CaMKII system analysis

using ModelingToolkit
using OrdinaryDiffEq, SteadyStateDiffEq, DiffEqCallbacks
using Plots
using LsqFit
using CaMKIIModel
using CaMKIIModel: μM, hil, second, Hz
Plots.default(lw=1.5)

CaMKII sensitivity to calcium

Model CaM/Calcium binding only. No phosphorylation or oxidation.

@parameters Ca = 0μM ROS = 0μM
sys = get_camkii_sys(; Ca, ROS, simplify=true)

\[\begin{split} \begin{align} \frac{\mathrm{d} \mathtt{CaMKOX}\left( t \right)}{\mathrm{d}t} &= \mathtt{kCaM4\_off} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) - \mathtt{k\_OX\_B} \mathtt{CaMKOX}\left( t \right) + \mathtt{k\_dephospho} \mathtt{CaMKPOX}\left( t \right) - \mathtt{kCaM4\_on} \mathtt{CaMKOX}\left( t \right) \mathtt{Ca4CaM}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKPOX}\left( t \right)}{\mathrm{d}t} &= \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) - \mathtt{k\_OXP\_P} \mathtt{CaMKPOX}\left( t \right) - \mathtt{k\_dephospho} \mathtt{CaMKPOX}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKP2}\left( t \right)}{\mathrm{d}t} &= \mathtt{k\_P1\_P2} \mathtt{CaMKP}\left( t \right) - \mathtt{k\_P2\_P1} \mathtt{CaMKP2}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKP}\left( t \right)}{\mathrm{d}t} &= \mathtt{kCaM0P\_off} \mathtt{CaM0\_CaMKP}\left( t \right) + \mathtt{kCaM2CP\_off} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) + \mathtt{kCaM2NP\_off} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) + \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) + \mathtt{k\_OXP\_P} \mathtt{CaMKPOX}\left( t \right) - \mathtt{k\_P1\_P2} \mathtt{CaMKP}\left( t \right) + \mathtt{k\_P2\_P1} \mathtt{CaMKP2}\left( t \right) - \mathtt{k\_dephospho} \mathtt{CaMKP}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right)}{\mathrm{d}t} &= - \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) - \mathtt{k\_OXP\_P} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) - \mathtt{k\_dephospho} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) + \mathtt{ROS} \mathtt{k\_P\_OXP} \mathtt{Ca4CaM\_CaMKP}\left( t \right) + \mathtt{k\_phosCaM} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca4CaM\_CaMKOX}\left( t \right)}{\mathrm{d}t} &= - \mathtt{kCaM4\_off} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) - \mathtt{k\_OX\_B} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) + \mathtt{k\_dephospho} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) + \mathtt{ROS} \mathtt{k\_B\_OX} \mathtt{Ca4CaM\_CaMK}\left( t \right) + \mathtt{kCaM4\_on} \mathtt{CaMKOX}\left( t \right) \mathtt{Ca4CaM}\left( t \right) - \mathtt{k\_phosCaM} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca4CaM\_CaMKP}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} - \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) + \mathtt{k\_OXP\_P} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) - \mathtt{k\_dephospho} \mathtt{Ca4CaM\_CaMKP}\left( t \right) - \mathtt{ROS} \mathtt{k\_P\_OXP} \mathtt{Ca4CaM\_CaMKP}\left( t \right) + \mathtt{k\_K2C\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) + \mathtt{k\_K2N\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) + \mathtt{k\_phosCaM} \mathtt{Ca4CaM\_CaMK}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{CaM0\_CaMKP}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} - \mathtt{kCaM2NP\_off} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) - \mathtt{k\_dephospho} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) + \mathtt{k\_K2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) \mathtt{Ca4CaM\_CaMKP}\left( t \right) - \mathtt{k\_K2N\_off} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) + \mathtt{k\_phosCaM} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{CaM0\_CaMKP}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} + \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} - \mathtt{kCaM2CP\_off} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) - \mathtt{k\_dephospho} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) + \mathtt{k\_K2C\_off} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) + \mathtt{k\_K2N\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) + \mathtt{k\_phosCaM} \mathtt{CaMKAct}\left( t \right) \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaM0\_CaMKP}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{CaM0\_CaMKP}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} + \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{CaM0\_CaMKP}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} - \mathtt{kCaM0P\_off} \mathtt{CaM0\_CaMKP}\left( t \right) + \mathtt{k\_K2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) + \mathtt{k\_K2N\_off} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) \\ \frac{\mathrm{d} \mathtt{Ca4CaM\_CaMK}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{Ca2CaM\_C\_CaMK}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} - \mathtt{kCaM4\_off} \mathtt{Ca4CaM\_CaMK}\left( t \right) + \mathtt{k\_OX\_B} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) + \mathtt{k\_dephospho} \mathtt{Ca4CaM\_CaMKP}\left( t \right) - \mathtt{ROS} \mathtt{k\_B\_OX} \mathtt{Ca4CaM\_CaMK}\left( t \right) + \mathtt{kCaM4\_on} \mathtt{CaMK}\left( t \right) \mathtt{Ca4CaM}\left( t \right) + \mathtt{k\_K2C\_off} \mathtt{Ca4CaM\_CaMK}\left( t \right) \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) + \mathtt{k\_K2N\_off} \mathtt{Ca4CaM\_CaMK}\left( t \right) \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) - \mathtt{k\_phosCaM} \mathtt{Ca4CaM\_CaMK}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{CaM0\_CaMK}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} - \mathtt{kCaM2N\_off} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) + \mathtt{k\_dephospho} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) + \mathtt{kCaM2N\_on} \mathtt{CaMK}\left( t \right) \mathtt{Ca2CaM\_N}\left( t \right) + \mathtt{k\_K2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) \mathtt{Ca4CaM\_CaMK}\left( t \right) + \mathtt{k\_K2N\_off} \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) - \mathtt{k\_phosCaM} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_C\_CaMK}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{Ca2CaM\_C\_CaMK}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{CaM0\_CaMK}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} - \mathtt{kCaM2C\_off} \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) + \mathtt{k\_dephospho} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) + \mathtt{kCaM2C\_on} \mathtt{Ca2CaM\_C}\left( t \right) \mathtt{CaMK}\left( t \right) + \mathtt{k\_K2C\_off} \left( -1 + \frac{\mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) + \mathtt{k\_K2N\_off} \mathtt{Ca4CaM\_CaMK}\left( t \right) \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) - \mathtt{k\_phosCaM} \mathtt{CaMKAct}\left( t \right) \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaM0\_CaMK}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1C\_on} \mathtt{k\_K2C\_on} \mathtt{CaM0\_CaMK}\left( t \right)}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} + \frac{ - \mathtt{Ca}^{2} \mathtt{k\_K1N\_on} \mathtt{k\_K2N\_on} \mathtt{CaM0\_CaMK}\left( t \right)}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} - \mathtt{kCaM0\_off} \mathtt{CaM0\_CaMK}\left( t \right) + \mathtt{kCaM0\_on} \mathtt{CaM0}\left( t \right) \mathtt{CaMK}\left( t \right) + \mathtt{k\_K2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2C\_on}}{\mathtt{k\_K1C\_off} + \mathtt{Ca} \mathtt{k\_K2C\_on}} \right) \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) + \mathtt{k\_K2N\_off} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_K2N\_on}}{\mathtt{k\_K1N\_off} + \mathtt{Ca} \mathtt{k\_K2N\_on}} \right) \\ \frac{\mathrm{d} \mathtt{Ca4CaM}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{Ca}^{2} \mathtt{k\_1C\_on} \mathtt{k\_2C\_on} \mathtt{Ca2CaM\_N}\left( t \right)}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_1N\_on} \mathtt{k\_2N\_on} \mathtt{Ca2CaM\_C}\left( t \right)}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} + \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKPOX}\left( t \right) + \mathtt{kCaM4P\_off} \mathtt{Ca4CaM\_CaMKP}\left( t \right) + \mathtt{kCaM4\_off} \mathtt{Ca4CaM\_CaMKOX}\left( t \right) + \mathtt{kCaM4\_off} \mathtt{Ca4CaM\_CaMK}\left( t \right) - \mathtt{kCaM4\_on} \mathtt{CaMKOX}\left( t \right) \mathtt{Ca4CaM}\left( t \right) - \mathtt{kCaM4\_on} \mathtt{CaMK}\left( t \right) \mathtt{Ca4CaM}\left( t \right) - \mathtt{k\_2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2C\_on}}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} \right) \mathtt{Ca4CaM}\left( t \right) - \mathtt{k\_2N\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2N\_on}}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} \right) \mathtt{Ca4CaM}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_N}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_1C\_on} \mathtt{k\_2C\_on} \mathtt{Ca2CaM\_N}\left( t \right)}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_1N\_on} \mathtt{k\_2N\_on} \mathtt{CaM0}\left( t \right)}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} + \mathtt{kCaM2NP\_off} \mathtt{Ca2CaM\_N\_CaMKP}\left( t \right) + \mathtt{kCaM2N\_off} \mathtt{Ca2CaM\_N\_CaMK}\left( t \right) - \mathtt{kCaM2N\_on} \mathtt{CaMK}\left( t \right) \mathtt{Ca2CaM\_N}\left( t \right) + \mathtt{k\_2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2C\_on}}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} \right) \mathtt{Ca4CaM}\left( t \right) - \mathtt{k\_2N\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2N\_on}}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} \right) \mathtt{Ca2CaM\_N}\left( t \right) \\ \frac{\mathrm{d} \mathtt{Ca2CaM\_C}\left( t \right)}{\mathrm{d}t} &= \frac{ - \mathtt{Ca}^{2} \mathtt{k\_1N\_on} \mathtt{k\_2N\_on} \mathtt{Ca2CaM\_C}\left( t \right)}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} + \frac{\mathtt{Ca}^{2} \mathtt{k\_1C\_on} \mathtt{k\_2C\_on} \mathtt{CaM0}\left( t \right)}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} + \mathtt{kCaM2CP\_off} \mathtt{Ca2CaM\_C\_CaMKP}\left( t \right) + \mathtt{kCaM2C\_off} \mathtt{Ca2CaM\_C\_CaMK}\left( t \right) - \mathtt{kCaM2C\_on} \mathtt{Ca2CaM\_C}\left( t \right) \mathtt{CaMK}\left( t \right) - \mathtt{k\_2C\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2C\_on}}{\mathtt{k\_1C\_off} + \mathtt{Ca} \mathtt{k\_2C\_on}} \right) \mathtt{Ca2CaM\_C}\left( t \right) + \mathtt{k\_2N\_off} \left( 1 + \frac{ - \mathtt{Ca} \mathtt{k\_2N\_on}}{\mathtt{k\_1N\_off} + \mathtt{Ca} \mathtt{k\_2N\_on}} \right) \mathtt{Ca4CaM}\left( t \right) \end{align} \end{split}\]

