User Manual MUscle SImulation COde
Contents
1. Std Dev Disp Same as above for the displacement of first myosin node the first myosin node Std Dev Disp Same as above for the displacement of last actin node the last actin node Number of at tached crossbridges per filament This number must be seen in relation to the total number of available cross bridges Number of cross bridges in state 1 If state 1 is a detached state this value will always be 0 Number of cross bridges in state n If state n is a detached state this value will always be 0 Displacement M Line This value is only of interest if the full sarcomer is simulated It displays the displacement of the M Line which con nects the left half sarcomer and the right half sarcomer Table 7 Columns in the result file 3 5 Output Files 31 3 5 5 distrib dat For some purposes it is quite useful to have the time course of the state distribution of the crossbridges That is the number of crossbridges in a certain state with a certain strain This information is quite bulky and should therefore not be written in every single timestep even when this is supported by the program Two different representations of the crossbridge distribution are mixed to gether in this file Due to the discrete nature of the model only the strain and state of the single crossbridges are known If a continuous distribution is needed for comparing the results with results of the continu2. which in a real setting would be bound to different actin fibers are bound to the A and fibers As those do not receive bindings from other myosin fibers the overall number of bindings on one actin fiber should be equivalent to the real case where one actin fiber receives bindings from three myosin fibers The currently implemented approach analyses the angles on a case by case basis Le for each combination of head and crown an explicit formula has been derived to determine the angles f and 6 Although this part of the code seems to work fine it is potentially prone to errors To verify the above mentioned code an alternative procedure was developed which computes the angles 5 and for a general case Then the different cases are casted into this general framework This general procedure looks at the geometric relations between a single myosin and a single actin fiber The setup of this general case is shown in Fig 12 The required parameters are the center coordinates of both fibers La Ya and m Ym the radius of the fibers ra rm and the angle of the 4 3 David Smith s Lattice 39 Actin H Actin H al A2 no D _ Symmetry Sector Hi i _ Symmetry Sector Hi H2 4 i Y Myosin ZH i a ai p wa l ARK A1 i f Y i d Hi Ha i o e ES ae Figure 13 Sketch of the Three Crowns on the myosin Fiber binding sites with respect to the X axis aq Am In a first step the algorit
3. 02 0 04 0 13 0 09 3 0 17 0 01 0 21 0 32 0 01 0 01 0 08 0 15 0 04 4 0 15 0 01 0 53 0 13 0 03 0 05 0 03 0 06 0 01 5 0 06 0 04 0 37 0 09 0 15 0 23 0 02 0 02 0 02 6 0 07 0 02 0 53 0 22 0 02 0 09 0 02 0 01 0 02 This is again data which was generated by a 9 state model The first number in the line identifies the myosin head As every single sarcomer is constructed in the same way this number uniquely identifies a myosin head The following numbers denote the fraction of these heads in a certain state Certainly the first number corresponds to the first state and so on Thus the number of columns is again n 1 3 5 6 displacement dat The displacement data file reveals the displacement field of the finite element models which are used to simulate the mechanics of the filaments Similar to the distribution output file it contains several data sets Each data set starts with the simulation time in a single line After that the displacement fields of the different filaments follow in separate blocks Hence the number of these blocks depends on the used lattice geometry which determines how many filaments are present Each displacement field is a set of lines where the first number in each line gives the original coordinate of the finite element node and the second number displays the displacement of that node At each time where an output is requested the block of data starts with a single floating point number which is the simulation time In
4. State Model 46 David Smith s 9 State Model 47 1 Introduction 4 1 Introduction The program is based on a modular object oriented code which implements a flexible framework for programming simulation codes which deal with bio logical systems in particular muscle cells Currently most of the code is focused on Monte Carlo simulations of muscle cells For a detailed introduction of the molecular structure of muscle cells refer to 3 Basically the muscle cells consist of thick myosin and thin actin fibers which are arranged in a comb like structure Between these fibers crossbridges are dynamically created by molecules which are attached to the myosin fibers and bind to the actin The Monte Carlo simulation uses a 1D finite element code cf 1 to model the fibers To resemble the chemical reactions which happen in the cross bridges the Monte Carlo simulation is used Thus the numerical proce dure can be seen as a staggering scheme After computing the mechanics the crossbridges may change their state which then influence the mechanics again Besides the Monte Carlo simulation three other numerical procedures for simulating the mechanical response of muscle fibers are implemented 2 Installation Build Currently the software can be installed in a Windows environment using the Cygwin environment under Linux and on Apple s OS X 2 1 Installation Windows XP Cygwin The code for simulating the muscle fibers its
5. a actin2 50 4 set actin period 13 set dx actin 2 77 set e actini 1 27e3 set e actin2 1 27e3 set length actin 1050 0 end subsection mvosin set a mvsoin 50 0 set dx_myosin 14 3 set e_mysoin 3 52e3 set myosin_heads 49 set myosin_offset 125 0 end subsection numerics set model C4 set xamax 15 0 end 4 4 Flight Muscle In insect flight muscle the 3D lattice structure differs from that found in striated muscle of vertebrates The myosin crowns have 4 instead of 3 myosin heads each Furthermore the angle between two crowns in not 40 but 45 4 4 Flight Muscle 41 Therefore the axial structure of the myosin crowns repeats at every second Crown Furthermore the different structural setup requires three actin fibers instead of two in the model of the vertebrae lattice structure Note Although this lattice model is implemented it has not yet been thoroughly verified Flight geom flight prm Listing of Parameters H zn 22222222 gt subsection actin set a actini 50 4 set a actin2 50 4 set a actin3 50 4 set actin period 13 set dx actin 2 77 set e actini 1 27e3 set e actin2 1 27e3 set e actin3 1 27e3 set length actin 1050 0 end subsection axial set d actin mvosin 30 0 set radius actin 3 5 set radius mvosin 7 0 end subsection mvosin set a mvsoin 50 0 set dx_myosin 14 5 set e_mysoin 3 52e3 set myosin_heads 49 set myosin_offset 125 0 end s
