SBML-PET-MPI: A parallel parameter estimation tool for SBML ...

SBML-PET-MPI: A parallel parameter estimation tool for SBML based models Zhike Zi

BIOSS Centre for Biological Signalling Studies University of Freiburg, 79104 Freiburg, Germany http://www.bioss.uni-freiburg.de/cms/sbml-pet-mpi.html http://sites.google.com/site/sbmlpetmpi/ E-mail:

December 2, 2010 SBML-PET-MPI version 1.1

Copyright © 2010, Zhike Zi

Table of Contents 1 INTRODUCTION .............................................................................................................................. 1 1.1 OVERVIEW OF SBML-PET-MPI ........................................................................................................... 1 1.2 METHODS IMPLEMENTED IN SBML-PET-MPI ........................................................................................ 1 1.2.1 Parameter estimation with global optimization algorithm................................................... 1 1.2.2 Parameter uncertainty analysis with profile likelihood exploit algorithm ............................ 1 1.2.3 Parameter uncertainty analysis with bootstrap method ...................................................... 2 2 QUICK START .................................................................................................................................. 3 2.1 SYSTEM AND PACKAGE REQUIREMENTS .................................................................................................. 3 2.2.1 Windows................................................................................................................................ 3 2.1.2 Linux ...................................................................................................................................... 3 2.2.3 Mac OS .................................................................................................................................. 3 2.2 INSTALLATION AND START INSTRUCTIONS ................................................................................................ 3 2.2.1 Windows................................................................................................................................ 3 2.2.2 Linux ...................................................................................................................................... 4 2.2.2 Mac OS .................................................................................................................................. 4 3 USING SBML-PET-MPI..................................................................................................................... 5 3.1 RUN SBML-PET-MPI ........................................................................................................................ 5 3.1.1 MPI daemon (mpd) launch and exit ...................................................................................... 5 3.1.2 Start SBML-PET-MPI .............................................................................................................. 5 3.1.3 Options for ODE Solver .......................................................................................................... 6 3.2 EXPERIMENTAL ) should not appear in the mathematical expression of the constant="false" in the SBML model.  Tip #3: An alternative to specify the constant="false". Then add new assignment rule for the new species. This trick can also speed up the optimization process. For the above example, one can add new species "totalEpo" and "totaEpoi", with the following assignment rule: totalEpo = Epo + dEpoe totalEpoi = Epo_EpoRi + dEpoi Then the mathematical expression of the id="ligand_washout"> time 60 0

10

Suppose the user has the following data set. Time 0 10 45 90 120 Normalized

data1 0.02 0.43 0.50 1.00 0.20 0

SD 0.02 0.05 0.10 0.10 0.05

Since the above data set doesn't include the event trigger time 60, the user can add a time point in the time course. Time 0 10 45 60 90 120 Normalized

data1 0.02 0.43 0.50 NaN 1.00 0.20 0

SD 0.02 0.05 0.10 NaN 0.10 0.05

For other trigger of events without time variable, SBML-PET-MPI will automatically detect the trigger time of the events.

3.3 Results display and save 3.3.1 Results displayed in the terminal and saved as files SBML-PET-MPI displays the result of optimization after each generation in the terminal as the following: -------------------- MESSAGE FROM SBML-PET-MPI ------------------->>> PARAMETER ESTIMATION with Experimental Data...... > generation: 1, best result from generation 1 > chisquare value = 2.874709e+04, penalty value for constraints = 0.000000 > best parameters = 7.208e-05 4.626e-05 1.698e-01 1.037e-04 8.938e+01 2.652e+02 1.941e+03 1.868e-06 1.498e+00 1.007e+03 > used time = 2.215 seconds > generation: 2, best result from generation 2 > chisquare value = 2.270294e+04, penalty value for constraints = 0.000000 > best parameters = 1.002e-04 2.092e-05 2.295e-01 7.647e-02 2.347e-02 7.488e-04 2.255e+03 4.425e-07 1.951e-06 9.985e+02 > used time = 2.496 seconds > generation: 3, best result from generation 3 > chisquare value = 1.221319e+04, penalty value for constraints = 0.000000 > best parameters = 4.048e-01 2.423e-04 1.325e-01 4.580e-04 2.467e+01 3.787e+01 1.940e+03 2.634e-06 2.700e+00 1.003e+03 > used time = 2.792 seconds ... ... When the running of SBML-PET-MPI is finished, the SBML models with the parameter set from the best fit, the optimization history and the summary of parameter estimation analysis are saved in "result" subdirectory of the installed SBML-PET-MPI directory (Figure 3.2).