prob = SteadyStateProblem(sys, [sys.k_phosCaM => 0])
alg = DynamicSS(Rodas5P())

ca = exp10.(range(log10(0.03μM), log10(10μM), 1001))
prob_func = (prob, i, repeat) -> begin
    remake(prob, p=[Ca => ca[i]])
end
trajectories = length(ca)
sol0 = solve(prob, alg; abstol=1e-10, reltol=1e-10) ## warmup
sim = solve(EnsembleProblem(prob; prob_func, safetycopy=false), alg; trajectories, abstol=1e-10, reltol=1e-10)

EnsembleSolution Solution of length 1001 with uType:
SciMLBase.NonlinearSolution{Float64, 1, Vector{Float64}, Vector{Float64}, SciMLBase.SteadyStateProblem{Vector{Float64}, true, ModelingToolkit.MTKParameters{Vector{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x34c7948e, 0x9460a6e0, 0xff82b077, 0x04630745, 0x444f8e46), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x211988d8, 0x86ae8a5c, 0x7d935e91, 0x2624d3ca, 0xd27c21ea), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, ModelingToolkit.MTKParameters{Vector{Float64}, StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 2, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x4a66ef23, 0xa7d777f5, 0xcb21e25f, 0xb620b783, 0xfe818c90), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x5e4298fb, 0x4b34b7b8, 0x1c7d1d5f, 0x04d86dbe, 0xc29af941), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb1e7009e, 0xa12019ca, 0x59b42dfc, 0x518f8bb4, 0x9b54abc3), Nothing}}}}, Returns{StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}}, ComposedFunction{typeof(identity), SymbolicIndexingInterface.MultipleGetters{SymbolicIndexingInterface.ContinuousTimeseries, Vector{Any}}}}, Nothing, ModelingToolkit.SetInitialUnknowns{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolic{Real}}}}}}, Val{true}}, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}, SteadyStateDiffEq.DynamicSS{OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoForwardDiff{nothing, ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, 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ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, ModelingToolkit.MTKParameters{Vector{Float64}, StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 2, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x4a66ef23, 0xa7d777f5, 0xcb21e25f, 0xb620b783, 0xfe818c90), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x5e4298fb, 0x4b34b7b8, 0x1c7d1d5f, 0x04d86dbe, 0xc29af941), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb1e7009e, 0xa12019ca, 0x59b42dfc, 0x518f8bb4, 0x9b54abc3), Nothing}}}}, Returns{StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}}, ComposedFunction{typeof(identity), SymbolicIndexingInterface.MultipleGetters{SymbolicIndexingInterface.ContinuousTimeseries, Vector{Any}}}}, Nothing, ModelingToolkit.SetInitialUnknowns{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolic{Real}}}}}}, Val{true}}, Nothing}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, ModelingToolkit.MTKParameters{Vector{Float64}, StaticArraysCore.SizedVector{0, 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Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb1e7009e, 0xa12019ca, 0x59b42dfc, 0x518f8bb4, 0x9b54abc3), Nothing}}}}, Returns{StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}}, ComposedFunction{typeof(identity), SymbolicIndexingInterface.MultipleGetters{SymbolicIndexingInterface.ContinuousTimeseries, Vector{Any}}}}, Nothing, ModelingToolkit.SetInitialUnknowns{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolic{Real}}}}}}, Val{true}}, Nothing}, Float64, ModelingToolkit.MTKParameters{Vector{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}}, LinearSolve.LinearCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, SciMLBase.NullParameters, LinearSolve.DefaultLinearSolver, LinearSolve.DefaultLinearSolverInit{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Tuple{LinearAlgebra.LU{Float64, Matrix{Float64}, Vector{Int64}}, Vector{Int64}}, Tuple{LinearAlgebra.LU{Float64, 