name("Node"));

    // Constraint: n ⊙ n = id (double negative = positive)
    cat.add_equation(
        compose(name("Negative"), name("Negative")),
        identity(name("Node"))
    );

    cat.into()
}

CatColab Implementation

Node Declaration

{
  "type": "ObDecl",
  "name": "p53",
  "theory_type": "Node",
  "description": "tumor suppressor protein"
}

Positive Regulation (Activation)

{
  "type": "MorDecl",
  "name": "activates_apoptosis",
  "dom": "p53",
  "cod": "Bax",
  "theory_type": "Positive",
  "description": "p53 promotes apoptosis via Bax"
}

Negative Regulation (Inhibition)

{
  "type": "MorDecl",
  "name": "inhibits_growth",
  "dom": "p53",
  "cod": "CyclinD",
  "theory_type": "Negative",
  "description": "p53 blocks cell cycle progression"
}

Network Motifs

Feedforward Loop (FFL)

     GeneA
    /     \
   +       +
  ↓         ↓
GeneB ──+──► GeneC

Type: Coherent (all positive)
Function: Noise filtering, delay

Negative Feedback Loop

GeneA ──+──► GeneB ──+──► GeneC
  ▲                        │
  └────────(-)─────────────┘

Function: Homeostasis, oscillation

Toggle Switch (Bistability)

GeneA ◄──(-)──► GeneB
         ⇅
       (-)

Function: Binary cell fate decision

Practical Examples

Example 1: p53 Tumor Suppressor Network

Nodes: p53, MDM2, ATM, Bax, p21

Edges:
  ATM ──(+)──► p53       (DNA damage activates p53)
  p53 ──(+)──► MDM2      (p53 induces its own inhibitor)
  MDM2 ──(-)──► p53      (MDM2 degrades p53)
  p53 ──(+)──► Bax       (p53 promotes apoptosis)
  p53 ──(+)──► p21       (p53 arrests cell cycle)

Motif: p53-MDM2 negative feedback loop

Example 2: Lac Operon

Nodes: LacI, LacZ, Lactose, Glucose

Edges:
  LacI ──(-)──► LacZ     (repressor blocks transcription)
  Lactose ──(-)──► LacI  (lactose inactivates repressor)
  Glucose ──(-)──► LacZ  (catabolite repression)

Function: Metabolic switch for sugar utilization

Analysis Capabilities

CatColab can analyze regulatory networks for:

• Steady states: Fixed points of the dynamics
• Stability: Eigenvalue analysis of Jacobian
• Motif enrichment: Statistical over-representation
• Boolean dynamics: Logical model simulation

GF(3) Triads

catcolab-regulatory-networks (-1) ⊗ topos-catcolab (0) ⊗ catcolab-stock-flow (+1) = 0 ✓
crn-topology (-1) ⊗ catcolab-regulatory-networks (0) ⊗ alife (+1) = 0 ✓

Commands

# Create regulatory network
just catcolab-new regulatory "p53-network"

# Analyze motifs
just catcolab-analyze p53-network --motifs

# Export to SBML
just catcolab-export p53-network --format=sbml

# Simulate Boolean dynamics
just catcolab-simulate p53-network --boolean

References

• Alon (2007) "Network motifs: theory and experimental approaches"
• Karlebach & Shamir (2008) "Modelling and analysis of gene regulatory networks"
• CatColab Regulatory Networks Help

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Skill Name: catcolab-regulatory-networks Type: Systems Biology / Gene Regulation Trit: -1 (MINUS) GF(3): Conserved via triadic composition