11

Figure 3.2 Final results of parameter estimation The estimated parameters are stored in SBML files: "BestFitnessSBML.xml" (SBML format level 2 version 4) and "BestFitnessSBML_L2V1.xml" (SBML format level 2 version 1), which can be imported to other SBML software for analysis. Even SBML-PET-MPI is suddenly stopped or aborted, an updated SBML model with the latest best fitted parameters will be automatically saved as well. The summary of the estimated parameter values are saved in the file of "ParameterAnalysis.txt", which looks like the following: ******************************************************************** Summary of Estimated Parameters with Profile Likelihood Exploit ******************************************************************** chisquare = 2.11705 total number of experimental data, Nd = 38 total number of estimated parameters, Np = 4 chisquare/Nd = 0.055712 ID (Name) Value from Best Fit 95% confidence interval E 5.007936e+00 (4.970768e+00; 5.045104e+00) k1 1.977730e+00 (1.896612e+00; 2.066574e+00) k2 9.822647e-01 (9.477320e-01; 1.019867e+00) k3 9.985696e-01 (9.943764e-01; 1.002763e+00) 3.3.2 Plot of data fitting and parameter analysis with MATLAB SBML-PET-MPI provides the plot of data fitting and parameter analysis from profile likelihood with MATLAB. If MATLAB is installed in the host computer, the summary of the data fitting, optimization history and the profile likelihood of the parameters will be automatically plotted at the end of parameter estimation analysis (Figure 3.3 and Figure 3.4). The user can also copy the "result" subdirectory from the host to the local computer with MATLAB, then run plotResults (included in SBML-PET-MPI) in the MATLAB of the local computer. Please note that the "result" directory should put as a subdirectory of the directory where the file "plotResults.m" locates.

12

Figure 3.3 Summary of data fitting plots from SBML-PET-MPI

Figure 3.4 Profile likelihood plots of parameters from SBML-PET-MPI. The gray dotted line in denotes the threshold for 95% confidence intervals of the parameters calculated with profile likelihood exploit algorithm. The red circle point refers to the parameter value from the best fit.  Tip #6: Do NOT modify the files saved in result directory. The change of

13

these files might affect the plot of data fitting and parameter analysis with MATLAB.

3.4 Specific Explanation for Data from Multiple Experimental Conditions If the experimental data are obtained from different experimental conditions, the user need modify some parameters defined in the model to distinguish the difference of the conditions. After SBML-PET-MPI started, a file named "ExpConditionsData.txt" will be produced at the "temp" sub-directory of SBML-PET-MPI. For different experimental conditions, the possible varied parameters values will be listed in this file. The user should modify the corresponding data for different conditions and save the file, then press ENTER key to continue. The following messages will display on the terminal: -------------------- MESSAGE FROM SBML-PET-MPI -------------------Please open the file temp/ExpConditionsData.txt and modify the data for different experimental conditions > If you have finished, press ENTER key to continue The value of the species listed in the file of "ExpConditionsData.txt" indicates the corresponding initial concentration/amount of the species. The user can also define some parameters to specify different experimental conditions, define them as global parameters rather than local parameters in the model.

14

4 Examples This section will present some examples showing the application and usage of SBML-PET-MPI. These examples cover different applications of SBML-PET-MPI.

