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Table 2 Parameter estimation with eSS (AMIGO implementation): settings and results

From: BioPreDyn-bench: a suite of benchmark problems for dynamic modelling in systems biology

Model ID B1 B2 B3 B4 B5 B6
p U p nom \(10\cdot p_{\textit {nom}}^{(ex)}\) \(10\cdot p_{\textit {nom}}^{(ex)}\) p nom varying varying
p L 0.2·p nom \(0.1\cdot p_{\textit {nom}}^{(ex)}\) \(0.1\cdot p_{\textit {nom}}^{(ex)}\) 0.2·p nom varying varying
Local method DHC FMINCON none FMINCON DHC FMINCON
CPU time ≈170 hours ≈3 hours ≈336 hours ≈1 hour ≈16 hours ≈24 hours
Evaluations 6.9678·105 9.0728·104 7.2193·106 1.6193·105 8.8393·104 2.0751·106
J 0 5.8819·109 3.1136·104 4.6930·1016 6.6034·108 3.1485·104 8.5769·105
J f 1.3753·104 2.3390·102 3.7029·10−1 4.5718·101 3.0725·103 1.0833·105
J nom 1.0846·106 0 3.9068·101 4.2737·103
\(\sum \textit {NRMSE}_{0}\) 3.5834·101 8.5995·10−2 3.5457·101 4.8005·101 4.0434·101 2.3808·102
\(\sum \textit {NRMSE}_{f}\) 5.7558 2.4921 2.9298·10−1 2.8010 2.7430·101 1.6212·102
\(\sum \textit {NRMSE}_{\textit {nom}}\) 3.8203 0 2.8273 3.0114·101
  1. Optimization settings and results obtained for each of the benchmarks with the eSS method, using the implementation provided in the AMIGO toolbox. In some cases the lower (p L) and upper (p U) bounds in the parameters are specified as a function of the nominal parameter vector, p nom . There may be exceptions to these bounds, in cases where it makes sense biologically to have a different range of values (e.g. Hill coefficients in the range of 1–12). Cases with exceptions are marked by (ex). In other cases all the parameters have specific bounds; this is marked as “varying”. The initial objective function value, J 0, corresponds to the parameter vector p 0 used as initial guess in the optimizations, which is randomly selected between the bounds p L and p U. The only exception is benchmark B2, where p 0 is the parameter vector reported in the original publication. The final value achieved in the optimizations is J f, and the value obtained with the nominal parameter vector is J nom . More details about the definition of the objective functions J are given in section “Problem statement”. \(\sum \)NRMSE is the cumulative normalized root-mean-square error as defined in eq. (12); the subscripts ( 0 , f , nom ) have the same meaning as in the objective functions J. Results obtained on a computer with Intel Xeon Quadcore processor, 2.50 GHz, using Matlab 7.9.0.529 (R2009b) 32-bit.