6. of different microstructures onto the mechanical prop erties models of different complexity were integrated into the program The following sections will give a brief overview about geometry models and the parameters which can be used to adjust these models 4 1 Linear Lattice The linear lattice does not resemble an actual muscle structure It mostly serves as a simple verification tool because it comes closest to the continuous 4 1 Linear Lattice 34 models which do not include any assumptions about the microstructure of the muscle Basically the muscle consists of a single actin and a single myosin fiber The actin fiber has a length which is specified by the parameter length actin There is no explicit parameter for the length of the myosin fiber It is de termined by the spacing between the heads dx_myosin and the number of myosin heads myosin_heads Thus the length is di myosin hEAdSmyosin The lattice structure does not influence the binding probabilities Both fibers do not start at the same position The offset between the two fibers is defined by myosin offset The mechanical properties which are used in the finite element model of the fibers are defined by a_actin e actin a myosin and e_myosin for the actin fiber and the myosin fiber respectively The parameter starting with a defines the crosssectional area of the fiber in nm while the parameter starting with e_ defines the elastic modulus in TODO The last parameter x
7. support points and either returns the value on those support points or determines a linear interpolation between the two neighbouring support points Clearly the accuracy of the approximation depends on the resolution of these support points The parameter dx determines the step size between two support points xs defines the first support point and xe the last one The default value should be sufficiently accurate It is selected by using the parameter David9 in the file simulation prm The model parameters can then be specified in model david prm which typically looks like 8 David Smith s 9 State Model 49 Listing of Parameters subsection david set model C4 end subsection misc set set set set set set set set set set end dco 7 0 etal 10 0 eta2 10 0 eta3 10 0 gl 314 g2 631 g3 1e9 gmax led kmax 1e6 lambda 1 0 subsection numerics set dx 0 02 set interpolation Linear set xe 20 0 set xs 20 0 end subsection powerstroke set h 4 0 set hd 4 0 set ht 1 0 end subsection rates set k_12 0 0 set k_13 100 0 set k_18 5000 0 set k_19 100 0 8 David Smith s 9 State Model 90 set set set set set set set set set set set set set set set set set set set set set set set set set set end k_21 k_23 k_26 k_28 k_31 k_32 k_34 k_43 k_45 k_49 k_53 k_54 k_56 k_57 k_62 k_65 k_68 k_73 k_75 k_78
8. the cygwin pack ages given in section 2 1 1 were selected If this doesn t help a look to the deal II mailing list which can be found on the same website as the source code might give some hints After a successful configuration the output should end with the following paragraph Please add the line export PATH PATH home okavserh deal II lib to vour bash profile file so that windows will be able to find the deal II shared libraries when executing your programs 2 2 Installation Linux 11 Finally the actual compilation has to be started by okayserh HP20439203671 deal II make j 4 1d epee Warnings that appear in the output can usually be ignored On a halfway modern computer the compilation will take between 30 and 60 minutes to finish A final step is to add the line given in the configuration output to the local configuration An arbitrary editor may be used to edit the file home okayserh bashrc the okayserh has to be replaced by the name found in the actual home di rectory If a windows editor is to be used the cygwin directories are usually located under c cygwin home okayserh for the example given above At the end of the file the following line or whatever is returned by the configuration script has to be appended export PATH PATH home okavserh deal II lib This should complete the installation of the deal II framework 2 1 3 Compiling the stochastic code 2 2 Installation Linux 2
9. 3 Macintosh OS X 3 Preliminaries As the primary purpose of the program is the comparison of results for dif ferent experimental protocols usually the first step is the definition of a protocol that defines the time course of the mechanical loads input file protocol dat After that the general simulation parameters and the output have to be specified input file simulation prm Finally specific files which provide the parameters for the selected models and muscle lattice have to be created the filenames depend on the selected model and lattice geometry 3 Preliminaries 12 Except for the file protocol dat all parameter files follow the conventions which come from the parmeter handler concept of the deal II library which provides the basis for the program The deal II parameter files are subdivided into sections which can contain one or more parameters Usually the sections divide the available parameters into logically related sets of parameters The following paragraph shows an example of the input file simulation prm Listing of Parameters H DI A i AT G l subsection global set K xb 1 0 set dt le 5 set model Huxley set numerics MonteCarlo set random_seed 0 end subsection output set distribution 99 set energy_distribution 99 set force_distribution 99 set filament 99 set macro 1e 10 set transition 99 end Each of the subsections starts with the keyword subsection followed by the name o
10. Model This was the first model which went away from the concept of simple me chanical elements and instead attributed the macrocopic behaviour of the muscle to the microscopic distribution of crossbridges One of the two states represents a detached myosin head while the other state is used for attached myosin heads The rate functions of the model are kio r ka1 x for O lt a lt h else for 0 lt x else 9 10 It is selected by using the parameter Huxley in the file simulation prm The model parameters can then be specified in model_huxley prm which typically looks like Listing of Parameters subsection huxley set fl 48 5 set gl 11 2 set g2 208 0 set h 15 6 end 6 Huxley s 1957 Model 45 Parameter Type Description actin_distance phi minus phi plus sd fac segment lengthi xi Floating Point gt 0 Floating Point Floating Point Floating Point Integer Floating Point Distance between two actin sites For accuracy this number should coincide with the number that is used in the lat tice geometry parameter file Tropomyosin chain angle in radians when the tropomyosin chain is pulled down by troponin Tropomyosin chain angle in radians when the tropomyosin chain is pulled up by a myosin head in R state Scaling factor for the standard devia tion that is computed with the numeri cal model For details refer to the tech nical report about the fl
11. User Manual M Uscle SImulation COde Dr Oliver Kayser Herold 06 17 2008 CONTENTS 2 Contents 1 Introduction 4 2 Installation Build 4 2 1 Installation Windows XP Cygwin 4 2 1 1 Installing Cygwin va et par 5 2 1 2 Compiling the deal II FEM framwork 8 2 1 3 Compiling the stochastic code 5 ooa ue 11 2 2 Installation Linux A du ee a 11 Dud Mae los ODER marins es A A et 2 a b ba 11 3 Preliminaries 11 3 1 The Protocol File ga ech IE a 13 3 2 Protocol File Examples lt 22 e De 15 3 3 Global Simulation Parameters 18 dl INUIN TIESS Ses daa io ia a 20 prox tp Blesa saes e A A da 27 3 98 trates dated oe ace wi vues et A MA as ee 28 310 2 states dat 4 Lime Li a A RNA 28 3 9 9 Versi n dat ies sares 2 ee RR fonte 29 IA POSTER de BAR Pr due da Aie een 29 3 00 idistrib dat Ka wk Mur oe Ma Dome Ae ns 31 3 5 6 displa enent Gab u eae LENS ee AP ee ba tak 32 3 1 filaments dat sure a da 32 4 Muscle Lattices 33 AE Linear Lattice A BE kee en we 33 4 2 Vertebrae Lattice a 52 A RR ee ee 35 4 3 David Smitb s Latti e a s 2 23206 oe wee we ta Ex 39 AA Pet Muscle gt 0 ea sas alt aaa 40 CONTENTS 3 Activation Models 42 5 1 Fixed Number of Blocked Sites 42 5 2 6 State Rigid Segment Model 42 5 3 2 State Rigid Segment Model 2 2 4 ico e 43 5 4 Flexible Chain Model 43 Huxley s 1957 Model 44 Duke 3