4.1 A Simple Model for Enzyme Substrate Reactions The well-known Michaelis-Menten equations describe enzyme kinetics, which include four molecular species, namely the enzyme, E, the substrate, S, the product, P and the intermediate, ES. The reactions are listed below: (E4.1)

k1 k3 E  S  ES  E  P k2 The ODE system consists of 4 ordinary differential equations.

d[ E]  k 2 [ ES ]  k3 [ ES ]  k1[ E ][ S ] dt

(E4.2)

d[S ]  k2 [ ES ]  k1[ E ][ S ] dt

(E4.3)

d [ ES ]  k1[ E ][ S ]  k2 [ ES ]  k3[ ES ] dt

(E4.4)

d [ P]  k3 [ ES ] dt

(E4.5)

To demonstrate the usage of SBML-PET-MPI, we assume that the true values of k1 = 2, k2 = 1, k3 = 1 and the initial concentration of [E] = 5 [S] = 10, [ES] = 0, [P] = 0. Then we run SBML-PET-MPI to estimate the values of k1, k2, k3 and the initial concentration of E and S within the following range Table 4.1 Range of parameters to be estimated Parameter ID

Minimum Value

Maximum Value

E

0.01

100

k1

0.001

1000

k2

0.001

1000

k3

0.001

1000

There are 4 parameters to be estimated. The initial concentration of E covers four magnitudes, while the value of kinetic parameters k1, k2 and k3 covers 6 magnitudes. The in silico experimental data is generated in the following way: we randomly sampled 20 experimental data sets from a normal distribution: the mean = the original simulation data and standard deviation = 5% corresponding data. The mean and corresponding standard deviations from these 20 artificial data sets are

15

used for parameter estimation. According the description in section "3.2 Experimental Data File for Parameter Estimation", the data file for this example is # data file for Michaelis-Menten Equations Model #*********************************************************** # PART I: PAREMETERS TO BE ESTIMATED #*********************************************************** PARAMETER ID MINIMUM VALUE MAXIMUM VALUE E 0.01 100 k1 0.001 1e3 k2 0.001 1e3 k3 0.001 1e3 #*********************************************************** # PART II: Experimental Data #*********************************************************** TOTAL NUMBER of EXPERIMENTAL CONDITIONS = 1 # The number of Time Courses at each Experimental Condition AT CONDITION 1, The TOTAL NUMBER of TIME COURSES is 1 DATA OF TIME COURSE 1 AT CONDITION 1 NUM_TIME_POINTS 19 NUM_EXP_DATA 2 Time S SD P SD 0 9.971476235 0.453141397 0 0 0.2 5.292235504 0.268280699 0.628160073 0.019133061 0.4 4.39726553 0.208274125 1.458557064 0.070669452 0.6 3.726573609 0.165386153 2.232857326 0.084280801 0.8 3.080806333 0.134350275 3.09237209 0.16668692 1 2.510472349 0.141611198 3.80900556 0.173941468 1.2 1.982882059 0.067026255 4.557345262 0.195609836 1.4 1.536801613 0.076873175 5.215831005 0.280492893 1.6 1.153292228 0.081686909 5.902796165 0.275531448 1.8 0.846651717 0.038701391 6.465823544 0.358515561 2 0.65483179 0.047542333 6.891697826 0.446185162 3 0.179484392 0.009625662 8.74544011 0.320153952 4 0.061421876 0.003239136 9.421806752 0.546102974 5 0.023203428 0.001171091 9.641262587 0.454142332 6 0.008947044 0.000525554 9.711707321 0.493301063 7 0.003764036 0.00012968 9.884865955 0.512322358 8 0.001521486 9.03E-05 9.941232427 0.46427798 9 0.000603876 2.71E-05 9.839203148 0.44095379 10 0.000251923 1.30E-05 10.06626382 0.314179785 Normalized 0 0 #*********************************************************** # PART III: Constraints Information #*********************************************************** # The number of Constraints at each Experimental Condition AT CONDITION 1, The TOTAL NUMBER of CONSTRAINTS is 0 Figure 4.1 Parameter estimation data file for the enzyme substrate model The fitting of the data and the profile likelihood of the parameters are shown in Figure 4.2 and Figure 4.3.

16

Figure 4.2 Summary of data fitting plots for the enzyme substrate model

Figure 4.3 Profile likelihood plots of parameters for the enzyme substrate model. The gray dotted line in denotes the threshold for 95% confidence intervals of the parameters calculated with profile likelihood exploit algorithm. The red circle point refers to the parameter value from the best fit.

