Morpheus User Reference Manual
Contents
1. Morpheus User manual Walter de Back and J rn Starru Center for Information Services and High Performance Computing Technische Universit t Dresden 01062 Dresden Germany http imc zih tu dresden de wiki morpheus DISCLAIMER NON COMMERCIAL USE Morpheus the Software is distributed for academic use and cannot be used for commercial gain without explicitly written agreement by the Developers No WARRANTY The Software is provided as is without warranty ofany kind either express or implied including without limitation any implied warranties of condition uninterrupted use merchantability fitness for a particular purpose or non infringement NO LIABILITY The Developers do not accept any liability for any direct indirect incidential special exemplary or consequential damages arising in any way out of the use of the Software This manual describes Morpheus version 1 1 0 revision 866 Development of Morpheus was supported by the German Ministry for Education and Research BMBF through grants 0315734 and 0316169 Department for Innovative Methods of Computing Center for Information Services and High Performance Computing Technische Universit t Dresden 01062 Dresden Germany http imc zih tu dresden de imc Contents 1 Introduction 1 1 About Morpheus 1 1 1 1 12 1 2 3 1 2 About this manual 1 2 1 1 2 2 2 1 1 2 1 2 2 13 2 1 4 2 1 9 2 1 6 21 7 2 1 8 2 2 Simulation and results 2 2 22 22. 223 2 2 4 2 23 2 2 0 2 232 2 3 9 Software Usability Developers Margin images More information Graphical user interface 2 1 Model construction and editing Examples Documents panel Element editor Attribute editor Expression editor Documentation Clipboard 2 24244 Fixboard Simulation execution Status and error messages Job queue and archive Simulation output Restoring and checkpointing SBML import 2 3 Parameter sweeps Multiple parameters Simulation and results Restoring parameter sweeps Contents WwW 0 0 0 NN RAE el CO CO 0 Y Y Y Y Y dd dd Odd OO OO 91 Rh Aa e e pa O QQ HH O O Contents 3 Model formalisms 15 3 1 The System construct 0 0 00 a ee 15 3 2 Differential equations 2 2222 oo on 15 3 2 1 Finite difference solvers 2 2 2 2 2 eee a 16 3 2 2 Stochastic differential equations 16 3 2 3 Delay differential equations 0 17 224 Initialbcenchiich lt s s i i i sds eara waaar sgal 17 3 3 Reaction diffusion models ooa a a a nn a 17 3 3 1 Diffusion u een bb Bebe ne dr ro 18 3 3 2 Boundary conditions aooo a a m m nn nn 18 333 Initial condition 2 2 scenes aaa ee ww es 18 3 4 Cellular Potts model aa aa aa a a 19 3 4 1 Modified Metropolis kinetics 19 J42 O soe os hee eRe ewe ewe we ee eS 20 3 4 3 Neighborhood m nn m nn 20 3 4 4
3. Functions Inlined in RateRule AssignmentRule Equation RateRule DiffEgn AlgebraicRule not supported Reaction Converted to RateRule Event Event Delays are not supported 2 2 5 Restoring and checkpointing Models in the job archive can be restored by opening the file model xml This opens the initial state of the simulation model as a new document When using checkpointing Time SaveInterval the complete simulation state is saved as a compressed XML file Title Time xml gz at the specified interval These files can be restored in the same way however in this case a new document is opened containing the complete simulation state at time Time 2 2 6 SBML import Morpheus supports importing SBML models for intracellular dynamics SBML Systems Biology Markup Language is a standard for describing models of bio chemical pathways SBML models can be generated using SBML simulators such as Copasi or downloaded from public repositories such as the BioModels database Upon importing an SBML model the SBML model is converted into systems of ordinary differential equations for intracellular dynamics in Morpheus model description language MDL Some SBML constructs can be translated in a one to one fashion into Morpheus MDL while others require a more elaborate conversion see table 2 2 In general SBML models are imported in the following way SBML constructs in red and Morpheus MDL constructs in blue 1 A new non spatial s
4. ordinary stochas tic delay and partial differential equations to solve initial value problems Cauchy 15 Model formalisms Table 3 1 Numerical solvers for ordinary and stochastic differential equations Method Numerical scheme Euler Unti Yn T Alfin Un Heun midpoint Yn 1 Yn sAtf tn Yn f tn 1 Yn 1 where Yn 1 Yn Atf tn Yn Runge Kutta RK4 Yn 1 Yn At k1 2k2 2k3 ka where k Fin Yn ko f tn 5At yn 5Atk1 k3 f tn 3At yn 5Atko al ka f tn At yn Atk3 Euler Maruyama Yn 1 Yn Atf tn yn VAtAW Heun Maruyama Yn 1 Yn tsAtf tn yn 5 AtAWnt f tr41 In where Yn 1 Un ar ISG bee Yn AF gt AtAW problems of the form a f t y t together with the initial condition y to yo Differential equations DiffEqn must be specified inside a System element that provides an environment for synchronous updating of tightly coupled updated dif ferential equations 3 2 1 Finite difference solvers Morpheus implements finite difference methods Euler Heun and Runge Kutta see table 3 1 The method to be used for a set of differential equations is specified in System solver All solvers have a fixed time step that must be specified by the user in SyStem time step Note that forward solver methods are not suitable for solving stiff systems and may require small time steps to guarantee stability and sufficient accuracy All solvers use fixed time
