BIOINFORMATICS Pages 1–7

Vol. 00 no. 00 2008 BIOINFORMATICS Pages 1–7

Dynamical Modeling and Multi-Experiment Fitting with PottersWheel Thomas Maiwald∗and Jens Timmer Freiburg Center for Data Analysis and Modeling, Freiburg University, Eckerstr. 1, 79104 Freiburg Institute of Physics, Freiburg University, Hermann Herder Str. 3, 79104 Freiburg, Germany Received on ; revised on ; accepted on

Associate Editor:

ABSTRACT 1 INTRODUCTION Motivation: Modelers in Systems Biology require a framework PottersWheel evolved within close collaborations of experimenta- to easily create new dynamic models, investigate their properties lists and modelers within the German HepatoSys initiative since and to apply multi-experiment fitting, where several experimental 2005. Therefore, modeling of experimental data has been the central data sets are fitted simultaneously. Multi-experiment fitting is a objective for its development from the beginning. Two experiences powerful approach to estimate parameter values, to check the were our main guidelines: validity of a given model, and to discriminate competing model hypotheses. The framework should be accessible via both a graphical 1. In order to identify models which are compliant with existing interface and an application programming interface, providing support biological knowledge and new laboratory measurements, within custom programs. High performance integration of ordinary the modeler requires easy interactive access to investigate differential equations and robust optimization is required for fitting of the dynamical properties of the model. To generate new experimental data. hypotheses about the reaction network or to postulate new Results: We here present a new comprehensive modeling system variables, intensive working with a model is crucial, framework called PottersWheel including novel methodologies to always in close relation to the measurements. The necessary satisfy these requirements with strong emphasis on the inverse functionalities ranges from real-time changing of parameter problem, i.e. data-based modeling of partially observed and noisy values and characteristics of driving input functions to efficient systems like signal transduction pathways and metabolic networks. refinement of the model structure itself. Simultaneously, we provide a detailed performance analysis of 2. Powerful fitting procedures are required to calibrate model integration and optimization approaches. PottersWheel is designed parameters in the context of several experimental data sets, as a MATLAB toolbox and includes numerous user interfaces. often under different experimental settings and with different Dynamically generated C MEX files and improved FORTRAN sets of measured species. Model-data-compliance and model integration approaches lead to up to 900 times faster integration discrimination should be quantified by statistical tests. than using pure MATLAB. Deterministic and stochastic optimization routines are combined by fitting in logarithmic parameter space In 2006, PottersWheel was released to the public as the first allowing for robust parameter calibration. Model investigation includes MATLAB toolbox to provide real-time, graphical user interface- statistical tests for model-data-compliance, model discrimination, based interactive modeling including multi-experiment fitting with identifiability analysis and calculation of Hessian- and Monte-Carlo- highly optimized model integration. Since then, the experiences and based parameter confidence limits. A rich application programming needs not only of HepatoSys members, but also from external users interface is available for customization and use within own MATLAB were incorporated into the software resulting in a very stable and code. rich modeling framework. The current release PottersWheel 1.6 is Availability: PottersWheel is freely available for academic usage up to 900 times faster than integration with MATLAB integrators. from http://www.PottersWheel.de/ for MATLAB 7.1 or higher. The web-site also contains a detailed documentation, examples, and Existing modeling software like the SBToolbox (Schmidt and introductory videos. The program has been intensively used since Jirstrand, 2006), the commercial MATLAB SymBiology toolbox, 2005 on Windows, Linux, and Intel Macintosh computers and does or COPASI (Hoops et al., 2006), originate from the direct problem not require special MATLAB toolboxes. where system properties are to be analyzed based on a given Contact: [email protected] model and parameter values. The analysis includes e.g. steady- state, stability, metabolic control, bifurcation and stoichiometric analysis. PottersWheel, on the other hand, originates from the inverse problem, where for existing data a model has to be identified (Winterhalder et al., 2006). This approach requires application of statistical concepts concerning calculation of confidence intervals ∗to whom correspondence should be addressed of calibrated parameters, identifiability analysis, and statistical tests

c Oxford University Press 2008. 1 Maiwald and Timmer

for model discrimination and model-data-compliance. In addition, PottersWheel introduced a new approach to save experimental Reaction scheme SBML model Raw ODE settings within data ﬁles. After coupling a model to a data set, the experimental condition will be translated into an external PW model PW ODE model driving input function and optionally the standard deviation of the measurements can be estimated. Network reconstruction

