BioPreDyn-bench B3 (E. coli Enzymatic and Transcriptional Regulation) Parameter Estimation Benchmark
Parameter estimation of the Kotte et al. E. coli carbon-source regulation model
This benchmark implements problem B3 from the BioPreDyn-bench suite (Villaverde et al. 2015), addressing SciMLBenchmarks.jl#555. B3 is a combined enzymatic-and-transcriptional model of how E. coli senses and adapts to its carbon source, due to Kotte, Zaugg and Heinemann (2010): 47 dynamic states (12 metabolite concentrations, 26 enzyme/transcription-factor protein and protein-complex concentrations, 3 phosphorylation states of the glucose PTS system, and the culture's optical density, glucose and acetate concentrations) connected by 109 reaction rates (enzyme kinetics, phosphorylation, transcription-factor/metabolite binding, gene expression, protein degradation/dilution, and environmental exchange), with 193 parameters, of which 178 are estimated (15 are known constants: Hill/oligomerization coefficients, two fixed operon-coupling factors, a universal protein degradation rate, and a gene expression rate constant). Unlike B2, which fits real experimental data, B3 fits noise-free pseudo-data simulated directly from the model's own nominal parameters: all 47 states, sampled every 1000 seconds over an "extended diauxic shift" scenario that switches the growth medium from glucose, to acetate, to a glucose/acetate mixture, for a total of 161 time points spanning about 45 hours. Because the 47 concentrations span five orders of magnitude, the benchmark's own objective function scales each state's squared residuals by (the inverse square of) that state's own maximum value across the time course.
The model equations, nominal parameters, parameter bounds, and simulated data below are transcribed directly from the benchmark's official MATLAB/AMIGO2 implementation, available as supplementary material to the paper (Additional files 2-3, directory BioPreDynBenchFiles/B3, file dynamics_in_Matlab/b3_dyn_p.m).
using OrdinaryDiffEq, Optimization, ForwardDiff
using OptimizationBBO, OptimizationNLopt, Plots, BenchmarkTools, DataFrames
using ParallelParticleSwarms
using LinearAlgebra
gr(fmt = :png)Plots.GRBackend()Model
Each state belongs to one of a few groups: 12 small-molecule/metabolite concentrations (GLC, ACT, ACoA, AKG, cAMP, FBP, G6P, GLX, ICT, MAL, OAA, PEP, PG3, PYR), the optical density OD tracking population size, the PTS phospho-relay (EIIA, EIIA_P, EIICB), 11 metabolic enzymes and their regulatory proteins (AceA, AceB, AceK, Acoa2act, Acs, Akg2mal, CAMPdegr, Cya, Emp, Eno, Fdp, GltA, Icd/Icd_P, Mdh, MaeAB, PckA, Pdh, PfkA, Ppc, PpsA, PykF), and 3 transcription factors and their metabolite-bound complexes (Cra/CraFBP, Crp/CrpcAMP, PdhR/PdhRPYR, plus the constitutively-expressed IclR). Each of the 109 reaction rates is one of: an enzyme kinetic rate (Michaelis-Menten, two-substrate Michaelis-Menten, or Monod-Wyman-Changeux/Hill allosteric regulation), a PTS phosphorylation rate, a transcription-factor/metabolite Hill-type binding rate, a gene expression rate (basal-plus-regulated, itself scaled by the instantaneous growth rate mu), a protein degradation/dilution rate, or an environmental exchange rate (glucose uptake, acetate uptake/excretion, population growth). The growth rate mu and 7 first-order "biomass" consumption rates are computed by linearly interpolating between two reference vectors of steady-state concentrations reached on glucose-only and acetate-only media (ssGLC, ssACT, both transcribed from the reference files below), weighted by how saturated the glucose and acetate uptake systems currently are; two further rates (MaeAB and Ppc gene expression) use the same interpolation directly, as those two proteins are assumed to be permanently at this growth-rate-dependent quasi-steady state. State derivatives are S * f, where S is the model's (fixed, mostly ±1) stoichiometric matrix and f is the vector of 109 reaction rates – reproduced here as literally as Julia syntax allows from the reference b3_dyn_p.m, including its original variable names, to keep the two directly comparable. A handful of Hill/MWC exponents in the transcription factor and gene-expression rates are themselves estimated parameters and can be non-integer; the few bare state^exponent terms that can encounter a state variable transiently perturbed slightly negative (e.g. during a stiff solver's initial step-size estimate) are floored at zero with max(..., 0.0), matching how the reference C implementation's floating-point pow silently saturates rather than raising a domain error.
# ============================================================
# B3 (Kotte et al. E. coli enzymatic + transcriptional regulation) model
# ============================================================
# state indices
const b3_i_OD = 1
const b3_i_ACT = 2
const b3_i_GLC = 3
const b3_i_ACoA = 4
const b3_i_AKG = 5
const b3_i_cAMP = 6
const b3_i_FBP = 7
const b3_i_G6P = 8
const b3_i_GLX = 9
const b3_i_ICT = 10
const b3_i_MAL = 11
const b3_i_OAA = 12
const b3_i_PEP = 13
const b3_i_PG3 = 14
const b3_i_PYR = 15
const b3_i_AceA = 16
const b3_i_AceB = 17
const b3_i_AceK = 18
const b3_i_Acoa2act = 19
const b3_i_Acs = 20
const b3_i_Akg2mal = 21
const b3_i_CAMPdegr = 22
const b3_i_Cya = 23
const b3_i_Emp = 24
const b3_i_Eno = 25
const b3_i_Fdp = 26
const b3_i_GltA = 27
const b3_i_Icd = 28
const b3_i_Icd_P = 29
const b3_i_Mdh = 30
const b3_i_MaeAB = 31
const b3_i_PckA = 32
const b3_i_Pdh = 33
const b3_i_PfkA = 34
const b3_i_Ppc = 35
const b3_i_PpsA = 36
const b3_i_PykF = 37
const b3_i_EIIA = 38
const b3_i_EIIA_P = 39
const b3_i_EIICB = 40
const b3_i_Cra = 41
const b3_i_CraFBP = 42
const b3_i_Crp = 43
const b3_i_CrpcAMP = 44
const b3_i_IclR = 45
const b3_i_PdhR = 46
const b3_i_PdhRPYR = 47
# rate indices
const b3_r_env_growth = 1
const b3_r_env_GLCup = 2
const b3_r_env_ACTup = 3
const b3_r_env_ACTex = 4
const b3_r_e_AceA = 5
const b3_r_e_AceB = 6
const b3_r_e_AceK_Ki = 7
const b3_r_e_AceK_Ph = 8
const b3_r_e_Acoa2act = 9
const b3_r_e_Acs = 10
const b3_r_e_Akg2mal = 11
const b3_r_e_CAMPdegr = 12
const b3_r_e_Cya = 13
const b3_r_e_Emp = 14
const b3_r_e_Eno = 15
const b3_r_e_Fdp = 16
const b3_r_e_GltA = 17
const b3_r_e_Icd = 18
const b3_r_e_MaeAB = 19
const b3_r_e_Mdh = 20
const b3_r_e_PckA = 21
const b3_r_e_Pdh = 22
const b3_r_e_PfkA = 23
const b3_r_e_Ppc = 24
const b3_r_e_PpsA = 25
const b3_r_e_PykF = 26
const b3_r_pts_r1 = 27
const b3_r_pts_r4 = 28
const b3_r_tf_Cra = 29
const b3_r_tf_Crp = 30
const b3_r_tf_PdhR = 31
const b3_r_g_aceA = 32
const b3_r_g_aceB = 33
const b3_r_g_aceK = 34
const b3_r_g_acoa2act = 35
const b3_r_g_acs = 36
const b3_r_g_akg2mal = 37
const b3_r_g_campdegr = 38
const b3_r_g_cra = 39
const b3_r_g_crp = 40
const b3_r_g_cya = 41
const b3_r_g_EIIA = 42
const b3_r_g_EIICB = 43
const b3_r_g_emp = 44
const b3_r_g_eno = 45
const b3_r_g_fdp = 46
const b3_r_g_gltA = 47
const b3_r_g_icd = 48
const b3_r_g_iclr = 49
const b3_r_g_mdh = 50
const b3_r_g_maeAB = 51
const b3_r_g_pckA = 52
const b3_r_g_pdh = 53
const b3_r_g_pdhr = 54
const b3_r_g_pfkA = 55
const b3_r_g_ppc = 56
const b3_r_g_ppsA = 57
const b3_r_g_pykF = 58
const b3_r_d_ACoA = 59
const b3_r_d_AKG = 60
const b3_r_d_cAMP = 61
const b3_r_d_FBP = 62
const b3_r_d_G6P = 63
const b3_r_d_GLX = 64
const b3_r_d_ICT = 65
const b3_r_d_MAL = 66
const b3_r_d_OAA = 67
const b3_r_d_PEP = 68
const b3_r_d_PG3 = 69
const b3_r_d_PYR = 70
const b3_r_d_AceA = 71
const b3_r_d_AceB = 72
const b3_r_d_AceK = 73
const b3_r_d_Acoa2act = 74
const b3_r_d_Acs = 75
const b3_r_d_Akg2mal = 76
const b3_r_d_CAMPdegr = 77
const b3_r_d_Cra = 78
const b3_r_d_CraFBP = 79
const b3_r_d_Crp = 80
const b3_r_d_CrpcAMP = 81
const b3_r_d_Cya = 82
const b3_r_d_EIIA = 83
const b3_r_d_EIIA_P = 84
const b3_r_d_EIICB = 85
const b3_r_d_Emp = 86
const b3_r_d_Eno = 87
const b3_r_d_Fdp = 88
const b3_r_d_GltA = 89
const b3_r_d_Icd = 90
const b3_r_d_Icd_P = 91
const b3_r_d_IclR = 92
const b3_r_d_Mdh = 93
const b3_r_d_MaeAB = 94
const b3_r_d_PckA = 95
const b3_r_d_Pdh = 96
const b3_r_d_PdhR = 97
const b3_r_d_PdhRPYR = 98
const b3_r_d_PfkA = 99