5. Monte Carlo Step MCS and time scaling 20 3 4 5 Cell shape interaction and motility 21 3 4 6 Initial condition of spatial configuration 22 3 5 Auxiliary model formalisms 00 00508 23 4 Model description language 25 4 1 Declarative ee Hee eho em 25 4 2 Domain specific oa aoa a a a a 25 4 3 Encapsulation ss m ese igi mea A aa a 26 4 4 Two tiered architecture a a a a a a 26 441 XML csi aaa a a 26 AAD Symbole gt s ss aA wohn au ee ows 27 4 5 Mathematical constructs 2 2 Coon 28 4 6 XML schema na ka Kahn euer 28 5 Model integration 33 DL Spatial mapping lt lt lt redresa ie ne EEE ee se 33 5 1 1 Automated mapping 424 rain nd 34 5 1 2 Explicit mapping with Reporters 34 5 2 SCEHedUlhE zu 2 8 2 6 En eee a ERE eS Ew dd oe 34 5 2 1 Simulation schedule 2 222 Lo m nn nn 34 5 2 2 Scheduling update order 222 2 mm m nn 36 523 Update interval s o sxi 4er Ee BH 36 ag Introduction 1 1 About Morpheus Morpheus is a modeling and simulation environment for the study of multiscale multicellular systems It supports the simulation of discrete cell based models as well as ordinary differential equations and reaction diffusion systems and facilitates the integration of these models into multiscale models It allows users to construct and simulate models of gene regulation signaling pathways tissue patterning and morphogenesis and exp
6. The lattice size Lattice Size and the physical interpretation of the size of a lattice site Lattice NodeLength are shared by both CPM and PDE models which simplifies their integration There fore in some cases the mapping is trivial enough to be handled automatically In other cases a more complex mapping is required such as calculating a sum or average which requires the explicit instructions by the user using a Reporter 33 34 Model integration 5 1 1 Automated mapping Mappings that present an unambiguous relationship are resolved automatically For instance when non spatial constants and variables are used within a spatial context the same value is used for all lattice sites over which is iterated The use of symbols referring to cell boundary Properties within reaction diffusion PDE models linking PDE to CPM models can also be handled automatically In that case for each lattice site for which the PDE Layer is defined the value of the cell property of the cell at that site is used The reverse case when PDE variables are used in CPM models can be automat ically mapped only in certain cases For instance in the Chemotaxis extension cell motility is biased by the gradient of concentration of a species in a reaction diffusion model Here the difference in concentrations of the PDE variable at the target and trial site is automatically resolved see 3 4 5 3 5 1 2 Explicit mapping w
7. and mathematical expressions Therefore syntactically correct models may still result in simulation errors despite validation by XML schema 4 6 XML schema Table 4 1 User defined and pre defined symbols in Morpheus model description lan guage Context Space Time Various Celltype CPM PDE cell vectors Description Simulation symbols Size of lattice Current location Spatial discretization Initial simulation time Termination time of simulation Interval between checkpointing Current time Model symbols Constant with global scope Constant with local scope Constant vector with local scope Mathematical expression Cell bound variable Cell bound variable vector Cell bound variable with delay Time of single Monte Carlo step Reaction diffusion species Predefined symbols Element Type Lattice Size vector SpaceSymbol vector NodeLength double StartTime double StopTime double Savelnterval double TimeSymbol double Global double Constant double ConstantVector vector Function double Property double PropertyVector vector DelayProperty double MCSDuration double Layer double cell id integer cell type integer cell volume integer cell surface integer cell center vector cell length double cell orientation vector symbol x y Zz double symbol abs double symbol phi theta double Unique cell index Cell type index Number of lattice sites cell occupies Number of lattice sites of cell bou
