DATA FOR PARAMETER ESTIMATION PROBLEM OF A 3-STEP PATHWAY ========================================================== This document describes the problem considered in: Moles, C. G., Pedro Mendes and Julio R. Banga (2003) Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Research, 13(11):2467-2474. SYSTEM'S DYNAMICS The nonlinear dynamic model is: $G1 = V1/(1+(P/Ki1)^ni1+(Ka1/S)^na1)- k_1*G1; $G2 = V2/(1+(P/Ki2)^ni2+(Ka2/M1)^na2) - k_2*G2; $G3 = V3/(1+(P/Ki3)^ni3+(Ka3/M2)^na3) - k_3*G3; $E1 = V4*G1/(K4+G1) - k_4*E1; $E2 = V5*G2/(K5+G2) - k_5*E2; $E3 = V6*G3/(K6+G3) - k_6*E3; $M1 = kcat1*E1*(1/Km1)*(S-M1)/(1+S/Km1+M1/Km2)-kcat2*E2*(1/Km3)*(M1-M2)/(1+M1/Km3+M2/Km4); $M2 = kcat2*E2*(1/Km3)*(M1-M2)/(1+M1/Km3+M2/Km4)-kcat3*E3*(1/Km5)*(M2-P)/(1+M2/Km5+P/Km6); where $G1 means time derivative of G1 (i.e. dG1/dt) etc. The initial conditions for the states (and for any experiment) are: G1 = 6.6667e-1 G2 = 5.7254e-1 G3 = 4.1758e-1 E1 = 4e-1 E2 = 3.6409e-1 E3 = 2.9457e-1 M1 = 1.419 M2 = 9.3464e-1 INTEGRATION SETTINGS The absolute and relative tolerances for the integration of the ODEs are both taken as 1e-8 in order to ensure high precision. UPPER AND LOWER BOUNDS FOR THE PARAMETERS The estimation problem is stated as finding the 36 parameters in order to get the best possible fit with respect a set of pseudo-experimental data (given below). The parameters to be estimated are: Parameter Lower bound Correct value Upper bound (nominal (*)) V1 1.000000E-12 1.000000E+00 1.000000E+06 Ki1 1.000000E-12 1.000000E+00 1.000000E+06 ni1 1.000000E-01 2.000000E+00 1.000000E+01 Ka1 1.000000E-12 1.000000E+00 1.000000E+06 na1 1.000000E-01 2.000000E+00 1.000000E+01 k_1 1.000000E-12 1.000000E+00 1.000000E+06 V2 1.000000E-12 1.000000E+00 1.000000E+06 Ki2 1.000000E-12 1.000000E+00 1.000000E+06 ni2 1.000000E-01 2.000000E+00 1.000000E+01 Ka2 1.000000E-12 1.000000E+00 1.000000E+06 na2 1.000000E-01 2.000000E+00 1.000000E+01 k_2 1.000000E-12 1.000000E+00 1.000000E+06 V3 1.000000E-12 1.000000E+00 1.000000E+06 Ki3 1.000000E-12 1.000000E+00 1.000000E+06 ni3 1.000000E-01 2.000000E+00 1.000000E+01 Ka3 1.000000E-12 1.000000E+00 1.000000E+06 na3 1.000000E-01 2.000000E+00 1.000000E+01 k_3 1.000000E-12 1.000000E+00 1.000000E+06 V4 1.000000E-12 1.000000E-01 1.000000E+06 K4 1.000000E-12 1.000000E+00 1.000000E+06 k_4 1.000000E-12 1.000000E-01 1.000000E+06 V5 1.000000E-12 1.000000E-01 1.000000E+06 K5 1.000000E-12 1.000000E+00 1.000000E+06 k_5 1.000000E-12 1.000000E-01 1.000000E+06 V6 1.000000E-12 1.000000E-01 1.000000E+06 K6 1.000000E-12 1.000000E+00 1.000000E+06 k_6 1.000000E-12 1.000000E-01 1.000000E+06 kcat1 1.000000E-12 1.000000E+00 1.000000E+06 Km1 1.000000E-12 1.000000E+00 1.000000E+06 Km2 1.000000E-12 1.000000E+00 1.000000E+06 kcat2 1.000000E-12 1.000000E+00 1.000000E+06 Km3 1.000000E-12 1.000000E+00 1.000000E+06 Km4 1.000000E-12 1.000000E+00 1.000000E+06 kcat3 1.000000E-12 1.000000E+00 1.000000E+06 Km5 1.000000E-12 1.000000E+00 1.000000E+06 Km6 1.000000E-12 1.000000E+00 1.000000E+06 IMPORTANT: (*) In order to perform fair comparisons, the nominal value of the parameters (i.e. the correct values which one should find by solving the inverse problem) are given here for the sake of completeness, but PLEASE NOTE that they should not be taken into account if you are trying to solve this problem with some other method. This is, in order to be fair, the problem should be solved in a blind way, without using