17

4.2 The Epo Model (with Real Experimental Data) Becker et al. developed a mathematical model for the erythropoietin (Epo) and Epo receptor (EpoR) interaction (Becker et al., 2010). This model was calibrated with multiple real experimental data sets. Here we use the model and the experimental data sets in this work as a benchmark example for SBML-PET-MPI. Since the data sets in this work are generated for two models (the core model and the auxiliary model), we merged the original two models in one model ("Examples/Epo.xml") in order to simultaneously fitting the data sets for the two models. The SBML models and data sets are obtained from the following website: http://webber.physik.uni-freiburg.de/~jeti/Science_Becker_data_models/ In SBML-PET-MPI, we set the same estimated parameters with the same ranges and use the same data sets as the original work. Detailed information is described in the file of "Examples/Epo_data.txt". As shown in Table 4.2, SBML-PET-MPI can reproduce a similar parameter estimation result as original work. It also obtained similar parameter intervals information. Table 4.2 Values and statistical interval of the estimated parameters from SBML-PETMPI and those from the original study Parameter ID

Estimated value from SBML-PET-MPI

Estimated value in Becker et al.

95% confidence interval from SBML-PET-MPI

95% confidence interval in Becker et al.

kt

0.03270

0.03294

(0.03259;0.03278)

(0.0300; 0.0365)

kon

1.0523×10-4

1.0496×10-4

(1.049×10-4; 1.055×10-4)

(1.003×10-4;1.097×10-4)

ke

0.07465

0.07483

(0.0744; 0.0749)

(0.0723; 0.0776)

kex

0.00982

0.00994

(0.00979; 0.0985)

(0.0082; 0.0119)

kdi

0.003150

0.003179

(0.00314; 0.00316)

(0.00272; 0.00365)

kde

0.01635

0.01640

(0.01630; 0.01640)

(0.01557; 0.01726)

Epo

2030.1

2030.19

(2023.79; 2036.47)

(2024.98; 2035.41)

kon_SAv

2.287×10-6

2.294×10-6

(2.280×10-6; 2.294×10-6)

(2.162×10-6; 2.430×10-6)

kex_SAv

0.0108

0.0110

(0.0107; 0.0109)

(0.0041; 0.0186)

SAv

999.291

999.293

(996.168; 1002.413)

(999.173; 999.413)

The fitting of all the data sets is shown in Figure 4.4. In this case, the SBMLPET-MPI was run with 7 CPUs and the chi-square objective function converged within one minute.

18

Figure 4.4 Data fitting and convergence curve of Epo model To evaluate the performance of the parallelized parameter estimation analysis algorithms, we recorded the running time of SBML-PET-MPI for global optimization (parameter estimation) and profile likelihood exploit analysis (parameter identifiability analysis) of the Epo model. The speed up scalability is good when the number of processors is up to 10, but the communication time between the processors decreases the speed up performance for this small example (Figure 4.5). The speed up scalability is in general better for complex models with more estimated parameters.

Figure 4.5 The running time and speed up for the optimization (1000 generations) and parameter identifiability analysis of the Epo model with real experimental data sets using 1-20 CPUs. The running time might slightly vary at different execution of the program.

19

4.3 E. Coli Tryptophan Operon Model (With Data from Different Conditions) Sharad Bhartiya et al. developed a mathematical model to study the effect of external tryptophan on the trp operon (Bhartiya et al., 2003). The model accounts for the effect of feedback repression by tryptophan with a Hill equation. The model describes such a process: (1) In the first step, tryptophan molecule, T, and the aporepressor molecule, R, form the intermediate, RT. (2) Then, another tryptophan molecule binds with RT yielding the holorepressor, RT2. (3) The holorepressor binds with the free operator, O, and forms operon–holorepressor complex, ORT2, which represses tryptophan synthesis. A schematic representation of the trp operon system is provided in Figure 4.6.