const b3_r_d_Ppc = 100
const b3_r_d_PpsA = 101
const b3_r_d_PykF = 102
const b3_r_bm_ACoA = 103
const b3_r_bm_AKG = 104
const b3_r_bm_G6P = 105
const b3_r_bm_OAA = 106
const b3_r_bm_PEP = 107
const b3_r_bm_PG3 = 108
const b3_r_bm_PYR = 109
# parameter indices
const b3_p_env_M_ACT = 1
const b3_p_env_M_GLC = 2
const b3_p_env_uc = 3
const b3_p_e_AceA_kcat = 4
const b3_p_e_AceA_n = 5
const b3_p_e_AceA_L = 6
const b3_p_e_AceA_Kict = 7
const b3_p_e_AceA_Kpep = 8
const b3_p_e_AceA_Kpg3 = 9
const b3_p_e_AceA_Kakg = 10
const b3_p_e_AceB_kcat = 11
const b3_p_e_AceB_Kglx = 12
const b3_p_e_AceB_Kacoa = 13
const b3_p_e_AceB_Kglxacoa = 14
const b3_p_e_AceK_kcat_ki = 15
const b3_p_e_AceK_kcat_ph = 16
const b3_p_e_AceK_n = 17
const b3_p_e_AceK_L = 18
const b3_p_e_AceK_Kicd = 19
const b3_p_e_AceK_Kicd_P = 20
const b3_p_e_AceK_Kpep = 21
const b3_p_e_AceK_Kpyr = 22
const b3_p_e_AceK_Koaa = 23
const b3_p_e_AceK_Kglx = 24
const b3_p_e_AceK_Kakg = 25
const b3_p_e_AceK_Kpg3 = 26
const b3_p_e_AceK_Kict = 27
const b3_p_e_Acoa2act_kcat = 28
const b3_p_e_Acoa2act_n = 29
const b3_p_e_Acoa2act_L = 30
const b3_p_e_Acoa2act_Kacoa = 31
const b3_p_e_Acoa2act_Kpyr = 32
const b3_p_e_Acs_kcat = 33
const b3_p_e_Acs_Kact = 34
const b3_p_e_Akg2mal_kcat = 35
const b3_p_e_Akg2mal_Kakg = 36
const b3_p_e_CAMPdegr_kcat = 37
const b3_p_e_CAMPdegr_KcAMP = 38
const b3_p_e_Cya_kcat = 39
const b3_p_e_Cya_KEIIA = 40
const b3_p_e_Emp_kcat_f = 41
const b3_p_e_Emp_kcat_r = 42
const b3_p_e_Emp_Kfbp = 43
const b3_p_e_Emp_Kpg3 = 44
const b3_p_e_Eno_kcatf = 45
const b3_p_e_Eno_kcatr = 46
const b3_p_e_Eno_Kpg3 = 47
const b3_p_e_Eno_Kpep = 48
const b3_p_e_Fdp_kcat = 49
const b3_p_e_Fdp_n = 50
const b3_p_e_Fdp_L = 51
const b3_p_e_Fdp_Kfbp = 52
const b3_p_e_Fdp_Kpep = 53
const b3_p_e_GltA_kcat = 54
const b3_p_e_GltA_Koaa = 55
const b3_p_e_GltA_Kacoa = 56
const b3_p_e_GltA_Koaaacoa = 57
const b3_p_e_GltA_Kakg = 58
const b3_p_e_Icd_kcat = 59
const b3_p_e_Icd_n = 60
const b3_p_e_Icd_L = 61
const b3_p_e_Icd_Kict = 62
const b3_p_e_Icd_Kpep = 63
const b3_p_e_Mdh_kcat = 64
const b3_p_e_Mdh_n = 65
const b3_p_e_Mdh_Kmal = 66
const b3_p_e_MaeAB_kcat = 67
const b3_p_e_MaeAB_n = 68
const b3_p_e_MaeAB_L = 69
const b3_p_e_MaeAB_Kmal = 70
const b3_p_e_MaeAB_Kacoa = 71
const b3_p_e_MaeAB_Kcamp = 72
const b3_p_e_PckA_kcat = 73
const b3_p_e_PckA_Koaa = 74
const b3_p_e_PckA_Kpep = 75
const b3_p_e_Pdh_kcat = 76
const b3_p_e_Pdh_n = 77
const b3_p_e_Pdh_L = 78
const b3_p_e_Pdh_Kpyr = 79
const b3_p_e_Pdh_KpyrI = 80
const b3_p_e_Pdh_Kglx = 81
const b3_p_e_PfkA_kcat = 82
const b3_p_e_PfkA_n = 83
const b3_p_e_PfkA_L = 84
const b3_p_e_PfkA_Kg6p = 85
const b3_p_e_PfkA_Kpep = 86
const b3_p_e_Ppc_kcat = 87
const b3_p_e_Ppc_n = 88
const b3_p_e_Ppc_L = 89
const b3_p_e_Ppc_Kpep = 90
const b3_p_e_Ppc_Kfbp = 91
const b3_p_e_PpsA_kcat = 92
const b3_p_e_PpsA_n = 93
const b3_p_e_PpsA_L = 94
const b3_p_e_PpsA_Kpyr = 95
const b3_p_e_PpsA_Kpep = 96
const b3_p_e_PykF_kcat = 97
const b3_p_e_PykF_n = 98
const b3_p_e_PykF_L = 99
const b3_p_e_PykF_Kpep = 100
const b3_p_e_PykF_Kfbp = 101
const b3_p_pts_k1 = 102
const b3_p_pts_km1 = 103
const b3_p_pts_k4 = 104
const b3_p_pts_KEIIA = 105
const b3_p_pts_Kglc = 106
const b3_p_tf_Cra_scale = 107
const b3_p_tf_Cra_kfbp = 108
const b3_p_tf_Cra_n = 109
const b3_p_tf_Crp_scale = 110
const b3_p_tf_Crp_kcamp = 111
const b3_p_tf_Crp_n = 112
const b3_p_tf_PdhR_scale = 113
const b3_p_tf_PdhR_kpyr = 114
const b3_p_tf_PdhR_n = 115
const b3_p_g_aceBAK_vcra_unbound = 116
const b3_p_g_aceBAK_vcra_bound = 117
const b3_p_g_aceBAK_Kcra = 118
const b3_p_g_aceBAK_aceBfactor = 119
const b3_p_g_aceBAK_aceKfactor = 120
const b3_p_g_aceBAK_KDNA = 121
const b3_p_g_aceBAK_KP = 122
const b3_p_g_aceBAK_KPprime = 123
const b3_p_g_aceBAK_KG = 124
const b3_p_g_aceBAK_L = 125
const b3_p_g_aceBAK_kcat_iclr = 126
const b3_p_g_aceBAK_DNA = 127
const b3_p_g_aceBAK_vcrp_bound = 128
const b3_p_g_aceBAK_vcrp_unbound = 129
const b3_p_g_aceBAK_Kcrp = 130
const b3_p_g_acs_vcrp_unbound = 131
const b3_p_g_acs_vcrp_bound = 132
const b3_p_g_acs_n = 133
const b3_p_g_acs_Kcrp = 134
const b3_p_g_akg2mal_vcrp_unbound = 135
const b3_p_g_akg2mal_vcrp_bound = 136
const b3_p_g_akg2mal_Kcrp = 137
const b3_p_g_akg2mal_n = 138
const b3_p_g_emp_vcra_unbound = 139
const b3_p_g_emp_vcra_bound = 140
const b3_p_g_emp_Kcra = 141
const b3_p_g_emp_vcrp_unbound = 142
const b3_p_g_emp_vcrp_bound = 143
const b3_p_g_emp_Kcrp = 144
const b3_p_g_eno_vcra_unbound = 145
const b3_p_g_eno_vcra_bound = 146
const b3_p_g_eno_Kcra = 147
const b3_p_g_fdp_vcra_unbound = 148
const b3_p_g_fdp_vcra_bound = 149
const b3_p_g_fdp_Kcra = 150
const b3_p_g_gltA_vcrp_unbound = 151
const b3_p_g_gltA_vcrp_bound = 152
const b3_p_g_gltA_Kcrp = 153
const b3_p_g_gltA_n = 154
const b3_p_g_icd_vcra_unbound = 155
const b3_p_g_icd_vcra_bound = 156
const b3_p_g_icd_Kcra = 157
const b3_p_g_mdh_vcrp_unbound = 158
const b3_p_g_mdh_vcrp_bound = 159
const b3_p_g_mdh_Kcrp = 160
const b3_p_g_pckA_vcra_unbound = 161
const b3_p_g_pckA_vcra_bound = 162
const b3_p_g_pckA_Kcra = 163
const b3_p_g_pdh_vpdhr_unbound = 164
const b3_p_g_pdh_vpdhr_bound = 165
const b3_p_g_pdh_Kpdhr = 166
const b3_p_g_pfkA_vcra_unbound = 167
const b3_p_g_pfkA_vcra_bound = 168
const b3_p_g_pfkA_Kcra = 169
const b3_p_g_ppsA_vcra_unbound = 170
const b3_p_g_ppsA_vcra_bound = 171
const b3_p_g_ppsA_Kcra = 172
const b3_p_g_pykF_vcra_unbound = 173
const b3_p_g_pykF_vcra_bound = 174
const b3_p_g_pykF_Kcra = 175
const b3_p_d_k_degr = 176
const b3_p_bm_k_expr = 177
const b3_p_bm_muACT = 178
const b3_p_bm_muGLC = 179
const b3_p_bm_GLC_ACoA = 180
const b3_p_bm_GLC_AKG = 181
const b3_p_bm_GLC_G6P = 182
const b3_p_bm_GLC_OAA = 183
const b3_p_bm_GLC_PEP = 184
const b3_p_bm_GLC_PG3 = 185
const b3_p_bm_GLC_PYR = 186
const b3_p_bm_ACT_ACoA = 187
const b3_p_bm_ACT_AKG = 188
const b3_p_bm_ACT_G6P = 189
const b3_p_bm_ACT_OAA = 190
const b3_p_bm_ACT_PEP = 191
const b3_p_bm_ACT_PG3 = 192
const b3_p_bm_ACT_PYR = 193
# stoichiometric matrix (47 states x 109 rates)
const b3_S = let
S = zeros(47, 109)
S[b3_i_OD,b3_r_env_growth] = 1
S[b3_i_GLC,b3_r_env_GLCup] = -1
S[b3_i_ACT,b3_r_env_ACTup] = -1
S[b3_i_ACT,b3_r_env_ACTex] = 1
S[b3_i_AceA,b3_r_g_aceA] = 1
S[b3_i_AceA,b3_r_d_AceA] = -1
S[b3_i_AceB,b3_r_g_aceB] = 1
S[b3_i_AceB,b3_r_d_AceB] = -1
S[b3_i_AceK,b3_r_g_aceK] = 1
S[b3_i_AceK,b3_r_d_AceK] = -1
S[b3_i_Acoa2act,b3_r_g_acoa2act] = 1
S[b3_i_Acoa2act,b3_r_d_Acoa2act] = -1
S[b3_i_Acs,b3_r_g_acs] = 1
S[b3_i_Acs,b3_r_d_Acs] = -1
S[b3_i_Akg2mal,b3_r_g_akg2mal] = 1
S[b3_i_Akg2mal,b3_r_d_Akg2mal] = -1
S[b3_i_CAMPdegr,b3_r_g_campdegr] = 1
S[b3_i_CAMPdegr,b3_r_d_CAMPdegr] = -1
S[b3_i_Cra,b3_r_g_cra] = 1
S[b3_i_Cra,b3_r_d_Cra] = -1
S[b3_i_CraFBP,b3_r_d_CraFBP] = -1
S[b3_i_Crp,b3_r_g_crp] = 1
S[b3_i_Crp,b3_r_d_Crp] = -1
S[b3_i_CrpcAMP,b3_r_d_CrpcAMP] = -1
S[b3_i_Cya,b3_r_g_cya] = 1
S[b3_i_Cya,b3_r_d_Cya] = -1
S[b3_i_Emp,b3_r_g_emp] = 1
S[b3_i_Emp,b3_r_d_Emp] = -1
S[b3_i_Eno,b3_r_g_eno] = 1
S[b3_i_Eno,b3_r_d_Eno] = -1
S[b3_i_Fdp,b3_r_g_fdp] = 1
S[b3_i_Fdp,b3_r_d_Fdp] = -1
S[b3_i_GltA,b3_r_g_gltA] = 1
S[b3_i_GltA,b3_r_d_GltA] = -1
S[b3_i_Icd,b3_r_g_icd] = 1
S[b3_i_Icd,b3_r_d_Icd] = -1
S[b3_i_Icd_P,b3_r_d_Icd_P] = -1
S[b3_i_IclR,b3_r_g_iclr] = 1
S[b3_i_IclR,b3_r_d_IclR] = -1
S[b3_i_Mdh,b3_r_g_mdh] = 1
S[b3_i_Mdh,b3_r_d_Mdh] = -1
S[b3_i_MaeAB,b3_r_g_maeAB] = 1
S[b3_i_MaeAB,b3_r_d_MaeAB] = -1
S[b3_i_PckA,b3_r_g_pckA] = 1
S[b3_i_PckA,b3_r_d_PckA] = -1
S[b3_i_Pdh,b3_r_g_pdh] = 1
S[b3_i_Pdh,b3_r_d_Pdh] = -1
S[b3_i_PdhR,b3_r_g_pdhr] = 1
S[b3_i_PdhR,b3_r_d_PdhR] = -1
S[b3_i_PdhRPYR,b3_r_d_PdhRPYR] = -1
S[b3_i_PfkA,b3_r_g_pfkA] = 1
S[b3_i_PfkA,b3_r_d_PfkA] = -1
S[b3_i_Ppc,b3_r_g_ppc] = 1
S[b3_i_Ppc,b3_r_d_Ppc] = -1
S[b3_i_PpsA,b3_r_g_ppsA] = 1
S[b3_i_PpsA,b3_r_d_PpsA] = -1
S[b3_i_PykF,b3_r_g_pykF] = 1
S[b3_i_PykF,b3_r_d_PykF] = -1
S[b3_i_Icd,b3_r_e_AceK_Ki] = -1
S[b3_i_Icd,b3_r_e_AceK_Ph] = 1
S[b3_i_Icd_P,b3_r_e_AceK_Ki] = 1
S[b3_i_Icd_P,b3_r_e_AceK_Ph] = -1
S[b3_i_EIIA,b3_r_pts_r1] = -1
S[b3_i_EIIA,b3_r_pts_r4] = 1
S[b3_i_EIIA,b3_r_g_EIIA] = 1
S[b3_i_EIIA,b3_r_d_EIIA] = -1
S[b3_i_EIIA_P,b3_r_pts_r1] = 1
S[b3_i_EIIA_P,b3_r_pts_r4] = -1
S[b3_i_EIIA_P,b3_r_d_EIIA_P] = -1
S[b3_i_EIICB,b3_r_g_EIICB] = 1
S[b3_i_EIICB,b3_r_d_EIICB] = -1