these nominal values to 'guess' good initial points for the method of choice. The above problem was initially stated by P. Mendes to J.R. Banga in such blind way, which is of course the type of situation that one will encounter in real practice. Needless to say, many methods will perform pretty well if they are initialised close to the correct solution, but again please bear in mind that this would be sort of 'cheating' ;-) since in real problems such knowledge would be very rare. Thus, please consider only the upper and lower bounds given above. Take random choices, inside those bounds, for the initial guesses of the parameters. We would appreciate if you keep us posted about your results with different methods, especially if they are very good! For those who would like to reproduce or cross-check the results from our papers, here is the initial vector considered for the computations of Figure 2 of the Genome Research paper (Moles et al, 2003, Genome Research, 13(11):2467-2474) which was also used for Figure 1 of the ICSB-2004 paper mentioned below: x0 = [500. 500. 5. 500. 5. 500. 500. 500. 5. 500. 5. ... 500. 500. 500. 5. 500. 5. 500. 500. ... 500. 500. 500. 500. 500. 500. 500. ... 500. 500. 500. 500. 500. 500. 500. 500. 500. 500.]; This example of initial vector has a value of the objective function (as defined by Eqn. 15 in the Genome Research paper) of J = 1180.0 EXPERIMENTAL DATA Next, the pseudo-experimental data set considered in the paper is given. Four S and P values were combined to generate a total of 16 sets of data, which are called here Experiments 1-16. The S and P values for each of these 16 experiments were: Experiment S-value P-value 1 1.00E-01 5.00E-02 2 1.00E-01 1.36E-01 3 1.00E-01 3.68E-01 4 1.00E-01 1.0 5 4.64E-01 5.00E-02 6 4.64E-01 1.36E-01 7 4.64E-01 3.68E-01 8 4.64E-01 1.0 9 2.15E+00 5.00E-02 10 2.15E+00 1.36E-01 11 2.15E+00 3.68E-01 12 2.15E+00 1.0 13 10.0 5.00E-02 14 10.0 1.36E-01 15 10.0 3.68E-01 16 10.0 1.0 The time-S-P-states data are given in the following 16 blocks (experiments), with a final time of 120.0 (the initial conditions for all the states given above are common for all the experiments): Time S-value P-value M1 M2 E1 E2 E3 G1 G2 G3 6.0000e+000 1.0000e-001 5.0000e-002 4.9836e-001 4.4611e-001 2.5444e-001 3.3628e-001 2.7428e-001 1.1529e-002 2.5706e-001 2.1153e-001 1.2000e+001 1.0000e-001 5.0000e-002 2.5018e-001 2.1588e-001 1.4416e-001 2.3680e-001 1.9387e-001 9.9048e-003 7.1813e-002 5.5386e-002 1.8000e+001 1.0000e-001 5.0000e-002 1.7091e-001 1.4480e-001 8.3541e-002 1.4937e-001 1.2103e-001 9.9008e-003 3.1492e-002 2.2900e-002 2.4000e+001 1.0000e-001 5.0000e-002 1.4057e-001 1.1872e-001 5.0272e-002 9.2729e-002 7.4231e-002 9.9007e-003 2.0419e-002 1.4658e-002 3.0000e+001 1.0000e-001 5.0000e-002 1.2664e-001 1.0703e-001 3.2013e-002 5.8798e-002 4.6438e-002 9.9007e-003 1.6240e-002 1.1648e-002 3.6000e+001 1.0000e-001 5.0000e-002 1.1923e-001 1.0093e-001 2.1992e-002 3.8956e-002 3.0308e-002 9.9007e-003 1.4254e-002 1.0249e-002 4.2000e+001 1.0000e-001 5.0000e-002 1.1476e-001 9.7337e-002 1.6493e-002 2.7432e-002 2.1010e-002 9.9007e-003 1.3143e-002 9.4832e-003 4.8000e+001 1.0000e-001 5.0000e-002 1.1177e-001 9.4987e-002 1.3475e-002 2.0727e-002 1.5646e-002 9.9007e-003 1.2436e-002 9.0064e-003 5.4000e+001 1.0000e-001 5.0000e-002 1.0958e-001 9.3289e-002 1.1818e-002 1.6791e-002 1.2529e-002 9.9007e-003 1.1937e-002 8.6750e-003 6.0000e+001 1.0000e-001 