BioPARKIN Release 1.2 — User Manual
Contents
1. A 24 8486 un 24 8109 a 24 7669 24 7252 oo LH bld dimensionless 3 4874 3 4874 3 4874 3 4879 3 4890 3 4904 3 4919 3 4937 3 4957 3 4999 3 5037 3 5080 3 5174 3 5272 3 5550 3 5732 3 5952 3 6168 LH bld dimensionless 3 4874 3 4874 3 4874 3 4879 3 4890 3 4904 3 4919 3 4937 3 4957 3 4999 3 5037 3 5080 3 5174 3 5272 3 5550 3 5732 3 5952 3 6168 Figure 11 Results Window Simulation Table 23 Within these tabs the figure is on the right and two subordinated tabs named Data Sources and Setting Es Actions are on the left In the tab Settings Actions the user can switch on a legend change the scaling of the y axis between linear and logarithmic and save the plot in a number of predefined formats e g png jpg eps or pdf In addition below the figure on the right there are icons for panning and zooming in and out of the graphic Note that typical machine precisions are about le 16 If Simulation Plot in the Results Window shows solution curves below machine precision there are two possible interpretations 1 The species is in fact zero so this has no effect on the correctness of the rest of solution Do not panic 2 It could be that the current model indeed induces such small values In this case the user should think about introducing different physical units to avoid this effect The solution however is still within the chosen to2. If changes are made the modified model can be saved by clicking on File Save or File Save as respec tively For these commands short cut icons are available in the icon list as well In addition to the standard SBML view BioPARKIN offers the possibility to view the generated differential equations by clicking on the 4th icon in the upper left corner The ODEs are displayed both in terms on IDs as well as names with reaction abbreviations or in full notation You can close a model by clicking on File Close In contrast to Close Quit will end the programm 4 2 Workbench Settings The tab Workbench Settings contains a number of specifications that are required to simulate the model 4 2 1 General Settings For parameter identification two different Gauf Newton codes can be used either Gaufi Newton NLSCON or Gaufi Newton PARKIN PARKIN is a variant of the standard NLSCON with a different damping strategy The user might wish to test and compare both variants If experimental data are loaded which shall be included in the simulation plots click the corresspond ing check box 2SBML uses a subset of MathML 2 0 standard www w3 org TR MathML BioPARKIN v1 1 12 Actions Tools Help Bd ETS Model Overview Workbench General Settings Settings Identification Backend Gauf Newton NLSCON Include Experimental Data in Simulation Plots pecies Time Settings Start Time 25 End Time 44 Plot Se
3. FSH bld m E2 pg mL P4 ng mL Ago_cc n Settin Data 4 9583 17 17 0 it Species 5 pecies 4 9792 15 13 Data Type experimental 5 0000 13 15 File 5 0069 25 14 Path naf105 100mcg csv Parameters S 5 0139 37 18 Filesize OkB 5 0208 40 19 5 0313 42 19 Last Modified 23 Jun 2011 15 43 26 5 0417 45 20 Parameter Sets 5 0833 77 27 5 1250 32 5 1667 36 5 2083 37 5 2500 35 y Data Browser 5 3333 34 EZ gyncycle ago sd naf105 100mcg csv Fit Sensitivit Plot All Data Sets Plot Selected Data Sets Figure 7 Workbench Data Browser tab Top Before loading data Bottom With loaded data 17 Results Data Sources Settings amp Actions ID Sensitivities Timepoint 9 5 01 g Show 02 Show Show Show Show SelectAll Deselect All Invert Selection h Sensitivities Timepoint 9 5 16 53 14 Subc CloseAll Close amp Tab Mode Results Data Sources Settings amp Actions Settings EZ Show Units in Table Headers Switch rows vs columns amp Coloring Threshold Actions Save Data Add Data to Data Browser gt Sensitivities Timepoint 9 5 16 53 14 Subc CloseAll Close amp Tab Mode 4 P2 None 2 P7 None 3 P6 None 4 P3 None s P1 None Ago_R 0 0003 0 0005 0 0001 0 0002 0 0024 4 P2 None 2 P7 None 3 P6 None 4 P3 None
4. the Results Window shows up 3 1 Model Overview This tab is devided into three parts Under Model List on the left the currently opened model is shown In future it will be possible to open several models at the same time and quickly switch between them perform simulations and compare results In the center under Entity Tree there is a tree like view of all the model s constituents i e compart ments species reactions etc Together they define the entire model When clicking on one entity from the list information about its properties are shown on the right Under Entity Details the attributes of the currently selected entity are shown together with their current values Details that are not modifiable by the user are greyed out Currently the Model Overview tab is meant to give an overview of the constituents of the opened model A future update of BioPARKIN will allow to build and edit a model e g add and remove species and reactions from within the Model Overview tab Due to the SBML principle of interoperatibility between software tools it is easily possible to use other programs to build and change a model At present whenever the user wants to modify species reactions rules or events we recommend to edit the model in CellDesigner and to reload it into BioPARKIN However changes in names of species parameters rules or events in formulas assignement rules reaction rates and event specifications can already
5. 2 3 4 5 6 7 8 9 134240 134240 2 09451 2 09451 51 5581 51 5581 y Data Browser Hd o 309343 309343 6960 53 6960 53 161849 161849 Fit Sensitivit 1713 71 1713 71 Duplicate Selected Remove Selected Figure 6 Workbench Parameter Sets tab 15 be removed by clicking on the corresponding button Remove Selected By a click on the button Duplicate Selected in the lower right corner a selected set can be duplicated to save intermediate parameter sets before further modification If no set is marked as selected then the active set is duplicated When a model is saved all parameter sets are stored They are reloaded when the model file is opened again If the SBML file is opened with a different software e g Copasi only that parameter set is loaded which was labeled as active at the timepoint of saving 4 6 Workbench Data Browser In the tab Workbench Data Browser the user has the possibility to browse through the file system click on Browse and import data files click on Import provided they are in the format described in 2 2 The loaded data are presented column wise in a table They can be plotted in a separate Results Window which comes up if one clicks on Plot Check boxes on top of each column allow for a selection of species for plotting or parameter identification Selected columns can be saved as new table button Save As A click on the button Plot in the lower right corner opens the