Figure 4.6 Schematic representation of Dynamic Model of E. coli Tryptophan Operon. Tryptophan concentration is influenced by (a) enzyme synthesis (E) with kinetic constant k1, (b) enzyme catalyzed reaction for tryptophan synthesis from a nitrogenous substrate (NS) with kinetic constant kd, and (c) instantaneous uptake from the environment. Sufficient availability of tryptophan, T, leads to binding with the aporepressor molecule R, with a dissociation constant K1. The resulting holorepressor next binds with the free operon, O, with a dissociation constant K2, resulting in transcriptional and translational repression of enzyme synthesis. (Figure is modified from Fig. 1 in Bhartiya et al., 2003) The ODE system for this model includes two species and 11 parameters. Detailed information about this model is described in Bhartiya et al., 2003. The SBML file provided in SBML-PET-MPI is modified from the model in JWS Online models (http://jjj.biochem.sun.ac.za/index.html). The data produced with different extracellular trypophan concentrations (0.01 μM, 0.1 μM, 1.0 μM and 10 μM) are used to estimate 8 parameters values: ki1, Ot, eval, fval, Tomax, gval, kg, mu. The necessary files for this problem are located in directory of Examples with names of "Ecoli.xml" and "Ecoli_data.txt". Note: After SBML-PET-MPI is started, please modify the corresponding data in the file "ExpConditionsData.txt" at the "temp" sub-directory of SBML-PET-MPI. Set the values of parameter "Text" to be 0.01, 0.1, 1.0 and 10 at condition 1, condition 2, condition 3 and condition 4 respectively, which indicates that the extra-cellular trypophan concentrations are 0.01 0.1, 1.0 and 10μM for different conditions. Don’t change other parameters values listed in the file "ExpConditionsData.txt". After modifying the experimental conditions file, press 20

ENTER key to continue. The modified "ExpConditionsData.txt" has the following contents: # DATA FOR EXPERIMENTAL CONDITION 1 DATA ID VALUE Enz 0 Ts 0 s 0 Text 0.01 # DATA FOR EXPERIMENTAL CONDITION 2 DATA ID VALUE Enz 0 Ts 0 s 0 Text 0.1 # DATA FOR EXPERIMENTAL CONDITION 3 DATA ID VALUE Enz 0 Ts 0 s 0 Text 1 # DATA FOR EXPERIMENTAL CONDITION 4 DATA ID VALUE Enz 0 Ts 0 s 0 Text 10 The fitting of the data and the profile likelihood of the parameters are shown in Figure 4.7 and Figure 4.8.

Figure 4.7 Result of Best Fit for E. coli. Tryptophan Operon Model

21

Figure 4.8 Profile likelihood plots of parameters for E. coli. Tryptophan Operon Model. The gray dotted line in denotes the threshold for 95% confidence intervals of the parameters calculated with profile likelihood exploit algorithm. The red circle point refers to the parameter value from the best fit.

22

5 FAQ Q1: What is the objective function ( χ2) defined in SBML-PET-MPI? A: SBML-PET-MPI defines the objective/cost function (chi-square, χ2) with the sum of the weighted least squares between model simulation data and the experimental data. The SD values are the weight factors, which can be set as the measured standard deviation or the maximum value of the data sets. More details are described in section of "3.2.5 An important note about the SD values in the data file". Q2: How to prepare SBML file for my model? A: There are many tools for create the model in SBML format, for example, Copasi or CellDesigner. If the model contains events, we recommend CellDesigner for defining events. Q3: How to prepare the data file for parameter estimation? A: Read the section of "3.4 Prepare the Data File for Parameter Estimation" in manual documentation. Q4: How to define events in the SBML-PET-MPI? A: We recommend the user to use CellDesigner to edit the model with events. If time will appear in the events, use it as "time". The user does not need set a parameter/species for time, it will be automatically recognized by SBML-PET-MPI. For the time-dependent events, please pay attention to the notes for the data file preparation at section "3.2.6 About trigger of events with time variable". For other variables that are controlled by events, the user should define them as global parameters or species variables. Q5: How much of memory needed for running SBML-PET-MPI? A: SBML-PET-MPI use dynamic memory allocation and free strategy. Therefore, the usage of memory in SBML-PET-MPI depends on the number of processors (CPUs) started, the complexity of the model, the number of data points and parameters to be used. We recommend 100 M bytes of memory per processor. Q6: What does it mean if the estimated parameters are marked as nonidentifiable with profile likelihood exploit analysis? A: In this case, you should check the plot of profile likelihood of the parameters. If the plot of profile likelihood of the parameter is concave, then you can increase number of sampling points for profile likelihood exploit (--with-ple=###). If the plot of profile likelihood of the parameter is concave and it hits the bound of parameters (min or max), then you can try to increase the range of the corresponding parameter. If the plot of profile likelihood of the parameter is flat or convex, it means that the parameter value cannot be identified with the data sets. The user may either get more informative data set or measure (or fix) some estimated parameter values from other data sets. In addition, if the number of SRES generations is too small, this may also happen. In this case, please increase the number of SRES generations for global optimization. Q7: What about the speed up performance of SBML-PET-MPI and how fast is SBML-PET-MPI compared with other tools? A: SBML-PET-MPI has good speedup scalability with the increasing number of processors. SBML-PET-MPI speeds up the process of prameter estimation and identifiability analysis when the number of precoessors is more than 3. It would