5.0000e-002 1.0784e-001 9.1956e-002 1.0909e-002 1.4441e-002 1.0693e-002 9.9007e-003 1.1554e-002 8.4224e-003 6.6000e+001 1.0000e-001 5.0000e-002 1.0639e-001 9.0835e-002 1.0411e-002 1.3001e-002 9.5849e-003 9.9007e-003 1.1240e-002 8.2151e-003 7.2000e+001 1.0000e-001 5.0000e-002 1.0513e-001 8.9847e-002 1.0137e-002 1.2084e-002 8.8921e-003 9.9007e-003 1.0973e-002 8.0358e-003 7.8000e+001 1.0000e-001 5.0000e-002 1.0401e-001 8.8949e-002 9.9865e-003 1.1471e-002 8.4371e-003 9.9007e-003 1.0738e-002 7.8752e-003 8.4000e+001 1.0000e-001 5.0000e-002 1.0299e-001 8.8116e-002 9.9040e-003 1.1037e-002 8.1195e-003 9.9007e-003 1.0528e-002 7.7281e-003 9.0000e+001 1.0000e-001 5.0000e-002 1.0205e-001 8.7336e-002 9.8587e-003 1.0711e-002 7.8825e-003 9.9007e-003 1.0338e-002 7.5917e-003 9.6000e+001 1.0000e-001 5.0000e-002 1.0119e-001 8.6600e-002 9.8339e-003 1.0452e-002 7.6940e-003 9.9007e-003 1.0164e-002 7.4641e-003 1.0200e+002 1.0000e-001 5.0000e-002 1.0039e-001 8.5903e-002 9.8203e-003 1.0236e-002 7.5358e-003 9.9007e-003 1.0004e-002 7.3444e-003 1.0800e+002 1.0000e-001 5.0000e-002 9.9651e-002 8.5241e-002 9.8128e-003 1.0050e-002 7.3975e-003 9.9007e-003 9.8560e-003 7.2317e-003 1.1400e+002 1.0000e-001 5.0000e-002 9.8959e-002 8.4613e-002 9.8087e-003 9.8857e-003 7.2731e-003 9.9007e-003 9.7196e-003 7.1255e-003 1.2000e+002 1.0000e-001 5.0000e-002 9.8314e-002 8.4016e-002 9.8064e-003 9.7374e-003 7.1590e-003 9.9007e-003 9.5931e-003 7.0253e-003 6.0000e+000 1.0000e-001 1.3572e-001 5.1902e-001 5.0414e-001 2.5444e-001 3.3672e-001 2.7891e-001 1.1527e-002 2.6640e-001 2.4441e-001 1.2000e+001 1.0000e-001 1.3572e-001 2.8614e-001 2.7933e-001 1.4416e-001 2.4240e-001 2.0835e-001 9.9032e-003 8.8662e-002 8.5025e-002 1.8000e+001 1.0000e-001 1.3572e-001 2.1015e-001 2.0645e-001 8.3540e-002 1.5877e-001 1.3907e-001 9.8992e-003 4.5808e-002 4.4126e-002 2.4000e+001 1.0000e-001 1.3572e-001 1.8025e-001 1.7901e-001 5.0270e-002 1.0346e-001 9.2232e-002 9.8992e-003 3.2736e-002 3.2194e-002 3.0000e+001 1.0000e-001 1.3572e-001 1.6621e-001 1.6634e-001 3.2012e-002 6.9706e-002 6.3452e-002 9.8992e-003 2.7462e-002 2.7443e-002 3.6000e+001 1.0000e-001 1.3572e-001 1.5851e-001 1.5944e-001 2.1991e-002 4.9639e-002 4.6267e-002 9.8992e-003 2.4825e-002 2.5071e-002 4.2000e+001 1.0000e-001 1.3572e-001 1.5367e-001 1.5514e-001 1.6492e-002 3.7779e-002 3.6075e-002 9.8992e-003 2.3268e-002 2.3676e-002 4.8000e+001 1.0000e-001 1.3572e-001 1.5024e-001 1.5214e-001 1.3473e-002 3.0731e-002 3.0002e-002 9.8992e-003 2.2214e-002 2.2745e-002 5.4000e+001 1.0000e-001 1.3572e-001 1.4757e-001 1.4987e-001 1.1817e-002 2.6475e-002 2.6330e-002 9.8992e-003 2.1421e-002 2.2060e-002 6.0000e+001 1.0000e-001 1.3572e-001 1.4533e-001 1.4803e-001 1.0908e-002 2.3834e-002 2.4056e-002 9.8992e-003 2.0774e-002 2.1517e-002 6.6000e+001 1.0000e-001 1.3572e-001 1.4336e-001 1.4647e-001 1.0409e-002 2.2128e-002 2.2596e-002 9.8992e-003 2.0218e-002 2.1065e-002 7.2000e+001 1.0000e-001 1.3572e-001 1.4159e-001 1.4510e-001 1.0135e-002 2.0966e-002 2.1614e-002 9.8992e-003 1.9725e-002 2.0673e-002 7.8000e+001 1.0000e-001 1.3572e-001 1.3996e-001 1.4387e-001 9.9849e-003 2.0126e-002 2.0917e-002 9.8992e-003 1.9278e-002 2.0326e-002 8.4000e+001 1.0000e-001 1.3572e-001 1.3846e-001 1.4276e-001 9.9025e-003 1.9480e-002 2.0392e-002 9.8992e-003 1.8869e-002 2.0015e-002 9.0000e+001 1.0000e-001 1.3572e-001 1.3705e-001 1.4174e-001 9.8572e-003 1.8955e-002 1.9976e-002 9.8992e-003 1.8492e-002 1.9731e-002 9.6000e+001 