6. Wegner M I Aladjem S M Wimalaratne et al The systems biology graphical notation Nature biotechnology 27 8 735 741 2009 R Machn A Finney S M ller J Lu S Widder and C Flamm The SBML ODE Solver Library a native API for symbolic and fast numerical analysis of reaction networks Bioinformatics 22 11 1406 2006 U Nowak and P Deuflhard Towards parameter identificatin for large chemical reaction systems In P Deuflhard and E Hairer editors Numerical Treatment of Inverse Problems in Differential and Integral Equations Birkhauser 1983 U Nowak and P Deuflhard Numerical identification of selected rate constants in large chemical reaction systems Applied Numerical Mathematics 1 1 59 75 1985 U Nowak and L Weimann A familiy of Newton codes for systems of highly nonlinear equations ZIB report 91 10 Zuse Institute Berlin ZIB 1991 L Stromback and P Lambrix Representations of molecular pathways an evaluation of SBML PSI MI and BioPAX Bioinformatics 21 24 4401 2005 26 A Problem Description A 1 Large Kinetic Networks in Systems Biology A major topic in systems biology is the study of the dynamical evolution of bio chemical mechanisms within a well defined biology related context The bio chemical mechanisms in such a compound under consideration are typically given as a possibly huge set of chemical reactions between numerous species Thus the set of chemical reactions are the buildin
7. Detailed sensivity analysis Subconditions 19 y y 1 The sensitivities are computed for the complete time interval I specified in the tab Workbench Settings The sensitivities over time are displayed in the Results Window 2 The sensitivities are computed at user specified time points 3 The sensitivities are computed at all time points where measurement values of certain species are available Case 1 is handled via the Button Sensitivity Overview below the two tables After a click on Compute Sensitivities the sensitivities are calculated over the whole time interval specified in Workbench Settings and the results are shown in the Results Window both as table as well as time plot Case 2 and 3 are handled under Detailed Sensitivities One can choose between Get Timepoints from Data or specifying time points by hand Time points can directly be edited in the window below Timepoints space separated or calculated from Start Time End Time and Intervals or Interval Size For every timepoint two seperate tabs open in the Results Window One tab shows the sensitivities of the selected species with respect to the selected parameters at that specific time point The other tab contains the subcondition numbers For more details we refer to 4 9 2 To keep an overview over all timepoints it is recommended not to use too many of them at the same time 4 8 Workbench Fit If experimental data are loaded BioPARKIN can be used for p
8. Os Wobei Gra EE del Ade do 11 AD MEME nn Gs s qem gd do PD Oe ESG EU EE a oe oe LD dot ee me me ed 11 ADA 2BX Orb SELLIMES a ds nue By Gey RS Did et EE uius deri et eret he tm betes ae rr a 11 AD SIMON Led Dane E DA es home ee Ens Gk OB Di ek ee de ee SS 12 4 9 Workbench Species uud ets s Rok GER omo 3 X X X RS EEE Ow MODO RO S RAR d em 13 AA Workbench Parimet 274 wt mech Es me db dE ol ee RC HABITUS ERE ES 13 4 5 Workbench Parameter Sets 4 5399 s e ew Bw vs ce E E ES gr UR V he 13 4 6 Workbench Data Browser LL LL 16 Z M orkbench SengtlViy at save BS aed Wem dE P d a ee AV ele ib we pa 16 LS VV OR OCMC ile Ib ss De atest ects es da Be ty he NS gh eo ee a ar OE 20 Zo SN CON at se MUN es dt et a eS os eee ee ua 20 Zo PARKIN cd scrips clos Dy ble Chae feo des Bip ia ads We Et tes aie Se d watt e Oh se dns 21 AO Results M dO a yaoa Be he Rete eer e ee eo CAT a SS SiS DE EG GS OU O a 21 CO Data Pio ELLE eS EOS Bw ES OBO amp ASS d ex 21 AO aa AO LCS ER ae es o eee UE Ll ae e c Pec m m Se ae i ee We ee a eee pee hes 24 References 24 A Problem Description 27 A 1 Large Kinetic Networks in Systems Biology 27 A 2 Multiple Experiments 3 tede Stok ea EEE b hcg ie es de ee ew a 2f As Breakporit Handling css dome S Redon dt dob e usb de ee ek eek mde DR ASS 27 AUI Sensitivity Matik uob debe EB Da m Som SEERA EES DANESE doses 28 Ao Taramerer TIdentiiesbloli ud eode e m p
9. Results Window with a plot of the data in the table currently shown on the left On the right hand side of the table there are two tabs named Information and Actions The tab Information displays information on the file and the data In the tab Actions the user can shift data points in time by entering the desired timeshift and clicking on Timeshift Data This tool is especially useful for periodic systems as e g the human menstrual cycle where changes in parameter values or initial values can shift the periodic solution along the time axis BioPARKIN offers the possibility to include data from several files On the right hand side of the header at the bottom of the last data table there is a plus sign A click on this sign opens the initial data browser view with the two buttons Browse and Import to include further data files Data sets are selected if they are checkmarked in the lower left corner of the corresponding table header Several data sets can be plotted in one figure by clicking on Plot All Data Sets or Plot Selected Data Sets Data sets are closed by clicking on the cross in the header below the table 4 7 Workbench Sensitivity The tab Workbench Sensitivity contains a list with all parameters on the left and a list with all species on the right In the parameter list the user can click check boxes to select specific parameters for which sensitivities shall be computed A similar selection can be made in the species list N
10. as 7 n Le 9 a 7 E o a R_LH_des efaul 0 88238 N A Sc1 jefa 1 43277e 10 N A Sc2 Fault 7 27795e 08 N A SeF1 lefault 0 800773 N A y Data Browser SeF2 ef ault 6 34484e 05 N A csa1 degraded efault 1 N A Fit Sensitivit s38 efaul 1 N A M s33 efault 1 N A M C1 Si Initialize Thresholds mulate Figure 4 Workbench Species tab e Numerical experiments in 13 have shown that it might sometimes be useful to scale the residuals during the Newton iteration Residual Scaling should by default be set to no extra scaling If the algorithm cannot be applied successfully it could be an option to perform relative or absolute scaling This only applies to NLSCON Note however that residual scaling is not covered by theory 4 2 5 Simulation A click on the button Simulate in the lower right corner starts the numerical integration of the ODE system with the current settings Results are displayed in the Results Window which is explained in detail in Section 4 9 12 4 3 Workbench Species On the tab Workbench Species you find the list of all species with properties and property values The rows can be sorted alphabetically or numerically by clicking on the header of a specific column Compared to the species list in the tab Model Overview an additional column Threshold appears These entries are important for sensitivity analysis and parameter identification since they account for different scale
11. parameters are identifiable wheras larger values are colored in red It might happen however that the relative mea surement error of experimental data is larger than RT OL In this case fewer parameters are identifiable which become visible by the changing coloring if the threshold Anticipated Relative Measurement Error is increased 24 Results Sensitivity Plot 13 58 58 Data Sources Settings amp Actions Sensitivity Plot 13 58 58 20000 30000 4000050 10 0 10 20 Time days Select All DeselectAll Invert Selection Q q muy Closeall close amp Tab Mode Results Sensitivity Plot 14 35 11 Data Sources Settings amp Actions ID amp p3 Ago_R E Show _ Hide _ Hide Hide Sensitivity Plot 14 35 11 Hide Hide Hide _ Hide Hide FAMIUULIL LUITIEICIEISIUI TIC DD Hide Hide Hide Hide 05 Hide 30 20 10 O 10 20 Time s Hide Select All Deselect All Invert Selection Q c E Tf CloseAll close amp Tab Mode Figure 12 Results Window with sensitivity plots for the species Ago R with small concentration Top Sensitivities computed by numerical differentiation Bottom Sensitivities computed by variational equa tions 25 References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 P Deuflhard Newton Me
