Edit on GitHub

Spatial Signaling Model from Sanft and Othmer (2015)

using Catalyst, JumpProcesses, BenchmarkTools, Plots, Random

Model description and setup

Here we implement the model from [1] (8 species and 12 reactions) for different mesh sizes, and benchmark the performance of JumpProcesses.jl's spatial stochastic simulation alorithms (SSAs). Below, the value N will denote the number of subvolumes along one dimension of a cubic grid, representing the reaction volume. In [1] this value ranges from 20 to 60.

We first define some helper functions to convert concentration units into number units, as needed for spatial SSAs.

invmicromolar_to_cubicmicrometer(invconcen) = invconcen / (6.02214076e2)
micromolar_to_invcubicmicrometer(concen) = (6.02214076e2) * concen
micromolar_to_invcubicmicrometer (generic function with 1 method)

Next we create a well-mixed model with the desired chemistry

rn = @reaction_network begin
    @parameters k₁ ka kd k₄
    k₁, EA --> EA + A
    k₁, EB --> EB + B
    (ka,kd), EA + B <--> EAB
    (ka,kd), EAB + B <--> EAB₂
    (ka,kd), EB + A <--> EBA
    (ka,kd), EBA + A <--> EBA₂
    k₄, A --> ∅
    k₄, B --> ∅
end
Model ##ReactionSystem#232:
Unknowns (8): see unknowns(##ReactionSystem#232)
  EA(t)
  A(t)
  EB(t)
  B(t)
  ⋮
Parameters (4): see parameters(##ReactionSystem#232)
  k₁
  ka
  kd
  k₄

Let's next make a function to calculate the spatial transport rates, mesh/graph that will represent our domain, and initial condition. We use a cubic lattice of size N by N by N with reflecting boundary conditions

# domain_len is the physical length of each side of the cubic domain
# units should be in μm (6.0 or 12.0 in Sanft)
# D is the diffusivity in units of (μm)^2 s⁻¹
function transport_model(rn, N; domain_len = 6.0, D = 1.0, rng = Random.default_rng())
    # topology
    h = domain_len / N
    dims = (N, N, N)
    num_nodes = prod(dims)

    # Cartesian grid with reflecting BC at boundaries
    grid = CartesianGrid(dims)

    # Cartesian grid hopping rate to neighbors
    hopping_rate = D / h^2

    # this indicates we have a uniform rate of D/h^2 along each edge at each site
    hopping_constants = hopping_rate * ones(numspecies(rn))

    # figure out the indices of species EA and EB
    @unpack EA, EB = rn
    EAidx = findfirst(isequal(EA), species(rn))
    EBidx = findfirst(isequal(EB), species(rn))

    # spatial initial condition
    # initial concentration of 12.3 nM = 12.3 * 1e-3 μM
    num_molecules = trunc(Int, micromolar_to_invcubicmicrometer(12.3*1e-3) * (domain_len^3))
    u0 = zeros(Int, 8, num_nodes)
    rand_EA = rand(rng, 1:num_nodes, num_molecules)
    rand_EB = rand(rng, 1:num_nodes, num_molecules)
    for i in 1:num_molecules
        u0[EAidx, rand_EA[i]] += 1
        u0[EBidx, rand_EB[i]] += 1
    end

    grid, hopping_constants, h, u0
end
transport_model (generic function with 1 method)

Finally, let's make a function to setup the well-mixed model from the reaction model in a cube of side length h:

function wellmixed_model(rn, u0, end_time, h)
    kaval = invmicromolar_to_cubicmicrometer(46.2) / h^3
    setdefaults!(rn, [:k₁ => 150, :ka => kaval, :kd => 3.82, :k₄ => 6.0])

    # well-mixed initial condition corresponding to the spatial initial condition
    u0wm = sum(u0, dims = 2)
    dprobwm = DiscreteProblem(rn, u0wm, (0.0, end_time))
    jprobwm = JumpProblem(rn, dprobwm, Direct(), save_positions = (false,false))
    majumps = jprobwm.massaction_jump
    majumps, dprobwm, jprobwm, u0wm
end
wellmixed_model (generic function with 1 method)