1.0000e-001 1.3572e-001 1.3574e-001 1.4079e-001 9.8324e-003 1.8510e-002 1.9631e-002 9.8992e-003 1.8143e-002 1.9472e-002 1.0200e+002 1.0000e-001 1.3572e-001 1.3452e-001 1.3992e-001 9.8187e-003 1.8120e-002 1.9333e-002 9.8992e-003 1.7819e-002 1.9233e-002 1.0800e+002 1.0000e-001 1.3572e-001 1.3337e-001 1.3911e-001 9.8113e-003 1.7771e-002 1.9071e-002 9.8992e-003 1.7519e-002 1.9012e-002 1.1400e+002 1.0000e-001 1.3572e-001 1.3229e-001 1.3835e-001 9.8071e-003 1.7454e-002 1.8835e-002 9.8992e-003 1.7240e-002 1.8807e-002 1.2000e+002 1.0000e-001 1.3572e-001 1.3128e-001 1.3764e-001 9.8049e-003 1.7162e-002 1.8620e-002 9.8992e-003 1.6979e-002 1.8617e-002 6.0000e+000 1.0000e-001 3.6840e-001 5.6443e-001 6.4173e-001 2.5444e-001 3.3476e-001 2.8603e-001 1.1516e-002 2.7964e-001 3.1155e-001 1.2000e+001 1.0000e-001 3.6840e-001 3.6947e-001 4.3973e-001 1.4415e-001 2.5242e-001 2.3978e-001 9.8917e-003 1.2992e-001 1.7256e-001 1.8000e+001 1.0000e-001 3.6840e-001 3.0830e-001 3.6914e-001 8.3531e-002 1.8090e-001 1.8710e-001 9.8877e-003 8.9357e-002 1.2250e-001 2.4000e+001 1.0000e-001 3.6840e-001 2.8647e-001 3.4373e-001 5.0260e-002 1.3310e-001 1.4810e-001 9.8877e-003 7.6284e-002 1.0579e-001 3.0000e+001 1.0000e-001 3.6840e-001 2.7897e-001 3.3372e-001 3.2001e-002 1.0392e-001 1.2299e-001 9.8877e-003 7.1883e-002 9.9503e-002 3.6000e+001 1.0000e-001 3.6840e-001 2.7736e-001 3.2976e-001 2.1980e-002 8.6987e-002 1.0775e-001 9.8877e-003 7.0784e-002 9.7031e-002 4.2000e+001 1.0000e-001 3.6840e-001 2.7835e-001 3.2844e-001 1.6480e-002 7.7603e-002 9.8830e-002 9.8877e-003 7.1096e-002 9.6168e-002 4.8000e+001 1.0000e-001 3.6840e-001 2.8048e-001 3.2836e-001 1.3462e-002 7.2728e-002 9.3781e-002 9.8877e-003 7.2033e-002 9.6043e-002 5.4000e+001 1.0000e-001 3.6840e-001 2.8305e-001 3.2888e-001 1.1806e-002 7.0484e-002 9.1045e-002 9.8877e-003 7.3226e-002 9.6274e-002 6.0000e+001 1.0000e-001 3.6840e-001 2.8570e-001 3.2968e-001 1.0897e-002 6.9738e-002 8.9668e-002 9.8877e-003 7.4490e-002 9.6672e-002 6.6000e+001 1.0000e-001 3.6840e-001 2.8827e-001 3.3060e-001 1.0398e-002 6.9820e-002 8.9080e-002 9.8877e-003 7.5731e-002 9.7144e-002 7.2000e+001 1.0000e-001 3.6840e-001 2.9067e-001 3.3154e-001 1.0124e-002 7.0335e-002 8.8939e-002 9.8877e-003 7.6904e-002 9.7637e-002 7.8000e+001 1.0000e-001 3.6840e-001 2.9287e-001 3.3246e-001 9.9737e-003 7.1056e-002 8.9046e-002 9.8877e-003 7.7987e-002 9.8125e-002 8.4000e+001 1.0000e-001 3.6840e-001 2.9487e-001 3.3334e-001 9.8912e-003 7.1852e-002 8.9283e-002 9.8877e-003 7.8976e-002 9.8590e-002 9.0000e+001 1.0000e-001 3.6840e-001 2.9667e-001 3.3416e-001 9.8459e-003 7.2652e-002 8.9581e-002 9.8877e-003 7.9872e-002 9.9027e-002 9.6000e+001 1.0000e-001 3.6840e-001 2.9828e-001 3.3492e-001 9.8211e-003 7.3418e-002 8.9901e-002 9.8877e-003 8.0679e-002 9.9430e-002 1.0200e+002 1.0000e-001 3.6840e-001 2.9972e-001 3.3561e-001 9.8075e-003 7.4133e-002 9.0220e-002 9.8877e-003 8.1403e-002 9.9800e-002 1.0800e+002 1.0000e-001 3.6840e-001 3.0101e-001 3.3623e-001 9.8000e-003 7.4788e-002 9.0526e-002 9.8877e-003 8.2052e-002 1.0014e-001 1.1400e+002 1.0000e-001 3.6840e-001 3.0216e-001 3.3680e-001 9.7959e-003 7.5383e-002 9.0813e-002 9.8877e-003 8.2633e-002 1.0044e-001 1.2000e+002 1.0000e-001 3.6840e-001 3.0319e-001 3.3731e-001 9.7936e-003 7.5920e-002 9.1078e-002 9.8877e-003 8.3152e-002 1.0072e-001 6.0000e+000 1.0000e-001 1.0000e+000 6.3928e-001 9.1619e-001 2.5441e-001 3.1433e-001 2.7571e-001 1.1432e-002 2.4734e-001 3.1943e-001 