12. the Results Window There are two different types of tabs corresponding to two presentation types for the results figures data plot and tables data table In both tab types the number of displayed species can be controled by clicking on Show or Hide in the subordinated tab Data Sources There are shortcuts for this selection process via the buttons Select All Deselect All and Invert Selection 4 9 1 Data Plots There are three types of data plots that are created as new tab in the Results Window after the corre sponding calculation has been performed 1 Simulation Plot 2 Experimental Plot 3 Sensitivities Plot 21 Results Data Sources Settings amp Actions ID simulation Results Mgyncycle_ago_sd_naf105 Hide N A Hide Hide Hide D o Hide Hide Hide Hide Hide eo El Hide Amount dimensionless Hide 8 Hide Hide Hide UA 0 l FO g 30 20 10 0 10 20 Time days SelectAll Deselect All Invert Selection Q cal e ef D Simulation Plot 11 23 04 Simulation Table 11 23 04 Simulation Plot 11 21 Closeall close amp Tab Mode Results Data Sources Settings amp Actions Show Legend Logarithmic Y Axis Save Plot D o Amount dimensionless eo o eo eo 8 NJ o 036 20 10 0 10 20 Time days oo r Br Simulation Plot 11 23 04 Simulation Table 11 23 0
13. the official software library libS BML to access SBML files and interact with them This library is maintained by a central core of the community and is vital to almost all other software projects that work with SBML On the other hand several dozen tools exist to work with SBML models Most of these tools are developed independently and serve a specialised niche use case Currently BioPARKIN supports a subset of SBML Level 2 Version 4 8 SBML s event definitions for example support more types of events than BioPARKIN currently can handle Situations that the software cannot handle are usually reported by errors or warnings within the user interface or the log file Regardless of which standard is used the researcher has to know the syntax and terminology of the format SBML defines important entities like Species Reactions Compartments Kinetic Laws Rate Rules Assignment Rules Events and others BioPARKIN has adopted these terms and we will come across them later in this manual On the upside almost all SBML compatible tools share the same terminology This makes it easier to use new tools and it also facilitates the communication between researchers that use different tools BioPARKIN ensures compatibility with other tools available for systems biology e g COPASI COm plex PAthway SImulator 7 which has a large variety of features for simulation and analysis of biological models Another useful general purpose SBML tools is Cell
14. with Feedback or without feedback or by solving the variational equation Varia tional Equations as described in A 4 It is well known that numerical differentiation is numerically unstable but in this version of BioPARKIN it is more likely to be faster especially for large systems Numerical differentiation with feedback however tries to maximize the achievable precision by us ing numerical safeguards Note that the method Gauf Newton PARKIN only uses variational equations e Problems involving chemical reaction kinetics are usually nonlinear Different degrees of nonlinear ity mildly highly extremely require different damping strategies in the Gauf Newton iteration Unfortunately the equations themselves do not reveal their degree of nonlinearity We recommend to change the Problem Type whenever NLSCON does not converge e g in case of too small damping factors PARKIN uses a fixed damping strategy independent of the degree of nonlinearity 11 BioPARKIN v1 1 12 File Actions Tools Help B Model Overview Workbench gs ID Compartment Initial Quantity Threshold Disable b c P4 efault 0 688328 N A Settin PrA1 efault 2 81137 N A Species PrA2 fault 27 6365 N A PrF lefault 0 3355 N A R FSH efaul 5 14066 N A R FSH des efaul 2 33004 N A R Foll lefaul 0 416986 N A R GnRH a Fault 0 00912138 N A R GnRH i efault 0 000989343 N A R LH efaul 8 1573 N A A S ET Ad 7 E o a A
15. 4 Simulation Plot 11 21 Closeall close amp Tab Mode Figure 10 Results Window Simulation Plot 22 Results t i Data Sources Settings amp Actions EZ Simulation Results Hide Hide Hide Hide Hide Hide Hide Hide Hide Hide InhA delay Hide InhB Hide Hide Hide Show Hide Select All Deselect All Invert Selection Simulation Plot 11 23 04 Simulation Table 11 23 04 Simulation Plot 11 21 Closeall close amp Tab Mode Results t Data Sources Settings amp Actions Settings EZ Show Units in Table Headers Switch rows vs columns Coloring Threshold 9 00000 Actions Save Data Add Data to Data Browser Simulation Plot 11 23 04 Simulation Table 11 23 04 Simulation Plot 11 21 Closeall Close amp Tab Mode Timepoint days v 25 0 25 0 25 0 24 999 24 9962 24 9929 24 9893 24 9851 Fo M am amp win 24 9802 24 9705 o 24 9616 24 9517 N 24 9304 w 24 9085 A 24 8486 un 24 8109 a 24 7669 24 7252 co Timepoint days v 25 0 25 0 25 0 24 999 24 9962 24 9929 24 9893 24 9851 24 9802 M om sli win 24 9705 o 24 9616 _ 24 9517 N 24 9304 w 24 9085
16. 5 Pl None Ago Ri 0 0003 0 0005 0 0001 0 0000 0 0026 Ago cc 0 0000 0 0000 0 0000 0 0001 0 0006 Ago dc 0 0000 0 0000 0 0000 0 0000 0 0000 Ago dc Ago pc 0 0000 0 0000 0 0000 0 0001 0 0006 Ago pc E2 0 0084 0 0148 0 0006 0 0131 0 1111 E2 FSH R 0 0020 0 0013 0 0001 0 0032 0 0082 FSH R Figure 8 Results Window detailed sensitivity analysis Matrices 18 FSH bld 0 0034 0 0022 0 0002 0 0056 0 0141 FSH bld Results Data Sources Settings amp Actions ID Eubconditions Timepoint 9 5 Show Show Show Show Show SelectAll Deselect All Invert Selection la i a Subconditions Timepoint 9 5 16 53 14 Closeall close amp Tab Mode Results Data Sources Settings amp Actions Settings EZ Show Units in Table Headers EZ Switch rows vs columns Coloring Anticipated Relative Measurement Error Actions Save Data Add Data to Data Browser a b l4 Subconditions Timepoint 9 5 16 53 14 Closeall close amp Tab Mode 4 P1 None 2 p7 None 3 P2 None 4 P3 None s p6 None 4 P1 None 2 p7 None 3 P2 None 4 P3 None s p6 None Subcondition as 96 of 5647 17 0 0002 0 0019 0 0190 0 1709 1 0000 Subcondition as 96 of 5647 17 0 0002 0 0019 0 0190 0 1709 1 0000 Subcondition abs Subcondition abs Figure 9 Results Window
17. BioPARKIN Release 1 2 User Manual Computational Systems Biology Group csb zib de 31st October 2012 Zuse Institute Berlin Takustr 7 14195 Berlin Abstract BioPARKIN is a software package for simulation and parameter identification in ordinary differen tial equation models with special emphasis on systems biology It combines state of the art numerical routines with a graphical user interface that allows intuitive handling of models including species parameters and mechanisms The use of the SBML file format ensures compatibility with other tools available for systems biology BioPARKIN is available for Windows Mac OS X and LINUX This document provides a user manual for release 1 2 x Contents 1 Introduction and Background 2 2 Formats 3 XL OBM CPD E tk eae deus m ds SD omne EE he eB ex sce Ne nn MR 3 2 2 Input and Output of Experimental or Simulated Data 4 3 Graphical User Interface Overview 5 o WWiGdeMOVErVview a ahd ees EL ee m mew YS EE Re er OOD ee ee 5 32 NO CNC ap a arene dy So Aika eo RUE CR UR mdi eee ee eo ee Be 5 Dado CSS VIU ON uuu do eaten ier EO on gre Unc re DE dE EL EE 7 4 Graphical User Interface Usage 7 4 1 Model Overview Working with a Model 7 AD Norkbench SELES ee al ds se One E Xe dos dd Gel et sg He ee is ts bese AN a 9 ADA General Sotes vu Xo ar Be die dene ECC pr I d au dC weg UE ET e e es 9 202 CS INC cuta mo ge d uere wea wr BR