Model Solution

Let's look at one example to check our model seems reasonable. We'll plot the total number of molecules in the system to verify we get around 28,000 molecules, as reported in Sanft [1], when using a domain length of 6 μm.

end_time = 3.0
grid, hopping_constants, h, u0 = transport_model(rn, 60)
majumps, dprobwm, jprobwm, u0wm = wellmixed_model(rn, u0, end_time, 6.0)
sol = solve(jprobwm, SSAStepper(); saveat = end_time/200)
Ntot = [sum(u) for u in sol.u]
plt = plot(sol.t, Ntot, label="Well-mixed", ylabel="Total Number of Molecules",
                        xlabel="time")

# spatial model
majumps, dprobwm, jprobwm, u0wm = wellmixed_model(rn, u0, end_time, h)
dprob = DiscreteProblem(u0, (0.0, end_time), copy(dprobwm.p))
jprob = JumpProblem(dprob, DirectCRDirect(), majumps; hopping_constants,
                    spatial_system = grid, save_positions = (false, false))
spatial_sol = solve(jprob, SSAStepper(); saveat = end_time/200)
Ntot = [sum(vec(u)) for u in spatial_sol.u]
plot!(plt, spatial_sol.t, Ntot, label="Spatial",
      title="Steady-state number of molecules is $(Ntot[end])")

Benchmarking performance of the methods

We can now run the solvers and record the performance with BenchmarkTools. Let's first create a DiscreteCallback to terminate simulations once we reach 10^8 events:

@Base.kwdef mutable struct EventCallback
    n::Int = 0
end

function (ecb::EventCallback)(u, t, integ)
    ecb.n += 1
    ecb.n == 10^8
end

function (ecb::EventCallback)(integ)
    # save the final state
    terminate!(integ)
    nothing
end

We next create a function to run and return our benchmarking results.

function benchmark_and_save!(bench_dict, end_times, Nv, algs, domain_len)
    @assert length(end_times) == length(Nv)

    # callback for terminating simulations
    ecb = EventCallback()
    cb = DiscreteCallback(ecb, ecb)

    for (end_time, N) in zip(end_times, Nv)
        names = ["$s"[1:end-2] for s in algs]

        grid, hopping_constants, h, u0 = transport_model(rn, N; domain_len)

        # we create a well-mixed model within a domain of the size of *one* voxel, h
        majumps, dprobwm, jprobwm, u0wm = wellmixed_model(rn, u0, end_time, h)

        # the spatial problem
        dprob = DiscreteProblem(u0, (0.0, end_time), copy(dprobwm.p))

        @show N

        # benchmarking and saving
        benchmarks = Vector{BenchmarkTools.Trial}(undef, length(algs))

        # callback for terminating simulations

        for (i, alg) in enumerate(algs)
            name = names[i]
            println("benchmarking $name")
            jp = JumpProblem(dprob, alg, majumps, hopping_constants=hopping_constants,
                             spatial_system = grid, save_positions=(false,false))
            b = @benchmarkable solve($jp, SSAStepper(); saveat = $(dprob.tspan[2]), callback) setup = (callback = deepcopy($cb)) samples = 10 seconds = 3600
            bench_dict[name, N] = run(b)
        end
    end
end
benchmark_and_save! (generic function with 1 method)

Finally, let's make a function to plot the benchmarking data.