1.2000e+001 1.0000e-001 1.0000e+000 5.1631e-001 7.9836e-001 1.4410e-001 2.4784e-001 2.5519e-001 9.8080e-003 1.7830e-001 2.8451e-001 1.8000e+001 1.0000e-001 1.0000e+000 5.0529e-001 7.6064e-001 8.3465e-002 2.0182e-001 2.3723e-001 9.8039e-003 1.6870e-001 2.6929e-001 2.4000e+001 1.0000e-001 1.0000e+000 5.2648e-001 7.6039e-001 5.0187e-002 1.7701e-001 2.2549e-001 9.8039e-003 1.7634e-001 2.6778e-001 3.0000e+001 1.0000e-001 1.0000e+000 5.6103e-001 7.7359e-001 3.1924e-002 1.6720e-001 2.1957e-001 9.8039e-003 1.9053e-001 2.7158e-001 3.6000e+001 1.0000e-001 1.0000e+000 6.0092e-001 7.9106e-001 2.1901e-002 1.6679e-001 2.1768e-001 9.8039e-003 2.0691e-001 2.7699e-001 4.2000e+001 1.0000e-001 1.0000e+000 6.4154e-001 8.0941e-001 1.6400e-002 1.7165e-001 2.1819e-001 9.8039e-003 2.2314e-001 2.8265e-001 4.8000e+001 1.0000e-001 1.0000e+000 6.8009e-001 8.2736e-001 1.3381e-002 1.7899e-001 2.1999e-001 9.8039e-003 2.3795e-001 2.8807e-001 5.4000e+001 1.0000e-001 1.0000e+000 7.1503e-001 8.4432e-001 1.1724e-002 1.8707e-001 2.2240e-001 9.8039e-003 2.5081e-001 2.9308e-001 6.0000e+001 1.0000e-001 1.0000e+000 7.4574e-001 8.5994e-001 1.0815e-002 1.9489e-001 2.2500e-001 9.8039e-003 2.6164e-001 2.9760e-001 6.6000e+001 1.0000e-001 1.0000e+000 7.7218e-001 8.7400e-001 1.0316e-002 2.0196e-001 2.2756e-001 9.8039e-003 2.7060e-001 3.0158e-001 7.2000e+001 1.0000e-001 1.0000e+000 7.9466e-001 8.8640e-001 1.0042e-002 2.0809e-001 2.2994e-001 9.8039e-003 2.7795e-001 3.0502e-001 7.8000e+001 1.0000e-001 1.0000e+000 8.1360e-001 8.9717e-001 9.8916e-003 2.1327e-001 2.3209e-001 9.8039e-003 2.8394e-001 3.0797e-001 8.4000e+001 1.0000e-001 1.0000e+000 8.2945e-001 9.0639e-001 9.8091e-003 2.1758e-001 2.3398e-001 9.8039e-003 2.8883e-001 3.1045e-001 9.0000e+001 1.0000e-001 1.0000e+000 8.4267e-001 9.1422e-001 9.7638e-003 2.2113e-001 2.3561e-001 9.8039e-003 2.9281e-001 3.1253e-001 9.6000e+001 1.0000e-001 1.0000e+000 8.5366e-001 9.2080e-001 9.7390e-003 2.2404e-001 2.3699e-001 9.8039e-003 2.9605e-001 3.1426e-001 1.0200e+002 1.0000e-001 1.0000e+000 8.6276e-001 9.2631e-001 9.7253e-003 2.2642e-001 2.3816e-001 9.8039e-003 2.9870e-001 3.1570e-001 1.0800e+002 1.0000e-001 1.0000e+000 8.7031e-001 9.3091e-001 9.7178e-003 2.2836e-001 2.3914e-001 9.8039e-003 3.0086e-001 3.1688e-001 1.1400e+002 1.0000e-001 1.0000e+000 8.7654e-001 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4.9689e-001 4.8347e-001 3.8418e-001 9.8765e-001 9.3600e-001 6.2385e-001 6.0000e+000 1.0000e+001 1.3572e-001 2.3585e+000 9.0016e-001 4.3680e-001 3.9161e-001 2.9708e-001 9.7161e-001 8.1872e-001 4.3479e-001 1.2000e+001 1.0000e+001 1.3572e-001 2.9252e+000 1.0644e+000 4.6214e-001 4.2281e-001 3.0848e-001 9.7236e-001 8.7528e-001 5.1277e-001 1.8000e+001 1.0000e+001 1.3572e-001 3.3110e+000 1.2055e+000 4.7606e-001 4.4444e-001 3.2954e-001 9.7237e-001 8.9853e-001 5.7769e-001 2.4000e+001 1.0000e+001 1.3572e-001 3.5729e+000 1.2945e+000 4.8370e-001 4.5833e-001 3.5004e-001 9.7237e-001 9.1035e-001 6.1475e-001 3.0000e+001 1.0000e+001 1.3572e-001 3.7452e+000 1.3440e+000 4.8789e-001 4.6703e-001 3.6587e-001 9.7237e-001 9.1686e-001 6.3391e-001 3.6000e+001 1.0000e+001 1.3572e-001 3.8545e+000 1.3697e+000 4.9020e-001 4.7239e-001 3.7682e-001 9.7237e-001 9.2058e-001 6.4348e-001 4.2000e+001 1.0000e+001 1.3572e-001 3.9217e+000 1.3826e+000 4.9146e-001 4.7567e-001 3.8393e-001 9.7237e-001 9.2272e-001 6.4817e-001 4.8000e+001 1.0000e+001 1.3572e-001 3.9620e+000 1.3890e+000 4.9215e-001 4.7767e-001 