# Optimization examples

Here we give some examples of the BioUML usage for solving the problem of parameter estimation applied to the models of biochemical pathways. For details about creation your oun optimization document in BioUML, see the chapter Optimization document. All information about the optimization methods implemented in BioUML is done in the chapter Optimization problem.

## Testing the convergence rate of the optimization methods

• Optimization document: data > Examples > Optimization > Data > Documents > test_case_1A
• Model: data > Examples > Optimization > Data > Diagrams > diagram_1A
• Experimental data: data > Examples > Optimization > Data > Experiments > exp_data_1

To analyze a convergence rate of the optimization methods implemented in BioUML [1], we considered a reaction chain extracted from the model by Neumann et al. [2] and representing activation of caspase-8 triggered by the receptor CD95 (APO-1/Fas).

The test model of caspase-8 activation

 ID Reactions Reaction rates Initial values r1 CD95L + FADD:CD95R → DISC k1 ⋅ [CD95L] ⋅ [CD95R:FADD] [CD95L]0 = 113.220, [CD95R:FADD]0 = 91.266 r2 DISC + pro8 → DISC:pro8 k2 ⋅ [DISC] ⋅ [pro8] [pro8]0 = 64.477, [DISC]0 = 0.0 r3 DISC:pro8 + pro8 → 2 · p43/p41 k3 ⋅ [DISC:pro8] ⋅ [pro8] [pro8]0 = 64.477, [DISC:pro8]0 = 0.0 r4 2 · p43/p41 → casp8 k4 ⋅ [p43/p41]2 [p43/p41]0 = 0.0 r5 casp8 → k5 ⋅ [casp8] [casp8]0 = 0.0

We performed estimation of parameters using the search space defined as:

where upper bounds were chosen based on the order of magnitude of parameter values proposed in [2].

Estimation was based on the experimental data obtained by Neumann et al. [2] for procaspase-8 and its cleaved products p43/p41 and caspase-8.

 Time (min-1) p43/p41 (nM) pro-8 (nM) casp-8 (nM) 0.0 0.058 59.963 0.000 10.0 0.268 57.565 0.041 20.0 4.760 58.590 0.316 30.0 8.252 59.422 1.397 45.0 16.144 48.190 3.520 60.0 17.021 38.950 3.947 90.0 15.269 23.502 4.871 120.0 12.530 13.127 4.878 150.0 10.335 10.703 4.228

We reviewed solutions obtained by all optimization methods for 100 runs. Each run was based on the generation of 107 different guesses. The best result was obtained by the particle swarm optimization (PSO) and the cellular genetic algorithm (MOCell). Methods SRES, MOCell and PSO found similar solutions. Methods ASA and glbSolve found other values for parameters k1 and k2 showing lower efficiency.

The objective function mean values dynamics for 100 runs. The best value obtained by PSO is marked by the red line.

The best guesses obtained by optimization methods for 100 runs

 Parameters SRES MOCell PSO ASA glbSolve k1 0.0004691 0.0004611 0.0004277 0.0001028 0.0020576 k2 0.0002059 0.0002046 0.0002155 0.0007875 0.0001228 k3 0.0009999 0.0010000 0.0009984 0.0009930 0.0009527 k4 0.0007915 0.0008225 0.0008419 0.0008117 0.0007790 k5 0.0325900 0.0336720 0.0334167 0.0334118 0.0313443

Values of the objective function for 100 runs

 Methods The best value The mean value The worst value PSO 11.787 13.164 14.703 MOCell 12.082 13.484 14.771 SRES 12.466 14.987 18.283 ASA 13.728 15.794 16.610 glbSolve 16.614 16.614 16.614

## Testing the computational speed of the optimization methods

• Optimization documents: data > Examples > Optimization > Data > Documents
(test cases 1A, 1B, 1C, 2, 3)
• Models: data > Examples > Optimization > Data > Diagrams
(diagrams 1A, 1B, 1C, 2, 3 for the corresponding test cases)
• Experimental data: data > Examples > Optimization > Data > Experiments
(exp_data_1 for the test cases 1A, 1B, 1C; exp_data_2 for the test case 2; exp_data_3 for the test case 3)

We tested a computational speed of such optimization methods in BioUML as particle swarm optimization, adaptive simulated annealing, stochastic ranking evolution strategy (SRES), and cellular genetic algorithm. For this purpose, we used biochemical models with the different number of parameters and species introduced in the following test cases. Firstly, we derived three models of CD95-induced activation of caspase-8 from the model by Neumann et al. [2] with varying degrees of detail (the test cases 1A, 1B and 1C). Secondly, we took the test case proposed by Mendes et al. [3] for the MAP kinase cascade model developed by Kholodenko et al. [4] (the test case 2). Finally, we tested the model by Bagci et al. [5] representing the mitochondria-depended apoptosis (the test case 3).

As expected, the computational speed for all test cases directly depended on the number of parameters and species in the model. The greatest computational speed was shown by the method of particle swarm optimization. The other methods showed about the same speed for running in one core, wherease, for running in several cores, the computational speed of PSO, SRES and MOCell was evidently higher compared to the simulated annealing.

## References

1. Kutumova E., Ryabova A., Valeev T., Kolpakov F. BioUML plug-in for nonlinear parameter estimation using multiple experimental data. Virtual Biology. 2013. 1:47-58.
2. Neumann L., Pforr C., Beaudouin J., Pappa A., Fricker N., Krammer P.H., Lavrik I.N., Eils R. Dynamics within the CD95 death-inducing signaling complex decide life and death of cells. Molecular Systems Biology. 2010. 6:352.
3. Mendes P., Hoops S., Sahle S., Gauges R., Dada J., Kummer U. Computational modeling of biochemical networks using COPASI. Methods in Molecular Biology. 2009. 500:17–59.
4. Kholodenko B.N. Negative feedback and ultrasensitivity can bring about oscillations in the mitogenactivated protein kinase cascades. European Journal of Biochemistry. 2000. 267(6):1583–1588.
5. Bagci E.Z., Vodovotz Y., Billiar T.R., Ermentrout G.B., Bahar I. Bistability in apoptosis: roles of bax, bcl-2, and mitochondrial permeability transition pores. Biophysical Journal. 2006. 90(5):1546–1559.