Deterministic modeling of dynamical systems involves the following steps: Compilation as C/FORTRAN MEX file

1. Model creation: Creation of one or more models, which are Model in repository list systems of differential equations representing hypotheses about a biological network. 2. Fitting to experimental data: Automatically or manually Model visualization adjusting model parameters until the distance between model Sensitivity analysis trajectory and experimental data points is minimized. Use of Inverse Problem statistical tests to quantify model-data-compliance. Simulation of data External data 3. Model refinement: Changing the model structure to minimize Estimation of discrepancies between the fitted model and the experimental standard deviations data. 4. Investigating the kinetic model properties: Comparing Model-data-couples in repository list systematically model trajectories for different parameter values or input functions. Combination of multiple model-data-couples 5. Analysis of fitted parameter values: Identifiability, correlation, and confidence intervals of fitted parameter values. 6. Model selection: Comparing competing models qualitatively Multi-experiment fitting and based on statistical tests.

7. Prediction and experimental verification: Generating Fit sequence analysis Residuals analysis Driving input designer experimentally testable predictions for new system inputs or Identifiability analysis Model selection Chi-square landscape parameter values. Stimulus dependent view Fits scatter plot Animated time course System properties distrib. Best fits trajectories Confidence intervals 8. Exchange of modeling results and reporting: Exchange and discussion of fits and models with colleagues and collaborators. 9. Exchange of models: Saving of the final model in standardized Report to html, doc, pdf format, e.g. SMBL.

PottersWheel is the only modeling software providing a Fig. 1. PottersWheel workflow. comprehensive set of functions for each of these steps within one framework. We here present shortly the key functionalities of the toolbox, quantify their performance, and discuss important approach the inverse problem, one ore more data sets have to be methodological concepts concerning parameter identifiability, attached to the model either by simulation or from an external data confidence intervals and statistical tests for model validation and source. Optionally, the standard deviation of the data points can be discrimination. Further details and figures are available in the estimated. One or more model-data-couples are combined for multi- supplemental text. experiment fitting. Afterwards, a variety of fit-based analyses are available investigating for example the identifiability and confidence intervals of the calibrated parameters. Finally, each analysis can be 2 APPROACH appended into a report saved as a html, doc, or pdf file. 2.1 Workflow The central graphical user interfaces to operate within the Fig. 1 illustrates the workflow of modeling with PottersWheel (PW). workflow are the main PottersWheel window including a list of Either a reaction scheme is implemented by the modeler into a model-data-couples (Supplement, Fig. 1) and the so called Equalizer PottersWheel model definition file, or an SMBL model is imported. providing direct access to functionalities concerning the inverse Alternatively, a raw set of ordinary differential equations (ODE) is problem (Fig. 2). used to create an PW ODE model for which the reaction network can optionally be reconstructed. Loading of the model into the so called repository list results in the compilation of a C/FORTRAN 2.2 Creating an apoptosis example model MEX file. Then, the model can be used for model visualization or A realistic, medium sized model with 13 species, 41 reactions, methods of the direct problem, e.g. sensitivity analysis. In order to and 13 kinetic parameters serves as an example to demonstrate

2 PottersWheel

Right button column: Fit sequence analysis Optimizers in normal/log parameter space: F1: Single fit, F2/3: Fit sequence Single fit analysis Direct search Levenberg-Marquardt FB: Boosted fit Residuals analysis Trust region Simulated Annealing A: Animated time course Standard deviation estimation T: Switch time/stimulus domain MOTA identifiability analysis Left slider: Fine tuning O: Figure overwriting (on/off) Fit scatter plots Right slider: Order of magnitude Fit only x0, k, or s; Cancel fit Chi-square landscapes Text box: Parameter value Save current values as fit Trajectories of best fits List box: Selected parameter Save values for simulations Reset/Disturb parameter values Open input designer

Left button column: Open main PW window Arrange figures on screen Integrate and draw trajectories Show model graph Save current figures Selected slider for key control Set of disturbed parameters Set of fixed parameters Information for model & last fit

Fig. 2. PottersWheel Equalizer. The PottersWheel Equalizer comprises 10 pairs of sliders. Each pair can be attached to one of the fitted parameters in the list box below the slider pair. The left slider is used for fine-tuning and the right slider changes the magnitude of the selected parameter. Text boxes on the one hand reflect the current parameter value and can on the other hand be used directly to specify a certain value. 33 buttons and combo boxes provide direct access to the most important functions concerning multi-experiment fitting and analysis.