8. discrete method that repre sents individual cell shapes as lattice domains and models cell motility in terms of energy minimization using modified Metropolis kinetics Table 3 2 provides an overview of how cellular Potts model parameters are encoded in Morpheus MDL 3 4 1 Modified Metropolis kinetics A CPM defines cell shape and motility constraints in terms of energetical con straints described in a Hamiltonian Motility arises from updating the lattice configuration according to energy minimization based on a modified Metropolis kinetics with the following steps First a lattice site target site is chosen at random with uniform distribution see 3 4 2 Second a lattice site trial site is chosen from the neighborhood Nx with uniform distribution see 3 4 3 Then the change in free energy AH is calculated for the case if the state at the trial site x would be copied to the target site a Finally whether or not this transition is accepted depends on AH according to a Boltzmann probability 1 ifAH Y lt 0 P oy gt 0 AQ T otherwise Here T for temperature modulates the probability of unfavorable updates to be accepted and represents local protrusions retractions of the cell membrane The parameter Y for yield accounts for dissipative effects and represents for example cytoskeletal resistance to membrane fluctuations Model formalisms Figure 3 1 Neighborhood Order Neighborhood of a
9. lattice site with dot numbered according to the minimum neighborhood order in which they are included for square left and hexagonal right lattices The parameters for the modified Metropolis kinetics are specified in CPM MetropolisKinetics 3 4 2 Stepper In the standard random stepper algorithm the target site x is selected at random while the trial site x is selected from its local neighborhood In certain cases this can be highly inefficient In sparsely populated lattices for instance there is a high likelihood of selecting sites with identical states that cannot change the configuration Selecting such sites is therefore redundant To prevent such meaningless updates Morpheus provides the edgelist step per This sampling algorithm tracks all lattice sites that can potentially lead to a change in configuration and selects the target site x from this list with uniform ran dom distribution This can yield major improvements in computational efficiency without affecting model results The stepper algorithm can be selected in CPM MetropolisKinetics stepper default edgelist 3 4 3 Neighborhood The size of the Neighborhood N can be specified using either Distance or Order The Distance specifies the maximum Euclidean distance within which lattice sites are considered neighboring sites The Order uses a labeling scheme to identify the neighboring sites These labels are integer values that alternate between axial odd nu
10. modeling approaches and illustrate all elements of the Morpheus model description language These example models can serve as templates for the construction of new models Detailed descriptions can be found on the website fig 5 1 2 1 2 Documents panel The Documents panel gives an overview of the opened documents and provides a means to switch the current model The main elements of each model are shown and can be edited via the context menu right click or ctrl click The context menu also allows the user to view the generated XML document 2 1 3 Element editor The Editor panel shows the editable tree like structure of elements of the currently selected element in the Documents view Elements can be added copied or re moved using the context menu Adding an element opens a window showing a list of elements that can be added in the selected parent element Required elements cannot be cut or removed as to ensure model validity An element can also be temporarily disabled which renders it ineffective without removing it from the model Disabled elements can afterwards be re enabled 2 1 4 Attribute editor The Attribute panel shows a table of attributes of the element that is currently selected in the Editor panel The first column shows attribute names and for op tional attributes a checkbox The second column allows users to specify parameter values depending on the type of attribute double integer vector string etc En
11. parameter The targets and scalars are written here as constants for simplicity but can be cell specific and time dependent i e Ve o t by linking them to cell bound variables or functions Hamiltonian terms for CPM models that constrain cell shape VolumeConstraint ShapeConstraint LengthConstraint can be specified in Cell Types Cell Type 3 4 5 2 Interaction energies The CPM has been originally developed to study the effects of differential adhesion on cell sorting Adhesion can be modeled using interaction energies that define a free energy penalty per interface of contact between cells Differential cell adhesion can be modeled by specifying different interaction energies for contacts between different cells o of different cell types 7 This extends the Hamiltonian with the term H gt interfaces i j J TG T 77 L do 0 where J is a matrix of inter action energies between different cell types r o and g is the cell at lattice site i The Kronecker delta 11 0 0 0 0 oj ensures only interfaces between cells are taken into account The matrix of interaction energies between cell types can be specified in CPM Interaction These energies are normalized by the number of neighbors such that the interaction energies are automatically rescaled when the lattice structure is changed Interaction energies can be altered based on the state of a cell property that represents e g cadherin expression AddonAdhesion