S[b3_i_CraFBP,b3_r_tf_Cra] = 1
S[b3_i_Cra,b3_r_tf_Cra] = -1
S[b3_i_CrpcAMP,b3_r_tf_Crp] = 1
S[b3_i_Crp,b3_r_tf_Crp] = -1
S[b3_i_PdhRPYR,b3_r_tf_PdhR] = 1
S[b3_i_PdhR,b3_r_tf_PdhR] = -1
S[b3_i_ACoA,b3_r_e_Acs] = 1
S[b3_i_ACoA,b3_r_e_Pdh] = 1
S[b3_i_ACoA,b3_r_e_Acoa2act] = -1
S[b3_i_ACoA,b3_r_e_GltA] = -1
S[b3_i_ACoA,b3_r_e_AceB] = -1
S[b3_i_ACoA,b3_r_d_ACoA] = -1
S[b3_i_ACoA,b3_r_bm_ACoA] = -1
S[b3_i_AKG,b3_r_e_Icd] = 1
S[b3_i_AKG,b3_r_e_AceA] = 1
S[b3_i_AKG,b3_r_e_Akg2mal] = -1
S[b3_i_AKG,b3_r_d_AKG] = -1
S[b3_i_AKG,b3_r_bm_AKG] = -1
S[b3_i_cAMP,b3_r_e_Cya] = 1
S[b3_i_cAMP,b3_r_e_CAMPdegr] = -1
S[b3_i_cAMP,b3_r_d_cAMP] = -1
S[b3_i_FBP,b3_r_e_PfkA] = 1
S[b3_i_FBP,b3_r_e_Emp] = -0.5
S[b3_i_FBP,b3_r_e_Fdp] = -1
S[b3_i_FBP,b3_r_d_FBP] = -1
S[b3_i_G6P,b3_r_pts_r4] = 1
S[b3_i_G6P,b3_r_e_Fdp] = 1
S[b3_i_G6P,b3_r_e_PfkA] = -1
S[b3_i_G6P,b3_r_d_G6P] = -1
S[b3_i_G6P,b3_r_bm_G6P] = -1
S[b3_i_GLX,b3_r_e_AceA] = 1
S[b3_i_GLX,b3_r_e_AceB] = -1
S[b3_i_GLX,b3_r_d_GLX] = -1
S[b3_i_ICT,b3_r_e_GltA] = 1
S[b3_i_ICT,b3_r_e_AceA] = -1
S[b3_i_ICT,b3_r_e_Icd] = -1
S[b3_i_ICT,b3_r_d_ICT] = -1
S[b3_i_MAL,b3_r_e_AceB] = 1
S[b3_i_MAL,b3_r_e_Akg2mal] = 1
S[b3_i_MAL,b3_r_e_MaeAB] = -1
S[b3_i_MAL,b3_r_e_Mdh] = -1
S[b3_i_MAL,b3_r_d_MAL] = -1
S[b3_i_OAA,b3_r_e_PckA] = -1
S[b3_i_OAA,b3_r_e_GltA] = -1
S[b3_i_OAA,b3_r_e_Ppc] = 1
S[b3_i_OAA,b3_r_e_Mdh] = 1
S[b3_i_OAA,b3_r_d_OAA] = -1
S[b3_i_OAA,b3_r_bm_OAA] = -1
S[b3_i_PEP,b3_r_e_PckA] = 1
S[b3_i_PEP,b3_r_e_PpsA] = 1
S[b3_i_PEP,b3_r_e_PykF] = -1
S[b3_i_PEP,b3_r_e_Ppc] = -1
S[b3_i_PEP,b3_r_e_Eno] = 1
S[b3_i_PEP,b3_r_pts_r1] = -1
S[b3_i_PEP,b3_r_d_PEP] = -1
S[b3_i_PEP,b3_r_bm_PEP] = -1
S[b3_i_PG3,b3_r_e_Emp] = 1
S[b3_i_PG3,b3_r_e_Eno] = -1
S[b3_i_PG3,b3_r_d_PG3] = -1
S[b3_i_PG3,b3_r_bm_PG3] = -1
S[b3_i_PYR,b3_r_e_MaeAB] = 1
S[b3_i_PYR,b3_r_e_PykF] = 1
S[b3_i_PYR,b3_r_e_PpsA] = -1
S[b3_i_PYR,b3_r_e_Pdh] = -1
S[b3_i_PYR,b3_r_pts_r1] = 1
S[b3_i_PYR,b3_r_d_PYR] = -1
S[b3_i_PYR,b3_r_bm_PYR] = -1
S
end
# steady-state reference vectors (pure-glucose / pure-acetate steady states,
# used to interpolate a few growth-coupled rates between the two carbon sources)
const b3_ssACT = let
ssACT = zeros(47)
ssACT[b3_i_ACoA] = 2.045970006
ssACT[b3_i_AKG] = 1.11161563
ssACT[b3_i_cAMP] = 4.021170806
ssACT[b3_i_FBP] = 0.272276143
ssACT[b3_i_G6P] = 1.145931991
ssACT[b3_i_GLX] = 1.322800153
ssACT[b3_i_ICT] = 1.480239304
ssACT[b3_i_MAL] = 6.396156035
ssACT[b3_i_OAA] = 0.064573174
ssACT[b3_i_PEP] = 0.557008135
ssACT[b3_i_PG3] = 1.293579693
ssACT[b3_i_PYR] = 0.037723095
ssACT[b3_i_AceA] = 0.101787757
ssACT[b3_i_AceB] = 0.030536327
ssACT[b3_i_AceK] = 0.003053633
ssACT[b3_i_Acoa2act] = 0.001
ssACT[b3_i_Acs] = 0.010144567*0.000036201/0.001096222
ssACT[b3_i_Akg2mal] = 0.002192398
ssACT[b3_i_CAMPdegr] = 0.001
ssACT[b3_i_Cya] = 0.001
ssACT[b3_i_Emp] = 0.009751833*0.011389032/0.011515593
ssACT[b3_i_Eno] = 0.006304314*0.011389032/0.011552813
ssACT[b3_i_Fdp] = 0.000513512*0.000074810/0.000157492
ssACT[b3_i_GltA] = 0.003539257*0.000292771/0.001029612
ssACT[b3_i_Icd] = 0.002404566
ssACT[b3_i_Icd_P] = 0.007516755
ssACT[b3_i_Mdh] = 0.010969029*0.000491491/0.00345727
ssACT[b3_i_MaeAB] = 0.003399346
ssACT[b3_i_PckA] = 0.018902966*0.000336947/0.002290892
ssACT[b3_i_Pdh] = 0.001760705*0.001/0.004647401
ssACT[b3_i_PfkA] = 8.89703e-05*0.000242131/0.000143816
ssACT[b3_i_Ppc] = 0.000279893*0.000377962/0.000999714
ssACT[b3_i_PpsA] = 0.012844496
ssACT[b3_i_PykF] = 0.001305745*0.002501893/0.005977168
ssACT[b3_i_Cra] = 0.007009039
ssACT[b3_i_CraFBP] = 0.000280931
ssACT[b3_i_Crp] = 0.001327161
ssACT[b3_i_CrpcAMP] = 0.005962839
ssACT[b3_i_IclR] = 0.00729
ssACT[b3_i_PdhR] = 0.005926738
ssACT[b3_i_PdhRPYR] = 0.001363262
ssACT[b3_i_EIIA] = 0.002631995
ssACT[b3_i_EIIA_P] = 0.097368005
ssACT[b3_i_EIICB] = 0.003
ssACT
end
const b3_ssGLC = let
ssGLC = zeros(47)
ssGLC[b3_i_ACoA] = 0.351972298
ssGLC[b3_i_AKG] = 0.191190619
ssGLC[b3_i_cAMP] = 0.202804098
ssGLC[b3_i_FBP] = 6.57504207
ssGLC[b3_i_G6P] = 1.908140784
ssGLC[b3_i_GLX] = 5.70593e-09
ssGLC[b3_i_ICT] = 0.001408116
ssGLC[b3_i_MAL] = 3.278779135
ssGLC[b3_i_OAA] = 0.050535354
ssGLC[b3_i_PEP] = 0.210455879
ssGLC[b3_i_PG3] = 5.720977255
ssGLC[b3_i_PYR] = 0.863278018
ssGLC[b3_i_AceA] = 0.00472323
ssGLC[b3_i_AceB] = 0.001416969
ssGLC[b3_i_AceK] = 0.000141697
ssGLC[b3_i_Acoa2act] = 0.001
ssGLC[b3_i_Acs] = 0.000036201
ssGLC[b3_i_Akg2mal] = 0.001026848
ssGLC[b3_i_CAMPdegr] = 0.001
ssGLC[b3_i_Cya] = 0.001
ssGLC[b3_i_Emp] = 0.011389032
ssGLC[b3_i_Eno] = 0.011389032
ssGLC[b3_i_Fdp] = 0.000074810
ssGLC[b3_i_GltA] = 0.000292771
ssGLC[b3_i_Icd] = 0.004290789
ssGLC[b3_i_Icd_P] = 0.000220477
ssGLC[b3_i_Mdh] = 0.000491491
ssGLC[b3_i_MaeAB] = 0.000999714
ssGLC[b3_i_PckA] = 0.000336947
ssGLC[b3_i_Pdh] = 0.001
ssGLC[b3_i_PfkA] = 0.000242131
ssGLC[b3_i_Ppc] = 0.000377962
ssGLC[b3_i_PpsA] = 0.000987493
ssGLC[b3_i_PykF] = 0.002501893
ssGLC[b3_i_Cra] = 0.000299098
ssGLC[b3_i_CraFBP] = 0.006990902
ssGLC[b3_i_Crp] = 0.005943273
ssGLC[b3_i_CrpcAMP] = 0.001346727
ssGLC[b3_i_IclR] = 0.00729
ssGLC[b3_i_PdhR] = 0.001163813
ssGLC[b3_i_PdhRPYR] = 0.006126187
ssGLC[b3_i_EIIA] = 0.09647707
ssGLC[b3_i_EIIA_P] = 0.00352292
ssGLC[b3_i_EIICB] = 0.003
ssGLC
end
function b3_kinetics!(du, u, p, t)
x = u
S = b3_S
ssACT = b3_ssACT
ssGLC = b3_ssGLC
f = zeros(eltype(u), 109)
# rates
# Interpolate between pure glucose and pure acetate
# growth conditions
alphaGLC = x[b3_i_GLC]/(x[b3_i_GLC]+p[b3_p_pts_Kglc]);
alphaACT = x[b3_i_ACT]/(x[b3_i_ACT]+p[b3_p_e_Acs_Kact])*(1-x[b3_i_GLC]/(x[b3_i_GLC]+p[b3_p_pts_Kglc]));
# Calculate the growth rate 'mu'
mu = alphaGLC*p[b3_p_bm_muGLC] + alphaACT*p[b3_p_bm_muACT];
# Calculate the first order rate constants of the seven biomass reactions
k_bm_ACoA = alphaGLC*p[b3_p_bm_GLC_ACoA] + alphaACT*p[b3_p_bm_ACT_ACoA];
k_bm_AKG = alphaGLC*p[b3_p_bm_GLC_AKG] + alphaACT*p[b3_p_bm_ACT_AKG];
k_bm_G6P = alphaGLC*p[b3_p_bm_GLC_G6P] + alphaACT*p[b3_p_bm_ACT_G6P];
k_bm_OAA = alphaGLC*p[b3_p_bm_GLC_OAA] + alphaACT*p[b3_p_bm_ACT_OAA];
k_bm_PEP = alphaGLC*p[b3_p_bm_GLC_PEP] + alphaACT*p[b3_p_bm_ACT_PEP];
k_bm_PG3 = alphaGLC*p[b3_p_bm_GLC_PG3] + alphaACT*p[b3_p_bm_ACT_PG3];
k_bm_PYR = alphaGLC*p[b3_p_bm_GLC_PYR] + alphaACT*p[b3_p_bm_ACT_PYR];
f[b3_r_bm_ACoA] = k_bm_ACoA * x[b3_i_ACoA];
f[b3_r_bm_AKG] = k_bm_AKG * x[b3_i_AKG];
f[b3_r_bm_G6P] = k_bm_G6P * x[b3_i_G6P];
f[b3_r_bm_OAA] = k_bm_OAA * x[b3_i_OAA];
f[b3_r_bm_PEP] = k_bm_PEP * x[b3_i_PEP];
f[b3_r_bm_PG3] = k_bm_PG3 * x[b3_i_PG3];
f[b3_r_bm_PYR] = k_bm_PYR * x[b3_i_PYR];
# Calculate biomass reaction rates with 1st order kinetics
# Calculate the actual steady state levels of MaeAB and Ppc
SS_MaeAB = alphaGLC*ssGLC[b3_i_MaeAB] + alphaACT*ssACT[b3_i_MaeAB];
SS_Ppc = alphaGLC*ssGLC[b3_i_Ppc] + alphaACT*ssACT[b3_i_Ppc];
# Protein phosphorylation rates
# PTS phosphorylation kinetics
f[b3_r_pts_r1] = p[b3_p_pts_k1]*x[b3_i_PEP]*x[b3_i_EIIA]-p[b3_p_pts_km1]*x[b3_i_PYR]*x[b3_i_EIIA_P];
f[b3_r_pts_r4] = p[b3_p_pts_k4]*x[b3_i_EIICB]*x[b3_i_EIIA_P]*x[b3_i_GLC]/((p[b3_p_pts_KEIIA]+x[b3_i_EIIA_P])*(p[b3_p_pts_Kglc]+x[b3_i_GLC]));
# AceK_ki kinetics: MWC, substrate: Icd, inhibitors: GLX, ICT, OAA,
# PEP, PG3, PYR, AKG
f[b3_r_e_AceK_Ki] = x[b3_i_AceK]*p[b3_p_e_AceK_kcat_ki]*x[b3_i_Icd]/p[b3_p_e_AceK_Kicd]*(1+x[b3_i_Icd]/p[b3_p_e_AceK_Kicd])^(p[b3_p_e_AceK_n]-1)/((1+x[b3_i_Icd]/p[b3_p_e_AceK_Kicd])^p[b3_p_e_AceK_n]+p[b3_p_e_AceK_L]*(1+x[b3_i_ICT]/p[b3_p_e_AceK_Kict]+x[b3_i_GLX]/p[b3_p_e_AceK_Kglx]+x[b3_i_OAA]/p[b3_p_e_AceK_Koaa]+x[b3_i_AKG]/p[b3_p_e_AceK_Kakg]+x[b3_i_PEP]/p[b3_p_e_AceK_Kpep]+x[b3_i_PG3]/p[b3_p_e_AceK_Kpg3]+x[b3_i_PYR]/p[b3_p_e_AceK_Kpyr])^p[b3_p_e_AceK_n]);
# AceK_ph kinetics: MWC, substrate: Icd_P, activators: OAA, PEP,
# PG3, PYR, AKG
f[b3_r_e_AceK_Ph] = x[b3_i_AceK]*p[b3_p_e_AceK_kcat_ph]*x[b3_i_Icd_P]/p[b3_p_e_AceK_Kicd_P]*(1+x[b3_i_Icd_P]/p[b3_p_e_AceK_Kicd_P])^(p[b3_p_e_AceK_n]-1)/((1+x[b3_i_Icd_P]/p[b3_p_e_AceK_Kicd_P])^p[b3_p_e_AceK_n]+p[b3_p_e_AceK_L]/(1+x[b3_i_OAA]/p[b3_p_e_AceK_Koaa]+x[b3_i_AKG]/p[b3_p_e_AceK_Kakg]+x[b3_i_PEP]/p[b3_p_e_AceK_Kpep]+x[b3_i_PG3]/p[b3_p_e_AceK_Kpg3]+x[b3_i_PYR]/p[b3_p_e_AceK_Kpyr])^p[b3_p_e_AceK_n]);