18. Designer CellDesigner is one of the most widely used SBML tools to model and display SBML models 6 It can also be used to simulate models using the SOSlib library 10 It adheres to the SBGN standard that defines the visual representation of SBML entities 9 Although it supports the simulation of models it cannot compete with the numerical features that BioPARKIN offers It is more suited to get a quick impression of the model s simulation results In this way CellDesigner is an ideal complement to BioPARKIN Whenever a new model has to be designed from scratch we recommend to first use CellDesigner to define the model entities Afterwards the model can be loaded into BioPARKIN for fine tuning i e to identify parameters and modify mechanisms 2 2 Input and Output of Experimental or Simulated Data A tab separated value pseudo file format with ending txt or csv was chosen to save experimental data or simulation results It uses tabulators to separate fields and needs to have the structure given in Table 1 The first column has to be named Timepoint unit with unit defining the actual time unit of the measured or simulated data The timepoint need not to be ordered or to be unique Of the following columns every column ending with unit or is treated as data of a specified substance species The headers of these data columns must correspond to species IDs within the model so that BioPARKIN can ma
19. IN 12 4 Numerical experiments in 13 have shown that it is recommended to activate Automatic Row Scaling of Jacobian when NLSCON is used A 6 Parameter Constraints In order to enforce contraints such as positivity or upper and lower bounds on the unkown parameters to be determined in the model a differentiable transformation y R gt R4 can be applied resulting in a different parametrisation u of the model ODE system p e u y f y plu fly u 22 A global positivity contraint on the parameter vector p can be achieved for example by the componen twise exponential transformation Pi exp ui i lesi 23 To impose an upper and a lower bound A and B respectively a sinusoidal transformation B A pi 1 sinu TE MEIST 24 can be used For a single bound C as last example in this section a root square transformation cs 1 Vie ju EET 25 with the upper sign for an upper bound and the lower sign for a lower bound is possible Note that the application of any transformation of the parameters obviously change the sensitivities of the parameters to the dynamical evolution of the ODE system as will be shown next Therefore it is strongly recommended that parameter constraints should only be applied in order to prevent the parameter vector components p from taking on really unphysical values 30
20. RKIN especially designed for applications in systems biology In particular we aimed at a comprehensive and flexible implementation by object oriented design To summarise BioPARKIN contains a stand alone library written in C which deals most efficiently with the following numerical tasks e fast expression evaluation e solver LIMEX for stiff differential algebraic systems e Gauf Newton methods NLSCON and PARKIN for parameter identification The current release of BioPARKIN is not able to detect if a network of reactions is too large i e there are redundant reaction links that could be eliminated without any change in the results However BioPARKIN is designed to indicate and to correctly deal with the situation where the parameter space contains a non trivial subspace of solutions to the inverse problem at hand rather than only exact one solution even if actually accurate measured data is available In this case the parameter identification can be improved and the determined parameters are no longer open to any interpretation If a network is too small i e important compartments species and or reactions are missing in order to explain the given data this eventually shows up in poor convergence of the Gauf Newton method during parameter identification BioPARKIN combines the state of the art numerical routines with a graphical user interface that allows intuitive handling of models including species parameters and mechanisms How to n
21. arameter identification That means the underlying algorithm NLSCON determines the subset of parameters that can actually be identified from the given data and changes these parameter values to obtain the best fit to the data in a least squares sense compare A 5 In the tab Workbench Fit the user can select parameters for optimization by clicking on the checkbox in the column Estimate Value It is recommended to first run a sensitivity analysis with the desired parameters and to select only these parameters who are indeed identifiable A click on Identify Parameters in the lower right corner starts the Gaufs Newton method The final values which are displayed in the tab Par Identification in the Results Window are stored as new set in Workbench Parameter Sets and marked as active set such that the next simulation would use these new values 4 8 1 NLSCON The output of this subroutine is displayed in the BioPARKIN command shell The settings for NLSCON are followed by a table of iteration steps containing 1 The iteration step number 2 The value of p 2 M Normf Weighted update norm D Ap Normx Aa WU The damping factor A Damp Fet 5 A monitor for rank one Newton update steps not used at the moment 6 The rank of the Jacobian in the current iteration step 20 The NLSCON options FCSTART and FCMIN are determined by the user defined non linearity of the problem Workbench Settings whereas COND is determine
22. avigate through this interface will be the main topic of this manual Section 2 explains the compatibility of BioPARKIN with SBML and other software Section 3 gives an overview of the GUI whereas model handling simulation and parameter identification are described in detail in Section 4 For the well disposed user the mathematical background is explained in Appendix A 2 Formats BioPARKIN uses SBML Systems Biology Markup Language to handle model files and a tab separated value format to load and store experimental data and simulation results The two formats will be explained in this section 2 1 SBML Compatibility During the last decade it became clear that well defined standards would benefit the systems biology community Independent from each other several projects arose to create a standard for modelling biological systems and sharing them with the community 1M Hucka et al The Systems Biology Markup Language SBML A medium for representation and exchange of biochemical network models 2003 www sbml org Timepoint unit ID1 unit ID2 unit Arbitrary Column Table 1 Structure of a BioPARKIN data file Neigbouring cells are separated by a tab character While other formats are still maintained and progressed SBML is the de facto standard especially in simulation centric workgroups and projects 14 The SBML standard is supported by a lively community of researchers and software engineers On one hand there is