function fetch_and_plot(bench_dict, domain_len)
    names = unique([key[1] for key in keys(bench_dict)])
    Nv = sort(unique([key[2] for key in keys(bench_dict)]))

    plt1 = plot()
    plt2 = plot()

    medtimes = [Float64[] for i in 1:length(names)]
    for (i,name) in enumerate(names)
        for N in Nv
            try
                push!(medtimes[i], median(bench_dict[name, N]).time/1e9)
            catch
                break
            end
        end
        len = length(medtimes[i])
        plot!(plt1, Nv[1:len], medtimes[i], marker = :hex, label = name, lw = 2)
        plot!(plt2, (Nv.^3)[1:len], medtimes[i], marker = :hex, label = name, lw = 2)
    end

    plot!(plt1, xlabel = "number of sites per edge", ylabel = "median time in seconds",
                xticks = Nv, legend = :bottomright)
    plot!(plt2, xlabel = "total number of sites", ylabel = "median time in seconds",
                xticks = (Nv.^3, string.(Nv.^3)), legend = :bottomright)
    plot(plt1, plt2; size = (1200,800), legendtitle = "SSAs",
                     plot_title="3D RDME, domain length = $domain_len", left_margin=5Plots.mm)
end
fetch_and_plot (generic function with 1 method)

We are now ready to run the benchmarks and plot the results. We start with a domain length of 12 μm, analogous to Fig. 6 in [1]:

bench_dict = Dict{Tuple{String, Int}, BenchmarkTools.Trial}()
algs = [NSM(), DirectCRDirect()]
Nv = [20, 30, 40, 50, 60, 90, 120, 240, 360]
end_times = 20000.0 * ones(length(Nv))
domain_len = 12.0
benchmark_and_save!(bench_dict, end_times, Nv, algs, domain_len)
N = 20
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 30
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 40
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 50
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 60
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 90
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 120
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 240
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 360
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
plt=fetch_and_plot(bench_dict, domain_len)

We next consider a domain of length 6 μm, analogous to Fig. 7 in [1].

bench_dict = Dict{Tuple{String, Int}, BenchmarkTools.Trial}()
domain_len = 6.0
benchmark_and_save!(bench_dict, end_times, Nv, algs, domain_len)
N = 20
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 30
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 40
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 50
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 60
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 90
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 120
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 240
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
N = 360
benchmarking JumpProcesses.NSM
benchmarking JumpProcesses.DirectCRDirect
plt=fetch_and_plot(bench_dict, domain_len)

References

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/Jumps","Spatial_Signaling_Sanft.jmd")

Computer Information:

Julia Version 1.10.7
Commit 4976d05258e (2024-11-26 15:57 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Jumps/Project.toml`
  [6e4b80f9] BenchmarkTools v1.6.0
  [479239e8] Catalyst v14.4.1
  [8f4d0f93] Conda v1.10.2
  [a93c6f00] DataFrames v1.7.0
  [0c46a032] DifferentialEquations v7.15.0
  [31c24e10] Distributions v0.25.117
  [86223c79] Graphs v1.12.0
  [faf0f6d7] JumpProblemLibrary v1.0.0
  [ccbc3e58] JumpProcesses v9.14.1
  [961ee093] ModelingToolkit v9.61.0
  [1dea7af3] OrdinaryDiffEq v6.90.1
⌅ [86206cdf] PiecewiseDeterministicMarkovProcesses v0.0.8
  [91a5bcdd] Plots v1.40.9
  [438e738f] PyCall v1.96.4
  [b4db0fb7] ReactionNetworkImporters v0.15.1
  [31c91b34] SciMLBenchmarks v0.1.3
  [860ef19b] StableRNGs v1.0.2
  [f3b207a7] StatsPlots v0.15.7
  [c3572dad] Sundials v4.26.1
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random
  [10745b16] Statistics v1.10.0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/Jumps/Manifest.toml`
  [47edcb42] ADTypes v1.12.0
  [621f4979] AbstractFFTs v1.5.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.41
  [79e6a3ab] Adapt v4.1.1
  [66dad0bd] AliasTables v1.1.3
  [a95523ee] AlmostBlockDiagonals v0.1.10
  [ec485272] ArnoldiMethod v0.4.0
  [7d9fca2a] Arpack v0.5.4
  [4fba245c] ArrayInterface v7.18.0
  [4c555306] ArrayLayouts v1.11.0
  [13072b0f] AxisAlgorithms v1.1.0
  [aae01518] BandedMatrices v1.9.0
  [6e4b80f9] BenchmarkTools v1.6.0
  [e2ed5e7c] Bijections v0.1.9
  [d1d4a3ce] BitFlags v0.1.9
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.3.0
  [764a87c0] BoundaryValueDiffEq v5.14.0
  [7227322d] BoundaryValueDiffEqAscher v1.3.0
  [56b672f2] BoundaryValueDiffEqCore v1.6.0
  [85d9eb09] BoundaryValueDiffEqFIRK v1.4.0
  [1a22d4ce] BoundaryValueDiffEqMIRK v1.4.0
  [9255f1d6] BoundaryValueDiffEqMIRKN v1.3.0
  [ed55bfe0] BoundaryValueDiffEqShooting v1.4.0
  [70df07ce] BracketingNonlinearSolve v1.1.0
  [fa961155] CEnum v0.5.0
  [2a0fbf3d] CPUSummary v0.2.6
  [00ebfdb7] CSTParser v3.4.3
  [479239e8] Catalyst v14.4.1
  [d360d2e6] ChainRulesCore v1.25.1
  [fb6a15b2] CloseOpenIntervals v0.1.13
  [aaaa29a8] Clustering v0.15.8
  [944b1d66] CodecZlib v0.7.6
  [35d6a980] ColorSchemes v3.28.0
  [3da002f7] ColorTypes v0.12.0
  [c3611d14] ColorVectorSpace v0.11.0
  [5ae59095] Colors v0.13.0
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.15
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.16.0
  [b152e2b5] CompositeTypes v0.1.4
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [f0e56b4a] ConcurrentUtilities v2.4.3
  [8f4d0f93] Conda v1.10.2
  [187b0558] ConstructionBase v1.5.8
  [d38c429a] Contour v0.6.3
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.16.0
  [a93c6f00] DataFrames v1.7.0
  [864edb3b] DataStructures v0.18.20
  [e2d170a0] DataValueInterfaces v1.0.0
  [bcd4f6db] DelayDiffEq v5.52.0
  [8bb1440f] DelimitedFiles v1.9.1
  [2b5f629d] DiffEqBase v6.161.0
  [459566f4] DiffEqCallbacks v4.2.2
  [77a26b50] DiffEqNoiseProcess v5.24.1
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [0c46a032] DifferentialEquations v7.15.0
  [a0c0ee7d] DifferentiationInterface v0.6.31
  [8d63f2c5] DispatchDoctor v0.4.19
  [b4f34e82] Distances v0.10.12
  [31c24e10] Distributions v0.25.117
  [ffbed154] DocStringExtensions v0.9.3
  [5b8099bc] DomainSets v0.7.15
  [7c1d4256] DynamicPolynomials v0.6.1
  [06fc5a27] DynamicQuantities v1.4.0
  [4e289a0a] EnumX v1.0.4
  [f151be2c] EnzymeCore v0.8.8