the functionalities of PottersWheel. The model has been suggested 3. The merit function of the optimizer is dynamically generated by Legewie et al. (2006) and describes the feedback control of containing no overhead or slow MATLAB functions. caspase-3 and caspase-9 in the intrinsic apoptosis signaling pathway 4. Calculation of observables and residuals is also based on (Supplement, Fig. 3). The authors provided an SBML model which dynamical C MEX files. can be imported into PottersWheel. The systems biology markup language (SBML) has been designed to enable a standardized Currently, six FORTRAN integrators are supported by Potters- way to express, store, and exchange kinetic models based on Wheel, being described in Hairer and Wanner (1996). We use reaction networks (Finney and Hucka, 2003). In order to investigate the MATLAB interface of Ludwig (2006), which we extended the increased statistical power to discriminate competing model in two cases to reduce overhead by circumventing calls between hypothesis and to calibrate unknown parameters, we extended the integrator and the model equations. Our modification improves model by two driving input functions representing an externally the integration time by an additional factor of 10-35 and requires specified concentration of cyto-c and SMAC. The structure of the either FORTRAN compilers for Linux/Mac or the lcc compiler final model definition file is explained in the supplement comprising for Windows computers. The integrators are RADAU5, RADAU, also an automatic model visualization. SEULEX, DOP853, DOPRI5, and ODEX. The first three integrators are applicable to stiff differential equations, where the time-scales of the variables have huge differences in their range. In order to 2.3 Integration performance compare the integration time and accuracy, PottersWheel supports During parameter calibration, the model trajectories have to be all MATLAB integrators ode45, ode13s, ode23, ode113, and the calculated thousands of times until an optimal parameter setting is stiff integrators ode15s, ode23s and ode23tb. A short description of found. Hence, high integration speed is a crucial prerequisite for each integrator can be found in the supplement. interactive dynamical modeling of experimental data. PottersWheel applies the following strategy to meet these requirements: The right hand side of the differential equations including algebraic equations, interpolation formulas and events is saved and 1. Use of fast and accurate FORTRAN integrators. compiled as a C MEX file when a model is loaded into Potters- 2. The differential equations are saved and compiled as C MEX Wheel. For the apoptosis model, this approach is 20 times faster files. If possible, the integrator, interface, and model are than calling a MATLAB ODE function. compiled into a single executable.

3 Maiwald and Timmer

Integration time Integration deviation

DOPRI5 incl. ODE (0.001, 1e−06) 918.79 times faster than reference (ODE15s, Matlab ODE) 0.74 times more accurate than reference (ODE15s) RADAU5 incl. ODE (0.001, 1e−06) 162.21 5.31 RADAU5, MEX ODE (0.001, 1e−06) 10.88 5.31 RADAU5, Matlab ODE (0.001, 1e−06) 1.02 5.31 RADAU, MEX ODE (0.001, 1e−06) 10.85 5.31 RADAU, MEX ODE (0.0001, 1e−07) 10.52 112.88 RADAU, MEX ODE (1e−05, 1e−08) 9.67 374.07 RADAU, MEX ODE (1e−06, 1e−09) 8.07 1076.99 RADAU, MEX ODE (1e−07, 1e−10) 6.58 3007.75 RADAU, MEX ODE (1e−08, 1e−11) 5.68 NaN (RADAU 1e-08, 1e-11 is here the gold standard) SEULEX, MEX ODE (0.001, 1e−06) 9.41 0.17 DOPRI853, MEX ODE (0.001, 1e−06) 17.48 0.37 DOPRI5, MEX ODE (0.001, 1e−06) 26.14 0.74 ODEX, MEX ODE (0.001, 1e−06) 19.02 0.26 ODE45, Matlab ODE (0.001, 1e−06) 1.63 6.57 ODE45, MEX ODE (0.001, 1e−06) 4.35 6.57 ODE23, MEX ODE (0.001, 1e−06) 3.66 0.32 ODE23s, MEX ODE (0.001, 1e−06) 1.02 0.16 ODE23t, MEX ODE (0.001, 1e−06) 1.09 0.25 ODE23tb, MEX ODE (0.001, 1e−06) 1.27 0.23 ODE113, MEX ODE (0.001, 1e−06) 1.34 0.56 ODE15s, MEX ODE (0.001, 1e−06) 1.39 1 ODE15s, Matlab ODE (0.001, 1e−06) reference reference