12. a result of the prescribed force 3 3 Global Simulation Parameters The next file which has to be present in all simulation runs is simulation prm Besides the model type and the lattice geometry type some numerical pa 3 3 Global Simulation Parameters 19 T T T switeh result txt u 1 4 HH Displacement nm 30 l 35 L 40 L 45 L L L L L L L L L 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 09 Time s Figure 10 Switch from prescribed displacements Isometric to prescribed forces Isotonic 3 4 Numerics 20 rameters and the structure of the output files are defined in this parameter file An example which contains all currently implemented parameters is shown below Listing of Parameters H ET subsection global set K xb 1 0 set dt le 5 set model Huxley set numerics MonteCarlo set random_seed 0 end subsection output set distribution 99 set energy_distribution 99 set force_distribution 99 set filament 99 set macro 1e 10 set transition 99 end The parameters which belong to the subsection global are explained in Tab 4 Those in subsection output can be found in Tab 5 3 4 Numerics Monte Carlo Simulation Code The Monte Carlo MC simulation is a computationally expensive but quite accurate way of simulating the biomechanical responses of muscle fibers Many of the components of the MC simulation can be easily exchanged by modifying the correspondi
13. ain x which is always in the first column The number of columns in a line depends on the model and is nx n 1 1 with n being the number of states Hence for a 3 state model a typical line looks like x ki kas koi k23 k31 k32 3 5 2 states dat Another interesting information about the used model is the stationary dis tribution of the crossbridge states When the rate functions are used in a continuous model this model will reach a stationary state at t oo If there s no convection the problem reduces to finding the solution of the linear system A K x T R x 1 for each strain x For a three state model the matrices are gt kiz koi kos k3o k12 K k23 k13 k13 kz k32 2 3 5 Output Files 29 with right hand side vector n qe a The stationary distribution of the third state can easily be found by using Sold 4 i 1 3 A line in the file has then the following content 3 5 3 version dat To keep track of the version and to have some information about the version of the program which was used to create specific results the version number of the program is written to this output file Unfortunatelv there is no CVS tag to automaticallv insert the date of the last commit command into the source code Therefore this version number may currently not always be accurate It looks like Id manual tex v 1 10 2008 08 14 19 31 50 okayserh Exp 3 5 4 res
14. amax defines the search interval for finding matching actin binding sites which are in the range of the myosin head If the myosin head is attached to the myosin fiber at position x it can potentially bind to all binding sites in the interval x Lamaz Y Zamar Clearly the strain dependent rate functions have to allow this binding Simple geom_simple prm Listing of Parameters H A A A AA SP subsection actin set a_actin 5 04 set dx_actin 2 77 set e actin 1 27e3 set length actin 1050 0 end subsection mvosin set a mvsoin 5 0 set dx mvosin 14 3 set e mvsoin 3 25e3 set myosin_heads 49 set myosin_offset 125 0 end 4 2 Vertebrae Lattice 35 subsection numerics set xamax 20 0 end 4 2 Vertebrae Lattice Most of the parameters which are offered in the Vertebrae lattice model have the same meaning as in the simple lattice Basically there are two differences Instead of one actin fiber in the simple model there are two actin fibers which need separate parameters for the elastic modulus and the crosssectional area The second new parameter is related to the axial structure of the fibers It defines the number of actin binding sites after which the actin the binding sites completed a full rotation The default value of 13 should be correct in most cases Other values do not lead to problems with the program but should only be used when it is clear what is done The same holds for the elastic modulu
15. and line starts in the home directory which is home okayserh in the example shown above Now it is time to unpack the compressed source file in the home directory okayserh HP20439203671 7 tar gzip extract file cygdrive c Docum ents and Settings okayserh Desktop deal nodoc 6 1 0 tar gz The part following the command file has to be changed to the place where the file was stored during the download Cygwin mounts the windows drives in a subdirectory cygdrive plus the letter describing the hard disk partition In the example the file was saved on the users desktop After unpacking the archive it is always good the verify that the everything is in place 2 1 Installation Windows XP Cygwin 10 okayserh0HP20439203671 7 ls deal II okayserh0HP20439203671 7 cd deal II The directory deal II should contain the following files and directories okayserh HP20439203671 deal II 1s Makefile Version in common contrib lac README aclocal m4 configure deal II lib README configure base configure in doc reconfigure If everything is in place the next step is the configuration of the library which should usually run fully automatically okayserh HP20439203671 deal II configure with umfpack Eseg Appending with umfpack is very important since the direct solver UMF PACK is used as primary solver for linear systems of equations in most parts of the program In case of error messages make sure that
16. cosine result txt u 1 4 2 l 4 3b 4 at al 5 l A 6 L al 7 l al L 1 1 1 1 L 1 0 0 1 0 2 0 3 04 05 0 6 0 7 0 8 0 9 Time s 3 2 Protocol File Examples 17 T T mult step result txt u 1 4 al A 10 4 12 l 4 14 L 1 l L 0 0 1 0 2 03 04 0 5 Time s Figure 8 Multiple Step 0 000 00 0 200 40 0 300 80 0 4 0 0 12 0 0 500 14 0 0 600 00 It is certainly possible to combine the multiple steps with the technique for gradual steps as shown in the previous paragraph Results of the protocol are displayed in Fig 8 Multiple Sine Functions Another option is having a sine function with differing amplitude 0 00000 0 2024 10 4 0 5 0 50 28 10 8 0 5 1 000000 3 3 Global Simulation Parameters 18 0 H 7 Jeosine2 resulttxt u 1 4 Displacement nm 16 L L L L L 0 0 2 0 4 0 6 0 8 1 12 Time s Figure 9 Sine function with diffing amplitude Again this is just the combination of two different sine functions Fig 9 shows the resulting displacement Switch from Isometric to Isotonic Finally another example for switching from prescribed boundary displacements isometric conditions to prescribed forces isotonic conditions 0 00000 0 5 1 0 500 0 1 000000 The results are shown in Fig 10 It can be clearly seen that initially the displacements are fixed while after 0 5 seconds the displacements are
17. d Integer gt 0 Type be found in the section about the spe cific geometries This parameter has two functions A value greater zero switches the code for full sarcomers on while the de fault value of zero leads to the stan dard simulation of a half sarcomere When the full sarcomere mode is used this parameter defines the number of full sarcomeres in series Le a num ber of one simulates one full sarcomer while a number of two simulates two full sarcomeres in series It should be noted that the computational effort ap proximately increases with filaments x full_sarcomer Description start_x end_x dx Floating Point Floating Point Floating Point Start point for distribution bins End point for distribution bins Size of distribution bin Table 6 Parameters for defining the bins that are used to evaluate head distributions 3 4 Numerics 25 Parameter Type Description dx Floating Point gt Spacing of discretisation points 0 x start Floating Point Initial start point of the computational domain x_end Floating Point Inital end point of the computational domain points back into the domain of interest the implemented algorithm avoids this problem Nonetheless in some cases this procedure may still lead to problems in cases where the detachment happen slowly over a huge domain A careful analysis of the results is always recommended when a new model is tested with