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RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x211988d8, 0x86ae8a5c, 0x7d935e91, 0x2624d3ca, 0xd27c21ea), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, ModelingToolkit.MTKParameters{Vector{Float64}, StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 2, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x4a66ef23, 0xa7d777f5, 0xcb21e25f, 0xb620b783, 0xfe818c90), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x5e4298fb, 0x4b34b7b8, 0x1c7d1d5f, 0x04d86dbe, 0xc29af941), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, 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LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, SciMLBase.OverrideInitData{SciMLBase.NonlinearProblem{Nothing, true, ModelingToolkit.MTKParameters{Vector{Float64}, StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}, SciMLBase.NonlinearFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 2, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x4a66ef23, 0xa7d777f5, 0xcb21e25f, 0xb620b783, 0xfe818c90), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x5e4298fb, 0x4b34b7b8, 0x1c7d1d5f, 0x04d86dbe, 0xc29af941), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb1e7009e, 0xa12019ca, 0x59b42dfc, 0x518f8bb4, 0x9b54abc3), Nothing}}}}, Returns{StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}}, ComposedFunction{typeof(identity), SymbolicIndexingInterface.MultipleGetters{SymbolicIndexingInterface.ContinuousTimeseries, Vector{Any}}}}, Nothing, ModelingToolkit.SetInitialUnknowns{SymbolicIndexingInterface.MultipleSetters{Vector{SymbolicIndexingInterface.ParameterHookWrapper{SymbolicIndexingInterface.SetParameterIndex{ModelingToolkit.ParameterIndex{SciMLStructures.Initials, Int64}}, SymbolicUtils.BasicSymbolic{Real}}}}}}, Val{true}}, Nothing}, Vector{Float64}, ModelingToolkit.MTKParameters{Vector{Float64}, Vector{Float64}, Tuple{}, Tuple{}, Tuple{}, Tuple{}}}, Vector{Float64}, ADTypes.AutoForwardDiff{nothing, ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}}, Float64, Tuple{}}, ForwardDiff.DerivativeConfig{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}, Float64, 1}}}, Tuple{}}, DifferentiationInterfaceForwardDiffExt.ForwardDiffTwoArgDerivativePrep{Tuple{SciMLBase.TimeGradientWrapper{true, SciMLBase.ODEFunction{true, true, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x34c7948e, 0x9460a6e0, 0xff82b077, 0x04630745, 0x444f8e46), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x211988d8, 0x86ae8a5c, 0x7d935e91, 0x2624d3ca, 0xd27c21ea), Nothing}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, 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Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, ModelingToolkit.ObservedFunctionCache{ModelingToolkit.System}, Nothing, ModelingToolkit.System, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", 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Tuple{}, @NamedTuple{}}, SciMLBase.StandardNonlinearProblem}, Nothing, Nothing, Nothing, ModelingToolkit.InitializationMetadata{ModelingToolkit.ReconstructInitializeprob{ModelingToolkit.var"#_getter#911"{Tuple{ComposedFunction{ModelingToolkit.PConstructorApplicator{typeof(identity)}, ModelingToolkit.ObservedWrapper{true, ModelingToolkit.GeneratedFunctionWrapper{(2, 3, true), RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0x1ec90857, 0x943f2754, 0xc098dff8, 0x640a5ed4, 0xf710a1ae), Nothing}, RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :__mtk_arg_1, :___mtkparameters___, :t), ModelingToolkit.var"#_RGF_ModTag", ModelingToolkit.var"#_RGF_ModTag", (0xb1e7009e, 0xa12019ca, 0x59b42dfc, 0x518f8bb4, 0x9b54abc3), Nothing}}}}, Returns{StaticArraysCore.SizedVector{0, Float64, Vector{Float64}}}, Returns{Tuple{}}, Returns{Tuple{}}, Returns{Tuple{}}}}, 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ForwardDiff.Tag{DiffEqBase.OrdinaryDiffEqTag, Float64}}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}, Nothing, Nothing, Nothing}