23. be done here and will be handeled and displayed consistently throughout the rest of the software 3 2 Workbench This tab hosts the settings and controls to start simulations including the generation of modifiable and saveable figures and tables to perform detailed sensitivity analysis or to identify parameters e Adjust global settings for the integrator these changes will impact all computations BioPARKIN v1 1 6 File Actions Tools Help La B amp e Model Overview Workbench ModelList Entity Tree Entity Details BioPARKIN v1 1 6 File Actions Tools Help sas EO 4 Model Overview Workbench Settings Species Parameters y Data Browser Parameter Sets Fit Sensitivit Initialize Thresholds Simulate Figure 1 Starting window of BioPARKIN Top Model Overview tab Bottom Workbench tab 6 e Modify names value or thresholds of species and select species that are treated as boundary condition e Modify names values or thresholds of parameters e Manage parameter sets e Manage experimental data Load select shift and save multiple data sets e Select parameters and species taken into account in the sensitivity analysis e Select parameters that are identified based on experimental data These tabs will be explained in more detail in Section 4 of the manual 3 3 Results Window In this window which opens automatically after performing a simulation a sensitivity analy
24. bration measurements might be necessary for example or data related to different initial conditions Yo 1 V0 2 Y0 v are given Numerically these situations can be handled by the concatenation of several IVPs i dol uud Yy tov Yo v p L32 s 3 very similar to the management of breakpoints events see below If required the solution y corre sponding to the virtual initial timepoint to can readily be shifted to the original initial time to for comparison or plotting purpose A 3 Breakpoint Handling A sudden event maybe from outside the biological system is handled by introducing a breakpoint tp gt to and subsequently splitting the ODE system into a y part for to lt t lt ty and a y part for ta lt t Yo 4 ts 9 y 5 p 5 21 where g IR x R gt R is a mapping of the initial conditions possibly dependent on the parameter vector p The approach of splitting the ODE system with respect to time also applies in the case of multiple experiments at least for the cases described above A 4 Sensitivity Matrix The dependence of the solution y t p on the parameters p is characterised by the sensitivity m q matrix S t defined by 40 EO 6 The matrix can readily be computed as solution to the variational equation S fyly p S foly p dg 0 7 Consequently the variational equation associated with a transformed system reads S J
25. cit Euler method with extrapolation which is especially suited for stiff systems From a mathematical point of view the main difficulty is not to simulate such systems i e to solve the differential equations but to determine the unknown parameters in such a way that the simulation results agree with experimental measurement values This is an inverse problem which can be formulated as a nonlinear least squares problem In biological models there are usually dependencies between parameters which leads to rank deficient problems For the solution of such problems efficient and reliable numerical algorithms Gauss Newton methods based on affine invariant Newton methods have been developed over the past decades 1 An error oriented algorithm with adaptive trust region and rank strategies has been implemented in the code NLSCON Nonlinear Least Squares with CONstraints 1 A precursor version of NLSCON had already been part of the software package PARKIN 12 4 11 a single shooting method for parameter identification in large chemical reaction kinetic networks The two methods differ in the damping strategy of the Newton steps NLSCON allows the user to control the algorithmic setting whereas PARKIN works much more in an automated way which might not always be suitable for the problem at hand Both methods have been integrated into BioPARKIN and form the numerical core of the software Thus BioPARKIN can be considered as a successor of PA
26. d as 1 RTOL Besides other criteria the iteration terminates as soon as Norma lt XTOL After NLSCON has come to an end the iterates p are displayed 4 8 2 PARKIN Amongst other things the command shell displays internal scaling values for measurements which are chosen automatically by PARKIN These values correspond to the measurement tolerances dy compare Section A 5 Moreover in each iteration the following output is produced 1 The iteration step number IT 2 The norm of the update step Ap LEVELX The damping factor Ax RELFC Aa WU The rank of the Jacobian in the current iteration step RANK ON The weighted least squares norm F LEVELF The largest subcondition number of the parameters to be estimated SUBCOND J The number ri1 from the QR decomposition SENS A large value indicates large sensitivity oo N O The inkompatibility factor KAPPA The identification result only makes sense if lt 5 If is larger especially amp gt 1 model and data do not fit together and the result should be discarded There two output lines The first line contains the values of some items at the beginning of a Newton iteration the second line contains the values of some items at the end depending on when they are computed In addition the results of a rank optimization step are displayed For details we refer to 1 4 9 Results Window Every performed calculation leads to a new tab in
27. eveals its properties and property values in the right column Entity Details Note that this list is not the complete SBML entity catalogue BioPARKIN supports only a selection of entities essential for the numerical computation tools Modifiable property values are displayed in black fixed ones are grey e Compartments are characterized by their volume size If the compartment ID occurs in the equa tions it is replaced by the volume during the computations The compartment name is optional e The details of the species appear as they have been specified in the software that was used to build the model The initial quantity can be changed in Workbench Species To edit other properties a different software e g CellDesigner has to be used BioPARKIN v1 1 12 File Actions Tools Help Bd ua Model Overview Workbench Model List Entity Tree Entity Details gyncycle ago sd naf105 100mcg sbml ID A Compartments Species Reactions Name ago admin dorian Target Assignment 1 Ago_dc Assignment Rules v Events ago_admin ago_admin Triggei eq time 5 Name Property Value Assignment 1 p101 Figure 2 Model Overview tab with a loaded model e The reaction list contains the mathematical descriptions of all interactions between species which build up the right hand side of the ODE system In particular under Entity Details the Math property describes the corresponding reaction rate i e the rate of change in am