23

be no apparent speed up between 1 processor and 2 processors due to the necesary model preocssing, results analysis designed in SBML-PET-MPI. Beacuse libSRES uses a different ODE solver (CVODE in C) from SBML-PET-MPI (ODEPACK in Fortran) and it doesn't need to decode the SBML model information. One should expect that after manually translating the specific model into ODE system, manually coding the estimated parameters and data information in libSRES, libSRES would be faster than SBML supported parameter estimation tools because our tool and other SBML tools need to process the SBML model, map the experimental data to the model simulation and update the model with the fitted parameters (write updated sbml model files during optimization). Therefore, it would be difficult to make a fair comparison of the absolute optimization time between compiled parameter estimation program with other SBML tools. Due to the above reason, it would be more reasonable to compare the speedup performance (not the absolute optimization time) of our tool to libSRES. We tested them with enzyme substrate reactions model at the same computer (same estimated parameters, generation, miu, lamada parameters). The speed up performance of both tools are similar (Figure 5.1). This is not a surprising result because our tool and libSRES used the same SRES algorithm and both are parallelized with MPI protocol.

Figure 5.1 Speed up performance of SBML-PET-MPI and libSRES Different SBML software tools might have different ODE solver, encoded in different programming language and use different optimization algorithms for parameter estimation. It would be difficult to compare the speed of optimization across different tools. Since both SBML-PET-MPI and COPASI are encoded in C. They have the same ODE solver from ODEPACK and also can use SRES for parameter estimation. We tested COPASI and SBML-PET-MPI with the enzyme substrate reactions model (details see Section 4.1). The estimated parameters and their range are set the same in COPASI as those in SBML-PET-MPI (details in Section 4.1). It takes about 14-15 minutes to finish the optimization with the following SRES algorithm setting: Number of Generation = 1000, Population Size = 238. With the same example running in the same computer system, we also implemented the optimization in SBML-PET-MPI with the same SRES algorithm setting as those set in COPASI. As shown in Table 5.1, SBML-PET-MPI has better performance than COPASI with 1 processor and it has good speed up scalability with the increasing number of processors. 24

Table 5.1 Performance of SBML-PET-MPI and its comparison to COPASI Number of Processors

Running time in COPASI 4.6

Running time in SBML-PET-MPI

1

890 seconds

370 seconds

2

NA

344 seconds

3

NA

188 seconds

4

NA

142 seconds

5

NA

120 seconds

6

NA

106 seconds

Note: The real running time might be slightly different at different execution of the programs