−4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 2 4 6 8 10 12 x 10 Integration time [sec] Absolute deviation from RADAU Integration (1e−8, 1e−11)

Fig. 3. Integration time and accuracy. Left: The mean integration time of all 13 supported integrators with specified relative and absolute tolerance is displayed. The integrator either includes the ODE (rows 1 and 2) or is attached to an ODE compiled as C MEX file or saved as a normal MATLAB function. The reference is the integration time using MATLAB integrator ODE15s for stiff systems with a MATLAB ODE (ca. 0.1 seconds, last row). DOPRI5 for non-stiff systems (first row) is 919 times faster if the ODE is included, else 26 times faster using a MEX ODE (row 13). RADAU5, a stiff integrator, is 162 times faster (second row) with included ODE and 11 times faster with a MEX ODE compared to the reference. Calling RADAU with increasing integration accuracy leads to a slightly longer integration time. Right: Integration with RADAU using high tolerances of 10−8 and 10−11 serves here as a gold standard to estimate the accuracy of all integrators by quantifying the mean deviation between the calculated trajectories. RADAU5 (10−3, 10−6)(row 2) is not only faster than ODE15s, but also 5 times more accurate.

We compare the 13 integrators on the basis of the medium sized smaller than for the reference. Simultaneously, RADAU5 reaches apoptosis example model. Their performance was determined by a 5 times smaller deviation than the reference compared to the four criteria: trajectory of the gold standard. Fig. 6 of the supplement compares the number of ODE calls and the time per call. 1. Total integration time. 2. Accuracy, measured as the averaged absolute distance of 2.4 Optimization performance the integrated trajectory to a highly accurate integration with The χ2 merit function which is optimized within PottersWheel to fit RADAU with 10−11 absolute and 10−8 relative tolerance. the model y = y(t; p) is 3. Number of calls of the right hand side of the ODE system. N 2 2 X yi − y(ti; p) 4. Time per call of the right hand side. χ (p) = , (1) σi i=1 The relative and absolute tolerances of the integration are usually set to 10−6 and 10−3 respectively. Only the RADAU integrator with yi being data point i with standard deviation σi and y(ti; p) is tested with a variety of tolerances, to illustrate the effect on being the model value at time point i for parameter values p. integration time and on the number of calls to the ODE and to serve Currently, five implementations of optimizers are available: Direct as a gold standard for the estimation of the integration accuracy. The search, trust region, Levenberg-Marquardt, genetic algorithm, and integrations were applied on a Macintosh laptop with Intel Core 2 simulated annealing. The direct search method is only useful Duo 2.4 GHz with 2 GB RAM. for illustration purposes or small models. The trust region and Levenberg-Marquardt algorithms are powerful deterministic least- Fig. 3 compares in detail the integration time and deviation for square optimizers. The simulated annealing and genetic algorithms 23 different integration strategies. In summary, a compiled DOPRI5 are stochastic approaches able to handle local minima, but requiring including the ODE is approximately 900 times faster than using the more time. We quantify the accuracy of the optimization by reference MATLAB ode15s with a MATLAB ODE. If the ODE is the average deviation D of the np fitted parameters to the true not compiled into the integrator executable, DOPRI5 is still 26 times parameters: faster. Using RADAU5 which is applicable to stiff systems, reaches np i i ! an integration time 160 (incl. ODE) or 11 (attached ODE) times 1 X pfit ptrue D = max i , i (2) np p p i=1 true fit

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4 2 10 Final iteration 10 A Dmax=1.3018 B Dmean=1.0994 101

103 100

10−1 102 10−2 Parameter value −3

Deviation (1=perfect ﬁt) 10 101

10−4

−5 100 10

0 50 100 150 200 250 300 0 50 100 150 200 250 300 Iteration Iteration 107 2 Fit: χ /N=0.832975 χ2=368.175 N=442 p=13 C9_binds_activeA 106 Method: smarquardt C9_releases_activeA Exit: Stopped by small x−step C3_act_via_C9 105 C Algorithm: Levenberg−Marquardt with Jacobian Approximation C9_act 4 Parameter space: log C9_releases_X 10 Function calls: 598 activeC3_binds_X

/N Iterations: 295 2 χ 103 TolX: 1e−010 activeC3_releases_X TolFun: 1e−010 A_prod MaxIter: 10000 102 C9_prod pValue(N): 0.984639 pValue(N−p): 0.995522 X_prod 101 AIC: 1498.7 C3_prod AICc: 1499.55 degradation 0 BIC: 1551.89 10 C9_binds_X 0 50 100 150 200 250 300 Iteration

Fig. 4. Optimization performance of Levenberg-Marquardt. A: Deviation compared to true parameters. The dashed red line displays the mean deviation over all parameters. B: Parameter values during fitting. C: χ2 value during fitting and fit settings.

