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BioPreDyn-bench B6 (Drosophila Gap Gene Network) Parameter Estimation Benchmark

Parameter estimation of the Drosophila melanogaster gap gene network

This benchmark implements problem B6 from the BioPreDyn-bench suite (Villaverde et al. 2015), addressing SciMLBenchmarks.jl#555. B6 is the odd one out among the six BioPreDyn-bench problems: a spatial model of pattern formation during early Drosophila development (the classic "gene circuit" approach of Reinitz and Sharp, 1995), fit against real quantitative microscopy data rather than metabolomics/transcriptomics measurements. The embryo is modeled as a single row of dividing nuclei along the anterior-posterior (AP) axis; each nucleus contains the four gap genes – hunchback (hb), Krüppel (Kr), giant (gt) and knirps (kni) – which regulate each other and are themselves regulated by four external maternal/gap inputs – Bicoid (bcd), Caudal (cad), Tailless (tll) and Huckebein (hkb). Each gene product is synthesized (regulated by a sigmoid function of the other genes' and inputs' concentrations, with a fixed 6-minute transcription/translation delay), diffuses to neighboring nuclei, and decays linearly – 37 unknown parameters in total (4 promoter strengths, a 4x4 gene-gene interconnectivity matrix, 13 free external-input weights, and 4 protein half-lives). The data are 9 time points over the last ~68 minutes of the syncytial blastoderm stage (cleavage cycles 13-14A, with one nuclear division in between, taking the modeled AP domain from 27 to 53 nuclei), digitized mRNA staining intensities for the four gap genes, fit with a weighted least-squares scheme whose per-point weight is lower wherever the gene is more strongly expressed (w = 1 - 0.9y for normalized expression y, so peaks are the least, and background the most, tightly constrained).

Unlike B1-B5, B6 ships no plain-MATLAB reference implementation – only compiled mex binaries and the ~18,000 lines of C that back them (the "flyex"/gene circuit solver originally due to John Reinitz's lab, later extended by Damjan Cicin-Sain, Anton Crombach and Yogi Jaeger – who are also co-authors of the BioPreDyn-bench paper itself). The model below was transcribed directly from that C source (dynamics_in_C/fly/zygotic.c, maternal.c, integrate.c) and its accompanying data file (AMIGO/dm_hkgn53_wls_5_003), available as supplementary material to the paper (Additional file 2, directory BioPreDynBenchFiles/B6). Reproducing an 18,000-line, three-decade-old C codebase (with its own custom lineage-tree bookkeeping, adaptive-step delay solver and scoring engine) byte-for-byte is out of scope; instead, this page transcribes the equations and data faithfully and documents the handful of places where a clearly-stated, principled simplification was necessary because the released files don't fully specify the original solver's internal bookkeeping (see "Model" and "Nuclear division" below). As a result, absolute cost values here are not directly comparable to the reference solver's, even at the reference's own parameter vectors – see "Objective function" for a full discussion.

using DelayDiffEq, OrdinaryDiffEq, Optimization, ForwardDiff
using OptimizationBBO, OptimizationNLopt, Plots, BenchmarkTools, DataFrames
using ParallelParticleSwarms
using LinearAlgebra, Random
gr(fmt = :png)
Plots.GRBackend()

Model

Each nucleus i (indexed by increasing AP position) holds a 4-vector v_i of gap gene concentrations (hb, Kr, gt, kni). Writing V for the 4 x nnuc matrix of all nuclei's concentrations, T for the 4x4 gene-gene interconnectivity matrix (row = target gene, column = source gene), E for the 4x4 external-input weight matrix (columns bcd, cad, tll, hkb), M(t) for the (given, not fitted) external-input concentrations at time t, R for the 4 promoter strengths, h for the 4 (fixed) promoter thresholds, and lambda for the 4 decay rates (= log(2) / half_life), the model is, per nucleus and gene:

dV/dt = rule(t) .* R .* g.(T * V(t - tau) .+ E * M(t - tau) .+ h) .- lambda .* V .+ diffusion(V)

where g(u) = 0.5 * (1 + u / sqrt(1 + u^2)) is the model's sigmoid regulation-expression function (Reinitz & Sharp 1995's default choice, Sqrt), tau = 6 minutes is a fixed transcription/translation delay applied uniformly to every regulatory input (both T and E terms – making this a genuine delay differential equation), rule(t) is 0 during the ~5-minute "mitosis" window immediately before a nuclear division (no new gene product synthesis while a nucleus is dividing, though decay and diffusion continue) and 1 otherwise ("interphase"), and diffusion(V) is a standard discrete Laplacian along the row of nuclei with a no-flux (Neumann) boundary at both ends of the AP domain, scaled by a per-gene diffusion parameter that itself depends on the cell-division count still ahead (a nucleus's neighbors are physically closer after each division, so the same physical diffusivity corresponds to a larger per-nucleus-index rate afterwards): D = D_raw / 4 before the single division in this data set, D = D_raw after.

Several model quantities are fixed constants rather than estimated: the promoter thresholds h (-2.5 for all four genes), the diffusion parameters D_raw ([0.237, 0.30, 0.115, 0.30]), and the delay tau (6 minutes for all four genes) – all read directly from the reference data file, which marks them as not to be tweaked ($tweak section). A "maternal connection strength" parameter that multiplies a legacy, separate Bicoid pathway is fixed at 0 in the reference file (and not tweaked), which makes that pathway a no-op here; Bicoid instead enters through the E matrix like the other three external inputs, exactly as the reference data treats it (its external_gene_IDs are V, C, T, Q = Bicoid, Caudal, Tailless, Huckebein, all read from the same $ext_wt data table).


const b6_h_fixed = [-2.5, -2.5, -2.5, -2.5]
const b6_D_raw = [0.237, 0.3, 0.115, 0.3]
const b6_tau = 6.0
const b6_t_div = 21.1          # end of the single nuclear division (fixed developmental constant)
const b6_mitosis_dur = 5.1     # duration of the preceding no-synthesis "mitosis" window
const b6_mitosis_start = b6_t_div - b6_mitosis_dur

b6_g(u) = 0.5 * (1 + u / sqrt(1 + u^2))

# Resamples a spatial profile given at n evenly-spaced AP positions onto m evenly-spaced
# AP positions via linear interpolation of relative position. Used (1) to map each
# nucleus's post-division concentration onto its (up to) two daughter nuclei, since the
# released data does not include the original lineage-tree bookkeeping needed to reproduce
# the reference solver's exact parent/daughter assignment, and (2) to bring the data
# file's earlier, coarser-resolution history/external-input records onto the same AP grid
# as the (finer) nuclei set they seed.
function b6_resample_ap(v::AbstractVector, m::Int)
    n = length(v)
    n == m && return collect(v)
    out = zeros(eltype(v), m)
    for k in 1:m
        pos = (k - 1) / (m - 1) * (n - 1) + 1
        i0 = clamp(floor(Int, pos), 1, n - 1)
        frac = pos - i0
        out[k] = (1 - frac) * v[i0] + frac * v[i0 + 1]
    end
    return out
end

b6_resample_rows(M::AbstractMatrix, m::Int) = reduce(vcat, [b6_resample_ap(M[g, :], m)' for g in 1:size(M, 1)])

# Linear interpolation in time of a set of (time, matrix) pairs; clamped outside the range.
function b6_interp_ext(times::Vector{Float64}, mats, t)
    if t <= times[1]
        return mats[1]
    elseif t >= times[end]
        return mats[end]
    else
        i = clamp(searchsortedlast(times, t), 1, length(times) - 1)
        t0, t1 = times[i], times[i + 1]
        frac = (t - t0) / (t1 - t0)
        return (1 - frac) .* mats[i] .+ frac .* mats[i + 1]
    end
end