  [460bff9d] ExceptionUnwrapping v0.1.11
  [d4d017d3] ExponentialUtilities v1.27.0
  [e2ba6199] ExprTools v0.1.10
⌅ [6b7a57c9] Expronicon v0.8.5
  [c87230d0] FFMPEG v0.4.2
  [7a1cc6ca] FFTW v1.8.0
  [9d29842c] FastAlmostBandedMatrices v0.1.4
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [442a2c76] FastGaussQuadrature v1.0.2
  [29a986be] FastLapackInterface v2.0.4
  [a4df4552] FastPower v1.1.1
  [1a297f60] FillArrays v1.13.0
  [64ca27bc] FindFirstFunctions v1.4.1
  [6a86dc24] FiniteDiff v2.26.2
  [53c48c17] FixedPointNumbers v0.8.5
  [1fa38f19] Format v1.3.7
  [f6369f11] ForwardDiff v0.10.38
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.5.2
  [46192b85] GPUArraysCore v0.2.0
  [28b8d3ca] GR v0.73.12
  [c145ed77] GenericSchur v0.5.4
  [d7ba0133] Git v1.3.1
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.12.0
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.10.15
  [eafb193a] Highlights v0.5.3
  [3e5b6fbb] HostCPUFeatures v0.1.17
  [34004b35] HypergeometricFunctions v0.3.25
  [7073ff75] IJulia v1.26.0
  [615f187c] IfElse v0.1.1
  [d25df0c9] Inflate v0.1.5
  [842dd82b] InlineStrings v1.4.2
  [18e54dd8] IntegerMathUtils v0.1.2
  [a98d9a8b] Interpolations v0.15.1
  [8197267c] IntervalSets v0.7.10
  [3587e190] InverseFunctions v0.1.17
  [41ab1584] InvertedIndices v1.3.1
  [92d709cd] IrrationalConstants v0.2.2
  [82899510] IteratorInterfaceExtensions v1.0.0
  [1019f520] JLFzf v0.1.9
  [692b3bcd] JLLWrappers v1.7.0
  [682c06a0] JSON v0.21.4
  [98e50ef6] JuliaFormatter v1.0.62
  [faf0f6d7] JumpProblemLibrary v1.0.0
  [ccbc3e58] JumpProcesses v9.14.1
  [ef3ab10e] KLU v0.6.0
  [5ab0869b] KernelDensity v0.6.9
  [ba0b0d4f] Krylov v0.9.9
  [7f56f5a3] LSODA v0.7.5
  [b964fa9f] LaTeXStrings v1.4.0
  [23fbe1c1] Latexify v0.16.5
  [10f19ff3] LayoutPointers v0.1.17
  [5078a376] LazyArrays v2.3.2
  [2d8b4e74] LevyArea v1.0.0
  [87fe0de2] LineSearch v0.1.4
  [d3d80556] LineSearches v7.3.0
  [7ed4a6bd] LinearSolve v2.38.0
  [2ab3a3ac] LogExpFunctions v0.3.29
  [e6f89c97] LoggingExtras v1.1.0
  [bdcacae8] LoopVectorization v0.12.171
  [d8e11817] MLStyle v0.4.17
  [1914dd2f] MacroTools v0.5.15
  [d125e4d3] ManualMemory v0.1.8
  [a3b82374] MatrixFactorizations v3.0.1
  [bb5d69b7] MaybeInplace v0.1.4
  [739be429] MbedTLS v1.1.9
  [442fdcdd] Measures v0.3.2
  [e1d29d7a] Missings v1.2.0
  [961ee093] ModelingToolkit v9.61.0
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.7
  [6f286f6a] MultivariateStats v0.10.3
  [ffc61752] Mustache v1.0.20
  [d8a4904e] MutableArithmetics v1.6.2
  [d41bc354] NLSolversBase v7.8.3
  [2774e3e8] NLsolve v4.5.1
  [77ba4419] NaNMath v1.0.3
  [b8a86587] NearestNeighbors v0.4.21
  [8913a72c] NonlinearSolve v4.3.0
  [be0214bd] NonlinearSolveBase v1.4.0
  [5959db7a] NonlinearSolveFirstOrder v1.2.0
  [9a2c21bd] NonlinearSolveQuasiNewton v1.1.0
  [26075421] NonlinearSolveSpectralMethods v1.1.0
  [510215fc] Observables v0.5.5
  [6fe1bfb0] OffsetArrays v1.15.0
  [4d8831e6] OpenSSL v1.4.3
  [429524aa] Optim v1.10.0
  [bac558e1] OrderedCollections v1.7.0
  [1dea7af3] OrdinaryDiffEq v6.90.1
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.1.0
  [6ad6398a] OrdinaryDiffEqBDF v1.2.0
  [bbf590c4] OrdinaryDiffEqCore v1.15.0
  [50262376] OrdinaryDiffEqDefault v1.2.0
  [4302a76b] OrdinaryDiffEqDifferentiation v1.3.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.2.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.3.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.6.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.2.0