# Metabolite- transcription factor binding rates
# The IclR-GLX-PYR binding state is incorporated into the gene
# expression kinetics of the aceBAK operon
# Cra-FBP binding kinetics: Hill
f[b3_r_tf_Cra] = p[b3_p_tf_Cra_scale]*((x[b3_i_Cra]+x[b3_i_CraFBP])*max(x[b3_i_FBP], 0.0)^p[b3_p_tf_Cra_n]/(max(x[b3_i_FBP], 0.0)^p[b3_p_tf_Cra_n]+p[b3_p_tf_Cra_kfbp]^p[b3_p_tf_Cra_n])-x[b3_i_CraFBP]);
# Crp-cAMP binding kinetics: Hill
f[b3_r_tf_Crp] = p[b3_p_tf_Crp_scale]*((x[b3_i_Crp]+x[b3_i_CrpcAMP])*max(x[b3_i_cAMP], 0.0)^p[b3_p_tf_Crp_n]/(max(x[b3_i_cAMP], 0.0)^p[b3_p_tf_Crp_n]+p[b3_p_tf_Crp_kcamp]^p[b3_p_tf_Crp_n])-x[b3_i_CrpcAMP]);
# PdhR-PYR binding kinetics: Hill
f[b3_r_tf_PdhR] = p[b3_p_tf_PdhR_scale]*((x[b3_i_PdhR]+x[b3_i_PdhRPYR])*max(x[b3_i_PYR], 0.0)^p[b3_p_tf_PdhR_n]/(max(x[b3_i_PYR], 0.0)^p[b3_p_tf_PdhR_n]+p[b3_p_tf_PdhR_kpyr]^p[b3_p_tf_PdhR_n])-x[b3_i_PdhRPYR]);
# Metabolic reaction rates
# AceA kinetics: MWC, substrate: ICT, inhibitors: PG3, PEP, AKG
f[b3_r_e_AceA] = x[b3_i_AceA]*p[b3_p_e_AceA_kcat]*x[b3_i_ICT]/p[b3_p_e_AceA_Kict]*(1+x[b3_i_ICT]/p[b3_p_e_AceA_Kict])^(p[b3_p_e_AceA_n]-1)/((1+x[b3_i_ICT]/p[b3_p_e_AceA_Kict])^p[b3_p_e_AceA_n]+p[b3_p_e_AceA_L]*(1+x[b3_i_PEP]/p[b3_p_e_AceA_Kpep]+x[b3_i_PG3]/p[b3_p_e_AceA_Kpg3]+x[b3_i_AKG]/p[b3_p_e_AceA_Kakg])^p[b3_p_e_AceA_n]);
# AceB kinetics: Two-substrate MM, substrates: GLX, ACoA
f[b3_r_e_AceB] = x[b3_i_AceB]*p[b3_p_e_AceB_kcat]*x[b3_i_GLX]*x[b3_i_ACoA]/(p[b3_p_e_AceB_Kglxacoa]*p[b3_p_e_AceB_Kacoa]+p[b3_p_e_AceB_Kacoa]*x[b3_i_GLX]+p[b3_p_e_AceB_Kglx]*x[b3_i_ACoA]+x[b3_i_GLX]*x[b3_i_ACoA]);
# Acoa2act kinetics: MWC, substrate: ACoA, activator: PYR
f[b3_r_e_Acoa2act] = x[b3_i_Acoa2act]*p[b3_p_e_Acoa2act_kcat]*x[b3_i_ACoA]/p[b3_p_e_Acoa2act_Kacoa]*(1+x[b3_i_ACoA]/p[b3_p_e_Acoa2act_Kacoa])^(p[b3_p_e_Acoa2act_n]-1)/((1+x[b3_i_ACoA]/p[b3_p_e_Acoa2act_Kacoa])^p[b3_p_e_Acoa2act_n]+p[b3_p_e_Acoa2act_L]/(1+x[b3_i_PYR]/p[b3_p_e_Acoa2act_Kpyr])^p[b3_p_e_Acoa2act_n]);
# Acs kinetics: MM, substrate: ACT
f[b3_r_e_Acs] = x[b3_i_Acs]*p[b3_p_e_Acs_kcat]*x[b3_i_ACT]/(x[b3_i_ACT]+p[b3_p_e_Acs_Kact]);
# Akg2mal kinetics: MM, substrate: AKG
f[b3_r_e_Akg2mal] = x[b3_i_Akg2mal]*p[b3_p_e_Akg2mal_kcat]*x[b3_i_AKG]/(x[b3_i_AKG]+p[b3_p_e_Akg2mal_Kakg]);
# CAMPdegr kinetics: MM, substrate: cAMP
f[b3_r_e_CAMPdegr] = p[b3_p_e_CAMPdegr_kcat]*x[b3_i_CAMPdegr]*x[b3_i_cAMP]/(x[b3_i_cAMP]+p[b3_p_e_CAMPdegr_KcAMP]);
# Cya kinetics: MM, substrate: Cya
f[b3_r_e_Cya] = p[b3_p_e_Cya_kcat]*x[b3_i_Cya]*x[b3_i_EIIA_P]/(x[b3_i_EIIA_P]+p[b3_p_e_Cya_KEIIA]);
# Emp kinetics: reversible MM, substrates: FBP, PG3
f[b3_r_e_Emp] = (x[b3_i_Emp]*p[b3_p_e_Emp_kcat_f]*x[b3_i_FBP]/p[b3_p_e_Emp_Kfbp]-x[b3_i_Emp]*p[b3_p_e_Emp_kcat_r]*x[b3_i_PG3]/p[b3_p_e_Emp_Kpg3])/(1+x[b3_i_FBP]/p[b3_p_e_Emp_Kfbp]+x[b3_i_PG3]/p[b3_p_e_Emp_Kpg3]);
# Eno kinetics: reversible MM, substrates: PG3, PEP
f[b3_r_e_Eno] = (x[b3_i_Eno]*p[b3_p_e_Eno_kcatf]*x[b3_i_PG3]/p[b3_p_e_Eno_Kpg3]-x[b3_i_Eno]*p[b3_p_e_Eno_kcatr]*x[b3_i_PEP]/p[b3_p_e_Eno_Kpep])/(1+x[b3_i_PG3]/p[b3_p_e_Eno_Kpg3]+x[b3_i_PEP]/p[b3_p_e_Eno_Kpep]);
# Fdp kinetics: MWC, substrate: FBP, activator: PEP
f[b3_r_e_Fdp] = x[b3_i_Fdp]*p[b3_p_e_Fdp_kcat]*x[b3_i_FBP]/p[b3_p_e_Fdp_Kfbp]*(1+x[b3_i_FBP]/p[b3_p_e_Fdp_Kfbp])^(p[b3_p_e_Fdp_n]-1)/((1+x[b3_i_FBP]/p[b3_p_e_Fdp_Kfbp])^p[b3_p_e_Fdp_n]+p[b3_p_e_Fdp_L]/(1+x[b3_i_PEP]/p[b3_p_e_Fdp_Kpep])^p[b3_p_e_Fdp_n]);
# GltA kinetics: Two-substrate MM, substrates: OAA, ACoA,
# competitive inhibitor: AKG
f[b3_r_e_GltA] = x[b3_i_GltA]*p[b3_p_e_GltA_kcat]*x[b3_i_OAA]*x[b3_i_ACoA]/((1+x[b3_i_AKG]/p[b3_p_e_GltA_Kakg])*p[b3_p_e_GltA_Koaaacoa]*p[b3_p_e_GltA_Kacoa]+p[b3_p_e_GltA_Kacoa]*x[b3_i_OAA]+(1+x[b3_i_AKG]/p[b3_p_e_GltA_Kakg])*p[b3_p_e_GltA_Koaa]*x[b3_i_ACoA]+x[b3_i_OAA]*x[b3_i_ACoA]);
# Icd kinetics: MWC, substrate: ICT, inhibitor: PEP
f[b3_r_e_Icd] = x[b3_i_Icd]*p[b3_p_e_Icd_kcat]*x[b3_i_ICT]/p[b3_p_e_Icd_Kict]*(1+x[b3_i_ICT]/p[b3_p_e_Icd_Kict])^(p[b3_p_e_Icd_n]-1)/((1+x[b3_i_ICT]/p[b3_p_e_Icd_Kict])^p[b3_p_e_Icd_n]+p[b3_p_e_Icd_L]*(1+x[b3_i_PEP]/p[b3_p_e_Icd_Kpep])^p[b3_p_e_Icd_n]);
# MaeAB kinetics: MWC, substrate: MAL, inhibitors: AcoA, cAMP
f[b3_r_e_MaeAB] = x[b3_i_MaeAB]*p[b3_p_e_MaeAB_kcat]*x[b3_i_MAL]/p[b3_p_e_MaeAB_Kmal]*(1+x[b3_i_MAL]/p[b3_p_e_MaeAB_Kmal])^(p[b3_p_e_MaeAB_n]-1)/((1+x[b3_i_MAL]/p[b3_p_e_MaeAB_Kmal])^p[b3_p_e_MaeAB_n]+p[b3_p_e_MaeAB_L]*(1+x[b3_i_ACoA]/p[b3_p_e_MaeAB_Kacoa]+x[b3_i_cAMP]/p[b3_p_e_MaeAB_Kcamp])^p[b3_p_e_MaeAB_n]);
# Mdh kinetics: Hill, substrate: MAL
f[b3_r_e_Mdh] = x[b3_i_Mdh]*p[b3_p_e_Mdh_kcat]*max(x[b3_i_MAL], 0.0)^p[b3_p_e_Mdh_n]/(max(x[b3_i_MAL], 0.0)^p[b3_p_e_Mdh_n]+p[b3_p_e_Mdh_Kmal]^p[b3_p_e_Mdh_n]);
# PckA kinetics: MM, substrate: OAA, competitive inhibitor: PEP
f[b3_r_e_PckA] = x[b3_i_PckA]*p[b3_p_e_PckA_kcat]*x[b3_i_OAA]/(x[b3_i_OAA]+p[b3_p_e_PckA_Koaa]*(1+x[b3_i_PEP]/p[b3_p_e_PckA_Kpep]));
# Pdh kinetics: MWC, substrate: PYR, inhibitors: GLX, PYR
f[b3_r_e_Pdh] = x[b3_i_Pdh]*p[b3_p_e_Pdh_kcat]*x[b3_i_PYR]/p[b3_p_e_Pdh_Kpyr]*(1+x[b3_i_PYR]/p[b3_p_e_Pdh_Kpyr])^(p[b3_p_e_Pdh_n]-1)/((1+x[b3_i_PYR]/p[b3_p_e_Pdh_Kpyr])^p[b3_p_e_Pdh_n]+p[b3_p_e_Pdh_L]*(1+x[b3_i_GLX]/p[b3_p_e_Pdh_Kglx]+x[b3_i_PYR]/p[b3_p_e_Pdh_KpyrI])^p[b3_p_e_Pdh_n]);
# PfkA kinetics: MWC, substrate: G6P, inhibitor: PEP
f[b3_r_e_PfkA] = x[b3_i_PfkA]*p[b3_p_e_PfkA_kcat]*x[b3_i_G6P]/p[b3_p_e_PfkA_Kg6p]*(1+x[b3_i_G6P]/p[b3_p_e_PfkA_Kg6p])^(p[b3_p_e_PfkA_n]-1)/((1+x[b3_i_G6P]/p[b3_p_e_PfkA_Kg6p])^p[b3_p_e_PfkA_n]+p[b3_p_e_PfkA_L]*(1+x[b3_i_PEP]/p[b3_p_e_PfkA_Kpep])^p[b3_p_e_PfkA_n]);
# Ppc kinetics: MWC, substrate: PEP, activator: FBP
f[b3_r_e_Ppc] = x[b3_i_Ppc]*p[b3_p_e_Ppc_kcat]*x[b3_i_PEP]/p[b3_p_e_Ppc_Kpep]*(1+x[b3_i_PEP]/p[b3_p_e_Ppc_Kpep])^(p[b3_p_e_Ppc_n]-1)/((1+x[b3_i_PEP]/p[b3_p_e_Ppc_Kpep])^p[b3_p_e_Ppc_n]+p[b3_p_e_Ppc_L]/(1+x[b3_i_FBP]/p[b3_p_e_Ppc_Kfbp])^p[b3_p_e_Ppc_n]);
# PpsA kinetics: MWC, substrate: PYR, inhibitor: PEP
f[b3_r_e_PpsA] = x[b3_i_PpsA]*p[b3_p_e_PpsA_kcat]*x[b3_i_PYR]/p[b3_p_e_PpsA_Kpyr]*(1+x[b3_i_PYR]/p[b3_p_e_PpsA_Kpyr])^(p[b3_p_e_PpsA_n]-1)/((1+x[b3_i_PYR]/p[b3_p_e_PpsA_Kpyr])^p[b3_p_e_PpsA_n]+p[b3_p_e_PpsA_L]*(1+x[b3_i_PEP]/p[b3_p_e_PpsA_Kpep])^p[b3_p_e_PpsA_n]);
# PykF kinetics: MWC, substrate: PEP, activator: FBP
f[b3_r_e_PykF] = x[b3_i_PykF]*p[b3_p_e_PykF_kcat]*x[b3_i_PEP]/p[b3_p_e_PykF_Kpep]*(1+x[b3_i_PEP]/p[b3_p_e_PykF_Kpep])^(p[b3_p_e_PykF_n]-1)/((1+x[b3_i_PEP]/p[b3_p_e_PykF_Kpep])^p[b3_p_e_PykF_n]+p[b3_p_e_PykF_L]/(1+x[b3_i_FBP]/p[b3_p_e_PykF_Kfbp])^p[b3_p_e_PykF_n]);
# Gene expression rates
# aceBAK expression: sum of the following three kinetics
# MM plus basal expression, substrate: Cra
# MM plus basal expression, substrate: Crpcamp
# MWC-like, substrate: IclR, activator: GLX, inhibitor: PYR
# aceB and aceK expression are coupled to aceA expression
# with constant factors
f[b3_r_g_aceA] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_aceBAK_Kcra]))*p[b3_p_g_aceBAK_vcra_unbound] +x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_aceBAK_Kcra])*p[b3_p_g_aceBAK_vcra_bound] +(1-x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_aceBAK_Kcrp]))*p[b3_p_g_aceBAK_vcrp_unbound] +x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_aceBAK_Kcrp])*p[b3_p_g_aceBAK_vcrp_bound] +p[b3_p_g_aceBAK_kcat_iclr]*x[b3_i_IclR]*(1-(p[b3_p_g_aceBAK_DNA]/p[b3_p_g_aceBAK_KDNA]) *(1+x[b3_i_PYR]/p[b3_p_g_aceBAK_KPprime])/(1+(x[b3_i_GLX]/p[b3_p_g_aceBAK_KG] *(1+x[b3_i_GLX]/p[b3_p_g_aceBAK_KG]))/p[b3_p_g_aceBAK_L] +p[b3_p_g_aceBAK_DNA]/p[b3_p_g_aceBAK_KDNA]+x[b3_i_PYR]/p[b3_p_g_aceBAK_KP] +p[b3_p_g_aceBAK_DNA]*x[b3_i_PYR]/p[b3_p_g_aceBAK_KDNA]/p[b3_p_g_aceBAK_KPprime])));
f[b3_r_g_aceB] = p[b3_p_g_aceBAK_aceBfactor]*f[b3_r_g_aceA];
f[b3_r_g_aceK] = p[b3_p_g_aceBAK_aceKfactor]*f[b3_r_g_aceA];
# acoa2act kinetics: constitutive expression
f[b3_r_g_acoa2act] = 0;
# acs kinetics: Hill plus basal expression, substrate: Crpcamp
f[b3_r_g_acs] = p[b3_p_bm_k_expr]*mu*((1-max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_acs_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_acs_n]+p[b3_p_g_acs_Kcrp]^p[b3_p_g_acs_n]))*p[b3_p_g_acs_vcrp_unbound]+max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_acs_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_acs_n]+p[b3_p_g_acs_Kcrp]^p[b3_p_g_acs_n])*p[b3_p_g_acs_vcrp_bound]);