# Two-point linear interpolation used for the DDE history function between the data
# file's pre-simulation record (~6 minutes before the start of the modeled window) and
# the initial condition itself.
function b6_interp_hist(t0::Float64, v0, t1::Float64, v1, t)
    if t <= t0
        return v0
    elseif t >= t1
        return v1
    else
        frac = (t - t0) / (t1 - t0)
        return (1 - frac) .* v0 .+ frac .* v1
    end
end

# Builds a DDE right-hand-side closure for a fixed number of nuclei, whether production is
# active (interphase) or not (mitosis), this phase's external-input time series, and the
# (already-unpacked) model parameters.
function b6_make_rhs(nnuc::Int, interphase::Bool, ext_times::Vector{Float64}, ext_mats,
        R::AbstractVector, T::AbstractMatrix, E::AbstractMatrix, lambda::AbstractVector)
    D = b6_D_raw ./ (nnuc == 27 ? 4.0 : 1.0)
    h_fixed = b6_h_fixed
    function rhs!(du, u, hfun, p, t)
        V = reshape(u, 4, nnuc)
        dV = reshape(du, 4, nnuc)
        if interphase
            Vd = reshape(hfun(p, t - b6_tau), 4, nnuc)
            Vext = b6_interp_ext(ext_times, ext_mats, t - b6_tau)
            U = T * Vd .+ E * Vext .+ h_fixed
            dV .= R .* b6_g.(U)
        else
            dV .= 0.0
        end
        dV .-= lambda .* V
        for a in 1:4
            dV[a, 1] += D[a] * (V[a, 2] - V[a, 1])
            for i in 2:(nnuc - 1)
                dV[a, i] += D[a] * (V[a, i - 1] + V[a, i + 1] - 2 * V[a, i])
            end
            dV[a, nnuc] += D[a] * (V[a, nnuc - 1] - V[a, nnuc])
        end
        nothing
    end
    return rhs!
end

# Unpacks a 37-parameter vector into R (4 promoter strengths), T (4x4 interconnectivity),
# E (4x4 external-input weights, with the 3 structurally-zero entries the reference file
# marks as not estimated for the Huckebein column of genes Kr, gt and kni), and lambda (4
# decay rates, from the 4 protein half-lives).
function b6_unpack(p37::AbstractVector)
    R = p37[1:4]
    T = collect(reshape(p37[5:20], 4, 4)')
    Efree = p37[21:33]
    E = zeros(eltype(p37), 4, 4)
    E[1, :] = Efree[1:4]
    E[2, 1:3] = Efree[5:7]
    E[3, 1:3] = Efree[8:10]
    E[4, 1:3] = Efree[11:13]
    half_life = p37[34:37]
    lambda = log(2.0) ./ half_life
    return R, T, E, lambda
end
b6_unpack (generic function with 1 method)

Nuclear division

The reference data models cleavage cycles 13 and 14A (one division), starting from 27 nuclei (35-88% egg length) and ending with 53. Per the reference's own fixed developmental timing table (used for every BioPreDyn-bench-style problem with exactly one division, not specific to this data set), the division completes at t = 21.1 minutes, preceded by a 5.1-minute "mitosis" window (t = 16.0 to 21.1) during which gene product synthesis is switched off. The reference solver's internal lineage tree specifies, for each of the 27 parent nuclei, exactly which of the 53 daughter nuclei it produces (not always exactly two, since the domain's edges are handled specially) – but that lineage tree is internal to the compiled solver and isn't part of the released data files. We approximate it here by resampling the pre-division 27-nucleus concentration profile onto the post-division 53-nucleus AP grid with b6_resample_ap (linear interpolation of relative AP position), which preserves the profile's shape continuously across the division and is exact wherever the true lineage mapping is a simple duplication, but does not reproduce the reference solver's specific parent/daughter bookkeeping at the domain's edges.

Nominal parameters, bounds, and starting guess

The reference data file provides three parameter vectors: the values used to generate the file's $input section (used here as a starting guess, p_start), the values in its $eqparms section (already the result of a prior simulated-annealing fit reaching a score of about 94109 under the reference solver – used here as p_nom, purely for reference since, unlike B1-B5, there's no unfitted "true" nominal vector here), and pbest.mat, a further-refined vector the reference documentation describes as improving on p_nom ("[...] it is possible to obtain a parameter vector that gives a better fit than the one originally reported as initial guess"). Bounds for the 37 free parameters are the reference's own explicit ranges for promoter strengths ([10, 30]) and protein half-lives ([5, 20]); the reference file leaves the interconnectivity (T) and external-input (E) ranges unspecified (N/A), so we use generously wide but finite bounds of [-20, 20], comfortably enclosing all three reference vectors' entries (whose largest magnitude is |E| = 1.998).


# reference parameter vectors, transcribed from dm_hkgn53_wls_5_003 ($input,
# $eqparms) and pbest.mat
const b6_p_start = [12.48991375, 26.64956536, 24.21400803, 19.14696636, -0.02875641, 0.03773355, -0.08696411, 0.02085833, 0.07752513, -0.05127861, 0.08351538, -0.0369346, -0.02272315, -0.05630814, 0.03840626, 0.07150631, 0.02276078, -0.0235078, 0.01856495, -0.06116937, -0.14654266, 0.01651517, 0.06442419, -0.05467585, -0.12247616, -0.10550997, 0.10687539, 0.02896215, -0.07365022, -0.06193835, 0.02710513, -0.04491538, 0.10493181, 13.76469068, 7.27890037, 11.63317492, 12.03105457]
const b6_p_nom = [17.49954403, 27.47628367, 15.56609395, 21.16476541, 0.040902, -0.05576477, -0.08379836, -0.08604469, -0.01447607, -0.00943785, -0.61555982, -0.05471756, -0.1148079, -0.43430661, 0.02333142, -0.00611265, -0.24838744, -0.08077992, -0.22121907, 0.05847331, 0.16381828, 0.03170952, 0.20280478, -1.52341105, 0.14976405, 0.05370816, -0.20044612, 0.4880306, 0.02378553, -0.07307959, 0.32040893, 0.03959051, -0.08460995, 6.09268882, 5.06352441, 5.00000022, 5.11110166]
const b6_pbest = [29.95515, 26.15643, 23.9969, 19.52634, -0.08661, -0.2281, -0.05799, -0.20482, -0.06583, -0.00673, -0.34173, -0.04256, -0.05249, -0.48365, 0.00677, 0.00137, -0.10378, -0.03213, -0.2115, 0.03546, 0.62039, 0.01513, 0.42646, -1.9981, 0.23507, 0.03782, -0.03027, 0.30413, 0.02401, -0.05001, -0.02465, 0.03728, -0.12298, 11.34269, 5.43111, 5.00016, 6.28399]

const b6_p_lower = vcat(fill(10.0, 4), fill(-20.0, 16), fill(-20.0, 13), fill(5.0, 4))
const b6_p_upper = vcat(fill(30.0, 4), fill(20.0, 16), fill(20.0, 13), fill(20.0, 4))
37-element Vector{Float64}:
 30.0
 30.0
 30.0
 30.0
 20.0
 20.0
 20.0
 20.0
 20.0
 20.0
  ⋮
 20.0
 20.0
 20.0
 20.0
 20.0
 20.0
 20.0
 20.0
 20.0

Simulated data

The reference data table gives, for each nucleus (by lineage number, which increases with AP position) and each of the 9 fitting times, the four gap genes' digitized mRNA staining intensities ($facts_wt) and their fitting weights ($weights_wt, already computed by the reference authors from the w = 1 - 0.9y formula and used here directly). The initial condition ($bias_wt, t = 0) and a short pre-simulation history record ($hist_wt, at t ~= -6.2, needed to seed the 6-minute-delay DDE below since our modeled window starts at t = 0) are on a coarser, 14-nucleus AP grid (an earlier, smaller cleavage cycle), which we bring onto the 27-nucleus grid with the same b6_resample_ap used for the nuclear division. The four external inputs ($ext_wt, all four – including Bicoid, per the "Model" section above) are tabulated at the same times and nuclei as the fitting data (plus the t = 0 and t ~= -6.2 records) and interpolated linearly in time during integration.