28. formed with the parameter values in the set labeled as Active Whenever parameter values are changed in the Model Overview or in Workbench Parameters these changes are displayed in the active set Vice versa changes made in the active parameter set are displayed in the other tabs as well If a set is labeled as Selected the buttons in the lower right corner can be applied A set can 13 BioPARKIN v1 1 12 Actions Tools Help Model Overview Workbench Scope ID Name Value From Set Initial Guess Threshold b LH syn 7309 92 N A Settings m E2 LH 7309 92 N A k PrA2 InB 447 467 N A Species b IhB 89 4935 N A T E2 LH 192 204 N A T P4 LH 2 3708 N A n E2 LH 10 N A n P4 LH 1 N A k Sc2 IhB 134240 N A k PrA2 E2 2 09451 N A vn fa aw E o a A a 7 n ET pu v E a b_E2 51 5581 N A k SeF1 E2 309343 N A k SeF2 E2 6960 53 N A k PrF E2 161849 N A y Data Browser k Lut1 E2 1713 71 N A k Lut4 E2 8675 14 N A cl E2 5 23501 N A Fit Sensitivit b IhA 1 44523 N A Initialize Thresholds Simulate Figure 5 Workbench Parameters tab 14 BioPARKIN v1 1 12 Actions Tools as amp Model Overview Workbench Set 1 Set 2 Settings Name Original Initial Guess Active amp Active pecies Selected pl 7309 92 7309 92 p2 7309 92 7309 92 447 467 447 467 Parameters S 89 4935 89 4935 192 204 192 204 2 3708 2 3708 p Parameter Sets 1 1 1
29. g blocks of the system model forming a large kinetic network Assuming the general principle of mass action kinectics this large network can readily be transformed to a system of n ordinary differential equations ODEs leading to the initial value problem IVP y fysp w to wo 1 where the right hand side f denotes the dependence of the change in the species vector y on both the species y and the parameter vector p of dimension q The initial condition vector yo has the same dimension as the species vector y In BioPARKIN the ODE systems are solved numerically with LIMEX a linear implicit Euler method with extrapolation 5 The first step in Systems Biology often involves developing a suitable model description f that will not be dealt with in detail here Instead it is assumed that some discrete experimental data in form of species concentration versus time T1 21 s TM Zu 2 is available Note that freqently only a certain amount of the n species concentration are measurable observables if at all The task at hand now reduces to quantify the q unknown components of the parameter vector p by comparison between model synthetic values and given measured data A 2 Multiple Experiments The design of experiments includes almost always different conditions such that the effects of these different conditions on the system under investigation can be observed and studied In the simplest case cali
30. g on Initialize Thresholds the values can be set to maschine accuracy e However a warning pops up which informs the user hat this might not be the right choice In addition the list contains variables defined by assignement rules Instead of a numeric value the column Value contains the word Assignement Rule for these variables A click on the button Simulate in the lower right corner starts the numerical integration of the ODE system with the current settings Results are displayed in the Results Window 4 5 Workbench Parameter Sets When the user is working on a model or exploring its parameter space parameter values are usually changed quite often The parameter sets feature of BioPARKIN tries to accommodate this behaviour In the tab Parameter Sets BioPARKIN offers the possibility of bookkeeping several parameter sets for a single set of equations Each parameter set includes all parameters and their values Parameter sets are displayed side by side in the tab and can easily be duplicated or removed Upon opening a SBML model an Original parameter set is created that allows the user to revert back to the initial parameter values at any time Another set is created automatically called Initial Guess This is meant to be used in the context of parameter identification The results of an identification run are also put into a parameter set and thus can quickly be plotted compared with measurements etc All simulations are per
31. lerance ATOL RT OL Explicitly forcing species to zero is not advisable since these two cases could not be distinguished by the integrator If a species is zero or very small it may happen that the sensitivity computed with numerical differentiations gets large due to numerical cancellation whereas the sensitivity computed by solving the variational equation is within the desired tolerance compare Fig 12 4 9 2 Data Tables There are three types of data tables that arise as new tab in the Results Window after the corresponding calculation has been performed 1 Simulation Table 2 Sensitivities 3 Subconditions 4 Par Identification Within these tabs the table is on the right and two tabs named Data Sources and Setting amp Actions are on the left The tab Settings Actions offers the possibility to change the layout of the table and to save the data In addition it is possible to add computed data from the table to experimental data Add Data to Data Browser In this case a new tab named Pseudo Experimental Data is added in Workbench Data Browser Moreover values in the table can be colored by activating the check box Coloring The user can specify a threshold below which the table entries are colored red whereas larger values are colored green For single time points parameters are ordered according to their subcondition number By default subconditions below 1 RT OL are colored green which means that the corresponding
32. max r max yj i vs Ti with some small safety factor 7 In NLSCON the measurement tolerances are computed as 02 max max yj t trsh y with some user specified threshold trsh Parameter identification is equivalent to solving the least squares minimization problem rok 3 PEE min 14 1 1 or alternatively Tp F p F p gt min 15 where F p is a vector of length M with entries yj ti zi F lp 16 i p En 16 That means we want to minimize the relative deviation of model and data at the measurement time points t The above problem which is usually highly nonlinear in p can be solved by affine covariant Gauss Newton iteration 1 where each iteration step k requires the solution of a linear least squares problem J p Ap F p min 17 po p d Ap kN PL 18 The update step uses a customized QR decomposition for solving the Jacobian system especially in the rank deficient case The i th row of the Jacobian M x q matrix J p has the form I p t Vy ti p 19 thus representing the sensitivity of the solution y with respect to the parameters p at the time points of measurements An analysis of the matrix J p gives some hints whether the current combination of model and data will permit an actual identification of the parameters Parameters with very small sensitivity have nearly no influence on the solution and can therefore not be estimated In this ca
33. oe B BMW de IN eee em DES E YA DUE A 28 B Parameter Const rait e qa dosi SE ESSE E OR Re RS e e USO SCR A Mb ee ie og 30 1 Introduction and Background Modelling parameter identification and simulation play an important role in systems biology In recent years various software packages have been established for scientific use in both licencing types open source as well as commercial Many of these codes are based on inefficient and mathematically outdated algorithms By introducing the package BioPARKIN we want to improve this situation significantly The development of the software BioPARKIN involves long standing mathematical ideas that however have not yet entered the field of systems biology as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks Systems biology aims at describing biological processes by mathematical models which permit bi ologically sound quatitative predictions To arrive at statements of this kind the time courses of bio logical processes are modeled by differential equations that describe the concentrations or amounts of the involved substances over time For the time being we consider only systems of ordinary differential equations ODEs Delay differential equations DDEs or differential algebraic systems DAEs are not supported until future releases of BioPARKIN In BioPARKIN the ODE systems are solved numerically with LIMEX 5 a linearly impli