"""Extract values from ensemble simulations by a symbol"""
extract(sim, k) = map(s -> s[k], sim)
"""Calculate Root Mean Square Error (RMSE)"""
rmse(fit) = sqrt(sum(abs2, fit.resid) / length(fit.resid))

Main.var"##230".rmse

Components

xopts = (xlabel="Ca (μM)", xscale=:log10, minorgrid=true, xlims=(ca[1], ca[end]))
plot(ca, extract(sim, sys.CaM0), lab="CaM0", ylabel="Conc. (μM)"; xopts...)
plot!(ca, extract(sim, sys.Ca2CaM_C), lab="Ca2CaM_C")
plot!(ca, extract(sim, sys.Ca2CaM_N), lab="Ca2CaM_N")
plot!(ca, extract(sim, sys.Ca4CaM), lab="Ca4CaM")
plot!(ca, extract(sim, sys.CaMK), lab="CaMK")
plot!(ca, extract(sim, sys.CaM0_CaMK), lab="CaM0_CaMK")
plot!(ca, extract(sim, sys.Ca2CaM_C_CaMK), lab="Ca2CaM_C_CaMK")
plot!(ca, extract(sim, sys.Ca2CaM_N_CaMK), lab="Ca2CaM_N_CaMK")
plot!(ca, extract(sim, sys.Ca4CaM_CaMK), lab="Ca4CaM_CaMK", legend=:topleft)

We excluded CaM0_CaMK (bound but lost all calcium) from the active CaMKII fraction

CaMKAct = 1 - (sys.CaMK + sys.CaM0_CaMK) / sys.CAMKII_T
println("Basal activity without ca is ", sol0[CaMKAct][end])
xdata = ca
ydata = extract(sim, CaMKAct)
plot(xdata, ydata, label=false, title="Active CaMKII", ylims=(0, 0.5); xopts...)

Basal activity without ca is 4.7188924878760474e-8

Least-square fitting of steady state CaMKII activity

@. model_camk(x, p) = p[1] * hil(x, p[2], p[4]) + p[3]
p0 = [0.4, 1μM, 0.0, 2.0]
lb = [0.0, 0.001μM, 0.0, 1.0]
fit = curve_fit(model_camk, xdata, ydata, p0; lower=lb, autodiff=:forwarddiff)