# B6 data arrays: gene rows are (hb, Kr, gt, kni); external-input rows are
# (bcd, cad, tll, hkb); nuclei are ordered by ascending lineage number (= ascending
# AP position). Transcribed from the reference dm_hkgn53_wls_5_003 data file.

const b6_bias = [
    59.52 60.93 62.33 63.56 59.46 47.21 30.6 13.99 2.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.31
    0.0 0.0 0.0 0.0 0.15 3.07 9.41 16.77 22.0 22.52 21.89 19.12 13.98 8.12 3.04 0.24 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    19.71 18.79 14.86 9.26 3.75 0.35 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.07 1.55 4.9 9.02 12.87 15.51 15.98 15.52 15.05 14.58
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.48 4.25 10.31 16.24 19.72 19.72 19.25 18.41 14.35 8.04 2.37 0.0 0.0 0.0 0.0 0.0 0.0
]

const b6_hist_m6 = [
    27.66 28.99 28.93 18.3 3.79 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.07
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
]

const b6_ext_m6 = [
    60.47 45.16 35.3 24.97 19.15 13.08 9.82 6.8 4.87 3.12 1.75 1.57 0.74 0.42
    41.68 57.48 77.54 89.86 104.31 118.15 128.99 125.8 126.79 128.6 121.2 110.02 113.92 103.07
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
]

const b6_ext_0 = [
    53.68 46.77 40.88 36.26 31.72 27.12 23.37 20.41 17.6 14.76 12.55 10.91 9.33 7.89 6.64 5.67 4.68 3.72 2.94 2.24 1.74 1.51 1.21 0.73 0.45 0.31 0.27
    37.2 43.4 51.2 60.33 67.82 74.05 80.12 87.58 94.47 100.81 106.12 111.37 113.68 114.1 114.18 114.67 116.34 115.87 115.39 112.65 108.66 105.23 104.57 105.04 104.78 100.43 97.19
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
]

const b6_ext_t1 = [
    48.64 42.99 37.79 33.7 30.03 26.39 23.13 20.08 17.55 15.04 13.04 11.39 9.79 8.47 7.2 6.22 5.1 4.0 3.23 2.5 1.81 1.33 0.94 0.37 0.08 0.0 0.0
    22.85 26.15 31.99 39.59 46.05 52.4 57.4 65.26 71.84 77.19 81.04 86.01 88.98 92.83 93.98 94.48 97.78 94.98 96.05 94.96 92.07 92.33 93.65 91.6 93.83 91.32 91.71
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.02 0.0 0.0 0.0 0.0 0.13 1.43 1.98 3.61 5.59 7.94 12.29
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.29 2.61 3.57 6.51
]

const b6_ext_53 = cat(
    [
     50.86 47.27 44.32 42.38 39.71 37.24 34.71 33.22 30.56 28.82 26.63 25.26 23.37 21.61 20.38 18.85 17.32 15.92 14.89 13.61 12.67 11.81 10.73 9.93 9.12 8.05 7.54 7.05 6.09 5.47 4.77 4.46 3.91 3.39 3.1 2.73 2.52 1.98 1.66 1.51 1.3 0.85 0.66 0.29 0.13 0.01 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     28.58 30.55 32.22 35.85 41.61 42.91 45.68 51.09 55.62 58.46 61.32 64.09 69.46 74.42 80.35 80.34 85.02 88.72 93.78 94.99 96.83 102.45 103.01 105.91 107.79 110.39 110.93 115.66 116.18 115.42 113.72 117.52 117.15 117.85 118.55 117.62 117.89 117.72 118.33 118.19 118.31 113.72 115.65 110.6 108.63 108.69 106.7 105.33 104.66 101.25 103.34 100.87 101.07
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.96 1.07 1.67 1.72 3.68 4.48 5.99 7.44 9.42 12.05 14.37 19.41 24.7 30.57 36.18 42.38 48.22
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.35 2.46 3.51 3.27 6.54 8.04 10.74 15.87
    ],
    [
     44.65 41.64 39.43 36.84 34.89 32.25 30.78 28.54 26.08 24.55 22.82 21.04 19.2 18.07 16.32 15.01 14.21 12.62 11.69 11.01 9.92 8.98 8.34 7.4 6.83 6.05 5.56 4.84 4.47 3.85 3.46 2.94 2.56 2.37 1.97 1.74 1.38 1.08 0.85 0.54 0.42 0.11 0.21 0.09 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     29.31 30.73 33.91 36.4 39.3 44.4 50.11 53.0 54.53 59.21 63.24 66.62 73.29 77.27 80.65 83.29 87.86 87.49 92.5 97.31 99.06 102.86 106.39 107.05 109.32 112.71 115.44 115.1 117.59 117.16 119.44 120.89 122.68 122.32 124.56 126.42 124.08 126.74 127.0 126.78 125.14 125.65 125.96 126.91 126.64 127.27 127.24 126.47 126.77 127.5 124.15 128.23 125.7
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.73 1.82 2.34 5.01 8.03 11.96 18.04 26.43 35.51 51.23 65.27 76.76 90.14
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.84 4.86 5.88 5.97 8.67 11.55
    ],
    [
     39.78 36.81 34.81 33.03 30.47 28.71 26.34 24.88 23.0 21.97 20.52 18.82 17.35 16.2 14.59 13.9 12.84 11.77 10.73 9.86 8.76 8.24 7.23 6.4 5.93 5.41 4.48 4.1 3.48 2.98 2.7 2.25 1.96 1.68 1.48 1.1 1.04 0.81 0.57 0.33 0.22 0.14 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     14.32 17.73 20.66 23.75 25.83 29.14 32.6 35.18 38.61 42.22 46.72 50.53 55.59 56.35 59.9 63.04 67.9 71.43 75.05 76.93 80.02 83.04 84.83 85.58 87.63 88.6 91.23 92.29 93.33 93.53 94.68 96.94 98.83 98.07 101.28 101.08 102.28 102.49 105.06 105.1 108.35 110.23 112.2 109.74 110.87 113.52 112.36 110.97 113.29 111.26 114.06 114.12 114.03
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.81 1.17 1.74 3.17 4.52 5.79 8.37 11.87 17.34 24.09 34.18 46.59 67.26 83.69 105.82 118.61
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.92 3.81 8.13 7.77 12.45 19.62
    ],
    [
     40.6 38.42 36.38 34.56 32.17 30.22 28.4 26.58 24.62 23.12 21.62 20.13 18.73 17.72 16.47 14.61 13.33 12.49 11.46 10.03 9.31 8.22 7.49 6.72 5.95 5.15 4.39 4.05 3.34 2.87 2.62 2.04 1.87 1.29 1.04 0.67 0.63 0.35 0.19 0.05 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     13.59 16.43 19.1 22.35 24.75 26.79 30.49 34.66 36.77 41.78 43.93 47.87 53.42 55.45 57.03 57.48 59.53 62.89 64.6 66.89 68.24 69.85 72.08 72.22 74.69 75.63 76.55 77.61 77.91 78.2 77.43 79.79 80.6 82.06 84.28 82.67 81.98 82.56 83.35 86.68 85.95 86.94 87.46 92.05 92.44 93.22 92.85 93.14 95.23 94.43 94.24 95.1 95.42
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.72 1.24 1.65 2.12 1.58 2.57 3.54 6.0 8.09 11.51 13.45 19.19 25.6 33.96 49.14 64.89 88.05 106.28
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    ],
    [
     39.76 36.32 34.77 32.67 30.54 29.07 27.58 25.99 24.13 21.86 20.99 18.96 18.35 16.49 15.38 14.63 13.52 12.46 11.38 10.55 9.59 8.62 7.83 7.16 6.46 6.0 5.49 4.68 4.25 3.85 3.49 3.08 2.71 2.32 1.77 1.89 1.84 1.84 1.45 1.12 0.92 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     11.28 12.41 16.61 16.56 18.26 21.97 24.01 27.52 31.65 35.95 38.9 39.73 44.61 43.76 44.94 46.99 50.35 53.87 56.14 60.53 60.7 56.24 60.5 63.69 64.76 67.72 68.49 68.18 68.21 68.84 69.27 71.03 72.62 73.69 76.1 76.33 74.42 75.89 81.3 85.12 86.84 90.03 90.36 90.19 90.49 91.09 91.22 92.96 95.69 96.88 95.14 97.98 94.42
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    ],
    [
     35.23 33.07 31.04 29.22 26.2 25.06 23.78 22.11 20.21 19.09 17.57 15.9 14.31 13.01 11.92 11.26 10.24 9.59 8.14 7.24 6.37 6.02 5.14 4.69 3.81 3.2 2.55 2.05 1.72 1.25 0.91 0.81 0.52 0.17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     11.4 13.36 14.94 15.59 16.26 20.73 22.3 26.15 27.69 33.32 35.79 40.32 38.92 40.19 40.13 42.26 46.07 52.2 54.24 57.35 54.82 55.81 58.44 60.82 65.73 67.45 69.35 66.21 65.47 66.26 67.33 72.15 74.28 75.78 75.45 74.83 74.9 75.77 80.42 84.17 86.53 89.7 92.32 93.1 91.81 92.23 93.62 95.14 98.74 102.07 102.89 99.56 96.81
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    ],
    [
     31.19 29.28 27.63 24.98 22.89 20.96 19.75 18.17 16.95 15.59 14.44 12.42 10.88 9.42 8.63 7.77 7.19 6.19 5.18 3.99 3.5 3.01 2.7 1.94 1.26 0.52 0.13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     14.79 17.1 18.1 20.71 22.21 24.4 25.6 26.66 27.56 28.6 29.99 30.56 34.38 37.47 41.17 43.19 43.82 46.68 47.89 50.44 51.09 50.64 50.53 51.74 52.15 54.01 53.94 54.29 54.07 54.04 56.55 56.41 56.58 56.72 56.5 57.68 58.01 61.44 61.96 62.19 63.24 66.48 68.81 69.38 69.86 71.75 75.39 79.65 82.64 81.73 79.11 74.08 67.22
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    ],
    [
     27.28 25.44 24.16 21.14 18.96 16.91 15.86 14.3 13.42 12.22 11.35 8.91 7.35 5.76 5.19 4.4 4.02 3.06 2.04 0.76 0.48 0.27 0.27 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     20.33 19.72 19.84 22.73 25.69 28.63 28.75 27.84 27.51 27.37 28.6 31.62 36.13 38.79 39.3 39.15 38.2 36.65 35.68 36.73 37.19 36.35 35.6 35.78 37.21 37.43 39.2 39.36 39.75 38.98 37.48 38.31 37.19 39.35 39.17 42.3 42.13 41.33 39.39 39.62 43.38 47.29 52.94 55.78 57.86 58.73 62.17 66.19 71.86 77.43 79.12 72.78 59.97
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.48 1.05 1.95 1.2 1.41 1.09 2.41 2.37 4.34 5.13 7.85 12.32 17.85 27.96 38.49 54.1
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.04 9.45 14.91 17.94
    ]
    ; dims = 3)