18. d Site Choose a site from this list or add pour own sites to the list Ayailable Download Sites http cyqwin elite systems org ftp ftp fedoramd org tepo fedoramd org http mirrors kemel org ftp ftp mirrorservice org http www mirrorservice org ftp cygwin osuosl org http evgwin osuosl org http kambing vism org ftp gd tuwien ac at User URL Figure 4 Initial windows of cvgwin setup program Cygwin Setup Select Packages Select Packages Select packages to install Category New Next gt Cancel B All amp Default Accessibility amp Default Admin amp Default Archive amp Default Audio amp Default Base amp Default Database 4 Default Devel amp Default FI Dar Default Hide obsolete packages Figure 5 Initial windows of cygwin setup program 2 1 Installation Windows XP Cygwin 8 Fig 5 shows the dialog that finally allows the user to select the packages that should be installed If a new installation is started several packages will automatically be selected These are crucial for the operation of the cygwin environment and should not be modified For the compilation of the deal II framework and the stochastic muscle code additional software developement packages have to be selected It might be easier to select these when the view is first switched to a list of all available packages The view mode can be changed by clicking
19. elf should be rather unprob lematic with respect to portability Unfortunately the deal II library uses many advanced language features which are not supported by the usual de velopment environments that are available under Windows Therefore the Cygwin library has to be used to provide Unix like API calls and the GNU compiler collection 2 1 Installation Windows XP Cygwin 5 Cygwin Setup Choose Installation Directory Select Root Install Directory Select the directory where you want to install Cygwin Also choose a few installation parameters Root Directory Browse Install For Default Text File Type All Users RECOMMENDED Unix binary RECOMMENDED Cygwin will be available to all users of No line translation done all files opened the system NOTE This is required if in binary mode Files on disk will have you wish to run services like sshd etc LF line endings O Just Me O DOS text Cygwin will only be available to the Line endings will be translated from unix current user Only select this if you lack LF to DOS CR LF on write and vice Admin privileges or you have specific versa on read needs Read more about file modes lt Back Next gt Cancel Figure 1 Initial dialog of cygwin setup program 2 1 1 Installing Cygwin If the cygwin environment isn t already installed on the target machine the first step consists of setting up the cygwin environment For this purpose the file setu
20. ether always varies If the filaments are assumed to be elastic the geometric setup also varies throughout the timecourse of the simulation Therefore the proposed model recomputes the geometric configuration of the binding sites in each time step It is assumed that the probability that a head on the myosin fiber can bind to a neighboring actin site is strongly influenced by the angle between the myosin head and the actin site which is called 9 Additionally also the angle between the actin site and the center of the myosin fiber is computed and will be denoted by B Fig 12 shows the basic geometric setup Looking at a crosssection which normal is aligned with the myosin fiber the structure shown in Fig 13 is found The three subfigures show the structure at the positions of the three different crowns This hexahedral structure 4 2 Vertebrae Lattice 38 N Binding Site Binding Site Myosin Figure 12 Sketch of the geometric setup used to compute the binding angles regularly continues in all directions Obviously the structure is rotationally symmetric with a rotation angle of 120 Although the state of each head is tracked separately the six actin fibers surrounding the myosin fiber are put together into two explicit actin fibers denoted by A and As This is reasonable as each actin fiber can be bound by one of the heads of one of the three surrounding myosin fibers Therefore the bindings of all three heads on one crown
21. exible chain model The number of actin sites between two troponin sites including the troponin sites Le a number of 7 means that there are 6 regular actins between two troponin sites Parameter in the original paper about the flexible chain model It defines the elasticity of the flexible chain kb_eff kt_eff km kps Floating Point Floating Point Floating Point Floating Point Rate for binding of the troponin site This is a scaling factor that yields the effective rate together with the tropomyosin chain position and stan dard deviation Rate for unbinding the troponin site Binding of Myosin This parameter has a slighly different meaning in the cou pled activation mechanics model It is a scaling factor that works on top of the regular myosin binding rate which is multiplied by another factor that represents the current state of the tropomyosin chain Scaling factor for the powerstroke state Since the actual strain dependent tran sition rate towards the powerstroke comes from the used model of the ac tomyosin cycle this factor will only be included into the overall reaction rate It can be used to tune the powerstroke transition to match the results without 7 Duke 3 State Model 46 7 Duke 3 State Model In Huxleys model the force originates in the unsymmetric rate functions which ensure that the Mysosin heads preferably attach on the actin fiber on positions where they immediatel
22. f the subsection The available section names will be discussed in detail in the other parts of this manual After that one or more lines defining the parameters in that section follow For each parameter the line starts with the keyword set and the subsequent name of the parameter Finally the value which should be assigned to that particular parameter comes behind an equal sign which separates the parameter value from its name Generally floating point numbers integer numbers and textual values are admissible as values for a parameter Which one of these three basic types is admissible depends on the parameter and will be discussed in the section about the parameter file The program looks for the parameter files in the directory from which is was started After each time step the program writes the current time step 3 1 The Protocol File 13 Column Description Remarks 1 Time Floating point 2 Boundary type Integer cf Tab 2 3 Function type Integer cf Tab 3 4 Function parameter a Floating point 5 Function parameter b Floating point optional 6 Function parameter c Floating point optional T Function parameter d Floating point optional Table 1 Columns in the protocol file Type Description 0 Prescribed displacments 1 Prescribed force Table 2 Available boundary conditions number to the console Error messages will be written to the console as well In case of a severe error the program w