  [521117fe] OrdinaryDiffEqLinear v1.1.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.2.1
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.3.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.2.0
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.4.0
  [2d112036] OrdinaryDiffEqSDIRK v1.2.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.2.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.2.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.1.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [79d7bb75] OrdinaryDiffEqVerner v1.1.1
  [90014a1f] PDMats v0.11.32
  [65ce6f38] PackageExtensionCompat v1.0.2
  [d96e819e] Parameters v0.12.3
  [69de0a69] Parsers v2.8.1
⌅ [86206cdf] PiecewiseDeterministicMarkovProcesses v0.0.8
  [b98c9c47] Pipe v1.3.0
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.3
  [91a5bcdd] Plots v1.40.9
  [e409e4f3] PoissonRandom v0.4.4
  [f517fe37] Polyester v0.7.16
  [1d0040c9] PolyesterWeave v0.2.2
  [2dfb63ee] PooledArrays v1.4.3
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.24
  [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
  [08abe8d2] PrettyTables v2.4.0
  [27ebfcd6] Primes v0.5.6
  [43287f4e] PtrArrays v1.3.0
  [438e738f] PyCall v1.96.4
  [1fd47b50] QuadGK v2.11.1
  [74087812] Random123 v1.7.0
  [e6cf234a] RandomNumbers v1.6.0
  [c84ed2f1] Ratios v0.4.5
  [b4db0fb7] ReactionNetworkImporters v0.15.1
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v3.27.4
  [f2c3362d] RecursiveFactorization v0.2.23
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.0
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.13
  [9dfe8606] SCCNonlinearSolve v1.0.0
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.43
  [0bca4576] SciMLBase v2.71.4
  [31c91b34] SciMLBenchmarks v0.1.3
  [19f34311] SciMLJacobianOperators v0.1.1
  [c0aeaf25] SciMLOperators v0.3.12
  [53ae85a6] SciMLStructures v1.6.1
  [6c6a2e73] Scratch v1.2.1
  [91c51154] SentinelArrays v1.4.8
  [efcf1570] Setfield v1.1.1
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.1.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [47a9eef4] SparseDiffTools v2.23.1
  [0a514795] SparseMatrixColorings v0.4.11
  [e56a9233] Sparspak v0.3.9
  [276daf66] SpecialFunctions v2.5.0
  [860ef19b] StableRNGs v1.0.2
  [aedffcd0] Static v1.1.1
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.10
  [1e83bf80] StaticArraysCore v1.4.3
  [82ae8749] StatsAPI v1.7.0
⌅ [2913bbd2] StatsBase v0.33.21
  [4c63d2b9] StatsFuns v1.3.2
  [f3b207a7] StatsPlots v0.15.7
  [9672c7b4] SteadyStateDiffEq v2.4.1
  [789caeaf] StochasticDiffEq v6.72.1
  [7792a7ef] StrideArraysCore v0.5.7
  [69024149] StringEncodings v0.3.7
  [892a3eda] StringManipulation v0.4.0
  [c3572dad] Sundials v4.26.1
  [2efcf032] SymbolicIndexingInterface v0.3.37
  [19f23fe9] SymbolicLimits v0.2.2
  [d1185830] SymbolicUtils v3.11.0
  [0c5d862f] Symbolics v6.24.0
  [ab02a1b2] TableOperations v1.2.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.0
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.2
  [a759f4b9] TimerOutputs v0.5.26
  [0796e94c] Tokenize v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [d5829a12] TriangularSolve v0.2.1
  [410a4b4d] Tricks v0.1.10
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.22.0
  [45397f5d] UnitfulLatexify v1.6.4