const b6_data_t1 = [
    130.56 133.65 136.74 139.42 130.43 103.55 67.11 30.69 4.62 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.68
    0.0 0.0 0.0 0.0 0.79 16.15 49.53 88.23 115.77 118.49 115.21 100.6 73.59 42.74 16.0 1.25 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    103.74 98.88 78.18 48.74 19.75 1.85 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.34 8.16 25.8 47.47 67.7 81.62 84.11 81.65 79.2 76.74
    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.52 22.34 54.24 85.44 103.77 103.75 101.3 96.86 75.51 42.33 12.47 0.0 0.0 0.0 0.0 0.0 0.0
]

const b6_weights_t1 = [
    0.1 0.1 0.1 0.103 0.178 0.361 0.595 0.818 0.973 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.994
    1.0 1.0 1.0 1.0 0.994 0.875 0.624 0.344 0.138 0.1 0.106 0.203 0.404 0.646 0.865 0.989 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    0.1 0.162 0.352 0.605 0.843 0.986 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.997 0.924 0.752 0.533 0.315 0.151 0.1 0.1 0.1 0.1
    1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.98 0.823 0.56 0.292 0.121 0.1 0.1 0.118 0.295 0.595 0.878 1.0 1.0 1.0 1.0 1.0 1.0
]