34. ote however that this selection is mainly for plotting purpose not for saving computing time If the Jacobian is computed by solving variational equations see Workbench Settings the computing time does not depend on the number of selected species or parameters since the full ODE system has to be solved in any case If numer ical differentiation is chosen the computing time depends linearly on the number of selected parameters If Gauf Neuton PARKIN is selected as identification backend in Workbench Settings sensitivities can only be computed if threshold for parameter values have been specified in Workbench Parameters For Gauf Newton PARKIN which is the default setting the following rule applies Sensitivities cannot be computed until thresholds for species Workbench Species and parameters Work bench Parameters have been defined The sensitivity of a species with respect to a parameter depends on time Thus to start the compu tation the user has to specify the time There are three possibilities 16 BioPARKIN v1 1 12 Tools Help Model Overview Workbench Species Settings Parameters S Import a Data File Browse Import y Data Browser Parameter Sets El Plot All Data Sets Plot Selected Data Sets Fit Sensitivit BioPARKIN v1 1 12 Actions Tools Help BSa EBE B Model Overview Workbench gs 1 g 2 g 3 MZ 4 g s5 Z 6 Information Actions Timepoint d LH_bld ml
35. ount or concentra tion of a species from Reactants to Products This expression has to be formulated in MathML and uses the IDs of the involved parameters and species e As for the species the details of the parameters appear as constructed in other software Parameter values can be changed in Workbench Parameters e Rate rules are used to modify species without introducing new reactions A rate rule sets the rate of change of a species to the value determined by the formula in Math Most models however do not use this possibility such that the rate rule list is often empty e Assignment rules are used as placeholder for certain algebraic expressions In contrast to Algebraic Rules which are not supported by the current BioPARKIN release Assignement Rules do not represent constraints on the systems The object defined by an assignment rule can either be constant when it depends only on constant entities e g parameters or dynamic i e time dependent when it depends on other dynamic entities e g species e Events take place at breakpoints t gt to defined by the Trigger eq time t Here time is the time variable i e the variable of integration At the time of an event new values can be assigned to species and parameters These assignments are defined by a Target specifying the species or parameter ID and an Expression that specifies the value to be assigned BioPARKIN does not yet support state dependent events
36. s caused by e g different physical units of the species compare Section A By clicking on Initialize Thresholds the values are set to the machine accuracy However a warning pops up which informs the user hat this might not be the right choice In the Species tab species names and initial quantities can be edited The column Disable b c has a check mark whenever the property Boundary Condition b c predefined SBML property name for the species in the Model Overview tab is true In this case a species is not integrated and handled as zero in the remaining differential equations which is usually desirable for e g degradation products If Boundary Condition is set true by mistake this results in a different model and thus a different simulation result A click on the button Simulate in the lower right corner starts the numerical integration of the ODE system with the current settings Results are displayed in the Results Window 4 4 Workbench Parameters The tab Workbench Parameters contains the list of all parameters with properties and property values The rows can be sorted alphabetically or numerically by clicking on the header of a specific column The Threshhold value gives a lower bound for the absolute value of each parameter Again these entries are important for sensitivity analysis and parameter identification since they account for different scales caused by e g different physical units of the parameters By clickin
37. se the entries of the corresponding column in J p and thus the weighted lo column norm are almost zero Furthermore some of the parameters might be linearly dependent which leads to nearly identical columns in J p In both cases the matrix J p will be singular or from a numerical point of view nearly singular Linearly independent parameters can be identified by analyzing their subcondition Let us consider the QR decomposition of J p By a suitable permutation of the matrix columns of J p the diagonal 29 elements of the upper triangular matrix can be ordered in the form ri1 gt r22 gt gt rqq The subcondition of parameter p is given by se riil Irul 20 Thus the permutation of matrix columns corresponds to a new ordering of parameters according to increasing subcondition The subcondition indicates whether a parameter can be estimated from the given data or not Only those parameters can be estimated for which sc 1 e 21 where e gt 0 is the relative precision of the Jacobian J p 3 Several safegurads and convergence monitors are in place for the gauss Newton approach e Natural monotonicity test via natural level function e Control of asymptotic convergence rate e Adaptive rank strategies by controlling the subcondition e Adaptive trust region strategies for relaxation estimation These techniques for solving a nonlinear least squares problem are implemented in the codes NLSCON 1 and PARK
38. sis or param eter estimation different tabs show the results of the numerical computations These tabs are explained in detail in Section 4 On the bottom of this window the user can change the view between tab mode and window mode close the current tab or close all tabs Note that the results are lost if a tab is closed without saving the data before 4 Graphical User Interface Usage This section illustrates the use of BioPARKIN in step by step descriptions from the perspective of a user aiming to guide through the features of the software 4 1 Model Overview Working with a Model When BioPARKIN is started in the upper left corner click on File Open to get access to your home directory Choose the desired SBML file and click on Open The file name now appears in the model list A short cut icon for this process is available right below File If you worked previously with that model it also appears in a list of at most 6 model files which opens when you click on File In this case you can directly click on the desired SBML file without opening the file browser The column Entity Tree in the middle now shows all model components Compartments species reactions parameters rate rules assignment rules and events By clicking on the plus sign or arrow in front of one of these components the tree opens up and shows the list of all objects belonging to a specific component e g the list of all species A click on one of these objects r
