Chapter 3 - MMSB

Figure 3.03¶

Michaelis-Menten kinetics

using OrdinaryDiffEq
using Catalyst
using Plots

Enzyme kinetics full model

@time "Build system" rn303 = @reaction_network begin
    k1, S + E --> ES
    km1, ES --> S + E
    k2, ES --> P + E
end

Build system: 14.594859 seconds (28.76 M allocations: 1.583 GiB, 3.46% gc time, 99.51% compilation time: <1% of which was recompilation)

up303 = Dict(:S => 5.0, :ES => 0.0, :P => 0.0, :E=>1.0, :k1 => 30.0, :km1 => 1.0, :k2 => 10.0)
tend = 1.0
@time "Build problem" prob303 = ODEProblem(rn303, up303, (0.0, tend); remove_conserved=true)

Build problem: 56.263388 seconds (58.07 M allocations: 3.255 GiB, 2.16% gc time, 99.90% compilation time: 47% of which was recompilation)

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
Initialization status: FULLY_DETERMINED
Non-trivial mass matrix: false
timespan: (0.0, 1.0)
u0: 2-element Vector{Float64}:
 5.0
 1.0

@time "Solve problem" sol303 = solve(prob303, Tsit5())
plot(sol303, idxs=[:S, :ES, :P, :E], title="Fig. 3.03 (Full model)", xlabel="Time", ylabel="Concentration")

Solve problem: 6.300355 seconds (12.51 M allocations: 709.643 MiB, 1.30% gc time, 99.92% compilation time)

QSSA of the ES complex¶

@time "Build system" rn303mm = @reaction_network begin
    mm(S, k2 * ET, (km1 + k2) / k1), S => P
end

Build system: 0.240911 seconds (741.73 k allocations: 41.408 MiB, 99.54% compilation time: 79% of which was recompilation)

up303mm = Dict(:k1 => 30.0, :km1 => 1.0, :k2 => 10.0, :ET => 1.0, :S => 5.0, :P => 0.0)
tend = 1.0
@time "Build problem" prob303mm = ODEProblem(rn303mm, up303mm, (0.0, tend); remove_conserved=true)
@time "Solve problem" sol303mm = solve(prob303mm, Tsit5())
plot(sol303mm, idxs=[:S, :P], labels=["S (QSSA)" "P (QSSA)"], title="Fig. 3.03 (QSSA vs Full)", xlabel="Time", ylabel="Concentration")
plot!(sol303, idxs=[:S, :P], labels=["S (full)" "P (full)"], linestyle=:dash, legend=:right)

Build problem: 0.737332 seconds (691.77 k allocations: 42.212 MiB, 97.04% compilation time)
Solve problem: 0.832993 seconds (976.00 k allocations: 53.483 MiB, 99.56% compilation time)

Fig 3.13¶

Generalized mass action (GMA) vs. Michaelis-Menten rate laws

using Plots
plot([t -> 2t / (1+t), t -> t^0.4], 0, 4, labels=["MM" "GMA"], title="Fig. 3.13", xlabel="Substrate concentration (S / Km)", ylabel="Reaction rate (AU)")

This notebook was generated using Literate.jl.