12. sharing a particular integer value will be added to the same cell The option keep_id ensures that this value is used as a cell ID internally i H O 6 dd 3 5 Auxiliary model formalisms 3 5 Auxiliary model formalisms The modular design of the formalisms allows a number of auxiliary formalisms to be constructed For instance Coupled ODE Lattice Coupled ordinary differential equation lattice models can be used to represent a regularly structured tissue in which each cell is repre sented by an intracellular ODE model and communicates with its neighboring cells Coupled ODE lattice models can be constructed by configuring a lattice of cells each occupying a single lattice site using InitRectangle A Cell Type with a System of DiffEqn describing the intracellular dynamics Intercellu lar communication is modeled using a NeighborReporter that reports the weighted average of the properties of directly adjacent neighbor cells Cellular Automata CA Cellular automata are a widely used discrete time discrete space discrete variable formalism to study the emergence of macroscopic behavior from microscopic local rules CA models can be constructed by configuring a lattice of cells each occupy ing a single lattice site using I
13. 14221 S4S812ms CellPopulations Name Type Date Modified y 1 2 ES Job 14220 311820 Analysis amp Job 14219 55s 827ms_ ParamSweep gnuplot_error0 log 0 bytes log File 10 25 13 1 51 PM amp Job 14218 54s 074ms CellcycleDelay xml logger_CDK1_Plk1_APC_p_ave 98 bytes log File 10 25 13 1 51 PM amp Job 14217 6 Description gt model xml 5 KB xml File 10 25 13 1 51 PM ES Job 14212 os 560ms Space model xml out 13KB out File 10 25 13 1 51 PM amp Job 14025 0s 403ms Time gnuplot_error1 log 119 bytes log File 10 25 13 1 51 PM amp Job 14024 25 026ms CellTypes cells_0 000 png 1 KB png File 10 25 13 1 51PM i amp Job 14023 3 032ms CellPopulations NP cells_0 050 png 3KB png File 10 25 13 1 52 PM amp Job 14019 BEE Analysis cells_0 100 png 3KB png File 10 25 13 1 52 PM amp Job 14018 em 7 ParamSweep E cells_0 150 png 3KB png File 1C 2 2 4 amp Job 14017 F cells_0 200 png 3KB png File 10 u amp Job 14016 ee ram 1 B cells 0 KB _pna File amp Job 14015 7 953ms amp Job 14014 em J7ms v Output Text amp Job 14013 15974ms m N gt ES Example CellCycl Time 0 93 n gt ES Example Dictyost Time 0 94 gt ES Example Excitabl GnuPlotter Saving cells_0 950 png gt 5 Example Gameof 2 2 3 FixBoard vw T me 0 95 gt ES Example Lateralsi mn A Time 0 96 v Text of node Title was set tc Time 0 97 gt E Example Pancreat Example CellCycle ame 0 90 ro nn en ae Time 0 99 St
14. Afterwards Reporters and Equations are run once to initialize all remaining symbols Finally the analysis and visualization tools Analysis are run and the simulation state is saved to file Simulation cycle The simulation cycle is iterated from StartTime to StopTime in steps that advance according to the smallest time step section 5 2 3 In each iteration the processes that are scheduled for the current time step are executed in the following order 1 Temporal models The time dependent models are executed A single Monte Carlo Step is performed for the CPM Afterwards Systems and Diffusion are updated 35 Model integration 2 Sequential processes Non time dependent models such as Equations Reporters and Events are updated Their order is computed from their symbolic interdependencies section 5 2 2 3 Analysis Finally visualization and data output is updated and the simula tion state is saved to file Termination At the end of simulation visualization and analysis tools specified in Analysis tools can be executed once more and the simulation state is written to file 5 2 2 Scheduling update order The order of execution is independent of the order in which model components are specified in the model description f le Rather updates of temporal processes CPM System Diffusion are scheduled according to the fixed schedule shown in figure 5 1 and updates of sequential processes such as Equations and Reporte
15. For the reaction step one of the finite difference methods is used as described above section 3 2 3 3 1 Diffusion The diffusion equation is solved using the central difference method The diffu sion coefficient D is specified for each species in the reaction diffusion systems in Layer Diffusion default 0 0 The spatial discretization h is specified in Space Lattice NodeLength default 1 0 During initialization the numerical time step for the reaction step is adopted by the diffusion problem If necessary the time step is automatically adjusted in order to satisfy the Courant Friedrichs Lewy CFL condition 27D lt 1 where d 1 2 3 is the number of dimensions 3 3 2 Boundary conditions Boundary conditions can be either periodic default constant Dirichlet or noflux Neumann These are specified in Space Lattice BoundaryConditions for each boundary separately x X y y Z Z Note that the type of boundary condition is shared for all spatial model components although values may differ 3 3 2 1 Constant boundary value In case of constant boundaries for each species in the reaction diffusion system a value for each boundary should be specified in PDE Layer BoundaryValue de fault 0 0 3 3 2 2 Irregular domains To solve reaction diffusion systems in irregular domains as often required in image based systems biology a shaped domain may be loaded from a TIFF image in Space Lattice Domain Image This TIFF m