39. tch model entities and data If a column called SD is preceded by a data column it holds the standard deviations of the measure ments in the preceding column Finally columns with arbitrary names can be appended at the end As long as they do not contain square brackets they are effectively ignored by BioPARKIN This can be helpful e g to keep track of patient IDs within files or to add any comments BioPARKIN not only reads files formatted this way but uses the same format to save simulation results to disk 3 Graphical User Interface Overview This section gives an overview of BioPARKIN s graphical user interface GUI When starting BioPARKIN the user is given a window with a menu bar and icons at the top Within the menu bar one can amongst others perform calculations and open the ODE Viewer The ODE Viewer shows the generated differential equations of the model in different forms i e with component IDs or names with reaction abbreviations or in full notation Within the icons one can amongst others switch to the Results Window If an error or a warning occurs a grey cloude appears on the right hand side of the icon list Once you click on this cloud a window with more specific information pops up and the cloud turns into a sun The main part of the window is taken up by two tabs that show the constituent parts of the current model Model Overview and the simulation control Workbench After computations a second window
40. thods for Nonlinear Problems Affine Invariance and Adaptive Algorithms Number 35 in Springer Series in Computational Mathematics Springer 2004 P Deuflhard and F Bornemann Scientific Computing with Ordinary Differential Equations Num ber 42 in Texts in Applied Mathematics Springer Verlag 1 edition 2002 P Deuflhard and A Hohmann Numerical analysis in Modern Scientific Computing An Introduc tion Number 43 in Texts in Applied Mathematics Springer Verlag 2 edition 2003 P Deuflhard and U Nowak Efficient numerical simulation and identification of large chemical reaction systems Berichte der Bunsengesellschaft f r physikalische Chemie 90 11 940 946 1986 P Deuflhard and U Nowak Extrapolation integrators for quasilinear implicit ODEs In P Deuflhard and B Engquist editors Large Scale Scientific Computing pages 37 50 Birkhauser 1987 A Funahashi N Tanimura M Morohashi and H Kitano CellDesigner a process diagram editor for gene regulatory and biochemical networks 2003 S Hoops S Sahle R Gauges C Lee J Pahle N Simus M Singhal L Xu P Mendes and U Kummer COPASI a complex pathway simulator Bioinformatics 22 24 3067 2006 M Hucka S Hoops S M Keating N Le Nov re S Sahle and D Wilkinson Systems biology markup language sbml level 2 structures and facilities for model definitions 2008 N Le Nov re M Hucka H Mi S Moodie F Schreiber A Sorokin E Demir K
41. ttings Time Unit days Expert Settings Relative Tolerance RTOL n uv s ET aus o E o Les a u a 7 n t D as v E o o a Absolute Tolerance ATOL Required Tolerance XTOL Max Newton Steps 50 Parameter Constraints strictly positive exponential transformation y Data Browser Jacobian Numerical Diff with Feedback Problem Type highly nonlinear Fit Sensitivit Residual Scaling no extra scaling Figure 3 Workbench Settings tab 10 4 2 2 Time Settings Start and end time for the simulation have to be defined Note that time is allowed to take negative values In particular for autonomous systems where the right hand side of the ODE system does not explicitly depend on time this does not change the solution but only the labeling of the time axis Note that time shifts can also be included when loading experimental data in the Data Browser 4 2 3 Plot Settings Here the user can directly edit the string of the time unit which is displayed on the x axis of any simulation plot 4 2 4 Expert Settings Numerical integration of the ODE system is performed with LIMEX a Linearly IMplicit Euler method with EXtrapolation 5 LIMEX is suitable for stiff differential equations as they often occur in chemical reaction kinetics The values RT OL relative tolerance and ATOL absolute tolerance are controling the adaptive time step selection of the integrator For n
42. umerical details we refer to 2 T here are model systems that are only solvable with very small absolute tolerance e g the Zhabotinski Belousov reaction desribed in the aforementioned book Negative solutions for species which are expected to stay positive can be an indicator for small absolute tolerance requirements Note however that a too small tolerance could cause the interator to interrupt before integration is completed because the desired tolerance might not be achievable Further settings refer to the damping strategy of the Gauf Newton method during parameter identi fication Some of these setting only apply to Gaufi Newton NLSCON If Gaufi Newton PARKIN is selected which works much more in an automated way the respective settings are greyed out e The tolerance XTOL refers to the the desired relative accuracy of the estimated parameters with respect to affine covariance 1 e Maz Newton Steps is the maximum allowed number of iterations in the Newton method When this number is reached the iterations are stoped even if convergence has not yet been achieved e Sometimes parameters might be required to be positive In this case chose strictly positive as Parameter Constraints At the moment this option can only be applied to all parameters or none but this will be improved in a future release e The Jacobian can be computed in three different ways either by computing difference quotients Numerical Diff
43. ur St fly wu fetus pu S flys e u eu S to 0 8 showing clearly the influence of the transformation v upon the sensitivity matrix S In the case that breakpoints events are specified the computation of the sensitivity matrix also splits into the computation of S and ST S fyy 3 DS foly D 9 S tg 0 10 SV fu pS four p 11 ST g pS to ge y p 12 respectively Often model species and model parameters cover a broad range of physical units and their values can vary over orders of magnitude To achieve comparability the absolute sensitivities have to be normalized by the absolute values of species and parameters to obtain relative sensitivities ne 2M gy maxlp ltrship Sas E 5 4 max maxser yi t trsh y 13 In this formula the user specified threshold values for parameters and species occur J is the integration time interval It is always the relative sensitivities that are displayed in the Results Window A 5 Parameter Identification Assume there are M experimental data points varying in the selected component at different time points ge Le rere E di 1 N 28 associated with corresponding measurement tolerances dz Here y r denotes the measurement of component yj at time 7 The M to N mapping 7 assigns to every measurement time point 7 one of the N components of y Remark In PARKIN the measurement tolerances are computed as 02
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