A value of 1 indicates a perfect fit. A higher value indicates that not capable to find an adequate minimum. They reach a deviation of on average the fitted value is D times higher than the true one or 165807 and 188 after 7651 and 40,000 function calls, respectively. visa versa. Fig. 4 exemplifies the result for the Levenberg-Marquardt In addition, the three last optimizers were started closer to the algorithm. The initial guess for all parameters is the default value 0.1 true values. Optimization in normal parameter space was in no (B) corresponding to a mean deviation over 1000. After 598 function case successful, stressing the importance of fitting functionalities calls the deviation is decreased to 1.0994, i.e. the parameters are in logarithmic space. Please see the supplement for a detailed on average only 10% higher or lower than the true values. This description of the performance analysis. performance is based on the optimizer properties and the Potters- Wheel framework allowing to fit in logarithmic parameter space, 2.5 Multi-experiment fitting because the true parameter values span five orders of magnitude. The trust region approach has a similar accuracy, but requires 2184 A key functionality of PottersWheel is multi-experiment fitting, function calls. The simulated annealing method fits 12 out of 13 where several data sets are fitted simultaneously. The data sets parameters correctly but result in a mean deviation of 12.62 after should derive preferably from different experimental conditions, 10323 function calls. Finally, direct search and genetic algorithm are e.g. different dose levels, pulses or ramp stimulations. The externally changed species, e.g. the ligand in models of signal

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2 CytoC: continuous 2 SMAC: continuous X_act 1.5 1.5 X_inhib A 1 1 B Apaf_deact ) 0.5 0.5 Apaf_act

0 0 C9_binds_X 0 50 100 150 200 0 50 100 150 200 degradation C9 obs X obs 30 60 C3_prod

20 40 X_prod C9_prod 10 20 A_prod 0 0 activeC3_releases_X −10 −20 0 50 100 150 200 0 50 100 150 200 activeC3_binds_X activeA C9 obs activeC3 ob C9_releases_X 4 8 C9_act 3 6 C3_act_via_C9 2 4 C9_releases_activeA 1 2

0 0 C9_binds_activeA

−1 −2 0 1 2 3 0 50 100 150 200 0 50 100 150 200 Parameter values of normal ﬁt sequence (median = 1)

1 2 SMAC: continuous X_act 0.8 CytoC: pulsed 1.5 X_inhib 0.6 1 Apaf_deact C 0.4 D 0.5 0.2 Apaf_act

0 0 C9_binds_X 0 50 100 150 200 0 50 100 150 200 degradation

2 1 SMAC: pulsed C3_prod CytoC: continuous 0.8 1.5 X_prod 0.6 1 C9_prod 0.4 0.5 0.2 A_prod

0 0 activeC3_releases_X 0 50 100 150 200 0 50 100 150 200 activeC3_binds_X

1 CytoC: pulsed 1 SMAC: pulsed C9_releases_X 0.8 0.8 C9_act 0.6 0.6 C3_act_via_C9 0.4 0.4

0.2 0.2 C9_releases_activeA

0 0 C9_binds_activeA 0 50 100 150 200 0 50 100 150 200 0 1 2 3 Parameter values of normal ﬁt sequence (median = 1)

Fig. 5. Application of multi-experiment fitting. If only one experiment with continuous stimulations for each driving input player cyto-c and SMAC (A, 4 observables, 10% rel. + 10% absolute error) is fitted 200 times with varying initial guess for the parameters, the distributions of the calibrated parameter values is rather broad: The parameters are not identifiable (B). C: Combination of all four experiments leads to significantly narrowed parameter distribution (D). Note that the distribution represented by the horizontal bars should not be mistaken for a confidence interval (see section ??). Therefore, true values (red stars) may lie outside of the parameter distributions.

