dual-site-phosphorylation.bngl

begin model
begin parameters
    # Dual-site phosphorylation: Requires two sequential modifications for activity.
    # Exhibits ultrasensitivity and bistability in specific contexts.
    
    # Kinase kinetics (Writing)
    k_bind_k 1.0       # Kinase binding to substrate
    k_unbind_k 0.1     # Kinase dissociation
    k_phos_1 0.5       # 1st phosphorylation rate (U -> P1)
    k_phos_2 0.8       # 2nd phosphorylation rate (P1 -> PP)
    
    # Phosphatase kinetics (Eraser)
    k_bind_p 1.0       # Phosphatase binding
    k_unbind_p 0.1
    k_dephos_1 0.4     # 1st dephosphorylation (PP -> P1)
    k_dephos_2 0.4     # 2nd dephosphorylation (P1 -> U)
    
    # Cooperative influence (Allosteric proxy)
    k_coop 2.0         # Speed-up of second modification
    
    # Initials
    Kinase_tot 50      # The writer
    Phosphatase_tot 50 # The eraser
    Substrate_tot 200  # The target
end parameters

begin molecule types
    Kinase(b,state~off~on)
    Phosphatase(b,state~off~on)
    Substrate(y1~U~P,y2~U~P,b)
end molecule types

begin seed species
    Kinase(b,state~on) Kinase_tot
    Phosphatase(b,state~on) Phosphatase_tot
    Substrate(y1~U,y2~U,b) Substrate_tot
end seed species

begin observables
    # KEY BIOLOGICAL OUTPUTS
    Molecules Unmodified     Substrate(y1~U,y2~U)       # Basal state
    Molecules Mono_Phos      Substrate(y1~P,y2~U),Substrate(y1~U,y2~P) # Primed
    Molecules Fully_Phos     Substrate(y1~P,y2~P)       # ACTIVE species
    Molecules Kinase_Bound   Substrate(b!+)              # Enzyme-substrate proxy
    Molecules Phos_Level     Substrate(y1~P),Substrate(y2~P) # Total stoichiometry
end observables

begin reaction rules
    ## PHOSPHORYLATION CHAIN
    # Kinase binds substrate
    Kinase(state~on,b) + Substrate(b) <-> Kinase(state~on,b!1).Substrate(b!1) k_bind_k,k_unbind_k
    
    # Phosphorylation at site y1
    Kinase(b!1).Substrate(b!1,y1~U) -> Kinase(b,state~on) + Substrate(b,y1~P) k_phos_1
    
    # Phosphorylation at site y2
    Kinase(b!1).Substrate(b!1,y2~U) -> Kinase(b,state~on) + Substrate(b,y2~P) k_phos_1
    
    # Accelerated second phosphorylation (The "Ultrasensitivity" requirement)
    Kinase(b!1).Substrate(b!1,y1~P,y2~U) -> Kinase(b,state~on) + Substrate(b,y1~P,y2~P) k_phos_2*k_coop
    Kinase(b!1).Substrate(b!1,y1~U,y2~P) -> Kinase(b,state~on) + Substrate(b,y1~P,y2~P) k_phos_2*k_coop
    
    ## DEPHOSPHORYLATION CHAIN
    # Phosphatase removes phosphates (Stochastic preference proxy)
    Phosphatase(state~on) + Substrate(y1~P) -> Phosphatase(state~on) + Substrate(y1~U) k_dephos_2
    Phosphatase(state~on) + Substrate(y2~P) -> Phosphatase(state~on) + Substrate(y2~U) k_dephos_2
end reaction rules

begin actions
    generate_network({overwrite=>1})
    simulate({method=>"ode",t_end=>5,n_steps=>300})
end actions
end model