23. hm computes the cartesian coordinates of the actin and myosin binding site denoted by Lra Yva and Lim Yom Then the co ordinate system is shifted to place the origin into the center of the myosin binding site For the angle between the X Axis and actin binding site we have 0 arctan um 5 Loa Lom Subtracting the angle am gives then the binding angle 9 The angle f is determined with a very similar procedure First the origin of the coordinate system is shifted to the center of the actin binding site After that the angle between the center of the myosin fiber and the X Axis is computed f arctan A 7 La Lom and corrected by the actin angle p a B 8 4 3 David Smith s Lattice This lattice model was originally used in D Smiths original 9 state model to include the lattice structure To be able to compare the results which were obtained with the original code with the results from the Monte Carlo sim ulation the algorithm was reimplemented It has many similarities with the 4 4 Flight Muscle 40 code for the vertebrae lattice The main difference is that it does not mod ify the binding probabilities according to the axial configuration but instead strictly differentiates the binding sites in those where the myosin head can bind and those where no binding is possible under any circumstances David geom_david prm Listing of Parameters H AA A subsection actin set a actini 50 4 set
24. ill usually abort the computation An error which can frequently be observed is Internal Error Detach Sum 1 662999e 00 This error indicates that time step size was to large for a rate If this error occurs only from time to time the results should usually not be significantly affected Otherwise the time step size should be reduced 3 1 The Protocol File As already mentioned the protocol file controls the type and time course of mechanical loads which will be prescribed during the simulation run It contains one or more lines which are composed of at least 4 entries The first parameter defines the time at which the simulation should switch from one mode to another mode Hence the first line should start with t 0 After that the type of boundary condition is defined Currently four boundary conditions are implemented While this field defines the type of boundary the value for this boundary condition can currently be defined by three different types of functions The 3 1 The Protocol File 14 Type Description 0 constant function f a 1 linear function f f axt b 2 sine function f a x sin b x 2r xt dxm c 3 twitch f c exp t a exp t b tp d Table 3 Available functions twitch is a special function which was introduced in the context of the ac tivation models It simulates a short stimulation of the muscle where the parameters control the time course and amplitude of the sti
25. ion Floating Point force distributip loating Point energv distribu filament macro transition tiowating Point Floating Point Floating Point Floating Point Time interval for writing the cross bridge distribution Time interval for writing the cross bridge force distribution The force re sults from the linear relationship be tween crossbridge strain and force Time interval for writing the cross bridge energy distribution Energy is the one half of the squared strain mul tiplied by the crossbridge stiffness Time interval for writing the individual forces of the sarcomers Time interval for writing the summa tion parameters Time interval for writing the transition matrix Table 5 Parameters in the simulation control which control the frequency for writing certain information to the output files 3 4 Numerics 23 subsection monte_carlo set activation None set filaments 100 set geometry Vertebrae set full_sarcomer O end subsection energy_dist set start_x 0 0 set end_x 200 0 set dx 1 0 end subsection force_dist set start_x 10 0 set end_x 50 0 set dx 0 5 end Besides the primary parameter section there are two additional parameter sections energy_dist and force_dist which offer the parameters shown in table 6 These parameters can be used to define the bins that are used to create outputs of the energy and force distribution of the attached myosin heads The di
26. kBT 4 14 set k 12 1000 0 set k 21 cap 100 0 set k_adp 71 0 set k_bind 170 0 end 8 David Smith s 9 State Model For a detailed description of the parameters in the 9 state model refer to 5 and 6 In the following parts only the sections which are specific to the Monte Carlo code will be discussed The first and most important selection is the type of model that should be used In the present version of the program the C3 and C4 version of D Smiths model are implemented They can be selected by setting the parameter model in the first subsection to either C3 for the C3 version or to 8 David Smith s 9 State Model 48 7 S are A Fr as g a mA i o LS AS Figure 14 Scheme of the biochemical states of the mvosin head in the 9 state model C4 Due to program internals the use of the exact rate functions is very slow Therefore it is recommended to use a linear or stepfunction approximation of the exact rate functions which is significantly faster The type of approx imation can be selected by setting the parameter interpolation to either Exact Linear or Step where the meaning of these values is obvious The algorithm for computing the approximate rate functions precomputes the exact rate functions on a set of support points before the actual computation starts When the Monte Carlo algorithm requests some rate function the algorithm looks for the closest precomputed
27. k_81 k_82 k_86 k_87 k_91 k_94 subsection set set set end cc dm dp subsection set set set cc dm dp state4 0 8 4 61 4 61 state5 0 25 7 35 3 8 David Smith s 9 State Model 51 end subsection set cc set dm set dp end subsection set cc set dm set dp end subsection set cc set dm set dp end subsection set cc set dm set dp end state6 0 25 7 20 6 state7 0 8 5 26 2 states 0 8 5 21 6 state9 0 8 4 61 4 61 REFERENCES 92 References BATHE KLAUS J RGEN Finite Element Procedures Prentice Hall En glewood Cliffs N J 1996 DUKE T A J Molecular model of muscle contraction Proc Natl Acad Sci USA 96 2770 2775 1999 MCMAHON THOMAS A Muscles refleres and locomotion Princeton University Press Princeton N J 1984 SMITH D A and M A GEEVES Cooperative Regulation of Myosin Actin Interaction by a Continuous Flexible Chain II Actin Tropomyosin Troponin and Regulation by Calcium Biophysical Journal 84 3168 3180 May 2003 SMITH D A M A GEEVES J SLEEP and S M MIJAILOVICH To wards a unified theory of muscle contraction I Foundations Biophys J 2007 submitted SMITH D A and S M MIJAILOVICH Towards a unified theory of mus cle contraction II Predictions with the mean field approximation Bio phys J 2007 submitted
28. mulation This function is of limited use for mechanical stimulus although it certainly can be used The parameter t in the above mentioned functions is not the absolute time It is the time from the beginning of that protocol step Thus when a new step in the protocol is reached t will start again from zero One example which defines isometric conditions followed by isotonic condi tions with a prescribes force per filament of 430pN looks like 30 0 O 1 30 ono BRO ooo PB O At t 0 5 the boundary mode is switched from prescibed displacements to prescribed forces The last line defines the end of the simulation Another example for a sinosoidal prescribed displacement at one end is 0 0000 0 1 0 2 50 10 0 0 0 5000 After a short initial phase the sinosoidal displacement is prescribed at one end of the filament Again the last line defines the end of the simulation run An abritrary number of lines is allowed in this protocol file But currently no comments may be added If the used numerical scheme is not the Monte Carlo code the force that is specified in the protocol file is a normalised force where a value of 1 corresponds to the maximum force under isometric conditions 3 2 Protocol File Examples 15 3 2 Protocol File Examples The following paragraphs display some of the protocols that can be imple mented by using the protocol file Gradual Step Instead of having an instant shortening the shor