# akg2mal kinetics: Hill plus basal expression, substrate: Crpcamp
f[b3_r_g_akg2mal] = p[b3_p_bm_k_expr]*mu*((1-max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_akg2mal_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_akg2mal_n]+p[b3_p_g_akg2mal_Kcrp]^p[b3_p_g_akg2mal_n]))*p[b3_p_g_akg2mal_vcrp_unbound]+max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_akg2mal_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_akg2mal_n]+p[b3_p_g_akg2mal_Kcrp]^p[b3_p_g_akg2mal_n])*p[b3_p_g_akg2mal_vcrp_bound]);
# campdegr kinetics: constitutive expression
f[b3_r_g_campdegr] = 0;
# cra kinetics: constitutive expression
f[b3_r_g_cra] = 0;
# crp kinetics: constitutive expression
f[b3_r_g_crp] = 0;
# cya kinetics: constitutive expression
f[b3_r_g_cya] = 0;
# emp kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_emp] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_emp_Kcra]))*p[b3_p_g_emp_vcra_unbound] +x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_emp_Kcra])*p[b3_p_g_emp_vcra_bound] +(1-x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_emp_Kcrp]))*p[b3_p_g_emp_vcrp_unbound] +x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_emp_Kcrp])*p[b3_p_g_emp_vcrp_bound]);
# eno kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_eno] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_eno_Kcra]))*p[b3_p_g_eno_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_eno_Kcra])*p[b3_p_g_eno_vcra_bound]);
# fdp kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_fdp] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_fdp_Kcra]))*p[b3_p_g_fdp_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_fdp_Kcra])*p[b3_p_g_fdp_vcra_bound]);
# gltA kinetics: Hill plus basal expression, substrate: Crpcamp
f[b3_r_g_gltA] = p[b3_p_bm_k_expr]*mu*((1-max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_gltA_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_gltA_n]+p[b3_p_g_gltA_Kcrp]^p[b3_p_g_gltA_n]))*p[b3_p_g_gltA_vcrp_unbound]+max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_gltA_n]/(max(x[b3_i_CrpcAMP], 0.0)^p[b3_p_g_gltA_n]+p[b3_p_g_gltA_Kcrp]^p[b3_p_g_gltA_n])*p[b3_p_g_gltA_vcrp_bound]);
# icd kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_icd] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_icd_Kcra]))*p[b3_p_g_icd_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_icd_Kcra])*p[b3_p_g_icd_vcra_bound]);
# iclr expression: constitutive
f[b3_r_g_iclr] = 0;
# mdh kinetics: MM plus basal expression, substrate: Crpcamp
f[b3_r_g_mdh] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_mdh_Kcrp]))*p[b3_p_g_mdh_vcrp_unbound]+x[b3_i_CrpcAMP]/(x[b3_i_CrpcAMP]+p[b3_p_g_mdh_Kcrp])*p[b3_p_g_mdh_vcrp_bound]);
# me kinetics: growth rate- dependent constitutive expression
f[b3_r_g_maeAB] = (mu+p[b3_p_d_k_degr])*SS_MaeAB;
# pckA kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_pckA] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pckA_Kcra]))*p[b3_p_g_pckA_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pckA_Kcra])*p[b3_p_g_pckA_vcra_bound]);
# pdh kinetics: MM plus basal expression, substrate: PdhR
f[b3_r_g_pdh] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_PdhR]/(x[b3_i_PdhR]+p[b3_p_g_pdh_Kpdhr]))*p[b3_p_g_pdh_vpdhr_unbound]+x[b3_i_PdhR]/(x[b3_i_PdhR]+p[b3_p_g_pdh_Kpdhr])*p[b3_p_g_pdh_vpdhr_bound]);
# pdhr expression: constitutive
f[b3_r_g_pdhr] = 0;
# pfkA kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_pfkA] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pfkA_Kcra]))*p[b3_p_g_pfkA_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pfkA_Kcra])*p[b3_p_g_pfkA_vcra_bound]);
# ppc kinetics: growth rate- dependent constitutive expression
f[b3_r_g_ppc] = (mu+p[b3_p_d_k_degr])*SS_Ppc;
# ppsA kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_ppsA] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_ppsA_Kcra]))*p[b3_p_g_ppsA_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_ppsA_Kcra])*p[b3_p_g_ppsA_vcra_bound]);
# pykF kinetics: MM plus basal expression, substrate: Cra
f[b3_r_g_pykF] = p[b3_p_bm_k_expr]*mu*((1-x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pykF_Kcra]))*p[b3_p_g_pykF_vcra_unbound]+x[b3_i_Cra]/(x[b3_i_Cra]+p[b3_p_g_pykF_Kcra])*p[b3_p_g_pykF_vcra_bound]);
# EIIA kinetics: constitutive expression
f[b3_r_g_EIIA] = 0;
# EIICB kinetics: constitutive expression
f[b3_r_g_EIICB] = 0;
# Protein degradation and dilution rates
# Constitutively produced proteins are neither produced,
# nor degraded, nor diluted, to keep their levels constant
# All other proteins degrade with a constant rate,
# and dilute with the growth rate
f[b3_r_d_AceA] = (mu+p[b3_p_d_k_degr])*x[b3_i_AceA];
f[b3_r_d_AceB] = (mu+p[b3_p_d_k_degr])*x[b3_i_AceB];
f[b3_r_d_AceK] = (mu+p[b3_p_d_k_degr])*x[b3_i_AceK];
f[b3_r_d_Acoa2act] = 0;
f[b3_r_d_Acs] = (mu+p[b3_p_d_k_degr])*x[b3_i_Acs];
f[b3_r_d_Akg2mal] = (mu+p[b3_p_d_k_degr])*x[b3_i_Akg2mal];
f[b3_r_d_CAMPdegr] = 0;
f[b3_r_d_Cra] = 0;
f[b3_r_d_CraFBP] = 0;
f[b3_r_d_Crp] = 0;
f[b3_r_d_CrpcAMP] = 0;
f[b3_r_d_Cya] = 0;
f[b3_r_d_Emp] = (mu+p[b3_p_d_k_degr])*x[b3_i_Emp];
f[b3_r_d_Eno] = (mu+p[b3_p_d_k_degr])*x[b3_i_Eno];
f[b3_r_d_Fdp] = (mu+p[b3_p_d_k_degr])*x[b3_i_Fdp];
f[b3_r_d_GltA] = (mu+p[b3_p_d_k_degr])*x[b3_i_GltA];
f[b3_r_d_Icd] = (mu+p[b3_p_d_k_degr])*x[b3_i_Icd];
f[b3_r_d_Icd_P] = (mu+p[b3_p_d_k_degr])*x[b3_i_Icd_P];
f[b3_r_d_IclR] = 0;
f[b3_r_d_Mdh] = (mu+p[b3_p_d_k_degr])*x[b3_i_Mdh];
f[b3_r_d_MaeAB] = (mu+p[b3_p_d_k_degr])*x[b3_i_MaeAB];
f[b3_r_d_PckA] = (mu+p[b3_p_d_k_degr])*x[b3_i_PckA];
f[b3_r_d_Pdh] = (mu+p[b3_p_d_k_degr])*x[b3_i_Pdh];
f[b3_r_d_PdhR] = 0;
f[b3_r_d_PdhRPYR] = 0;
f[b3_r_d_PfkA] = (mu+p[b3_p_d_k_degr])*x[b3_i_PfkA];
f[b3_r_d_Ppc] = (mu+p[b3_p_d_k_degr])*x[b3_i_Ppc];
f[b3_r_d_PpsA] = (mu+p[b3_p_d_k_degr])*x[b3_i_PpsA];
f[b3_r_d_PykF] = (mu+p[b3_p_d_k_degr])*x[b3_i_PykF];
f[b3_r_d_EIIA] = 0;
f[b3_r_d_EIIA_P] = 0;
f[b3_r_d_EIICB] = 0;
# Metabolite dilution rates
# Intracellular metabolites do not degrade,
# only dilute with the growth rate
f[b3_r_d_ACoA] = mu*x[b3_i_ACoA];
f[b3_r_d_AKG] = mu*x[b3_i_AKG];
f[b3_r_d_cAMP] = mu*x[b3_i_cAMP];
f[b3_r_d_FBP] = mu*x[b3_i_FBP];
f[b3_r_d_G6P] = mu*x[b3_i_G6P];
f[b3_r_d_GLX] = mu*x[b3_i_GLX];
f[b3_r_d_ICT] = mu*x[b3_i_ICT];
f[b3_r_d_MAL] = mu*x[b3_i_MAL];
f[b3_r_d_OAA] = mu*x[b3_i_OAA];
f[b3_r_d_PEP] = mu*x[b3_i_PEP];
f[b3_r_d_PG3] = mu*x[b3_i_PG3];
f[b3_r_d_PYR] = mu*x[b3_i_PYR];
# Environmental interaction
# Let the cell population grow with the actual growth rate
f[b3_r_env_growth] = x[b3_i_OD]*mu;
# Scale glucose uptake from the environment with the
# actual population size
f[b3_r_env_GLCup] = p[b3_p_env_uc]*p[b3_p_env_M_GLC]*x[b3_i_OD]*f[b3_r_pts_r4];
# Scale acetate uptake from the environment with the
# actual population size
f[b3_r_env_ACTup] = p[b3_p_env_uc]*p[b3_p_env_M_ACT]*x[b3_i_OD]*f[b3_r_e_Acs];
# Scale acetate excretion to the environment with the
# actual population size for all scenarios except 1 and 4,
# for which glucose should remain the sole carbon source -
# in these cases, the excreted acetate is directed to nowhere
f[b3_r_env_ACTex] = p[b3_p_env_uc]*p[b3_p_env_M_ACT]*x[b3_i_OD]*f[b3_r_e_Acoa2act];
mul!(du, S, f)
nothing
endb3_kinetics! (generic function with 1 method)Nominal parameters, bounds, and starting guess
Of the model's 193 parameters, 15 are known constants (the Hill/oligomerization coefficients embedded in the b3_fixed_idx/b3_fixed_val pairs below, plus the aceB/aceK operon-coupling factors, the universal protein degradation rate k_degr, and the gene expression rate constant k_expr); the other 178 are estimated. Bounds are the reference's own 0.1 x-10 x around each nominal value, except the seven Hill coefficients among the 178 (tighter, fixed [0.5, 4] bounds) and parameters smaller than 1e-8 (given a fixed upper bound of 1e-7 instead of 10x a near-zero number) – all transcribed directly from the reference b3_bounds.mat/b3_amigo.m. The starting guess p_start is the reference's own saved random point inside these bounds (b3_bounds.mat); following the same convention as B5, and per the reference documentation's explicit warning, p_nom is used only for post-estimation comparison, never as an optimization starting point.
