Quorum Sensing Work-Precision Diagrams

Quorum Sensing

Here we test a model of quorum sensing of Pseudomonas putida IsoF in continuous cultures with constant delay which was published by K. Buddrus-Schiemann et al. in "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, 2014.

using DelayDiffEq, DiffEqDevTools, DDEProblemLibrary, Plots
import DDEProblemLibrary: prob_dde_qs
gr()

sol = solve(prob_dde_qs, MethodOfSteps(Vern9(); fpsolve = NLFunctional(; max_iter = 1000)); reltol=1e-14, abstol=1e-14)
plot(sol)

Particularly, we are interested in the third, low-level component of the system:

sol = solve(prob_dde_qs, MethodOfSteps(Vern9(); fpsolve = NLFunctional(; max_iter = 1000)); reltol=1e-14, abstol=1e-14, save_idxs=3)
test_sol = TestSolution(sol)
plot(sol)

Qualitative comparisons

First we compare the quality of the solution's third component for different algorithms, using the default tolerances.

RK methods

sol = solve(prob_dde_qs, MethodOfSteps(BS3()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(Tsit5()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(RK4()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(DP5()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(DP8()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(OwrenZen3()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(OwrenZen4()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(OwrenZen5()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

Rosenbrock methods

sol = solve(prob_dde_qs, MethodOfSteps(Rosenbrock23()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(Rosenbrock32()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(Rodas4()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(Rodas5()); reltol=1e-4, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

Lazy interpolants

sol = solve(prob_dde_qs, MethodOfSteps(Vern7()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

sol = solve(prob_dde_qs, MethodOfSteps(Vern9()); reltol=1e-3, abstol=1e-6, save_idxs=3)
p = plot(sol);
scatter!(p,sol.t, sol.u)
p

Qualitative comparisons

Now we compare these methods quantitatively.

High tolerances

RK methods

We start with RK methods at high tolerances.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
          Dict(:alg=>MethodOfSteps(Tsit5())),
          Dict(:alg=>MethodOfSteps(RK4())),
          Dict(:alg=>MethodOfSteps(DP5())),
          Dict(:alg=>MethodOfSteps(OwrenZen3())),
          Dict(:alg=>MethodOfSteps(OwrenZen4())),
          Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

We also compare interpolation errors:

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
          Dict(:alg=>MethodOfSteps(Tsit5())),
          Dict(:alg=>MethodOfSteps(RK4())),
          Dict(:alg=>MethodOfSteps(DP5())),
          Dict(:alg=>MethodOfSteps(OwrenZen3())),
          Dict(:alg=>MethodOfSteps(OwrenZen4())),
          Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

And the maximal interpolation error:

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
          Dict(:alg=>MethodOfSteps(Tsit5())),
          Dict(:alg=>MethodOfSteps(RK4())),
          Dict(:alg=>MethodOfSteps(DP5())),
          Dict(:alg=>MethodOfSteps(OwrenZen3())),
          Dict(:alg=>MethodOfSteps(OwrenZen4())),
          Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L∞)
plot(wp)

Since the correct solution is in the range of 1e-7, we see that most solutions, even at the lower end of tested tolerances, always lead to relative maximal interpolation errors of at least 1e-1 (and usually worse). RK4 performs slightly better with relative maximal errors of at least 1e-2. This matches our qualitative analysis above.

Rosenbrock methods

We repeat these tests with Rosenbrock methods, and include RK4 as reference.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rosenbrock32())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(RK4()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rosenbrock32())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(RK4()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rosenbrock32())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(RK4()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L∞)
plot(wp)

Out of the tested Rosenbrock methods Rodas4 and Rosenbrock23 perform best at high tolerances.

Lazy interpolants

Finally we test the Verner methods with lazy interpolants, and include Rosenbrock23 as reference.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
          Dict(:alg=>MethodOfSteps(Vern7())),
          Dict(:alg=>MethodOfSteps(Vern8())),
          Dict(:alg=>MethodOfSteps(Vern9())),
          Dict(:alg=>MethodOfSteps(Rosenbrock23()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
          Dict(:alg=>MethodOfSteps(Vern7())),
          Dict(:alg=>MethodOfSteps(Vern8())),
          Dict(:alg=>MethodOfSteps(Vern9())),
          Dict(:alg=>MethodOfSteps(Rosenbrock23()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
          Dict(:alg=>MethodOfSteps(Vern7())),
          Dict(:alg=>MethodOfSteps(Vern8())),
          Dict(:alg=>MethodOfSteps(Vern9())),
          Dict(:alg=>MethodOfSteps(Rosenbrock23()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L∞)
plot(wp)

All in all, at high tolerances Rodas5 and Rosenbrock23 are the best methods for solving this stiff DDE.

Low tolerances

Rosenbrock methods

We repeat our tests of Rosenbrock methods Rosenbrock23 and Rodas5 at low tolerances:

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(Rodas5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(Rodas5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(Rosenbrock23())),
          Dict(:alg=>MethodOfSteps(Rodas4())),
          Dict(:alg=>MethodOfSteps(Rodas5()))]
wp = WorkPrecisionSet(prob_dde_qs,abstols,reltols,setups;
                      save_idxs=3,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L∞)
plot(wp)

Thus at low tolerances Rodas5 outperforms Rosenbrock23.

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/StiffDDE","QuorumSensing.jmd")

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffDDE/Project.toml`
  [f42792ee] DDEProblemLibrary v0.1.2
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  [f3b72e0c] DiffEqDevTools v2.32.0
  [91a5bcdd] Plots v1.34.0
  [31c91b34] SciMLBenchmarks v0.1.1

And the full manifest:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffDDE/Manifest.toml`
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  [e9f186c6] Libffi_jll v3.2.2+1
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  [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+1
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  [38a345b3] Libuuid_jll v2.36.0+0
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  [2381bf8a] Wayland_protocols_jll v1.25.0+0
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  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
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  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
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  [76f85450] LibGit2
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  [a4e569a6] Tar v1.10.0
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  [29816b5a] LibSSH2_jll v1.10.2+0
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  [14a3606d] MozillaCACerts_jll v2022.2.1
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  [bea87d4a] SuiteSparse_jll v5.10.1+0
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  [8e850b90] libblastrampoline_jll v5.1.1+0
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  [3f19e933] p7zip_jll v17.4.0+0