16. The URL of the website is http imc zih tu dresden de wiki morpheus Graphical user interface The graphical user interface GUI provides tools to construct and edit models execute simulations browse results and perform parameter sweeps 2 1 Model construction and editing The GUI provides a series of editing tools that facilitate the construction and modification of computational models and performs validation of the generated XML models see figure 2 1 2 1 1 La Open bel Save Morpheus CellCycle xml Y Y Y Documents o x CellCycle xml Description Space Time CellTypes j CPM CellPopulations Analysis ParamSweep gt CellCycleDelay xml Description Space Time CellTypes CellPopulations Analysis ParamSweep 2 1 2 FixBoard JO Text of node Title was set td Example CellCycle v CellType Property Property Property System v DiffEqn Expression __ DiffEqn DiffEqn Constant Constant Constant Constant Constant Constant Constant Constant Property Property Property Property Property VolumeConstraint SurfaceConstraint Event Proliferation Condition DivisionPlane v Triggers gt Equation gt Equation gt Equation CellType Functions Starting Job 14219 Starting Job 14220 3 JobQueue o x Name Symbol Expression Q Da cells Process v Progress al Bl CDK1 APC n Kon CDK1 B gt ES Example Autoc
17. and restore jobs in the job archive and perform parameter sweeps see figure 2 2 2 2 1 Simulation execution Simulations are executed using the Start button in the simulation toolbar Jobs can be executed in interactive local and remote mode Interactive mode is the most verbose mode and useful for model testing Graph ical output generated by Gnuplot is displayed on screen overriding the set tings in analysis plugins And error messages are displayed in a pop up win dow The Stop button in the toolbar terminates the most recently started running interactive simulation Local mode is the standard non verbose execution mode for simulation on a lo cal machine Graphical output is stored in files Error messages are displayed below the job queue The Stop button in the toolbar is disabled Remote mode is th execution mode for large scale simulation or batch processing Jobs are executed on a remote high performance computing resource via ssh using the remote batch system currently only LSF is supported See File Settings Remote Graphical user interface Me 2 2 1 Morpheus CellCycle xml y Y E File Examples About lloren bad sve Bez ur Job 14222 Example CellCycle nme Zi Cellcycle xml Process v Progress gt Description stop ES Output Folder BN Terminal 3 ES Example Autocrin Space v E Example CellCycle Time CellTypes lt Output Folder amp Job 1422 555 362m8 CPM amp Job
18. arting Job 14218 A GnuPlotter Saving cells_1 000 png Starting Job 14219 l Time 1 00 l Starting Job 14220 finished simulation Starting Job 14221 2 2 2 y Coa cpuTime spend on simulation 61 54 sec Starting Job 14222 EE C E Model loaded successfully Figure 2 2 Results view This view is opened at simulation start or upon selection a job in the job archive JobQueue Numbers correspond to sections in the text 2 2 2 Status and error messages Status and error messages of simulations executed in local mode are non intrusively displayed in the message box below the JobQueue panel In interactive mode these error messages are displayed in a pop up window 2 2 3 Job queue and archive The JobQueue panel provides an overview of pending running and completed simulation jobs and shows the progress and execution time Jobs can be stopped removed and debugged requires GNU debugger gdb using the context menu Jobs can be grouped and sorted according to model job ID state or sweep The result browser panel displays the content of the output folder of the cur rently selected job The toolbar allows jobs to be stopped when running and to open the output folder in a system dependent file browser or a command line terminal 2 2 4 Simulation output 2 2 4 1 Visualization Simulation data can be visualized in various ways The method of visualization can be configured in the Analysis element Gnuplotter is the most versatile too