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.71
  [81def892] VersionParsing v1.3.0
  [19fa3120] VertexSafeGraphs v0.2.0
  [897b6980] WeakValueDicts v0.1.0
  [44d3d7a6] Weave v0.10.12
  [cc8bc4a8] Widgets v0.6.7
  [efce3f68] WoodburyMatrices v1.0.0
  [ddb6d928] YAML v0.4.12
  [c2297ded] ZMQ v1.4.0
⌅ [68821587] Arpack_jll v3.5.1+1
  [6e34b625] Bzip2_jll v1.0.8+4
  [83423d85] Cairo_jll v1.18.2+1
  [ee1fde0b] Dbus_jll v1.14.10+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.4+3
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [f5851436] FFTW_jll v3.3.10+3
  [a3f928ae] Fontconfig_jll v2.15.0+0
  [d7e528f0] FreeType2_jll v2.13.3+1
  [559328eb] FriBidi_jll v1.0.16+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.12+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.47.1+0
  [7746bdde] Glib_jll v2.82.4+0
  [3b182d85] Graphite2_jll v1.3.14+1
  [2e76f6c2] HarfBuzz_jll v8.5.0+0
⌅ [1d5cc7b8] IntelOpenMP_jll v2024.2.1+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.2+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.7+0
  [aae0fff6] LSODA_jll v0.1.2+0
  [dd4b983a] LZO_jll v2.10.3+0
⌅ [e9f186c6] Libffi_jll v3.2.2+2
  [d4300ac3] Libgcrypt_jll v1.11.0+0
  [7e76a0d4] Libglvnd_jll v1.7.0+0
  [7add5ba3] Libgpg_error_jll v1.51.1+0
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.40.3+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.40.3+0
⌅ [856f044c] MKL_jll v2024.2.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v3.0.15+3
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.3.3+0
  [36c8627f] Pango_jll v1.55.5+0
⌅ [30392449] Pixman_jll v0.43.4+0
⌅ [c0090381] Qt6Base_jll v6.7.1+1
  [629bc702] Qt6Declarative_jll v6.7.1+2
  [ce943373] Qt6ShaderTools_jll v6.7.1+1
  [e99dba38] Qt6Wayland_jll v6.7.1+1
  [f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+2
  [2381bf8a] Wayland_protocols_jll v1.36.0+0
  [02c8fc9c] XML2_jll v2.13.5+0
  [aed1982a] XSLT_jll v1.1.42+0
  [ffd25f8a] XZ_jll v5.6.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.1+0
  [c834827a] Xorg_libSM_jll v1.2.4+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+3
  [0c0b7dd1] Xorg_libXau_jll v1.0.12+0
  [935fb764] Xorg_libXcursor_jll v1.2.3+0
  [a3789734] Xorg_libXdmcp_jll v1.1.5+0
  [1082639a] Xorg_libXext_jll v1.3.6+3
  [d091e8ba] Xorg_libXfixes_jll v6.0.0+0
  [a51aa0fd] Xorg_libXi_jll v1.8.2+0
  [d1454406] Xorg_libXinerama_jll v1.1.5+0
  [ec84b674] Xorg_libXrandr_jll v1.5.4+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.11+1
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.2+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+1
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [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 v1.4.6+1
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.1+0
  [8f1865be] ZeroMQ_jll v4.3.5+3
  [3161d3a3] Zstd_jll v1.5.7+0
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.56.3+0
  [1a1c6b14] gperf_jll v3.1.1+1
  [a4ae2306] libaom_jll v3.11.0+0
  [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.45+1
  [a9144af2] libsodium_jll v1.0.20+3
  [f27f6e37] libvorbis_jll v1.3.7+2
  [009596ad] mtdev_jll v1.1.6+0
  [1317d2d5] oneTBB_jll v2021.12.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+2
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
  • 1Sanft, Kevin R and Othmer, Hans G. Constant-complexity stochastic simulation algorithm with optimal binning. J. Chem. Phys., 143(7), 11 pp. (2015).