transduction pathways, is called driving input. Fig. 5 illustrates the 2.6 Confidence intervals power of multi-experiment fitting when applied to the apoptosis Calculation of confidence intervals on the estimated parameter model. If only one experiment is available for model fitting with values requires that the parameters are identifiable. If a Maximum 4 observables and continuous stimulation by cyto-c and SMAC Likelihood estimator is used to calibrate the parameters, the (A), the calibrated parameters possess a broad distribution - they confidence intervals can be determined based on the Hessian of share non-identifiabilities (B). Including additional experiments the objective function at the optimum. Since PottersWheel uses a with different combinations of pulsed and continuous stimulations weighted least-square optimization, this is the case for normally (C) resolves the non-identifiability for many parameters (D). In distributed errors. Otherwise, a Monte-Carlo approach is to be order to identify groups of parameters which are involved in a preferred, where new data sets have to be generated based on the functional relationship, first a fit sequence analysis reveals linear fitted model and an adequate observation error model. PottersWheel correlated pairs of parameters (see Supplement, Fig. 16). In a second supports both strategies, which are described in the supplement. step, the MOTA algorithm of Hengl et al. (2007) is applied, which is able to detect nonlinear dependencies between an arbitrary number 2.7 Statistical tests of parameters. In summary, 9 parameters are affected by a non- identifiability, distributed in 6 groups with three to five parameters. Two questions arise when experimental data is modeled: Please, see the supplement for further details. 1. Is the model statistically compliant with the data, i.e. what is the probability that the model produces the data set? 2. If two models are compliant with the data, which one should be taken?

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In order two answer the first question, usually the goodness- and which features are missing or should be improved. Many of-fit is determined, i.e. the distance between model and data is researchers appreciated MATLAB as a framework which can be related to the expected value if the model were true: a χ2 test extended with custom programs easily. In general, they expected is applied (Press et al., 1999). For the second question, it is a good software to be easy to install, learn and use, flexible, important to verify whether the models are nested, i.e. whether fast and efficient, can be obtained for free, is equipped with one model is an extension of the other model. In this case it may powerful analysis methods and good graphics capabilities, is well be permitted to apply a likelihood-ratio test with high statistical documented, possesses good support, and is SBML-compatible. power (Lehmann, 1986). Criteria like AIC and BIC are suggested PottersWheel is designed to meet these requirements in the to establish a ranking of models (Akaike, 1973; Schwartz, 1978). In academic community setting. Being the key modeling framework the supplement, the methods are described in more detail and are for several Systems Biology projects within the German HepatoSys applied to a reduced and enlarged apoptosis model, showing that initiative, PottersWheel has been intensively used by modelers and the reduced model is significantly not compliant with the data. The experimentalists since 2005. The reasoning to build upon MATLAB extended model by construction sufficiently describes the data with is to enable the user to customize and automate functionalities a lower χ2 value than the original model. However, a likelihood as much as possible. Numerical functions implemented within ratio test reveals that the improvement is statistically not significant, MATLAB which are too slow were consequently substituted by as expected. The involved statistical functions are available in the high performance FORTRAN or C code, including dynamically PottersWheel framework. generated ODE C files. Graphical user interface provide an intuitive approach to modeling. 2.8 Advanced modeling techniques We shortly summarize which further modeling techniques are available within PottersWheel. Please see the supplement for ACKNOWLEDGEMENTS detailed descriptions. Model families allows for creating basis and We are thankful for suggestions and feedback by Eva Balso-Canto, dependent models in order to reduce redundant work, when the basis Julie Blumberg, Stefan Hengl, Clemens Kreutz, Stefan Legewie, model is changed. Rules, start value assignments and events are David Liffmann, Christian Ludwig, Peter Nickel, Tatsunori algebraic equations evaluated during or before integration or when Nishimura, Martin Peifer, Andreas Raue, and Marcel Schilling. certain conditions are fulfilled. Basal states depending on parameter This work was supported by the HepatoSys initiative of the German values require either sufficient integration before stimulation or Federal Ministry of Education and Research (BMBF, 0313074D), should be set directly by start value assignments. Rule based the European project of Computational Systems Biology of modeling is useful to cope with combinatorial complexity. Derived Cell Signalling (COSBICS, LSHG-CT-2004-512060), the German variables and parameters help to focus on important subsystems Federal Ministry for Economy, and the European Social Fund or to analyze functions of parameters. 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