19. chema Table 4 3 Mathematical constructs Overview of the mathematical constructs available in model description language e symbol definition o symbol reference Element Constant Global Property PropertyVector DelayProperty Layer Function Equation Rule DiffEgn Reporter NeighborsReporter PDEReporter System Event Description Containers Constant value of type double with local scope i e valid within the CellType or System it is defined in Variable value of type double with global scope Cell bound variable Property and DelayProperty are of type double DelayProperty has attribute delay to set the lag between assignment and return of value PropertyVector defines Euclidean vector in space delimited format x y z PDE model variable i e species in reaction diffusion system Diffusivity of a Layer is specified in attribute diffusion Expressions Mathematical expression Computes a value double for the output symbol it defines but does not assign it to a variable Updated whenever when output symbol is referenced May not contain algebraic loop Mathematical expression Computes a value double and assigns it to the variable it references Updates are scheduled depending on its symbol dependencies May not contain algebraic loop Mathematical expression that defines a recurrence equation for use in environments such as System and Event Scheduled according to System time s
20. cting models performing simulation experiments and understanding how sim ulation results are achieved 1 2 1 Margin images Images in the page margins refer to example models that are associated with the text and contain hyperlinks to the example website The example models illustrate the main modeling features of Morpheus introduce the model description language and can be used as templates for new models These example models are embedded in the application see 2 1 1 and are described on the website see figure 5 1 1 2 2 More information 1 2 2 1 Documentation in GUI Further documentation on individual elements of a Morpheus model is available in the graphical user interface in the Documentation panel see section 2 1 6 and figure 2 1 Introduction 1 2 2 2 Website The Morpheus website provides additional up to date information It presents a growing number of example models that represent common use cases The website also answers frequently asked questions mentions how to submit bug reports presents a comparison to related software and a detailed performance analysis Published articles on Morpheus and an up to date list of research papers in which the software has been used are listed on the publication page When pub lishing results that were obtained with the help of Morpheus please cite the article on Morpheus as indicated on the publication page This helps to maintain Mor pheus and support its further development
21. diversity of cellular behaviors Chemotaxis Proliferation Sets the time scale of a Monte Carlo step MCSDuration the parameters for the cellular Potts model MetropolisKinetics and the parameters of interactions between cells Interaction Optionally for constant boundary condition sets a cell type at a boundary BoundaryValue Sets the symbol and diffusion coeffients for species Layer in a reaction diffusion model for use in mathematical expressions Equation May set a system of differential equations System DiffEqn for reactions Allows multiple populations to be defined Each population sets a cell type and size Population May set initializers e g Initrectangle May explicitly specify multiple cells with properties and positions Cell When saving the simulation state state of each cell is specified here Sets the visualization and analysis tools May contain various loggers and plotters Gnuplotter Logger Executed at user specified intervals A summary of the main elements of the Required Sub elements Title Details Lattice SpaceSymbol StartTime StopTime TimeSymbol Savelnterval RandomSeed Property System Constant Function Equation Event Reporter Chemotaxis Proliferation Interaction MetropolisKin MCSDuration BoundaryValue Layer Constant System Function Equation Cell Initrectangle TIFFReader Gnuplotter TIFFPlotter Logger HistogramLogger 4 6 XML s
22. ed in a straightforward fashion The main elements of the XML structure as shown in table 4 2 are used to describe both the structure and parameterization of the model and the details of its numerical simulation The spatio temporal aspects of the simulation are specified in the required Space and Time elements the initial conditions or simulation state are described in CellPopulations data output and visualization is configured in Analysis The title and annotation of a model are added in the Description element The model itself is configured using the CellTypes CPM and PDE elements The properties behavior and dynamics of cells including intracellular dynamics are specified in the CellTypes element The parameters of the cellular Potts model are configured in the CPM element and the reaction diffusion models are defined and configured in the PDE element These elements are all optional such that the various model formalisms can be used in isolation as well as in combination The XML represents this information in a hierarchical tree like structure that reflects the structure of the modeled biological system For instance the main ele ment CellPopulations can contain a Population that contains multiple Cells 4 4 Two tiered architecture Cells CPM System ODE PDE Analysis Figure 4 1 Schematic representation of the two tier architecture The hierarchical XML tree provides information on the structure of the model and its compo