LsqFit.LsqFitResult{Vector{Float64}, Vector{Float64}, Matrix{Float64}, Vector{Float64}, Vector{LsqFit.LMState{LsqFit.LevenbergMarquardt}}}([0.42985863566610194, 0.9629481615705772, 0.005187221731402542, 2.2985816361147173], [0.0047668809984537305, 0.0047623068479205185, 0.0047576842635910265, 0.004753012593260384, 0.004748291508315916, 0.004743520506801628, 0.004738698858466773, 0.00473382638761024, 0.004728902413106349, 0.004723926338661389  …  0.0061555732461918145, 0.0061738716859804366, 0.00619221374343254, 0.006210307477971411, 0.0062281780714761314, 0.006245529763521995, 0.006262975993014508, 0.006280192457009559, 0.0062971637516352486, 0.0063137044095691874], [0.00034441283959658843 -0.00035327459840129975 1.0 -0.0005133752446202099; 0.00034904093675432716 -0.0003580201191894054 1.0 -0.0005194001002440866; … ; 0.9953494647544906 -0.004749645624211468 1.0 0.004645194993542614; 0.9954108665430759 -0.004687224395670468 1.0 0.004595553468490025], true, Iter     Function value   Gradient norm 
------   --------------   --------------
, Float64[])

Parameters

pestim = coef(fit)

4-element Vector{Float64}:
 0.42985863566610194
 0.9629481615705772
 0.005187221731402542
 2.2985816361147173

println("Basal activity: ", pestim[3])
println("Maximal activated activity: ", pestim[1])
println("Half-saturation Ca concentration: ", pestim[2], " μM")
println("Hill coefficient: ", pestim[4])
println("RMSE: ", rmse(fit))

Basal activity: 0.005187221731402542
Maximal activated activity: 0.42985863566610194
Half-saturation Ca concentration: 0.9629481615705772 μM
Hill coefficient: 2.2985816361147173
RMSE: 0.004825706994775833

Fit result and the original model

yestim = model_camk.(xdata, Ref(pestim))

p1 = plot(xdata, [ydata yestim], lab=["Full model" "Fitted"], line=[:dash :dot], title="CaMKII activity", legend=:topleft; xopts...)

The simplified CaMKII system

simpsys = CaMKIIModel.get_camkii_simp_sys(;Ca, ROS, simplify=true)

\[\begin{split} \begin{align} \frac{\mathrm{d} \mathtt{CaMKOX}\left( t \right)}{\mathrm{d}t} &= \mathtt{kdeph\_CaMK} \mathtt{CaMKAOX}\left( t \right) - \mathtt{krd\_CaMK} \mathtt{CaMKOX}\left( t \right) + \left( 1 - \mathtt{kfb\_CaMK} + \frac{ - \mathtt{kfa\_CaMK} pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)}{pow\left( \mathtt{kmCa\_CaMK}, \mathtt{nCa\_CaMK} \right) + pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)} \right) \mathtt{r\_CaMK} \mathtt{CaMKBOX}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKAOX}\left( t \right)}{\mathrm{d}t} &= \mathtt{kb\_CaMKP} \mathtt{CaMKPOX}\left( t \right) - \mathtt{kdeph\_CaMK} \mathtt{CaMKAOX}\left( t \right) - \mathtt{krd\_CaMK} \mathtt{CaMKAOX}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKA2}\left( t \right)}{\mathrm{d}t} &= \mathtt{k\_P1\_P2} \mathtt{CaMKA}\left( t \right) - \mathtt{k\_P2\_P1} \mathtt{CaMKA2}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKA}\left( t \right)}{\mathrm{d}t} &= - \mathtt{k\_P1\_P2} \mathtt{CaMKA}\left( t \right) + \mathtt{k\_P2\_P1} \mathtt{CaMKA2}\left( t \right) + \mathtt{kb\_CaMKP} \mathtt{CaMKP}\left( t \right) - \mathtt{kdeph\_CaMK} \mathtt{CaMKA}\left( t \right) + \mathtt{krd\_CaMK} \mathtt{CaMKAOX}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKPOX}\left( t \right)}{\mathrm{d}t} &= - \mathtt{kb\_CaMKP} \mathtt{CaMKPOX}\left( t \right) - \mathtt{kdeph\_CaMK} \mathtt{CaMKPOX}\left( t \right) - \mathtt{krd\_CaMK} \mathtt{CaMKPOX}\left( t \right) + \mathtt{ROS} \mathtt{kox\_CaMK} \mathtt{CaMKP}\left( t \right) + \mathtt{kphos\_CaMK} \mathtt{CaMKBOX}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKP}\left( t \right)}{\mathrm{d}t} &= - \mathtt{kb\_CaMKP} \mathtt{CaMKP}\left( t \right) - \mathtt{kdeph\_CaMK} \mathtt{CaMKP}\left( t \right) + \mathtt{krd\_CaMK} \mathtt{CaMKPOX}\left( t \right) - \mathtt{ROS} \mathtt{kox\_CaMK} \mathtt{CaMKP}\left( t \right) + \mathtt{kphos\_CaMK} \mathtt{CaMKAct}\left( t \right) \mathtt{CaMKB}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKBOX}\left( t \right)}{\mathrm{d}t} &= \mathtt{kdeph\_CaMK} \mathtt{CaMKPOX}\left( t \right) - \mathtt{krd\_CaMK} \mathtt{CaMKBOX}\left( t \right) + \mathtt{ROS} \mathtt{kox\_CaMK} \mathtt{CaMKB}\left( t \right) + \left( -1 + \mathtt{kfb\_CaMK} + \frac{\mathtt{kfa\_CaMK} pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)}{pow\left( \mathtt{kmCa\_CaMK}, \mathtt{nCa\_CaMK} \right) + pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)} \right) \mathtt{r\_CaMK} \mathtt{CaMKBOX}\left( t \right) - \mathtt{kphos\_CaMK} \mathtt{CaMKBOX}\left( t \right) \mathtt{CaMKAct}\left( t \right) \\ \frac{\mathrm{d} \mathtt{CaMKB}\left( t \right)}{\mathrm{d}t} &= \mathtt{kdeph\_CaMK} \mathtt{CaMKP}\left( t \right) + \mathtt{krd\_CaMK} \mathtt{CaMKBOX}\left( t \right) - \mathtt{ROS} \mathtt{kox\_CaMK} \mathtt{CaMKB}\left( t \right) + \left( \mathtt{kfb\_CaMK} + \frac{\mathtt{kfa\_CaMK} pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)}{pow\left( \mathtt{kmCa\_CaMK}, \mathtt{nCa\_CaMK} \right) + pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)} \right) \mathtt{r\_CaMK} \mathtt{CaMK}\left( t \right) + \left( -1 + \mathtt{kfb\_CaMK} + \frac{\mathtt{kfa\_CaMK} pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)}{pow\left( \mathtt{kmCa\_CaMK}, \mathtt{nCa\_CaMK} \right) + pow\left( \mathtt{Ca}, \mathtt{nCa\_CaMK} \right)} \right) \mathtt{r\_CaMK} \mathtt{CaMKB}\left( t \right) - \mathtt{kphos\_CaMK} \mathtt{CaMKAct}\left( t \right) \mathtt{CaMKB}\left( t \right) \end{align} \end{split}\]