29. ng parameters numeric_mc prm Listing of Parameters 3 4 Numerics 21 Parameter Type Description K xb Floating Point The crossbridge stiffness It defines the stiffness of the linear element which is used to represent the elastic cross bridge which connects the thin with the thick filament dt Floating Point Time step size model One of Huxley This parameters defines the model for Duke Daniel the state transitions More detailed de Pate David9 scriptions can be found in the sections about specific models numerics One of So far four numerical procedures for random_seed MonteCarlo Char Integral FEM Integer simulating muscle fibers are imple mented One of these can be selected by this parameter Although the algorithm can be seen as a Monte Carlo simulation the results are supposed to be completely deter ministic This is due to the pseudoran dom generator which generates always the same sequence of random numbers when it is started with the same ini tial seed In some rare cases it might be useful to start the random gener ator with a different seed which can be specified in this field Generally it seems to be wise to start all runs with the default random seed 0 to get com parable results Table 4 General simulation parameters which define the used actomyosin cycle the lattice model and other components 3 4 Numerics 22 Parameter Type Description distribut
30. nts about the algorithm to determine the angles will be given On the actin the binding sites are equally spaced along the fiber But looking into the direction of the fiber the binding sites rotate around this fiber cf Fig 11 After 13 binding sites the binding sites have again the same angular positions In literature a value of 5 5nm can be found for the spacing along the axis The binding sites on the myosin fiber have a different structure Angularly they have a structure which is similar to a 3 blade fan I e they are separated by an angle of 120deg In the following these three binding sites which have the same longitudinal position along the myosin fiber will be called head Each triple of heads will be called a crown Looking along the myosin fiber the crowns are placed in a distance of 14 3nm apart Starting with crown 1 at position Onm crown 2 follows at position 14 3nm and is rotated 4 2 Vertebrae Lattice 37 Figure 11 Sketch of the 3D structure of the muscle lattice 40 clockwise while crown 3 follows at position 28 6nm and is rotated 40 counterclockwise At 43 9nm again a crown 1 follows which has the same angular orientation as the one on position Onm cf Fig 13 This geometrical setup of the muscle structure leads to the situation that the spatial periodicity along the fibers differs between the myosin and actin fiber Hence the angles under which the myosin heads and the actin binding sites will come tog
31. on the view button The following packages are needed for a successful compilation and later use of the program e emacs e gcc core e gcc g e gcc g77 e gnuplot e perl e openssh e cvs e make e lapack If additional packages of the cygwin system should be installed or updated it usually suffices to start the setup program again and select or deselect the required software cf also the cygwin website for further information 2 1 2 Compiling the deal Il FEM framwork Before the deal II framework can be installed the source code has to be downloaded from the homepage of the deal IT project www dealii org On the website a link to the download page can be found The recommended 2 1 Installation Windows XP Cygwin 9 version is 6 1 0 without documentation and examples It can be saved at any place in the filesystem Once the cygwin environment has been successfully installed the cygwin environment will give the following output when started for the first time Copying skeleton files These files are for the user to personalise their cygwin experience These will never be overwritten bashrc gt home okayserh bashrc bash_profile gt home okayserh bash_profile inputrc gt home okayserh inputrc Subsequent starts won t give this output but instead show a user prompt like the following okayserh HP20439203671 7 pwd home okayserh Usually the comm
32. ons A value of 0 2 corresponds to 20 percent detached heads This parameter can be used to adjust the ini tial slope This difference is not a numerical artifact but turned out to be the result of a slight difference in the model assumptions The integral method tries to imitate the behaviour observed in the MC code with a continuous approach to replicate the MC results with less computational effort A detailed report about the underlying ideas is in preparation The basis of this algorithm is still similar to the method of characteristics numeric_char_int prm Listing of Parameters subsection characteristics_integral set dx set x end set x start 0 1 20 0 20 0 set r 0 1876 end Finite Element Method The finite element method FEM is an other possible numerical scheme to solve the hyperbolic equations that describe the biochemical processes in 3 9 Output Files 27 Parameter Type Description global refine p order xend xstart Integer Integer Floating Point Floating Point Number of global refinement steps Ini tiallv the finite element mesh is ini tialised with a single element that goes from xstart to xend In each refine ment step everv element is split into two equal halfs Thus one refinement step leads to two elements two lead to four and so on Polvnomial order of the ansatzfunc tions used in the element One denotes linear shape functi
33. ons two quadratic etc Inital end point of the computational domain End of the FE domain Has to be larger than xstart muscle fibers It is implemented in the muscle simulation code but should be considered largelv experimental Furthermore a proper stabilization ap proach for the convective terms is currently missing in the implementation numeric_fem prm Listing of Parameters subsection fem set global_refine 6 set p_order 3 set xend 44 set xstart 44 end 3 5 Output Files During a program several output files will be created Some of them serve mostly the purpose of verfication while others contain the valueable results 3 5 Output Files 28 of the simulation runs All files contain the data in ASCII format Although these files consume more space then binary formats they have several ad vantages e Easier to process with scripting languages like Python e No problems with big little endianess e Can be edited if necessary e Enables preliminary views of data while simulation is still running In the following paragraphs the structure of the different output files will be shortly explained 3 5 1 rates dat When the program is started the rate functions which are the basis for the Monte Carlo state transitions are computed for a certain number of discrete strains in the range from x 15 to x 15 with a spacing of Ar 0 1 Each line of the file contains all rate functions for a single str