23. ee chapter 4 For the user a link between sub models is established by simply defining a symbol in one sub model and using it as an input in another sub model providing a convenient way to construct and explore complex multiscale biological systems using integrative models During simulation Morpheus makes the data accessible between sub models and if necessary mapping or transforming it to make it suit able for the target sub model Moreover the updates of the various sub models are appropriately scheduled by determining the correct order and the frequency of up dates as to guarantee that up to date data is used in all computations Both tasks are handled automatically as far as possible based on user specified time steps and symbolic interdependencies 5 1 Spatial mapping Integration of spatial model formalisms i e cell based and reaction diffusion mod els requires that the data from one sub model is accessible to the other sub model In Morpheus data is not copied between sub models but is directly accessible through symbolic identifiers Yet the data must be accessed in a way that is appropriate for the model that uses the symbol The model that uses a symbol determines the lattice sites for which the symbol is resolved whereas the model that defined it determines the value of the symbol at those lattice sites Morpheus uses the convention that the spatial discretization in cell based and reaction diffusion models is identical
24. erbolic cosine Hyperbolic tangens Arc hyperbolic sine Arc hyperbolic cosine Arc hyperbolic tangens Logarithm base 2 Logarithm base 10 Natural logarithm Exponent Power Square root Sign 1 if x lt 0 1 if x gt 0 Round nearest integer Absolute Minimum of arguments Maximum of arguments Sum of arguments Average of arguments Modulus remainder Uniform distribution Normal distribution Gamma distribution Boolean 0 or 1 Conditional statement Syntax and or xor l lt or amp lt gt or amp gt gt or amp gt sin cos tan asin acos atan Sinh cosh tanh asinh acosh atanh log2 log In exp pow base sqrt sign rint abs Miras dar sr e o ee u exponent MU ee SUM a aag ay Ka NH ee mod numer denom rand_uni min max rand_norm mean stdev rand_gamma shape scale rand_bool if condition then else Model integration Morpheus supports the integration of time discrete cell based models with time continuous models for intra and extracellular dynamics In particular cellular Potts models CPM can be linked to ordinary stochastic or delay differential equations as well as to reaction diffusion PDE models The Morpheus model description language facilitates the specification of links between sub models with the help of symbolic identifiers s
25. erval of a System is determined by its time step divided by time scaling see 3 1 Similarly the intervals at which Analysis processes are executed to do visualization and data analysis also depend on a user specified interval 5 2 Scheduling The update intervals of other processes such as Equations Reporters Events and Diffusion are automatically determined by propagation of the intervals of their input and output symbols according to the following rules e The process is updated as often as its output symbol s are used e The process is not updated more often than its input symbol s can change The former ensures that up to date data is used in all processes while the latter improves computational performance by preventing redundant computations Note that processes are not scheduled and computed if their output symbol s is are not used The interval of Diffusion is likewise determined by the interval of its output symbol but it may iterate multiple times with smaller intervals as to satisfy the CFL stability criterion see 3 3 1 Functions are not explicitly scheduled and do not have an update interval Instead they are updated whenever their output symbol is used but at most once per minimal time step User specified intervals for Analysis processes smaller than the minimal time step are ignored since the state of the simulation cannot change within this interval The final simulation schedule including the computed order o