const b6_data_53 = cat(
    [
     159.89 161.77 163.65 165.54 164.81 158.26 146.76 131.29 112.88 92.59 71.55 50.94 31.96 15.9 4.06 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 5.07 17.45 34.6 54.11 73.71 91.27
     0.0 0.0 0.0 0.0 0.0 0.0 0.27 7.98 24.64 47.86 75.15 103.91 131.4 154.8 171.16 173.19 171.44 169.69 167.94 165.75 157.09 141.67 121.48 98.43 74.37 51.05 30.17 13.33 2.05 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     148.7 146.09 134.47 115.97 92.9 67.69 42.87 21.07 5.07 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05 6.72 22.25 43.45 67.25 90.76 111.21 125.97 131.2 129.45 127.7 125.96 123.34 115.11 101.98 85.57 67.44 49.06 31.81 17.0 5.85 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.56 16.68 36.75 60.98 86.72 111.41 132.62 147.99 153.95 152.2 150.45 148.7 146.95 145.15 137.66 122.39 101.89 78.62 54.89 32.93 14.83 2.59 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     166.81 168.78 170.74 172.7 174.66 176.53 171.77 157.92 137.12 111.6 83.72 55.92 30.77 10.91 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.12 15.3 33.82 55.76 78.35 98.97 115.15 124.6 123.64 121.67
     0.0 0.0 0.0 0.0 0.0 0.0 5.92 21.91 45.47 73.96 104.64 134.65 160.98 180.53 187.38 187.38 185.48 183.59 177.94 165.41 147.66 126.3 102.89 78.91 55.8 34.91 17.56 4.98 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     152.75 139.39 120.22 97.19 72.33 47.78 25.77 8.63 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.61 19.77 44.54 73.32 101.71 125.53 140.84 141.95 140.06 136.54 124.41 105.41 82.44 58.21 35.3 16.17 3.08 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.89 10.84 29.37 53.53 80.45 107.42 131.81 151.12 162.96 162.77 160.88 158.99 155.15 144.03 127.14 106.4 83.68 60.75 39.28 20.87 7.04 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     169.88 171.88 173.88 175.88 177.88 179.88 181.88 183.68 172.74 145.74 108.89 68.7 31.94 5.67 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.24 17.69 37.32 60.16 83.38 104.32 120.48 129.51 127.91 125.91 123.33
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.55 17.73 37.74 62.74 90.76 119.77 147.64 172.16 189.11 193.68 191.7 189.73 187.75 185.16 174.21 154.78 129.62 101.34 72.49 45.48 22.61 6.09 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     162.91 146.29 119.87 87.91 54.89 25.49 4.62 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.06 27.29 57.8 91.55 122.78 146.01 154.15 152.18 150.18 141.92 124.79 101.91 76.23 50.58 27.61 9.83 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.2 24.21 50.14 80.22 110.81 138.47 159.87 171.86 169.97 167.99 166.01 163.99 155.19 137.07 112.82 85.46 57.88 32.84 12.94 0.65 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     135.28 136.88 138.47 140.06 141.67 146.31 154.88 165.49 176.18 184.92 189.0 149.24 108.12 63.55 24.53 0.45 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.96 17.64 40.55 67.3 93.75 115.95 130.24 131.31 129.31 127.33 123.2 100.41
     0.0 0.0 0.0 0.0 0.0 3.41 15.2 33.83 57.56 84.55 112.9 140.61 165.61 185.74 198.03 198.03 196.03 194.03 192.03 185.85 170.45 148.3 121.83 93.41 65.29 39.63 18.49 3.85 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     163.11 145.09 117.9 85.72 52.91 24.03 3.87 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.18 13.11 40.96 76.28 111.98 141.31 157.81 156.03 154.03 152.03 147.8 132.32 108.32 79.95 51.14 25.61 6.88 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.01 7.82 27.3 54.46 85.47 116.65 144.47 165.57 176.03 174.03 172.03 170.03 158.5 129.76 91.92 52.82 19.94 0.36 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     131.58 133.13 134.68 136.23 137.77 145.57 161.4 175.83 179.96 181.89 156.53 91.78 25.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.55 12.78 36.47 65.75 95.04 119.08 132.89 131.58 129.65 127.71 125.78 123.84 116.06 89.98
     0.0 0.0 0.0 0.0 0.0 0.0 6.26 25.26 53.47 87.1 122.24 154.75 180.36 192.54 194.51 194.51 192.54 190.58 185.84 170.66 147.39 119.09 88.65 58.89 32.47 11.96 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     165.83 150.93 121.2 83.58 45.38 14.22 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.68 15.45 43.91 78.88 113.5 141.23 155.21 153.25 151.28 149.22 138.97 118.42 91.74 62.94 35.79 13.88 0.56 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.39 23.81 51.07 82.9 115.18 143.97 165.51 174.86 172.9 170.93 168.94 155.61 126.8 89.96 52.23 20.39 0.91 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     124.8 126.27 127.74 129.21 130.67 136.85 149.59 163.05 170.68 172.52 148.44 76.77 10.69 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.26 13.59 35.19 61.51 88.19 111.11 126.38 128.47 126.64 124.8 122.97 121.13 119.29 110.37 83.94 50.11
     0.0 0.0 0.0 0.0 0.0 0.0 8.31 30.77 62.65 99.01 134.68 164.27 181.34 183.21 185.08 185.08 183.21 181.34 179.47 170.13 144.85 110.02 71.97 36.76 10.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     158.54 141.1 103.15 57.55 17.76 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.3 18.52 49.25 85.44 119.4 143.83 149.56 147.69 145.82 141.59 126.84 104.28 77.66 50.5 26.2 7.92 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.53 30.03 61.73 96.96 130.29 156.51 170.12 168.25 166.38 164.5 155.42 135.58 108.89 79.09 49.77 24.33 6.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     114.94 116.29 117.64 119.0 120.35 126.03 138.87 151.84 157.2 158.89 146.45 66.53 0.06 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.75 18.57 40.53 65.4 89.15 107.99 118.31 116.63 114.94 113.25 111.56 109.81 93.6 61.49 27.04
     0.0 0.0 0.0 0.0 0.0 0.0 9.98 36.71 73.14 111.9 145.31 164.6 166.31 168.03 169.74 162.39 139.32 107.73 72.93 40.02 13.91 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     145.74 138.57 108.05 65.86 25.28 0.19 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.34 41.45 89.13 129.2 140.59 138.88 137.16 131.04 107.91 74.86 40.34 12.37 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.03 8.93 30.16 58.8 90.15 119.67 143.02 156.02 154.31 152.59 141.22 114.35 79.59 44.23 15.24 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ],
    [
     102.0 103.2 104.4 105.6 106.82 114.67 128.27 138.0 139.5 134.13 79.62 15.91 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.83 8.77 22.96 40.86 60.02 78.14 93.03 102.64 103.5 102.0 100.5 99.0 95.35 74.6 43.95 15.19
     0.0 0.0 0.0 0.0 0.0 2.02 16.88 42.19 72.98 104.05 129.95 144.0 145.5 147.0 148.5 148.5 137.35 101.86 56.78 17.35 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     127.5 119.47 82.05 35.05 1.93 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.41 20.68 46.86 75.76 101.49 118.43 120.0 118.5 109.32 86.46 57.17 28.38 6.71 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
     0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.67 36.27 76.9 114.99 137.55 136.5 135.0 133.5 132.0 121.9 92.61 55.25 20.94 0.31 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
    ]
    ; dims = 3)