# expand the 178 estimated parameters into the full 193-parameter vector used by
# the kinetics function, re-inserting the 15 known/fixed constants at their
# original positions (Hill coefficients, two scaling factors, the protein
# degradation rate, and the gene-expression rate constant)
const b3_fixed_idx = [5, 17, 29, 50, 60, 68, 77, 83, 88, 93, 98, 119, 120, 176, 177]
const b3_fixed_val = [4.0, 2.0, 2.0, 4.0, 2.0, 1.33, 2.65, 4.0, 3.0, 2.0, 4.0, 0.3, 0.03, 2.8e-5, 2.0e4]
function b3_expand_params(p178)
p193 = zeros(eltype(p178), 193)
j = 0
for i in 1:193
if i in b3_fixed_idx
continue
end
j += 1
p193[i] = p178[j]
end
for (idx, val) in zip(b3_fixed_idx, b3_fixed_val)
p193[idx] = val
end
return p193
end
# nominal parameters, bounds, and starting guess for the 178 estimated
# B3 parameters, transcribed from the reference pnom.mat / b3_bounds.mat
const b3_p_nom = [
60.05, 180.156, 9.5e-07, 614.0, 50100.0, 0.022, 0.055, 0.72, 0.827, 47.8, 0.95, 0.755,
0.719, 3400000000000.0, 1700000000.0, 100000000.0, 0.043, 0.643, 0.539, 0.038, 0.173,
0.866, 0.82, 1.57, 0.137, 3079.0, 639000.0, 0.022, 0.022, 10295.72332256015, 0.001, 1530.0,
0.548, 1000.0, 0.1, 993.0, 0.0017, 1011.1125335322615, 857.4234284353578, 5.92, 16.6,
704.9945100689857, 529.5066679942597, 4.76, 1.11, 404.2035022055875, 4000000.0, 0.003, 0.3,
5676.0873447165195, 0.029, 0.212, 0.029, 0.63, 695.0, 127.0, 0.00016, 0.334,
5437.474358635255, 1.7, 10.1, 1879.0, 104000.0, 0.00624, 3.64, 6.54, 377.34274529822204,
0.184, 1000.0, 5479.285779, 3.4, 0.128, 0.231, 0.218, 539315.197145347, 95000000.0, 0.022,
0.138, 14904.6422391669, 5200000.0, 0.048, 0.408, 1.32, 1e-79, 0.00177, 0.001,
13734.69562127557, 100000.0, 5.0, 0.413, 116.0, 46.3, 2520.0, 0.0085, 0.0012, 100.0, 1.36,
2.0, 100000000.0, 0.895, 1.0, 100.0, 0.164, 1.0, 1.9e-09, 2e-06, 0.00365, 2.19, 0.897,
0.00301, 0.00488, 923.0, 0.00093, 1.0, 2.3e-10, 2e-08, 0.341, 0.0, 3.962810452627297e-08,
2.31, 0.0047, 0.0, 1.4e-06, 0.091, 0.74, 6.131859505628587e-07, 0.0, 0.09, 0.0, 4.7e-07,
0.012, 6.703598301123717e-07, 0.0, 0.016, 0.0, 2.13753714474386e-08, 0.00118, 0.0,
6.540068491820218e-07, 0.04, 1.07, 1.1e-07, 8.5e-07, 0.00117, 0.0, 1.2936704683174875e-06,
0.06, 0.0, 3.6770284238628443e-07, 0.00535, 7.74626506298897e-08, 2.797262383857128e-10,
0.0034, 1.380565583801524e-06, 1.1111869333036659e-08, 6.3e-07, 0.0, 3.3e-06, 0.017,
1.6324424376226333e-07, 8.790074664121873e-10, 0.0023, 5.6e-05, 0.00018, 1.88, 0.978,
0.154, 6.4, 0.423, 0.049, 0.553, 0.108, 0.056, 0.076, 1.43, 0.047, 0.066, 5.185
]
const b3_p_start = [
51.9872906889458, 761.6372473233096, 4.259319313214376e-06, 1093.5368179193445,
107602.48608345521, 0.12856327802490058, 0.35115086815391283, 1.367540794523535,
7.1291370173004225, 282.35998973420334, 1.512704967288441, 5.868498764277723,
4.893534020169673, 2480961073516.458, 10722519207.942562, 529526474.2185608,
0.0337670534804412, 4.689030575165757, 2.744805411137642, 0.30612708022470975,
0.2364511871216731, 1.338822559818776, 6.835380736104081, 7.200646034781244,
0.5117736406914741, 7489.119309735761, 1579798.089737093, 0.12091190728211798,
0.051005136021099656, 75667.03540669332, 0.009996239720450498, 507.1702996948929,
2.0086469566199767, 9742.597256139654, 0.8411808627229941, 2438.43463289049,
0.010446479058245566, 1830.4388677638158, 2557.7465381315074, 13.321171713997638,
165.56775370159167, 4173.7214323588505, 515.0360706943621, 32.37654019854651,
5.944775178949121, 1996.1615126943157, 36149566.14653705, 0.015326867540911813,
0.2826367800934886, 9939.710450787376, 0.05873391194167871, 0.1855681758846465,
0.18314779862524344, 6.2564240949126795, 4111.260304238739, 303.4217410898057,
2.5947967885760953e-05, 3.1890181214304074, 53103.125641475235, 0.6202144599860606,
3.8157650593330805, 4414.650828918563, 321100.8329489221, 0.011471969346310007,
1.9559752286652528, 13.894635186803768, 2960.475025096943, 0.7465710384619573,
1923.9293224618646, 14280.00683915794, 13.958901884433882, 1.2087313867321907,
1.9012665002860087, 0.9590967279283514, 5294736.118925223, 516874475.237177,
0.009076246135215845, 0.7500891004589059, 59890.44699564487, 5194927.46250496,
0.08349184149952518, 3.7926813614790076, 0.7994130369805195, 9.485617857371937e-08,
0.005111721177102413, 0.0014788792437894997, 68941.78008826809, 224020.51796748457,
10.629432440187573, 1.3928828377322677, 457.3427523308425, 151.18302929886178,
9100.754781009497, 0.009154114676870422, 0.0014676353351049183, 536.2670424020977,
13.069436643555605, 1.2332353067053212, 699138399.2224176, 1.0468834307366839,
3.357012534136147, 782.6945078153142, 0.6911961869779446, 1.6761266446784837,
2.360499249630532e-08, 1.1035593243190558e-05, 0.02406502549730533, 17.059091440658946,
7.47855712384406, 0.00869998704177407, 0.022199720717597068, 8058.863930428368,
0.003732109075993981, 6.845942354323129, 9.885393785738345e-08, 9.856244888052659e-08,
3.050267721805182, 9.228613110508363e-08, 3.6097253389449444e-07, 1.8838054845637102,
0.024338102779919265, 4.526787374869987e-08, 6.620511136826136e-06, 0.2835177972554397,
1.4852252118165121, 4.7341732114946925e-06, 8.00580575978308e-08, 0.7345404035726253,
2.484594252058977e-08, 1.960397879296566e-06, 0.012470924479408996, 2.7130914700039367e-06,
2.9466741628850323e-09, 0.11063498799091412, 6.336488991467392e-09, 3.9822656127133824e-08,
0.001463676078913, 6.402509795363276e-10, 5.14345263519968e-06, 0.1964569797861901,
0.9793935583183078, 2.227656879959588e-07, 6.051095202454897e-06, 0.002864531281511599,
7.24772013734526e-08, 3.035922872455849e-06, 0.4303770891907133, 8.996358588412889e-09,
1.341771913344561e-06, 0.008669381686404125, 7.187378447761685e-07, 2.914088294684804e-08,
0.02491122406230101, 9.033567216666716e-06, 3.049189199357003e-08, 8.698432835807478e-07,
9.515248117460242e-08, 2.2073703919870035e-05, 0.012819456779348282, 9.006671447937795e-07,
8.70339230376299e-08, 0.005663951885287251, 0.00013617690111761754, 0.0010264172333100672,
2.3924876488382307, 0.8576470862919916, 0.6014338667131146, 60.21156237769469,
0.7226916254649123, 0.19379433950168284, 2.31705269115718, 0.8589324633418385,
0.535137089631316, 0.3283539443238837, 6.866019560866234, 0.17884707414171855,
0.19469023453543735, 21.5413172931563
]
const b3_p_lower = [
6.005, 18.015600000000003, 9.5e-08, 61.400000000000006, 5010.0, 0.0022,
0.0055000000000000005, 0.072, 0.0827, 4.78, 0.095, 0.07550000000000001, 0.0719,
340000000000.0, 170000000.0, 10000000.0, 0.0043, 0.06430000000000001, 0.0539, 0.0038,
0.0173, 0.08660000000000001, 0.082, 0.15700000000000003, 0.013700000000000002,
307.90000000000003, 63900.0, 0.0022, 0.0022, 1029.572332256015, 0.0001, 153.0,
0.05480000000000001, 100.0, 0.010000000000000002, 99.30000000000001, 0.00017,
101.11125335322616, 85.74234284353578, 0.592, 1.6600000000000001, 70.49945100689858,
52.950666799425974, 0.476, 0.11100000000000002, 40.42035022055875, 400000.0,
0.00030000000000000003, 0.03, 567.6087344716519, 0.0029000000000000002, 0.0212,
0.0029000000000000002, 0.063, 69.5, 12.700000000000001, 1.6000000000000003e-05,
0.033400000000000006, 543.7474358635255, 0.5, 1.01, 187.9, 10400.0, 0.000624,
0.36400000000000005, 0.654, 37.73427452982221, 0.0184, 100.0, 547.9285779, 0.34, 0.0128,
0.023100000000000002, 0.0218, 53931.519714534705, 9500000.0, 0.0022, 0.013800000000000002,
1490.4642239166901, 520000.0, 0.0048000000000000004, 0.0408, 0.132, 1e-80,
0.00017700000000000002, 0.0001, 1373.469562127557, 10000.0, 0.5, 0.0413,
11.600000000000001, 4.63, 252.0, 0.0008500000000000001, 0.00011999999999999999, 10.0,
0.136, 0.5, 10000000.0, 0.08950000000000001, 0.5, 10.0, 0.0164, 0.5,
1.9000000000000002e-10, 2e-07, 0.00036500000000000004, 0.219, 0.0897,
0.00030100000000000005, 0.000488, 92.30000000000001, 9.300000000000001e-05, 0.1, 2.3e-11,
2e-09, 0.034100000000000005, 0.0, 3.962810452627297e-09, 0.5, 0.00047000000000000004, 0.0,
1.4e-07, 0.0091, 0.5, 6.131859505628586e-08, 0.0, 0.009, 0.0, 4.7000000000000004e-08,
0.0012000000000000001, 6.703598301123717e-08, 0.0, 0.0016, 0.0, 2.1375371447438603e-09,
0.00011800000000000001, 0.0, 6.540068491820219e-08, 0.004, 0.5, 1.1000000000000001e-08,
8.500000000000001e-08, 0.00011700000000000001, 0.0, 1.2936704683174875e-07, 0.006, 0.0,
3.677028423862845e-08, 0.000535, 7.74626506298897e-09, 2.797262383857128e-11, 0.00034,
1.380565583801524e-07, 1.111186933303666e-09, 6.300000000000001e-08, 0.0, 3.3e-07,
0.0017000000000000001, 1.6324424376226334e-08, 8.790074664121873e-11, 0.00023,
5.600000000000001e-06, 1.8e-05, 0.188, 0.0978, 0.0154, 0.6400000000000001,
0.042300000000000004, 0.004900000000000001, 0.05530000000000001, 0.0108,
0.005600000000000001, 0.0076, 0.143, 0.0047, 0.006600000000000001, 0.5185
]
const b3_p_upper = [
600.5, 1801.56, 9.5e-06, 6140.0, 501000.0, 0.21999999999999997, 0.55, 7.199999999999999,
8.27, 478.0, 9.5, 7.55, 7.1899999999999995, 34000000000000.0, 17000000000.0, 1000000000.0,
0.42999999999999994, 6.43, 5.390000000000001, 0.38, 1.73, 8.66, 8.2, 15.700000000000001,
1.37, 30790.0, 6390000.0, 0.21999999999999997, 0.21999999999999997, 102957.2332256015,
0.01, 15300.0, 5.48, 10000.0, 1.0, 9930.0, 0.016999999999999998, 10111.125335322615,
8574.234284353577, 59.2, 166.0, 7049.945100689857, 5295.066679942597, 47.599999999999994,
11.100000000000001, 4042.035022055875, 40000000.0, 0.03, 3.0, 56760.8734471652,
0.29000000000000004, 2.12, 0.29000000000000004, 6.3, 6950.0, 1270.0, 0.0016,
3.3400000000000003, 54374.74358635255, 4.0, 101.0, 18790.0, 1040000.0, 0.0624, 36.4, 65.4,
3773.4274529822205, 1.8399999999999999, 10000.0, 54792.857789999995, 34.0, 1.28, 2.31,
2.18, 5393151.97145347, 950000000.0, 0.21999999999999997, 1.3800000000000001,
149046.422391669, 52000000.0, 0.48, 4.08, 13.200000000000001, 1e-07, 0.0177, 0.01,
137346.9562127557, 1000000.0, 50.0, 4.13, 1160.0, 463.0, 25200.0, 0.085,
0.011999999999999999, 1000.0, 13.600000000000001, 4.0, 1000000000.0, 8.95, 4.0, 1000.0,
1.6400000000000001, 4.0, 1e-07, 1.9999999999999998e-05, 0.0365, 21.9, 8.97,
0.030100000000000002, 0.048799999999999996, 9230.0, 0.009300000000000001, 10.0, 1e-07,
2e-07, 3.41, 1e-07, 3.962810452627297e-07, 4.0, 0.047, 1e-07, 1.4e-05, 0.9099999999999999,
4.0, 6.131859505628587e-06, 1e-07, 0.8999999999999999, 1e-07, 4.7e-06, 0.12,
6.703598301123717e-06, 1e-07, 0.16, 1e-07, 2.13753714474386e-07, 0.011800000000000001,
1e-07, 6.540068491820219e-06, 0.4, 4.0, 1.1e-06, 8.5e-06, 0.0117, 1e-07,
1.2936704683174875e-05, 0.6, 1e-07, 3.6770284238628445e-06, 0.0535, 7.746265062988969e-07,
1e-07, 0.033999999999999996, 1.3805655838015239e-05, 1.111186933303666e-07, 6.3e-06, 1e-07,