3.8838e-001 9.7237e-001 9.2395e-001 6.5047e-001 5.4000e+001 1.0000e+001 1.3572e-001 3.9856e+000 1.3921e+000 4.9253e-001 4.7888e-001 3.9108e-001 9.7237e-001 9.2466e-001 6.5159e-001 6.0000e+001 1.0000e+001 1.3572e-001 3.9991e+000 1.3937e+000 4.9274e-001 4.7961e-001 3.9269e-001 9.7237e-001 9.2506e-001 6.5213e-001 6.6000e+001 1.0000e+001 1.3572e-001 4.0067e+000 1.3944e+000 4.9286e-001 4.8004e-001 3.9363e-001 9.7237e-001 9.2529e-001 6.5240e-001 7.2000e+001 1.0000e+001 1.3572e-001 4.0109e+000 1.3947e+000 4.9292e-001 4.8030e-001 3.9418e-001 9.7237e-001 9.2541e-001 6.5252e-001 7.8000e+001 1.0000e+001 1.3572e-001 4.0131e+000 1.3949e+000 4.9295e-001 4.8046e-001 3.9449e-001 9.7237e-001 9.2548e-001 6.5258e-001 8.4000e+001 1.0000e+001 1.3572e-001 4.0143e+000 1.3950e+000 4.9297e-001 4.8055e-001 3.9467e-001 9.7237e-001 9.2551e-001 6.5260e-001 9.0000e+001 1.0000e+001 1.3572e-001 4.0149e+000 1.3950e+000 4.9298e-001 4.8060e-001 3.9477e-001 9.7237e-001 9.2553e-001 6.5261e-001 9.6000e+001 1.0000e+001 1.3572e-001 4.0152e+000 1.3950e+000 4.9299e-001 4.8063e-001 3.9483e-001 9.7237e-001 9.2554e-001 6.5261e-001 1.0200e+002 1.0000e+001 1.3572e-001 4.0154e+000 1.3950e+000 4.9299e-001 4.8065e-001 3.9486e-001 9.7237e-001 9.2555e-001 6.5261e-001 1.0800e+002 1.0000e+001 1.3572e-001 4.0154e+000 1.3950e+000 4.9299e-001 4.8065e-001 3.9488e-001 9.7237e-001 9.2555e-001 6.5261e-001 1.1400e+002 1.0000e+001 1.3572e-001 4.0154e+000 1.3949e+000 4.9299e-001 4.8066e-001 3.9488e-001 9.7237e-001 9.2555e-001 6.5260e-001 1.2000e+002 1.0000e+001 1.3572e-001 4.0154e+000 1.3949e+000 4.9299e-001 4.8066e-001 3.9489e-001 9.7237e-001 9.2555e-001 6.5260e-001 6.0000e+000 1.0000e+001 3.6840e-001 2.3971e+000 1.0394e+000 4.2603e-001 3.8351e-001 3.0049e-001 8.7230e-001 7.4993e-001 4.7181e-001 1.2000e+001 1.0000e+001 3.6840e-001 3.0112e+000 1.2416e+000 4.4407e-001 4.0824e-001 3.1839e-001 8.7281e-001 7.9793e-001 5.4937e-001 1.8000e+001 1.0000e+001 3.6840e-001 3.4327e+000 1.3997e+000 4.5399e-001 4.2591e-001 3.4035e-001 8.7281e-001 8.1711e-001 6.0105e-001 2.4000e+001 1.0000e+001 3.6840e-001 3.7239e+000 1.5047e+000 4.5943e-001 4.3742e-001 3.5921e-001 8.7281e-001 8.2676e-001 6.3037e-001 3.0000e+001 1.0000e+001 3.6840e-001 3.9231e+000 1.5703e+000 4.6241e-001 4.4470e-001 3.7324e-001 8.7281e-001 8.3217e-001 6.4679e-001 3.6000e+001 1.0000e+001 3.6840e-001 4.0577e+000 1.6105e+000 4.6405e-001 4.4924e-001 3.8297e-001 8.7281e-001 8.3539e-001 6.5617e-001 4.2000e+001 1.0000e+001 3.6840e-001 4.1478e+000 1.6352e+000 4.6495e-001 4.5206e-001 3.8947e-001 8.7281e-001 8.3737e-001 6.6167e-001 4.8000e+001 1.0000e+001 3.6840e-001 4.2077e+000 1.6505e+000 4.6544e-001 4.5382e-001 3.9373e-001 8.7281e-001 8.3862e-001 6.6499e-001 5.4000e+001 1.0000e+001 3.6840e-001 4.2473e+000 1.6602e+000 4.6571e-001 4.5492e-001 3.9648e-001 8.7281e-001 8.3942e-001 6.6704e-001 6.0000e+001 1.0000e+001 3.6840e-001 4.2734e+000 1.6664e+000 4.6586e-001 4.5560e-001 3.9825e-001 8.7281e-001 8.3993e-001 6.6834e-001 6.6000e+001 1.0000e+001 3.6840e-001 4.2907e+000 1.6704e+000 4.6594e-001 4.5603e-001 3.9939e-001 8.7281e-001 8.4026e-001 6.6918e-001 7.2000e+001 1.0000e+001 3.6840e-001 4.3020e+000 1.6730e+000 4.6599e-001 4.5631e-001 4.0013e-001 8.7281e-001 8.4048e-001 6.6972e-001 7.8000e+001 1.0000e+001 3.6840e-001 4.3096e+000 1.6748e+000 4.6601e-001 4.5648e-001 4.0060e-001 8.7281e-001 8.4062e-001 6.7008e-001 8.4000e+001 1.0000e+001 3.6840e-001 4.3145e+000 1.6759e+000 4.6603e-001 