34. ous model an additional processing step is required In this step the relevant domain of the crossbridge strains is discretised into small segments Then the attached crossbridges are sorted into one of the small segments and counted for that segment When all crossbridges have been processed this gives a discrete approximation of the continuous crossbridge distribution field In this file the data is organised in datasets which are written whenever the specified output time interval parameter distribution in parameter file simulation prm is completed The dataset starts with a line where the simulation time of the following dataset is found Then several lines with the discrete approximation of the distribution field follow These lines consist of the strain x in the first column and the crossbridge distribution for the states 1 to n in the second to n 1th column A typical section in that part looks like Piet 4 50001 1 4 70 150 0 4 0 0 0 0 4 6 127 194 4 3 5000 0 14 25 108 278 10 3 0 0 0 0 48 70 133 277 10 2 5 0003 117 207 99 223 13 2 0 0 0 6 195 378 66 109 22 Ese for a 9 state model After that the accumulated raw data can be found in the dataset Here the single myosin heads on the thick filament are evaluated For each head a statistic over the available sarcomers is created 3 5 Output Files 32 0 0 01 0 02 0 02 0 21 0 35 0 38 0 01 0 0 1 0 12 0 06 0 35 0 01 0 04 0 05 0 13 0 21 0 03 2 0 02 0 04 0 14 0 5 0 02 0
35. p exe from the website www cygwin com has to be downloaded and started It automatically connects to the internet if such a connection is available and download other packages that are required for the installation Fig 1 shows the dialog that appears after starting the setup program It enables the user to change certain parameters like the default installation directory etc The default parameters are usually perfectly fine and shouldn t be modified The next dialog Fig 2 allows to specify a directory where the packages that will be downloaded are to be stored Usually the default suggestion will work Since the setup program tries to automatically download the packages from the internet a working internet connection has to be specified the corre sponding dialog is shown in Fig 3 If anything other than the default setting direct connection needs to be used consult the documentation of the cygwin setup program The next step is to specify a download site from which the packages should be downloaded cf Fig 4 Any choice should be fine here but might lead to differing installation speeds 2 1 Installation Windows XP Cygwin Cygwin Setup Select Local Package Directory Figure 2 Initial windows of cygwin setup program Cygwin Setup Select Connection Type Figure 3 Initial windows of cygwin setup program 2 1 Installation Windows XP Cygwin Cygwin Setup Choose Download Site s Choose A Downloa
36. s of the two actin fibers Although the program supports different elastic moduli for the two actin fibers which are used in the simulation but in most scenarios it does not seem sensible to use this freedom Vertebrae geom_vertebrae prm Listing of Parameters H AE EA subsection actin set a actini 50 4 set a actin2 50 4 set actin period 13 set dx actin 2 77 set e actini 1 27e3 set e actin2 1 27e3 set length actin 1050 0 end subsection axial set d actin mvosin 30 0 set radius actin 3 5 4 2 Vertebrae Lattice 36 set radius_myosin 7 0 end subsection myosin set a_mysoin 50 0 set dx_myosin 14 3 set e_mysoin 3 52e3 set myosin_heads 49 set myosin_offset 125 0 end subsection numerics set xamax 15 0 end In the section axial the axial configuration of the actin and myosin fibers can be defined The parameter d actin mvosin defines the distance between the center of the myosin fiber m Ym and the center of the actin fiber a Ya cf Fig 12 As the name suggests radius actin and radius myosin defines the radii of the actin and myosin fiber respectively These radii corre spond to the radii of the circles draw around a Ya and m Ym in Fig 12 respectively Hence they do not have to correspond to the real radii of in the muscle fibers but should rather be considered tools to define a proper dependence of the binding rates on the axial angles In the following a few comme
37. screte structure of the MC code makes it impossible to directly output distributions Therefore the discrete heads have to be sorted into bins according to a certain scheme Depending on the other parameters of the simulation and the desired result different sizes of bins may be more appropriate in one or the other situation Method of Characteristics Simulation Code Another well established numerical scheme for solving the hyperbolic equa tions that were developed to describe the biochemical and mechanical re sponses of muscle fibers is the method of characteristic This numerical scheme does not show the problems that appear in schemes like the finite dif ference or the finite element method when large convective terms are present But this advantage comes with disadvantages too The convective terms move the discretisation points in the domain Therefore the points will wander out of the domain of interest after some time By discarding solution points at the end of the domain and shifting the 3 4 Numerics 24 Parameter Type Description activation One of None Selects whether an activation model Fixed Geeves6S should be used and if what type of ac Lumped2S FChain tivation model filaments Integer gt 0 Defines the number of filaments in par allel geometry One of Simple Defines what geometry model should Vertebrae be used for the muscle The details can full sarcomer Parameter Flight Davi
38. tening is spread over a short time interval during which the displacement follows a linear function 000 1 40000 O 100 40 0000 0 w NNO OOOO The crucial part is in the second and third line of the protocol file where the function type is switched to a linear function with negative slope of 40000 After 1 10000s the function is switched back to a constant function which realises the final value of the displacement The resulting displacement curve is shown in Fig 6 Sine Cosine Function To have a continuous first time derivative of the displacement at the point where the protocol switches from constant displacement to sinosoidal dis placement we change the sine function to a cosine by adding an offset of 1 27 At time t 0 6s the frequency is reduced to half of what it was before 0 00000 0 2024 10 4 0 5 0 60245 40 5 1 000000 Results of this protocol are shown in Fig 7 Multiple Steps Having multiple steps is quite simple It just requires changing the constant displacement at certain timesteps as shown in the following protocol file 3 2 Protocol File Examples Displacement nm Displacement nm Figure 7 Changing frequency 0 t T result txt u 1 4 0 5 F 4 L 1 F a 15 4 2e 4 25 l 3 L 4 3 5 F 4 1 4 1 L l 0 199 0 1995 0 2 0 2005 0 201 Time s Figure 6 Gradual step f T T A N J
39. the next line the displacement field of the first fiber follows Each of the followings displacement fields for the other fibers is separated by an empty line 3 5 7 filaments dat For some simulations it is desireable to no only have the condensed informa tion of the full ensemble of sarcomers but also to have the mechanical data 4 Muscle Lattices 33 for each single sarcomer As this data can again be quite large it should usually only be written in selected time intervals The format of this output file is quite simple Currently each line has 3 columns which contain time force and displacement of a single sarcomer The number of sarcomers which were used for the simulation run determines the number of lines when the next data set for the next simulation time follows A typical file looks like EERI 1 000000e 04 0 000000e 00 0 000000e 00 1 000000e 04 2 527154e 01 0 000000e 00 1 000000e 04 0 000000e 00 0 000000e 00 1 000000e 04 0 000000e 00 0 000000e 00 1 010000e 02 1 816975e 02 0 000000e 00 1 010000e 02 1 817469e 02 0 000000e 00 1 010000e 02 1 927373e 02 0 000000e 00 1 010000e 02 1 881731e 02 0 000000e 00 ERN 4 Muscle Lattices On the molecular level the muscle fibers exhibit a complicated microstruc ture which has a significant influence on the mechanical properties of muscle This muscle structure strongly influences the possibilities of the myosin head to bind to the actin fiber To evaluate the effect