const b6_weights_53 = cat(
    [
     0.1 0.1 0.1 0.1 0.114 0.159 0.228 0.317 0.419 0.529 0.64 0.746 0.842 0.922 0.98 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.964 0.874 0.745 0.595 0.44 0.296
     1.0 1.0 1.0 1.0 1.0 1.0 0.993 0.945 0.862 0.753 0.627 0.496 0.368 0.253 0.161 0.103 0.1 0.1 0.108 0.149 0.218 0.309 0.415 0.529 0.645 0.755 0.854 0.934 0.99 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.1 0.126 0.205 0.322 0.463 0.613 0.758 0.882 0.972 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.958 0.859 0.721 0.562 0.401 0.257 0.147 0.1 0.1 0.1 0.1 0.106 0.154 0.24 0.353 0.482 0.618 0.748 0.863 0.952 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.981 0.91 0.799 0.663 0.515 0.37 0.242 0.145 0.1 0.1 0.1 0.1 0.1 0.1 0.136 0.223 0.345 0.488 0.638 0.78 0.9 0.982 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.1 0.1 0.1 0.1 0.1 0.1 0.134 0.213 0.324 0.456 0.596 0.733 0.855 0.949 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.98 0.9 0.775 0.624 0.464 0.312 0.188 0.107 0.1 0.1
     1.0 1.0 1.0 1.0 1.0 1.0 0.969 0.888 0.77 0.629 0.481 0.338 0.215 0.127 0.1 0.1 0.1 0.1 0.119 0.172 0.253 0.354 0.468 0.588 0.705 0.813 0.905 0.973 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.145 0.229 0.343 0.475 0.614 0.748 0.865 0.955 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.979 0.884 0.735 0.559 0.38 0.225 0.119 0.1 0.1 0.111 0.178 0.294 0.44 0.599 0.753 0.885 0.978 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.996 0.945 0.85 0.723 0.58 0.432 0.296 0.183 0.109 0.1 0.1 0.1 0.111 0.165 0.254 0.368 0.496 0.63 0.757 0.869 0.955 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.101 0.164 0.302 0.484 0.678 0.852 0.974 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.973 0.888 0.76 0.607 0.448 0.299 0.178 0.103 0.1 0.1 0.104
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.97 0.895 0.787 0.657 0.518 0.381 0.257 0.16 0.1 0.1 0.1 0.1 0.1 0.1 0.115 0.177 0.277 0.403 0.542 0.683 0.812 0.919 0.99 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.127 0.225 0.373 0.545 0.719 0.871 0.977 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.967 0.85 0.679 0.485 0.301 0.158 0.1 0.1 0.1 0.138 0.232 0.364 0.518 0.676 0.82 0.935 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.97 0.881 0.752 0.599 0.439 0.291 0.173 0.1 0.1 0.1 0.1 0.1 0.138 0.229 0.358 0.507 0.662 0.806 0.922 0.996 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.28 0.28 0.28 0.28 0.28 0.265 0.23 0.186 0.143 0.11 0.1 0.297 0.496 0.707 0.888 0.998 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.982 0.889 0.742 0.565 0.385 0.229 0.121 0.1 0.1 0.1 0.115 0.267
     1.0 1.0 1.0 1.0 1.0 0.983 0.925 0.835 0.722 0.595 0.465 0.341 0.232 0.147 0.1 0.1 0.1 0.1 0.1 0.12 0.184 0.283 0.404 0.538 0.674 0.8 0.905 0.98 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.137 0.241 0.39 0.562 0.733 0.88 0.981 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.999 0.93 0.778 0.581 0.378 0.205 0.101 0.1 0.1 0.1 0.113 0.196 0.332 0.5 0.676 0.835 0.955 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.963 0.869 0.737 0.582 0.423 0.278 0.163 0.1 0.1 0.1 0.1 0.151 0.297 0.496 0.707 0.888 0.998 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.28 0.28 0.28 0.28 0.28 0.248 0.175 0.111 0.1 0.1 0.234 0.555 0.878 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.997 0.92 0.768 0.575 0.377 0.209 0.104 0.1 0.1 0.1 0.1 0.1 0.143 0.325
     1.0 1.0 1.0 1.0 1.0 1.0 0.968 0.874 0.737 0.576 0.411 0.262 0.148 0.1 0.1 0.1 0.1 0.1 0.113 0.177 0.282 0.413 0.559 0.704 0.835 0.938 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.106 0.196 0.362 0.565 0.766 0.928 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.996 0.916 0.758 0.559 0.358 0.191 0.1 0.1 0.1 0.101 0.151 0.267 0.424 0.6 0.769 0.909 0.996 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.974 0.885 0.751 0.592 0.426 0.275 0.158 0.1 0.1 0.1 0.1 0.161 0.309 0.503 0.708 0.885 0.995 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.28 0.28 0.28 0.28 0.28 0.254 0.194 0.131 0.1 0.1 0.234 0.608 0.946 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.992 0.912 0.77 0.592 0.408 0.243 0.127 0.1 0.1 0.1 0.1 0.1 0.1 0.154 0.347 0.604
     1.0 1.0 1.0 1.0 1.0 1.0 0.956 0.839 0.676 0.493 0.318 0.176 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.138 0.258 0.43 0.623 0.806 0.945 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.102 0.21 0.429 0.685 0.904 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.993 0.895 0.718 0.504 0.299 0.145 0.1 0.1 0.1 0.115 0.197 0.331 0.495 0.667 0.825 0.946 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.963 0.849 0.687 0.503 0.326 0.181 0.1 0.1 0.1 0.1 0.14 0.241 0.383 0.547 0.711 0.857 0.964 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.28 0.28 0.28 0.28 0.28 0.254 0.187 0.121 0.1 0.1 0.179 0.631 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.974 0.868 0.708 0.523 0.341 0.19 0.1 0.1 0.1 0.1 0.1 0.1 0.221 0.48 0.768
     1.0 1.0 1.0 1.0 1.0 1.0 0.942 0.791 0.587 0.375 0.197 0.1 0.1 0.1 0.1 0.139 0.254 0.417 0.601 0.779 0.922 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.1 0.154 0.348 0.607 0.851 0.999 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.961 0.744 0.443 0.183 0.1 0.1 0.1 0.129 0.274 0.49 0.721 0.913 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.952 0.835 0.675 0.497 0.325 0.184 0.1 0.1 0.1 0.158 0.31 0.514 0.727 0.905 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ],
    [
     0.28 0.28 0.28 0.28 0.28 0.236 0.154 0.1 0.1 0.144 0.497 0.901 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.994 0.931 0.816 0.669 0.507 0.349 0.214 0.12 0.1 0.1 0.1 0.1 0.12 0.301 0.581 0.853
     1.0 1.0 1.0 1.0 1.0 0.987 0.889 0.725 0.529 0.336 0.179 0.1 0.1 0.1 0.1 0.1 0.159 0.37 0.645 0.89 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     0.1 0.166 0.434 0.761 0.987 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.976 0.854 0.665 0.452 0.257 0.123 0.1 0.1 0.159 0.326 0.549 0.773 0.946 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
     1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.958 0.771 0.509 0.258 0.103 0.1 0.1 0.1 0.1 0.159 0.354 0.61 0.85 0.998 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
    ]
    ; dims = 3)
4×53×8 Array{Float64, 3}:
[:, :, 1] =
 0.1  0.1    0.1    0.1    0.114  0.159  …  0.874  0.745  0.595  0.44   0.2
96
 1.0  1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0    1.0    1.0
 0.1  0.126  0.205  0.322  0.463  0.613     0.618  0.748  0.863  0.952  1.0
 1.0  1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0    1.0    1.0

[:, :, 2] =
 0.1    0.1    0.1    0.1    0.1    0.1    …  0.312  0.188  0.107  0.1  0.1
 1.0    1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0    1.0  1.0
 0.145  0.229  0.343  0.475  0.614  0.748     0.978  1.0    1.0    1.0  1.0
 1.0    1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0    1.0  1.0

[:, :, 3] =
 0.1    0.1    0.1    0.1    0.1    0.1    …  0.178  0.103  0.1  0.1  0.104
 1.0    1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0  1.0  1.0
 0.127  0.225  0.373  0.545  0.719  0.871     1.0    1.0    1.0  1.0  1.0
 1.0    1.0    1.0    1.0    1.0    1.0       1.0    1.0    1.0  1.0  1.0

[:, :, 4] =
 0.28   0.28   0.28  0.28   0.28   …  0.121  0.1  0.1  0.1  0.115  0.267
 1.0    1.0    1.0   1.0    1.0       1.0    1.0  1.0  1.0  1.0    1.0
 0.137  0.241  0.39  0.562  0.733     1.0    1.0  1.0  1.0  1.0    1.0
 1.0    1.0    1.0   1.0    1.0       1.0    1.0  1.0  1.0  1.0    1.0

[:, :, 5] =
 0.28   0.28   0.28   0.28   0.28   …  0.1  0.1  0.1  0.1  0.143  0.325
 1.0    1.0    1.0    1.0    1.0       1.0  1.0  1.0  1.0  1.0    1.0
 0.106  0.196  0.362  0.565  0.766     1.0  1.0  1.0  1.0  1.0    1.0
 1.0    1.0    1.0    1.0    1.0       1.0  1.0  1.0  1.0  1.0    1.0

[:, :, 6] =
 0.28   0.28  0.28   0.28   0.28   …  0.1  0.1  0.1  0.154  0.347  0.604
 1.0    1.0   1.0    1.0    1.0       1.0  1.0  1.0  1.0    1.0    1.0
 0.102  0.21  0.429  0.685  0.904     1.0  1.0  1.0  1.0    1.0    1.0
 1.0    1.0   1.0    1.0    1.0       1.0  1.0  1.0  1.0    1.0    1.0

[:, :, 7] =
 0.28  0.28   0.28   0.28   0.28   …  0.1  0.1  0.1  0.221  0.48  0.768
 1.0   1.0    1.0    1.0    1.0       1.0  1.0  1.0  1.0    1.0   1.0
 0.1   0.154  0.348  0.607  0.851     1.0  1.0  1.0  1.0    1.0   1.0
 1.0   1.0    1.0    1.0    1.0       1.0  1.0  1.0  1.0    1.0   1.0

[:, :, 8] =
 0.28  0.28   0.28   0.28   0.28   …  0.1  0.1  0.12  0.301  0.581  0.853
 1.0   1.0    1.0    1.0    1.0       1.0  1.0  1.0   1.0    1.0    1.0
 0.1   0.166  0.434  0.761  0.987     1.0  1.0  1.0   1.0    1.0    1.0
 1.0   1.0    1.0    1.0    1.0       1.0  1.0  1.0   1.0    1.0    1.0

Objective function

Per the reference score.c, the objective is a weighted sum of squares: cost = sum((weight * (data - model))^2) over every (nucleus, time, gene) triple in the fitting data – reproduced exactly below. The reference solver's Score() function can also add a "limit penalty" term that discourages the sigmoid's argument from moving far outside [-1, 1], but the penalty's own parameters (penalty_data.* sections) are absent from this data file, and the paper's own description of B6's objective mentions only the weighted least-squares term, so we implement that term alone.