4.5659e-001 4.0090e-001 8.7281e-001 8.4072e-001 6.7032e-001 9.0000e+001 1.0000e+001 3.6840e-001 4.3178e+000 1.6767e+000 4.6603e-001 4.5666e-001 4.0110e-001 8.7281e-001 8.4078e-001 6.7048e-001 9.6000e+001 1.0000e+001 3.6840e-001 4.3200e+000 1.6772e+000 4.6604e-001 4.5670e-001 4.0123e-001 8.7281e-001 8.4082e-001 6.7058e-001 1.0200e+002 1.0000e+001 3.6840e-001 4.3214e+000 1.6776e+000 4.6604e-001 4.5673e-001 4.0132e-001 8.7281e-001 8.4085e-001 6.7065e-001 1.0800e+002 1.0000e+001 3.6840e-001 4.3224e+000 1.6778e+000 4.6604e-001 4.5675e-001 4.0137e-001 8.7281e-001 8.4087e-001 6.7070e-001 1.1400e+002 1.0000e+001 3.6840e-001 4.3231e+000 1.6779e+000 4.6604e-001 4.5676e-001 4.0141e-001 8.7281e-001 8.4088e-001 6.7073e-001 1.2000e+002 1.0000e+001 3.6840e-001 4.3235e+000 1.6780e+000 4.6604e-001 4.5677e-001 4.0143e-001 8.7281e-001 8.4089e-001 6.7075e-001 6.0000e+000 1.0000e+001 1.0000e+000 2.4306e+000 1.3026e+000 3.7377e-001 3.4123e-001 2.8175e-001 4.9793e-001 4.5638e-001 3.7779e-001 1.2000e+001 1.0000e+001 1.0000e+000 3.0702e+000 1.5881e+000 3.5504e-001 3.3091e-001 2.8341e-001 4.9751e-001 4.7320e-001 4.1330e-001 1.8000e+001 1.0000e+001 1.0000e+000 3.5219e+000 1.7936e+000 3.4475e-001 3.2738e-001 2.8983e-001 4.9751e-001 4.7987e-001 4.3066e-001 2.4000e+001 1.0000e+001 1.0000e+000 3.8565e+000 1.9424e+000 3.3910e-001 3.2639e-001 2.9613e-001 4.9751e-001 4.8331e-001 4.4027e-001 3.0000e+001 1.0000e+001 1.0000e+000 4.1110e+000 2.0517e+000 3.3600e-001 3.2639e-001 3.0119e-001 4.9751e-001 4.8536e-001 4.4614e-001 3.6000e+001 1.0000e+001 1.0000e+000 4.3077e+000 2.1335e+000 3.3430e-001 3.2672e-001 3.0497e-001 4.9751e-001 4.8670e-001 4.4999e-001 4.2000e+001 1.0000e+001 1.0000e+000 4.4613e+000 2.1958e+000 3.3336e-001 3.2712e-001 3.0772e-001 4.9751e-001 4.8762e-001 4.5265e-001 4.8000e+001 1.0000e+001 1.0000e+000 4.5822e+000 2.2440e+000 3.3285e-001 3.2750e-001 3.0971e-001 4.9751e-001 4.8828e-001 4.5456e-001 5.4000e+001 1.0000e+001 1.0000e+000 4.6780e+000 2.2816e+000 3.3257e-001 3.2782e-001 3.1114e-001 4.9751e-001 4.8876e-001 4.5598e-001 6.0000e+001 1.0000e+001 1.0000e+000 4.7542e+000 2.3114e+000 3.3241e-001 3.2808e-001 3.1219e-001 4.9751e-001 4.8913e-001 4.5705e-001 6.6000e+001 1.0000e+001 1.0000e+000 4.8151e+000 2.3350e+000 3.3233e-001 3.2828e-001 3.1297e-001 4.9751e-001 4.8940e-001 4.5788e-001 7.2000e+001 1.0000e+001 1.0000e+000 4.8638e+000 2.3539e+000 3.3228e-001 3.2845e-001 3.1354e-001 4.9751e-001 4.8962e-001 4.5852e-001 7.8000e+001 1.0000e+001 1.0000e+000 4.9030e+000 2.3691e+000 3.3226e-001 3.2858e-001 3.1398e-001 4.9751e-001 4.8979e-001 4.5903e-001 8.4000e+001 1.0000e+001 1.0000e+000 4.9345e+000 2.3813e+000 3.3224e-001 3.2868e-001 3.1431e-001 4.9751e-001 4.8992e-001 4.5943e-001 9.0000e+001 1.0000e+001 1.0000e+000 4.9599e+000 2.3911e+000 3.3224e-001 3.2875e-001 3.1457e-001 4.9751e-001 4.9002e-001 4.5974e-001 9.6000e+001 1.0000e+001 1.0000e+000 4.9804e+000 2.3990e+000 3.3223e-001 3.2882e-001 3.1477e-001 4.9751e-001 4.9011e-001 4.6000e-001 1.0200e+002 1.0000e+001 1.0000e+000 4.9969e+000 2.4055e+000 3.3223e-001 3.2887e-001 3.1493e-001 4.9751e-001 4.9017e-001 4.6020e-001 1.0800e+002 1.0000e+001 1.0000e+000 5.0103e+000 2.4106e+000 3.3223e-001 3.2890e-001 3.1505e-001 4.9751e-001 4.9023e-001 4.6036e-001 1.1400e+002 1.0000e+001 1.0000e+000 5.0212e+000 2.4148e+000 3.3223e-001 3.2893e-001 3.1515e-001 4.9751e-001 4.9027e-001 4.6049e-001 1.2000e+002 1.0000e+001 1.0000e+000 5.0299e+000 2.4182e+000 3.3223e-001 