3.3e-05, 0.17, 1.6324424376226333e-06, 1e-07, 0.023, 0.00056, 0.0018000000000000002,
18.799999999999997, 9.78, 1.54, 64.0, 4.2299999999999995, 0.49, 5.53, 1.08, 0.56, 0.76,
14.299999999999999, 0.47, 0.66, 51.849999999999994
]178-element Vector{Float64}:
600.5
1801.56
9.5e-6
6140.0
501000.0
0.21999999999999997
0.55
7.199999999999999
8.27
478.0
⋮
0.49
5.53
1.08
0.56
0.76
14.299999999999999
0.47
0.66
51.849999999999994Simulated data: an extended diauxic shift
The pseudo-data are generated (in the reference implementation, with no noise added, since this is a black-box mex simulation rather than a noised sampling step) by simulating the model through three consecutive growth media, matching the reference b3_dyn.m/b3_amigo.m scenario 6, an "extended diauxic shift":
- Glucose only (
t = 0-8.15 h): starting from the glucose-only steady statessGLCwithOD = 0.03,ACT = 0,GLC = 4.8. - Acetate only (next
19.7 h): glucose is exhausted and acetate is added (ACT = 5,GLC = 0),ODis reset to0.03(a fresh inoculation into the new medium), and every other state carries over from the end of phase 1. - Glucose + acetate mixture (final
16.45 h): both carbon sources are present (ACT = 3,GLC = 3),ODis reset to5e-4, and the other states carry over from the end of phase 2.
All 47 states are sampled every 1000 s. Following the reference implementation exactly (including its abrupt medium switches, where the state used to seed the next phase is resampled at the last 1000 s grid point at or before the nominal phase boundary, not the boundary itself), the three phases contribute 30 + 71 + 60 = 161 rows in total – matching the paper's stated 161 time points over "45 hours" (44.3 h of nominal phase lengths). The data below is the reference b3_data.mat (xnom), transcribed verbatim.
# pseudo-experimental data (161 time points x 47 states), transcribed from
# the reference b3_data.mat (xnom), generated by simulating the model's
# extended-diauxic-shift scenario at the reference nominal parameters
const b3_data = [
0.03 0.0 4.8 0.351972298 0.191190619 0.202804098 6.57504207 1.908140784 5.70593e-09 0.001408116 3.278779135 0.050535354 0.210455879 5.720977255 0.863278018 0.00472323 0.001416969 0.000141697 0.001 3.6201e-05 0.001026848 0.001 0.001 0.011389032 0.011389032 7.481e-05 0.000292771 0.004290789 0.000220477 0.000491491 0.000999714 0.000336947 0.001 0.000242131 0.000377962 0.000987493 0.002501893 0.09647707 0.00352292 0.003 0.000299098 0.006990902 0.005943273 0.001346727 0.00729 0.001163813 0.006126187
0.03591513887781556 0.002173705488592223 4.787542569869577 0.3937323031626563 0.19775330727334195 0.20278547958047943 6.57373709992828 1.9081242484876217 5.442491737624688e-09 0.0014344846119460411 3.3400752998118795 0.04943411043198247 0.2104528927930759 5.7206951667121935 0.8629910088531018 0.004723391413659777 0.0014170174240979343 0.00014170182363362345 0.001 3.6200361353548564e-05 0.0010268433206936383 0.001 0.001 0.011390731782032731 0.011389042547057065 7.48129049898295e-05 0.000292768813524334 0.004290711964554366 0.00022065940449863633 0.0004914876070117683 0.0009997427569533228 0.0003369624936783816 0.0009999919623509543 0.0002421289586853418 0.0003779465971074153 0.0009875402470903045 0.002501884071115333 0.09647739829090965 0.0035225917090903496 0.003 0.00029921218363362456 0.0069907878163663765 0.005943374294304737 0.0013466257056952702 0.00729 0.0011641387214118802 0.006125861278588119
0.042996748789676395 0.004682775920234055 4.772628311467069 0.4025210394026368 0.19902017621230567 0.20279795235311818 6.573398510240928 1.9082074767072301 5.389155008641637e-09 0.0014395119596082768 3.351919834954686 0.04923262210748054 0.21046319672527075 5.721096572423753 0.8630227488577467 0.0047236106610227735 0.0014170831983068351 0.00014170838580351803 0.001 3.6200493350658034e-05 0.0010268458906959083 0.001 0.001 0.01139212362497318 0.01138905083432014 7.481722546896487e-05 0.00029276963130925326 0.004290903193744218 0.00022062306944355104 0.0004914889539635439 0.0009998133285186418 0.0003369855282499827 0.000999984103911385 0.00024212584023264392 0.00037793550911686225 0.0009876103816980825 0.002501869033416597 0.09647717842675316 0.003522811573246822 0.003 0.00029924174360883705 0.006990758256391164 0.005943306767893953 0.001346693232106128 0.00729 0.0011641027444993192 0.0061258972555006804
0.05147474349426925 0.007652064869961389 4.754772558248599 0.4063651933402864 0.19956503138475168 0.20280940041651496 6.5731991451195775 1.9082867272889348 5.366117375671636e-09 0.0014416626819609127 3.3569798102339785 0.049147982051912086 0.2104729765342868 5.7214910533993395 0.8630771662374145 0.004723814306147605 0.00141714429184428 0.00014171448276981333 0.001 3.620132788887083e-05 0.0010268547723950412 0.001 0.001 0.011393262218225552 0.011389056498769007 7.482158005615867e-05 0.000292773122469642 0.004291106132355681 0.00022057566497874384 0.0004914945059989107 0.0009998854484826564 0.0003370087479604471 0.000999980110154198 0.00024212265317932204 0.0003779269074677759 0.0009876810577089216 0.0025018532306891684 0.09647697665544708 0.003523013344552919 0.003 0.00029925915079722395 0.006990740849202778 0.005943244790511866 0.0013467552094882356 0.00729 0.0011640410671315826 0.006125958932868418
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2.602938372995805 1.7022351519811474 3.972142345803358e-27 0.8973776544657532 0.7700169489354316 3.9788509222913744 0.171010286797442 0.7278065203564876 8.990029461390174 0.9103673708013883 6.127032799188223 0.05959714019290858 0.30430608030071976 0.7885430084427291 0.03222146421322788 0.06646109247863395 0.019938327743590088 0.0019938327743590236 0.001 0.00021369243762736005 0.0017188172822040315 0.001 0.001 0.010349712140905673 0.008266690960092618 0.00017625277799597952 0.000716095507379397 0.0015933819787645818 0.006164307882947672 0.001124887069124135 0.002432890264705651 0.0018159951369560562 0.0006179189436623075 0.00018702064820410517 0.00021531245081254512 0.008228301710950668 0.001328356420645895 0.0040549073753854735 0.09594508262461426 0.003 0.007176530190226524 0.00011346980977347079 0.0013386847697957132 0.005951315230207469 0.00729 0.006092911852370674 0.0011970881476293053
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2.91117500277854 0.9970035189580678 9.347760042585584e-27 0.954827337887316 0.8423256433062201 3.9889179876067824 0.20056904087703437 0.8096308652836799 6.733799399021804 1.0448709474998001 6.25336783361835 0.06342191817377758 0.3580959758745102 0.9167675859094035 0.034661774735661795 0.07289931040811247 0.021869793122433652 0.002186979312243375 0.001 0.00023236137119547222 0.0017915847626784947 0.001 0.001 0.010241425760344062 0.007943311487304271 0.00018681672847200952 0.0007606894636588482 0.0017125592352418128 0.0063831042494354 0.0011915993972030853 0.002581882140013508 0.001968790936285012 0.0005801836485605773 0.00018125614539826727 0.00019837515392181947 0.00897053965091208 0.0012062184772577709 0.0037197349418109896 0.09628025505818875 0.003 0.0071348211625618515 0.00015517883743814567 0.001335925381351614 0.0059540746186515685 0.00729 0.0060180679743313545 0.0012719320256686258
3.0786310299847144 0.5923481493253293 6.458116131630219e-26 0.9865807331793094 0.8729441735471806 3.9927608264553243 0.2132704431780748 0.8438459268543125 5.824593742475117 1.1028826203145845 6.298502023941357 0.0647877271192822 0.3817650762441209 0.9718347619176325 0.035644414564931746 0.0757217533970205 0.022716526019106086 0.0022716526019106115 0.001 0.0002405834548353829 0.001823628061978964 0.001 0.001 0.01019383505247907 0.007801493371250839 0.00019145266483709417 0.0007803331174968727 0.0017653100672902025 0.006478612646982511 0.0012209852741266974 0.002647338303306716 0.002035615758022572 0.0005637215818466351 0.00017871646999086303 0.00019091319190731683 0.009294022203015607 0.001152570309191762 0.003592760669153083 0.09640722933084664 0.003 0.007115031500926062 0.00017496849907393605 0.001334875054643859 0.005955124945359326 0.00729 0.005988447280110411 0.00130155271988957
3.255395252204509 0.15170515365205345 -1.7327044021035927e-26 1.0086376868589608 0.8945843325185975 3.9957963361168742 0.2242343760386862 0.8758635879264122 5.084117453705945 1.1507906562182988 6.318145308665829 0.06552520285906764 0.40236052155308044 1.0192440555215312 0.03658138645403371 0.0782990590275354 0.02348971770826066 0.0023489717708260484 0.001 0.00024812245632600957 0.0018529704778888372 0.001 0.001 0.0101497941827794 0.007671347917893717 0.00019569127635384867 0.0007983383402601034 0.0018105378677839292 0.006568816647663078 0.001247917459299656 0.0027069645704189548 0.002096598785911442 0.0005486728121813882 0.00017637699298713596 0.00018404441857815382 0.009588539607875386 0.0011033563285176666 0.0035017675388078974 0.09649822246119183 0.003 0.0070970675841359795 0.0001929324158640185 0.0013340465543609978 0.005955953445642186 0.00729 0.005960473650279704 0.0013295263497202755
]161×47 Matrix{Float64}:
0.03 0.0 4.8 … 0.00729 0.00116381 0.00612619
0.0359151 0.00217371 4.78754 0.00729 0.00116414 0.00612586
0.0429967 0.00468278 4.77263 0.00729 0.0011641 0.0061259
0.0514747 0.00765206 4.75477 0.00729 0.00116404 0.00612596
0.0616244 0.0111865 4.7334 0.00729 0.00116399 0.00612601
0.0737755 0.0154034 4.7078 … 0.00729 0.00116394 0.00612606
0.0883224 0.0204404 4.67716 0.00729 0.00116391 0.00612609
0.105738 0.0264611 4.64048 0.00729 0.00116388 0.00612612
0.126587 0.0336606 4.59656 0.00729 0.00116387 0.00612613
0.151547 0.0422727 4.54399 0.00729 0.00116386 0.00612614
⋮ ⋱ ⋮
2.08078 2.73189 -2.37308e-26 0.00729 0.00631951 0.00097049
2.20058 2.51906 -9.16548e-26 0.00729 0.00625171 0.00103829
2.32727 2.27728 7.5839e-27 0.00729 0.00619183 0.00109817
2.46125 2.0054 1.8261e-26 … 0.00729 0.00613912 0.00115088
2.60294 1.70224 3.97214e-27 0.00729 0.00609291 0.00119709
2.75276 1.36654 1.55424e-26 0.00729 0.00605271 0.00123729
2.91118 0.997004 9.34776e-27 0.00729 0.00601807 0.00127193
3.07863 0.592348 6.45812e-26 0.00729 0.00598845 0.00130155
3.2554 0.151705 -1.7327e-26 … 0.00729 0.00596047 0.00132953Objective function
Following the reference b3_obj.m exactly, the objective is a scaled least-squares cost: each of the 47 states' squared residuals (model minus data, at all 161 time points) is divided by the square of that state's own maximum value across the time course (a per-state scale factor Q, reproduced below), except where that maximum is too small to divide by safely (below 1e-7), in which case no scaling is applied (Q = 1) – exactly matching the reference's Q_expmax/atol guard.
