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

BMC Systems Biology

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

Model ID	B1	B2	B3	B4	B5	B6
p ^U	5·p _nom	\(10\cdot p_{\textit {nom}}^{(ex)}\)	\(10\cdot p_{\textit {nom}}^{(ex)}\)	5·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	−

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.

ISSN: 1752-0509