3.2896e-001 3.1523e-001 4.9751e-001 4.9030e-001 4.6060e-001 Finally, the objective function to minimize is equation (15) in the Gen. Res. paper. You can choose to use scaling weights w_ij, but in fact we have solved the problem without scaling (w_ij=1 for all i,j), concluding that it does not make a difference for stochastic methods. BEST KNOWN SOLUTIONS (updated on November-2006) ===================================================================== In the Genome Research paper cited above, Evolution Strategies (ES) were the only method to solve this problem, with computation times of about 40 h using a PC Pentium-III/866 MHz. Shortly after this paper was submitted for publication, we were able to improve that solution by means of a hybrid stochastic-deterministic method. Preliminary results are mentioned in: Banga, J.R., Moles, C.G., Alonso, A.A. (2003) Global optimization of bioprocesses using stochastic and hybrid methods. In "Frontiers In Global Optimization ", C.A. Floudas and P. M. Pardalos, (Eds.), Nonconvex Optimization and Its Applications, vol. 74, pp. 45-70, Kluwer Academic Publishers, ISBN 1-4020-7699-1. Basically, we were able to recover *almost exactly* the original set of parameters with a much better computation times, about 4 hours using a PC Pentium-III/800 MHz, which is almost an order of magnitude less than using ES alone. Later, we were able to solve this problem with noise (i.e. errors in data) as well (3 and 5 % tested so far). These new results were presented at ICSB 2004: Rodríguez, M., C.G. Moles, P. Mendes, J. R. Banga (2004) A hybrid approach for efficient and robust parameter estimation in biochemical pathways. Oral presentation at 5th International Conference on Systems Biology (ICSB 2004), October 9 - 13, 2004, Heidelberg, Germany. An extended version of this paper has appeared in BioSystems: Rodriguez-Fernandez, M., P. Mendes and J. R. Banga (2006) A hybrid approach for efficient and robust parameter estimation in biochemical pathways. BioSystems 83(2-3):248-265. More recently, we have developed and implemented a new metaheuristic for global optimization which could solve this benchmark problem in just a *few minutes* using the same PC mentioned above. This results were presented at: Rodríguez-Fernández, M., J. A. Egea and J. R. Banga, (2005). Novel metaheuristics for parameter estimation in nonlinear dynamic biological systems. European Conference on Mathematical and Theoretical Biology (ECMTB05), July 18 - 22, 2005, Dresden, Germany. An extended paper describing this novel technique in detail appeared as: Rodriguez-Fernandez, M., J. A. Egea and J. R. Banga (2006) Novel Metaheuristic for Parameter Estimation in Nonlinear Dynamic Biological Systems. BMC Bioinformatics 7:483. Available for free at http://www.biomedcentral.com/1471-2105/7/483/abstract Please contact julio@iim.csic.es if you want updated info on this. Similarly, please let us know about your experiences with this problem. ===================================================================== Dr. Julio R. Banga Process Engineering Group Instituto de Investigaciones Marinas (C.S.I.C.) Spanish Council for Scientific Research C/Eduardo Cabello 6 36208 Vigo (SPAIN) E-mail: julio@iim.csic.es http://www.iim.csic.es/~julio/ http://www.iim.csic.es/~gingproc/ ph: +34 986 214 473 (direct) fax: +34 986 292 762 ---------------------------------------------------------------