DiffEqProblemLibrary.jl/lib/DDEProblemLibrary/src/qs.jl at 94ae6bdbcc7d173b17c0bfba89ab32d16839ad44 · ChrisRackauckas/DiffEqProblemLibrary.jl · GitHub

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"""
    prob_dde_qs

Delay differential equation model of Quorum Sensing (QS) of Pseudomonas putida IsoF in
continuous cultures.

# References

Buddrus-Schiemann et al. (2014). Analysis of N-Acylhomoserine Lactone Dynamics in Continuous
Cultures of Pseudomonas Putida IsoF By Use of ELISA and UHPLC/qTOF-MS-derived Measurements
and Mathematical Models, Analytical and Bioanalytical Chemistry.
"""
const prob_dde_qs = let S₀ = 1, τ = 2, D = 0.1, γₛ = 1.3e-12, Kₘ = 0.38, nₛ = 1.3, a = 0.66,
    αₐ = 2.3e-19, βₐ = 2.3e-18, C₁ = 70e-9, n₁ = 2.3, γₐ = 0.05, αC = 4e4, R = 5e-7,
    γ₃ = 0.08, Kₑ = 1.5e-4, αₗ = 1.1e-8, C₂ = 70e-9, n₂ = 2.5, γₗ = 0.005

    function hill_growth(x, K, n)
        if x > K
            return 1 / (1 + (K / x)^n)
        end

        if K == 0
            return one(x)
        end

        y = (x / K)^n
        return y / (1 + y)
    end

    global function f_dde_qs!(du, u, h, p, t)
        if u[1] < 0
            # eq. 1 and 2 not defined for u[1] < 0
            du[1] = 0
            du[2] = 0
        else
            tmp = u[2] * hill_growth(u[1], Kₘ, nₛ)
            du[1] = D * (S₀ - u[1]) - γₛ * tmp
            du[2] = a * tmp - D * u[2]
        end

        if u[4] < 0
            # eq. 3 not defined for u[4] < 0
            du[3] = 0
            du[4] = max(0, αC * (R - u[4]) * u[3] - γ₃ * u[4])
        else
            du[3] = (αₐ + βₐ * hill_growth(u[4], C₁, n₁)) * u[2] -
                    (γₐ + D) * u[3] - αC * (R - u[4]) * u[3] + γ₃ * u[4] -
                    Kₑ * u[3] * u[5]
            du[4] = αC * (R - u[4]) * u[3] - γ₃ * u[4]
        end

        Cₜ = h(p, t - τ; idxs = 4)
        if Cₜ < 0
            # eq. 5 not defined for Cₜ < 0
            du[5] = 0
        else
            du[5] = αₗ * hill_growth(Cₜ, C₂, n₂) * u[2] - (γₗ + D) * u[5]
        end

        nothing
    end

    function h_dde_qs(p, t; idxs::Union{Nothing, Int} = nothing)
        t ≤ 0 || error("history function is only implemented for t ≤ 0")

        if idxs === nothing
            [1.0, 8.4e8, 2.5e-9, 7.6e-8, 5e-15]
        elseif idxs == 1
            1.0
        elseif idxs == 2
            8.4e8
        elseif idxs == 3
            2.5e-9
        elseif idxs == 4
            7.6e-8
        elseif idxs == 5
            5e-15
        else
            error("delay differential equation consists of five components")
        end
    end

    DDEProblem(f_dde_qs!, h_dde_qs, (0.0, 45.0); constant_lags = [τ])
end