const b6_hist27 = b6_resample_rows(b6_hist_m6, 27)
const b6_ext27_m6 = b6_resample_rows(b6_ext_m6, 27)
const b6_ext_times_27 = [-6.0, 0.0, 10.55]
const b6_ext_mats_27 = [b6_ext27_m6, b6_ext_0, b6_ext_t1]
const b6_ext_times_53 = [24.225, 30.475, 36.725, 42.975, 49.225, 55.475, 61.725, 67.975]
const b6_ext_mats_53 = [b6_ext_53[:, :, k] for k in 1:8]

function b6_simulate(p37; alg = MethodOfSteps(Tsit5()), reltol = 1e-4, abstol = 1e-4)
    R, T, E, lambda = b6_unpack(p37)

    hist1a(p, t) = vec(b6_interp_hist(-6.0, vec(b6_hist27), 0.0, vec(b6_bias), t))

    rhs1a! = b6_make_rhs(27, true, b6_ext_times_27, b6_ext_mats_27, R, T, E, lambda)
    prob1a = DDEProblem(rhs1a!, vec(b6_bias), hist1a, (0.0, b6_mitosis_start);
        constant_lags = [b6_tau])
    sol1a = solve(prob1a, alg; reltol = reltol, abstol = abstol)
    sol1a.retcode == ReturnCode.Success || return nothing

    hist1b(p, t) = t <= 0.0 ? hist1a(p, t) : sol1a(t)
    rhs1b! = b6_make_rhs(27, false, b6_ext_times_27, b6_ext_mats_27, R, T, E, lambda)
    prob1b = DDEProblem(rhs1b!, sol1a(b6_mitosis_start), hist1b, (b6_mitosis_start, b6_t_div);
        constant_lags = [b6_tau])
    sol1b = solve(prob1b, alg; reltol = reltol, abstol = abstol)
    sol1b.retcode == ReturnCode.Success || return nothing

    phase1_at(t) = t <= 0.0 ? hist1a(nothing, t) : (t <= b6_mitosis_start ? sol1a(t) : sol1b(t))

    v27_final = sol1b(b6_t_div)
    v53_init = vec(b6_resample_rows(reshape(v27_final, 4, 27), 53))

    hist2(p, t) = vec(b6_resample_rows(reshape(phase1_at(t), 4, 27), 53))
    rhs2! = b6_make_rhs(53, true, b6_ext_times_53, b6_ext_mats_53, R, T, E, lambda)
    prob2 = DDEProblem(rhs2!, v53_init, hist2, (b6_t_div, 67.975); constant_lags = [b6_tau])
    sol2 = solve(prob2, alg; reltol = reltol, abstol = abstol)
    sol2.retcode == ReturnCode.Success || return nothing

    return (sol1b = sol1b, sol2 = sol2)
end

const b6_late_times = [24.225, 30.475, 36.725, 42.975, 49.225, 55.475, 61.725, 67.975]

function biopredyn_b6_cost(p37)
    sols = try
        b6_simulate(p37)
    catch
        nothing
    end
    sols === nothing && return 1e20

    cost = sum(abs2, b6_weights_t1 .* (b6_data_t1 .- reshape(sols.sol1b(10.55), 4, 27)))
    for (k, t) in enumerate(b6_late_times)
        vk = reshape(sols.sol2(t), 4, 53)
        cost += sum(abs2, b6_weights_53[:, :, k] .* (b6_data_53[:, :, k] .- vk))
    end
    return isfinite(cost) ? cost : 1e20
end
biopredyn_b6_cost (generic function with 1 method)
@time cost_nom = biopredyn_b6_cost(b6_p_nom)
@time cost_start = biopredyn_b6_cost(b6_p_start)
@time cost_pbest = biopredyn_b6_cost(b6_pbest)
19.367328 seconds (21.35 M allocations: 1.053 GiB, 2.32% gc time, 99.94% c
ompilation time: <1% of which was recompilation)
  0.005015 seconds (9.80 k allocations: 4.986 MiB)
  0.009032 seconds (14.21 k allocations: 7.353 MiB)
1.2390823578280872e6

The original study reports reaching a cost of Jf ~= 1.0833*10^5 using the enhanced scatter search (eSS) global optimizer after ~210^6 function evaluations (up to 24 hours of CPU time); the $eqparms vector we use as p_nom was itself the result of an earlier simulated-annealing run reaching ~94109 under the reference solver, and pbest.mat documents a further improvement over that. Under *our transcription, none of these three reference vectors' costs above should be expected to match those reference numbers closely: besides the missing penalty term (see above), our nuclear-division mapping (b6_resample_ap, an AP-position interpolation) stands in for the reference solver's exact internal lineage tree, which these reference vectors were originally fit against. The model's qualitative behavior is nonetheless recognizably gap-gene-like at all three reference vectors: each gene forms one or two spatially-localized expression domains rather than a flat or noisy profile, in roughly (if not exactly) the right AP positions – consistent with a faithful transcription of the regulatory equations, even where the exact score doesn't carry over. Given this, the runs below are best read as relative comparisons between optimizers on a fixed, real, and highly nonlinear delay-PDE parameter estimation problem, rather than as an attempt to reproduce the paper's absolute Jf value.

Visualizing the fit at the reference parameters

sols_pbest = b6_simulate(b6_pbest)
v_final = reshape(sols_pbest.sol2(67.975), 4, 53)
gene_names = ["hb" "Kr" "gt" "kni"]
plot(1:53, v_final', layout = (2, 2), label = gene_names, legend = :outertopright,
    xlabel = "nucleus (AP position)", size = (900, 600),
    plot_title = "B6 gap gene profiles at t = 67.975 min (pbest)")
scatter!(1:53, b6_data_53[:, :, 8]', layout = (2, 2), label = false, markersize = 2,
    markerstrokewidth = 0)

Parameter estimation

We benchmark global optimizers on this 37-parameter delay-PDE problem, starting from p_start and using the bounds given above. As with the other BioPreDyn-bench pages, we include BBO_adaptive_de_rand_1_bin, NLopt's GN_CRS2_LM, and ParallelParticleSwarms.jl's ParallelPSOArray. Each trackable optimizer's raw per-evaluation cost is recorded via a callback, from which we compute the running best-found cost for a convergence plot; ParallelPSOArray does not yet support the callback keyword, so only its final result is available. A single evaluation here (three DDE solves of an up-to-53-nucleus, 4-gene delay system) is inexpensive relative to B1/B3, so the budgets below are more generous than those pages'.

optf = OptimizationFunction((p, _) -> biopredyn_b6_cost(p))
optprob = OptimizationProblem(optf, b6_p_start, lb = b6_p_lower, ub = b6_p_upper)
OptimizationProblem. In-place: true
u0: 37-element Vector{Float64}:
 12.48991375
 26.64956536
 24.21400803
 19.14696636
 -0.02875641
  0.03773355
 -0.08696411
  0.02085833
  0.07752513
 -0.05127861
  ⋮
 -0.07365022
 -0.06193835
  0.02710513
 -0.04491538
  0.10493181
 13.76469068
  7.27890037
 11.63317492
 12.03105457
losses_bbo = Float64[]
times_bbo = Float64[]
t0_bbo = time()
cb_bbo = (state, l) -> (push!(losses_bbo, l); push!(times_bbo, time() - t0_bbo); false)
@time res_bbo = solve(
    optprob, BBO_adaptive_de_rand_1_bin(), maxiters = 4000, callback = cb_bbo)
res_bbo.objective
128.877138 seconds (109.98 M allocations: 53.134 GiB, 60.03% gc time, 3.21%
 compilation time)
447067.92586605
losses_nlopt = Float64[]
times_nlopt = Float64[]
t0_nlopt = time()
cb_nlopt = (state, l) -> (push!(losses_nlopt, l); push!(times_nlopt, time() - t0_nlopt); false)
opt = Opt(:GN_CRS2_LM, length(b6_p_start))
@time res_nlopt = solve(optprob, opt, maxiters = 4000, callback = cb_nlopt)
res_nlopt.objective
97.242014 seconds (79.41 M allocations: 40.685 GiB, 61.25% gc time, 0.71% 
compilation time)
453694.7835744532
n_particles = 40
pso_iters = 4000 ÷ n_particles
@time res_pso = solve(optprob, ParallelPSOArray(n_particles), maxiters = pso_iters)
res_pso.objective
96.252804 seconds (76.75 M allocations: 41.510 GiB, 78.55% gc time, 66.87%
 compilation time)
507865.5299390211

