hub:
  repo_type: generator
  builtin: hill-kinetics
  tags:
  - banzhaf-yamamoto
chemistries:
- id: hill-kinetics
  name: Hill kinetics (cooperative binding)
  intuition: 'Cooperative binding, and the standard way a smooth input becomes a switch. n copies of a
    protein bind a gene together, so the fraction of bound genes follows P^n/(K^n + P^n) - a curve that
    sharpens into a step as n grows. It is an idealisation on purpose: n molecules never actually arrive
    simultaneously, and with a single gene copy and a handful of proteins a differential equation is the
    wrong tool. Useful precisely because it compresses a mechanism into one parameter.

    '
  book: 18.2.2; used by the Repressilator 19.3.2
  refs:
  - '[741]'
  - '[916]'
  family: systems-biology
  kind: generator
  constructive: false
  S:
    repr: gene G, protein P, complex C, gene product X
    definition: explicit
  R:
    definition: explicit
    arity: n+1
    scheme: G + n P <-kf/kr-> C ; activation C -> C + X ; repression G -> G + X ; lumped rates H1 = P^n/(K^n
      + P^n), H2 = K^n/(K^n + P^n)
  A:
    reactor:
    - ode
    - ssa
    dilution: none
  params:
  - name: form
    type: enum
    default: elementary
    choices:
    - elementary
    - lumped
    role: structural
    meaning: elementary binding reactions (eqs. 18.5, 18.12, 18.15) or a single production reaction with
      a Hill rate law (eqs. 18.14, 18.16)
  - name: regulation
    type: enum
    default: activation
    choices:
    - activation
    - repression
    role: structural
    meaning: P activates (bound gene expresses) or represses (free gene expresses) X
  - name: n
    type: int
    default: 2
    min: 1
    max: 10
    role: kinetic
    meaning: Hill coefficient; better read as the DEGREE OF COOPERATIVITY than as a literal molecule count.
      n=1 gives no sigmoid, so no switching
  - name: K
    type: float
    default: 1.0
    min: 0
    role: kinetic
    meaning: K^n = K_d = kr/kf, the dissociation constant
  - name: kf
    type: float
    default: 1.0
    min: 0
    role: kinetic
    meaning: binding rate; the unbinding rate is kf K^n
  - name: k_expr
    type: float
    default: 1.0
    min: 0
    role: kinetic
    meaning: expression rate of X (k1 or k2 in the book)
  - name: G0
    type: float
    default: 1.0
    min: 0
    role: population
    meaning: total gene = free + complexed
  - name: P0
    type: float
    default: 1.0
    min: 0
    role: population
    meaning: initial protein
  provides:
  - topology
  - stoichiometry
  - catalysts
  - rate-law
  - rate-constants
  - initial-state
  fidelity: book+decisions
  decisions:
  - 'The book derives both forms; `form` lets the user pick. Defaults follow figure 18.5 (G(0) = P(0)
    = 1) with unit rates. In the lumped form the rate dict names the regulator species (''regulator'':
    ''P'') and ''mode''.'
  notes: 'The book flags the idealisation explicitly: n molecules never bind simultaneously, and with
    one gene copy and few proteins an ODE is a poor approximation.

    '