Q8: Can I run SBML-PET-MPI in computer with one processor? A: Yes, please download the single version of SBML-PET-MPI for running with one processor. In this case, there would be no speed up. The number of processors for SBML-PET-MPI should be more than 3 for speed up. Q9: What are the principles used to divide the global optimization, the profile likelihood method and the bootstrap method among multiple processors? A: The SRES global optimization process employs the classic (μ, λ)-ES evolution strategy algorithm, in which the selection is taken from the λ offspring only, whereas their μ parents are ignored even the fitness of the parents are better than that of the new generation. For each geneartion, SBML-PET-MPI distribute the evluations of the objective functions for λ offsprings into other n-1 processors with MPI protocol. The first processor is used to collecting the results from other processors and coordinate the overal algorithm. For the profile likelihood method, since the forward and backward profile likelihood can be independently calculated from the starting point of the best fitted parameter value. For m number of parameters, there are 2 × m profile likelihood calculations (forward and backward). Therefore, we distributed 2 × m profile likelihood calculations into n-1 processors. The first processor is used for result updating and the coordination of the whole algorithms. The bootstrap method with new synthetic data sets is based on the principle of parallel distribution as the SRES global optimization. The difference is that the bootstrap method requires many different runs of optimizations with new synthetic data sets. Q10: Why I get this message: "./SBML-PET-MPI: error while loading shared libraries: libsbml.so.4: cannot open shared object file: No such file or directory" A:The message comes out because the dynamic-link library (DLL) was not loaded in your system. The necessary DLL libraries are located at lib subdirectory of SBML-PET-MPI, to solve it, there are two solutions: (1) Permanent solution: you can export the LD_LIBRARY_PATH (for Linux system) or DYLD_LIBRARY_PATH (for Mac system) in the .bashrc or .bashr_profile file. Details please see the installation instruction notes in Page 4. (2) Temporary solution: After start a new terminal, first change to SBML-PET-MPI directory and then input the following command: export LD_LIBRARY_PATH=./lib

(for Linux system)

export DYLD_LIBRARY_PATH=./lib

(for Mac system)

25

Please note that solution 2 should be repeated for every new started terminal. If you stays in the same terminal, "export LD_LIBRARY_PATH=./lib" only need to do one time. Q11: How can I set the parameters for SRES global optimization algorithm? A: By default, SBML-PET-MPI uses an adaptive method for setting SRES algorithm parameters: lambda (offspring population size) and miu (parent population size), which depend on the number of parameters to be estimated: lambda = NumParameterToBeEstimated*7+210 miu = NumParameterToBeEstimated+30 The default maximum number of evolutionary generations is 2000. The uses can specify these parameter values by the following options: --with-gen=### --with-lambda=### --with-miu=###

set the number of evolutionary generations set the number of offspring population size set the number of parent population size

The ratio between parents and offspring population size is recommended to be about 1/3~1/10. According to the previous experiences reported, the ratio around 1/7 is good for most of the problems. Q12: More questions about SBML-PET-MPI? A: Please contact Zhike Zi by email:

.

26

6 References Becker, V., Schilling, M., Bachmann, J., Baumann, U., Raue, A., Maiwald, T., Timmer, J., and Klingmuller, U. (2010). Covering a broad dynamic range: information processing at the erythropoietin receptor. Science 328, 1404-1408. Bhartiya, S., Rawool, S., and Venkatesh, K.V. (2003). Dynamic model of Escherichia coli tryptophan operon shows an optimal structural design. Eur J Biochem 270, 26442651. Funahashi, A., Tanimura, N., Morohashi, M., and Kitano, H (2003). CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1, 159. Hindmarsh, A.C. (1983). ODEPACK, A Systematized Collection of ODE Solvers. In Scientific Computing, R.S. Stepleman, M. Carver, R. Peskin, W.F. Ames, and R. Vichnevetsky, eds. (Amsterdam, North-Holland), pp. 55-64. Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P., and Kummer, U. (2006). COPASI--a COmplex PAthway SImulator. Bioinformatics 22, 3067-3074. Hucka, M., Finney, A., Sauro, H.M., Bolouri, H., Doyle, J.C., Kitano, H., Arkin, A.P., Bornstein, B.J., Bray, D., Cornish-Bowden, A., et al. (2003). The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19, 524-531. Ji, X., and Xu, Y. (2006). libSRES: a C library for stochastic ranking evolution strategy for parameter estimation. Bioinformatics 22, 124-126. Press, W.H. (1992). Numerical recipes in C : the art of scientific computing, 2nd ed. edn (Cambridge University Press). Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmuller, U., and Timmer, J. (2009). Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 1923-1929. Runarsson, T.P., and Yao, X. (2000). Stochastic ranking for constrained evolutionary optimization. Ieee Transactions on Evolutionary Computation 4, 284-294. Zi, Z., and Klipp, E. (2006). SBML-PET: a Systems Biology Markup Language-based parameter estimation tool. Bioinformatics 22, 2704-2705.

27