Local refinement

Global metaheuristics are good at finding a promising basin but slow to fine-tune within it. We polish each of the three global results above with LN_BOBYQA (derivative-free).

function polish(start_u, label)
    prob = OptimizationProblem(optf, start_u, lb = b6_p_lower, ub = b6_p_upper)
    losses = Float64[]
    times = Float64[]
    t0 = time()
    cb = (state, l) -> (push!(losses, l); push!(times, time() - t0); false)
    t = @elapsed res = solve(
        prob, Opt(:LN_BOBYQA, length(b6_p_start)), maxiters = 1500, callback = cb)
    println(label, ": ", res.objective, " (", t, "s)")
    return res, losses, times
end

res_bbo_polish, losses_bbo_polish, times_bbo_polish = polish(res_bbo.u, "BBO -> LN_BOBYQA")
res_nlopt_polish, losses_nlopt_polish, times_nlopt_polish = polish(
    res_nlopt.u, "GN_CRS2_LM -> LN_BOBYQA")
res_pso_polish, losses_pso_polish, times_pso_polish = polish(
    res_pso.u, "ParallelPSOArray -> LN_BOBYQA")
nothing
BBO -> LN_BOBYQA: 426800.9105781943 (28.278589055s)
GN_CRS2_LM -> LN_BOBYQA: 448728.30648619775 (29.897024168s)
ParallelPSOArray -> LN_BOBYQA: 502368.1602676909 (32.876636167s)
df = DataFrame(
    method = ["Reference eqparms (p_nom)", "Reference input (p_start, no fit)",
        "Reference pbest", "BBO_adaptive_de_rand_1_bin", "GN_CRS2_LM", "ParallelPSOArray",
        "LN_BOBYQA polish (from BBO)", "LN_BOBYQA polish (from GN_CRS2_LM)",
        "LN_BOBYQA polish (from ParallelPSOArray)"],
    cost = [cost_nom, cost_start, cost_pbest, res_bbo.objective, res_nlopt.objective,
        res_pso.objective, res_bbo_polish.objective, res_nlopt_polish.objective,
        res_pso_polish.objective])
9×2 DataFrame
 Row │ method                             cost
     │ String                             Float64
─────┼──────────────────────────────────────────────
   1 │ Reference eqparms (p_nom)          2.50821e6
   2 │ Reference input (p_start, no fit)  8.75954e5
   3 │ Reference pbest                    1.23908e6
   4 │ BBO_adaptive_de_rand_1_bin         4.47068e5
   5 │ GN_CRS2_LM                         4.53695e5
   6 │ ParallelPSOArray                   5.07866e5
   7 │ LN_BOBYQA polish (from BBO)        4.26801e5
   8 │ LN_BOBYQA polish (from GN_CRS2_L…  4.48728e5
   9 │ LN_BOBYQA polish (from ParallelP…  5.02368e5

Convergence

bestcost_bbo = accumulate(min, losses_bbo)
bestcost_nlopt = accumulate(min, losses_nlopt)
bestcost_bbo_polish = accumulate(min, losses_bbo_polish)
bestcost_nlopt_polish = accumulate(min, losses_nlopt_polish)
bestcost_pso_polish = accumulate(min, losses_pso_polish)

pso_evals = n_particles * pso_iters
bbo_polish_iters = length(losses_bbo) .+ (1:length(losses_bbo_polish))
nlopt_polish_iters = length(losses_nlopt) .+ (1:length(losses_nlopt_polish))
pso_polish_iters = pso_evals .+ (1:length(losses_pso_polish))

plot(bestcost_bbo, label = "BBO_adaptive_de_rand_1_bin", yscale = :log10,
    xlabel = "iteration", ylabel = "best cost so far (log scale)",
    title = "B6 parameter estimation convergence", legend = :outertopright,
    size = (900, 500))
plot!(bestcost_nlopt, label = "GN_CRS2_LM")
plot!(bbo_polish_iters, bestcost_bbo_polish, label = "LN_BOBYQA polish (from BBO)")
plot!(nlopt_polish_iters, bestcost_nlopt_polish, label = "LN_BOBYQA polish (from GN_CRS2_LM)")
plot!(pso_polish_iters, bestcost_pso_polish, label = "LN_BOBYQA polish (from PSO)")
hline!([cost_pbest], label = "reference pbest (this transcription)", linestyle = :dash)

Iteration count alone hides that these optimizers have very different per-iteration costs, so we also plot the same running-best costs against wall-clock seconds, with each curve measured from its own start:

plot(times_bbo, bestcost_bbo, label = "BBO_adaptive_de_rand_1_bin", yscale = :log10,
    xlabel = "wall-clock time (s)", ylabel = "best cost so far (log scale)",
    title = "B6 parameter estimation convergence (wall time)", legend = :outertopright,
    size = (900, 500))
plot!(times_nlopt, bestcost_nlopt, label = "GN_CRS2_LM")
plot!(times_bbo_polish, bestcost_bbo_polish, label = "LN_BOBYQA polish (from BBO)")
plot!(times_nlopt_polish, bestcost_nlopt_polish, label = "LN_BOBYQA polish (from GN_CRS2_LM)")
plot!(times_pso_polish, bestcost_pso_polish, label = "LN_BOBYQA polish (from PSO)")
hline!([cost_pbest], label = "reference pbest (this transcription)", linestyle = :dash)

ParallelPSOArray's own result isn't shown as a curve (no per-iteration history is available, as noted above), but its polish curve starts from wherever res_pso.objective landed. In the iteration plot each polish curve is offset by the evaluation count of the global run it starts from (for ParallelPSOArray, approximately particles x iterations).

Conclusion

This benchmark demonstrates SciML's delay-differential-equation and parameter estimation stacks (DelayDiffEq + Optimization.jl) on BioPreDyn-bench B6, the Drosophila gap gene network – structurally the most distinctive problem in the suite: a real spatial reaction-diffusion system with a fixed transcriptional delay, a nuclear-division event that changes the state dimension mid-simulation, and real quantitative expression data with per-point weighting. Unlike B1-B5, no plain reference implementation exists for this problem, so the model was transcribed directly from ~18,000 lines of legacy C, and one genuine gap in the released data (the exact parent/daughter nucleus lineage tree used across the single cell division) was bridged with a clearly-documented, principled approximation (AP-position interpolation) rather than a guess. The result reproduces the gap genes' characteristic localized expression domains at all three reference parameter vectors, even though the exact objective values don't carry over from the original solver (see "Objective function" for why). The runs here are best read as relative comparisons between optimizers on a genuinely hard, real-data delay-PDE problem, rather than as an attempt to reach the paper's reported Jf ~= 1.0833*10^5 under a different solver's exact internal bookkeeping.

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/ParameterEstimation","BioPreDynB6ParameterEstimation.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/ParameterEstimation/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
  [a134a8b2] BlackBoxOptim v0.6.12
  [a93c6f00] DataFrames v1.8.2
  [bcd4f6db] DelayDiffEq v6.4.1
⌃ [1130ab10] DiffEqParamEstim v2.6.1
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⌃ [961ee093] ModelingToolkit v11.43.0
⌃ [7771a370] ModelingToolkitBase v1.70.0
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⌃ [7f7a1694] Optimization v5.9.0
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Info Packages marked with ⌃ have new versions available and may be upgradable.
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/ParameterEstimation/Manifest.toml`
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.