function b3_simulate(p178; rtol = 1e-6, atol = 1e-6)
p193 = b3_expand_params(p178)
# phase 1: growth on glucose alone, t = 0 .. 29000 s (8.15 h nominal, truncated to
# the last 1000 s sample, matching the reference AMIGO2 implementation)
y0 = copy(b3_ssGLC)
y0[b3_i_OD] = 0.03
y0[b3_i_ACT] = 0.0
y0[b3_i_GLC] = 4.8
t1 = collect(0.0:1000.0:29340.0)
prob1 = ODEProblem(b3_kinetics!, y0, (t1[1], t1[end]), p193)
sol1 = solve(prob1, FBDF(); reltol = rtol, abstol = atol, saveat = t1, maxiters = 20_000)
sol1.retcode == ReturnCode.Success || return nothing
y1 = Array(sol1)'
# phase 2: glucose exhausted, acetate added, t = 29000 .. 99000 s (19.7 h)
ic2 = copy(y1[end, :])
ic2[b3_i_ACT] = 5.0
ic2[b3_i_GLC] = 0.0
ic2[b3_i_OD] = 0.03
tf2 = (t1[end] / 3600 + 19.7) * 3600
t2 = collect(t1[end]:1000.0:tf2)
prob2 = ODEProblem(b3_kinetics!, ic2, (t2[1], t2[end]), p193)
sol2 = solve(prob2, FBDF(); reltol = rtol, abstol = atol, saveat = t2, maxiters = 20_000)
sol2.retcode == ReturnCode.Success || return nothing
y2 = Array(sol2)'
# phase 3: mixed glucose + acetate, t = 99000 .. 158000 s (16.45 h)
ic3 = copy(y2[end, :])
ic3[b3_i_ACT] = 3.0
ic3[b3_i_GLC] = 3.0
ic3[b3_i_OD] = 5e-4
tf3 = (t2[end] / 3600 + 16.45) * 3600
t3 = collect(t2[end]:1000.0:tf3)
prob3 = ODEProblem(b3_kinetics!, ic3, (t3[1], t3[end]), p193)
sol3 = solve(prob3, FBDF(); reltol = rtol, abstol = atol, saveat = t3, maxiters = 20_000)
sol3.retcode == ReturnCode.Success || return nothing
y3 = Array(sol3)'
return vcat(y1, y2, y3)
end
const b3_times = let
t1 = collect(0.0:1000.0:29340.0)
tf2 = (t1[end] / 3600 + 19.7) * 3600
t2 = collect(t1[end]:1000.0:tf2)
tf3 = (t2[end] / 3600 + 16.45) * 3600
t3 = collect(t2[end]:1000.0:tf3)
vcat(t1, t2, t3)
end
const b3_Q = let
expDataMax = maximum(b3_data, dims = 1)
Q = 1.0 ./ expDataMax
Q[.!isfinite.(Q) .| (Q .> 1e7)] .= 1.0
Q
end
function biopredyn_b3_cost(p178)
ms = try
b3_simulate(p178)
catch
nothing
end
(ms === nothing || size(ms) != size(b3_data)) && return 1e20
cost = sum(abs2, (ms .- b3_data) .* b3_Q)
return isfinite(cost) ? cost : 1e20
endbiopredyn_b3_cost (generic function with 1 method)@time cost_nominal = biopredyn_b3_cost(b3_p_nom)
@time cost_start = biopredyn_b3_cost(b3_p_start)6.414377 seconds (11.52 M allocations: 523.591 MiB, 5.63% gc time, 97.16%
compilation time: 2% of which was recompilation)
0.184876 seconds (261.60 k allocations: 26.721 MiB, 44.61% compilation ti
me: 100% of which was recompilation)
1.0e20The original study reports reaching a cost of Jf ≈ 0.3703 using the enhanced scatter search (eSS) global optimizer after ~7.2·10⁶ function evaluations (up to 336 hours of CPU time), starting from a random point inside the bounds; that is the reference cost to reach here. Unlike B2, B4 and B5, here p_nomis (as confirmed above) essentially the exact parameter vector that generated the data – the reference AMIGO2 script builds this benchmark's exp_data directly from a black-box simulation at p_nom, with no noise step. cost_nominal above is therefore tiny (limited only by the mismatch between our solver's tolerances and whatever tolerance the reference CVODES mex file used to generate xnom), not the ~900-ish residual level noise would produce. That makes Jf ≈ 0.3703 a very different kind of target than in B2/B4/B5: it does not reflect a fundamental data/model mismatch, but eSS's own difficulty fully converging this particular 178-parameter, highly nonlinear, occasionally stiff landscape within its evaluation budget – the paper itself singles out B1 and B3 as the two most computationally expensive problems in the suite. cost_start, by contrast, is typically 1e20: with such wide (0.1x-10x) bounds, most random points (including the reference's own saved starting guess) make the model's population size or metabolite pools diverge before the 45-hour scenario finishes integrating, so the optimizers below have to first find a basin where the model is even numerically well-posed.
Visualizing the fit at nominal parameters
ms_nom = b3_simulate(b3_p_nom)
t_h = b3_times ./ 3600
obs = [b3_i_OD, b3_i_GLC, b3_i_ACT, b3_i_FBP, b3_i_CrpcAMP, b3_i_PYR]
labels = ["OD" "GLC" "ACT" "FBP" "CrpcAMP" "PYR"]
plot(t_h, ms_nom[:, obs], layout = (3, 2), label = labels, legend = :outertopright,
xlabel = "time (h)", size = (900, 650),
plot_title = "B3 dynamics at nominal parameters (glucose -> acetate -> mixture)")
scatter!(t_h, b3_data[:, obs], layout = (3, 2), label = false, markersize = 2, markerstrokewidth = 0)
The three growth phases are visible as the abrupt jumps in OD, GLC and ACT at t ≈ 8.1 h and t ≈ 27.9 h; FBP, CrpcAMP and PYR show the regulatory response (via the Cra/FBP and Crp/cAMP transcription factors and the PdhR/pyruvate binding) to each change in carbon source. The model trajectory overlays the pseudo-data essentially exactly, as expected from cost_nominal above.
Parameter estimation
We benchmark global optimizers on this 178-parameter problem, starting from p_start (not p_nom, per the discussion above) and using the bounds given above. As with B2/B4/B5, we include BBO_adaptive_de_rand_1_bin, GN_CRS2_LM, and ParallelPSOArray from ParallelParticleSwarms.jl. 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. Given the cost of a single evaluation here (up to three stiff solves of a 47-state system), and per the paper's own note that B3 is one of the two most expensive problems in the suite, the evaluation budgets below are a small fraction of the reference eSS run's ~7.2 million evaluations.
optf = OptimizationFunction((p, _) -> biopredyn_b3_cost(p))
optprob = OptimizationProblem(optf, b3_p_start, lb = b3_p_lower, ub = b3_p_upper)OptimizationProblem. In-place: true
u0: 178-element Vector{Float64}:
51.9872906889458
761.6372473233096
4.259319313214376e-6
1093.5368179193445
107602.48608345521
0.12856327802490058
0.35115086815391283
1.367540794523535
7.1291370173004225
282.35998973420334
⋮
0.19379433950168284
2.31705269115718
0.8589324633418385
0.535137089631316
0.3283539443238837
6.866019560866234
0.17884707414171855
0.19469023453543735
21.5413172931563losses_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 = 5000, callback = cb_bbo)
res_bbo.objective242.040595 seconds (79.35 M allocations: 36.883 GiB, 6.96% gc time, 1.71% c
ompilation time)
1423.7977992574256losses_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(b3_p_start))
@time res_nlopt = solve(optprob, opt, maxiters = 5000, callback = cb_nlopt)
res_nlopt.objective514.125296 seconds (140.51 M allocations: 82.895 GiB, 8.63% gc time, 0.15%
compilation time)
1983.8590496423888n_particles = 50
pso_iters = 5000 ÷ n_particles
@time res_pso = solve(optprob, ParallelPSOArray(n_particles), maxiters = pso_iters)
res_pso.objective57.787770 seconds (56.75 M allocations: 26.148 GiB, 59.56% gc time, 85.14%
compilation time)
1664.0261735076992Local refinement
Global metaheuristics are good at finding a promising basin but slow to fine-tune within it; the original study's own eSS method is itself a hybrid global+local algorithm for exactly this reason. 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 = b3_p_lower, ub = b3_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(b3_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")
nothingBBO -> LN_BOBYQA: 999.4624914688662 (85.360101268s)
GN_CRS2_LM -> LN_BOBYQA: 1571.2843311905162 (52.703990126s)
ParallelPSOArray -> LN_BOBYQA: 1517.4261901081813 (57.499213815s)df = DataFrame(
method = ["Reference nominal parameters", "Starting guess (no fit)",
"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_nominal, cost_start, res_bbo.objective, res_nlopt.objective,
res_pso.objective, res_bbo_polish.objective, res_nlopt_polish.objective,
res_pso_polish.objective])8×2 DataFrame
Row │ method cost
│ String Float64
─────┼──────────────────────────────────────────────────
1 │ Reference nominal parameters 4.72667e-5
2 │ Starting guess (no fit) 1.0e20
3 │ BBO_adaptive_de_rand_1_bin 1423.8
4 │ GN_CRS2_LM 1983.86
5 │ ParallelPSOArray 1664.03
6 │ LN_BOBYQA polish (from BBO) 999.462
7 │ LN_BOBYQA polish (from GN_CRS2_L… 1571.28
8 │ LN_BOBYQA polish (from ParallelP… 1517.43Convergence
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)
# each polish curve starts where its own global run ended; `ParallelPSOArray` records no
# per-iteration history, so its (approximate) evaluation count, particles x iterations,
# is used as the offset instead
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 = "B3 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!([0.3703], label = "paper reference (eSS, ~7.2x10^6 evals)", 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 = "B3 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!([0.3703], label = "paper reference (eSS, ~7.2x10^6 evals)", 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, plotted alongside the other two polish curves for direct comparison. 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 parameter estimation stack (OrdinaryDiffEq + Optimization.jl) on BioPreDyn-bench B3, a 178-parameter combined enzymatic and transcriptional model of E. coli's adaptation to changing carbon sources, spanning a 45-hour, three-phase "extended diauxic shift" scenario – structurally closer to B1 than to B2/B4/B5, both in scale (178 unknowns across 47 stiff, tightly-coupled states) and in the paper's own assessment that B1 and B3 are the two most computationally demanding problems in the suite. Unlike B1, however, no structural discontinuity was found in the published model; the practical difficulty here is instead the sheer dimensionality and the ease with which parameter draws far from the nominal region (inevitable during global search over 0.1x-10x bounds) make the model's population size or metabolite pools diverge numerically, which the objective function handles by returning a large finite penalty for any point where the ODE integration fails or produces a non-finite cost. Because the pseudo-data were generated directly from p_nom with no added noise, the reference cost Jf ≈ 0.3703 (eSS, ~7.2 million evaluations, up to 336 CPU-hours) does not reflect a fundamental data/model mismatch, but rather how far a much larger optimization budget than used here can close the gap to the (near-zero) attainable minimum. The runs here are best read as relative comparisons between optimizers given a fixed, modest evaluation budget rather than as fully converged fits.
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","BioPreDynB3ParameterEstimation.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
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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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[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
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.