40. this algorithm Another problem is the introduction of numerical oscillations when discon tinuous rate functions are used Most recently proposed schemes of the ac tomyosin cycle use rather smooth rate functions Thus this problem is best seen with Huxleys two state model Since the results of interest are usually some integrals over the state distribution the practical implications of these oscillations are usually rather benign numeric_char prm Listing of Parameters H 2 Sue ss ee 28 subsection characteristics set dx 0 1 set x_end 50 0 set x_start 50 0 end The initial start and end points are moved in each time step when convective terms are present Thus the coordinates of these points change over the simulation As already mentioned in the previous section the algorithm shifts the points back into the original domain Integral Method Careful analysis of the results obtained with the method of characteristics and those from the MC simulation revealed a subtle difference in the resulting 3 4 Numerics 26 Parameter Type Description dx Floating Point gt Spacing of discretisation points 0 x start Floating Point Initial start point of the computational domain x_end Floating Point Inital end point of the computational domain r Floating Point The new approach needs an additional head populations parameter that describes the percent age of detached heads under isometric conditi
41. ubsection numerics 5 Activation Models 42 set xamax 15 0 end 5 Activation Models So far the biochemical mechanisms of muscle activation haven t been under stood completely Due to the accurate model of the molecular structure of muscle fibers the Monte Carlo simulation code provides an excellent testbed for examining different hypotheses of muscle activation One common property of the activation models is the protocol file which prescribes a time course of the parameter that regulates the activation While this is usually the calcium concentration other possibilities may appear in some of the simpler models The filename of the protocol file for the activation is ca protocol dat and the general structure is equivalent to that of the protocol file for mechan ics cf subsection 3 1 In contrast to the protocol file for the mechanical loading the calcium protocol file does not have different boundary condi tions Hence the functions always prescribe the time course of the calcium concentration or an alternative activation parameter 5 1 Fixed Number of Blocked Sites This model of activation blocks or opens a prescribed percentage of the actin binding sites It does not use any parameter file The values in the activation protocol file should take values between zero and one They prescribe the ratio of blocked actin sites to open actin sites A random process is used to assign the state open or closed to the actin sites E
42. ult txt In the result file the macroscopic mechanical properties of the complete sys tem are summarised Thus it is probably the most important piece of in formation which is generated by a simulation run The length of the lines varies again with the used model as it also contains some statistics about the crossbridges Tab 7 explains the different columns in the output file Note If either the multiple sarcomere mode or an activation model is used for a simulation run additional columns will appear in the result file Note The standard deviation is computed bya fast algorithm which might be plagued by round off errors For a precise standard deviation create the full distribution output to write the raw data and use a postprocessing script to accurately determine the standard deviation 3 5 Output Files 30 Parameter Description Time Simulation time Force The force per filament which is gener ated by the model Due to the Monte Carlo simulation its smoothness de pends strongly on the number of simu lated sarcomers Displacement first myosin node The displacements on this node display either the prescribed displacements or the displacements without the effects of the extensible filaments Displacement last actin node Basically the same as the previous value This time the extensibility of the filaments can be seen Std Dev Force Standard deviation of the force per fil ament
43. ven when myosin heads are still bound to an actin site the actin site can be switched to closed state 5 2 6 State Rigid Segment Model The 6 state model with rigid segments is based on a biochemical model of the actin sites that was proposed by D Smith and M Geeves in 4 5 3 2 State Rigid Segment Model 43 Listing of Parameters subsection six_state set e0 set ep set kit set k2t set k_21 set k_25 set k_32 set k_36 set k_65 set lam end 0 0 0 001 1 24e6 0 16e6 100 0 10000 0 100 0 100000 0 10 0 10 0 subsection structure set segment_length set strict_ end not e N closing 5 3 2 State Rigid Segment Model 5 4 Flexible Chain Model The most advanced activation model is the flexible chain model It incorpo rates the flexibility of the tropomyosin chain which covers the actin binding sites One of the drawbacks of this model is the tight coupling between the biochemical reactions of the myosin head and the actin binding site This coupling requires specifically designed models of the actomyosin cycle The parameters are defined in the file mg old prm Listing of Parameters subsection chain set actin_distance 5 5e 9 set phi_minus 0 3 set phi_plus 0 2 set sd_fac 0 6 6 Huxley s 1957 Model 44 set segment_length set xi end subsection rates set kb_eff 25 0 set km 2 4e6 set kps 4820 set kt_eff 50000 0 end 6 Huxley s 1957
44. y generate force through the crossbridge elas ticity A more realistic model introduces an additional attached state where the myosin head attaches in a neutral position and the force is generated by a swinging arm motion of the myosin head One of these 3 state model was proposed by Duke et al in 2 It uses the following rate functions kwolr cue Fer 11 2k8T os for Kastrj SEP a x l kz exp Ga x Gz z kgT else 9 karla kappexpl Ki r x0 kgT 13 The remaining reverse reaction rates k21 and k32 are determined by Gibbs relation kij g exp Gi z G5 hoP 15 ji and the following energy levels G 0 16 Go Ctr Kx 2 17 G K x d 2 18 The crossbridge stiffness K is equal to K_xb in the global parameter file simulation prm Finally the relation between the parameter file and the mathematical expressions above is given in Tab 8 It is selected by using the parameter Duke in the file simulation prm The model parameters can then be specified in model_duke prm which typically looks like Listing of Parameters 8 David Smith s 9 State Model 47 kgT KBT d D Cy GBind k_BT Ca C1 GStroke k_BT delta y gammal k1_ k_12 Kvina k_bind kapp k adp k32 k_21_cap Table 8 Relation between mathematical expressions and entries in parameter file A AAA subsection duke set D 10 5 set GBind 6 4 set GStroke 15 0 set delta 1 0 set
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