Comparing original and simplified model

sys = build_neonatal_ecc_sys(simplify=true, reduce_iso=true)
sys_simp = build_neonatal_ecc_sys(simplify=true, reduce_iso=true, reduce_camk=true)

println("# of state variable in the original model: ", length(unknowns(sys)))
println("# of state variable in the simplified model: ", length(unknowns(sys_simp)))

# of state variable in the original model: 84
# of state variable in the simplified model: 75

tend = 500.0second
stimstart = 100.0second
stimend = 300.0second
alg = KenCarp47()
@unpack Istim = sys
callback = build_stim_callbacks(Istim, stimend; period=1second, starttime=stimstart)
prob = ODEProblem(sys, [], tend);
prob_simp = ODEProblem(sys_simp, [sys_simp.kphos_CaMK => 5Hz], tend);

@time sol = solve(prob, alg; callback)
@time sol_simp = solve(prob_simp, alg; callback)

plot(sol, idxs=sys.CaMKAct, label="Full model", title="Active CaMKII")
plot!(sol_simp, idxs=sys_simp.CaMKAct, label="Simplified model", xlabel="Time (ms)")

 10.058189 seconds (15.62 M allocations: 750.066 MiB, 1.26% gc time, 84.95% compilation time)
  7.774051 seconds (12.68 M allocations: 607.943 MiB, 1.91% gc time, 82.18% compilation time)

@unpack CaMKB, CaMKP, CaMKA, CaMKA2, CaMK = sys_simp
plot(sol_simp, idxs=[CaMKB, CaMKP, CaMKA, CaMKA2, CaMK], legend=:right)

plot(sol, idxs=sys.CaMKAct, label="Full model", title="Active CaMKII")
plot!(sol_simp, idxs=sys_simp.CaMKAct, label="Simplified model", xlabel="Time (ms)", tspan=(295second, 300.0second))

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This notebook was generated using Literate.jl.