26. eters The number of jobs is calculated for the current configuration and displayed above the parameters To have multiple parameters set simultaneously changing sets a parameter can be dragged on another to couple them in a pairwise fashion Note that these parameter sets should be of the same length otherwise the lowest number of parameter values is used for the paired set 11 12 Graphical user interface Eg wo Morpheus CellCycle xml File Examples About a ES Process v Progress gt ES Example Autocrin v Example CellCycle 4 Job 14221 E Job 14220 HE Job 14219 amp Job 14218 licye JS 2 val C 1 253 amp Job 14217 Description 1 System Constantisymbol a3 value Double 2 3 E Job 14212 Space 4 Job 14025 Time 4 Job 14024 CellTypes amp Job 14023 CellPopulations amp Job 14019 Analysis amp Job 14018 ParamSweep z z MB Job 14017 ROCA amp Job 14016 amp Job 14015 amp Job 14014 amp Job 14013 gt Example CellCycl gt ES Example Dictyost gt ES Example Excitabl 5 Example GameOF gt Example LateralSi Text of node Title was set tc gt Amo Ani e Example CellCycle lt gt an S Pr Figure 2 3 Parameter sweep view This view is opened when selecting the ParamSweep element in the Documents panel Table 2 3 Syntax for list expansion in parameter sweep view Syntax Examples Resu
27. f execution and update intervals is calculated at the end of initialization and printed to the standard output This is written to the output file model xml out and is displayed in the output text box of results view in the GUI see 2 2 37 Table 5 1 Example models are available in the application and described on the website Ordinary differential equations u N PR PR PN a Y y La Y A y y VW pl M VIV AN fi Y JENS VIRINAIA Reaction diffusion systems ae as Cellular Potts models Multi scale models Miscellaneous models lt MorpheusModel gt minimal gt lt MorpheusModel gt
28. her as raw data or in processed form The Logger plugin writes cell property values or PDE values optionally reduced after summing or averaging The SpaceTimeLogger provides a way to write values of a linear PDE layer or a slice of a 2D 3D PDE layer to construct space time plots The HistogramLogger constructs and outputs frequency distributions of cell properties in a population The result files are standard tab delimited text files that can be imported and processed by external statistical software Additionally these logging tools include a Plot function to visualize the data The frequency of data output is set by the interval attribute of the particular Analysis element 2 2 4 3 Output folder Simulation data is written to a folder Title _ JoblD within the simulation folder configured in Settings Local Before execution the model XML file i e model xml and dependent files e g TIFF images are written copied to this folder During execution simulation output is written to the same folder The contents of the output folder can be browsed by selecting the job in the JobQueue panel Table 2 1 gives an overview of the files in the output folder Graphical user interface Table 2 2 SBML import Translation of SBML concepts red into Morpheus model description language blue SBML Morpheus MDL Comment Species Property Units are discarded Parameter Property Constant Depends on constant attribute InitialAssignment InitProperty
29. hrough the installation process Gnuplot for windows should be installed separately for which a recent version can be downloaded from the gnuplot sourceforge website Introduction 1 1 1 2 External software Morpheus is implemented in C and Qt and depends on a number of exter nal libraries and software packages The simulator morpheus is implemented in C It uses XMLParser to read model description files and write simulation states checkpointing Pseudo random numbers are generated using Mersenne Twister 19937 from the C 11 standard library Mathematical expressions are parsed and evaluated using the math parser library muparser TIFF image stacks are read and written using libIIFF For data visualization it depends on the graphing utility Gnuplot and its C interface Morpheus uses openMP for shared memory parallel processing The graphical user interface morpheus gui is implemented in the cross platform framework Qt C It uses a SQLite database to store an archive of jobs and parameter sweeps It depends on libSSH for secure communication to re mote high performance computing resources for remote job submission ssh and synchronization of simulation results sftp Furthermore it depends on libSBML to import biochemical network models written in SBML format 1 1 2 Usability Improving usability and transparency of computational tools in multicellular sys tems biology is a prime objective in the design of Morpheus This wide
30. ingle cell Morpheus model is created This model specifies a single lattice site and a population of 1 cell A CellType is created to contain the system of ordinary differential equations 2 Matching concepts such as Species Events InitialAssignments etc are translated from SBML into the Morpheus MDL concepts of Property Constants Events InitProperty etc 3 Reactions inthe SBML model are together with the associated KineticLaws and Functions converted into SBML RateRules These RateRules are 2 3 Parameter sweeps subsequently assembled into a System of differential equations DiffEqn such that a single differential equation describes the temporal evolution of a Species or Parameter 4 Unlike in SBML the symbolic identifier of a Property or a Constant must be unique in Morpheus MDL Therefore if necessary Reaction parameters are renamed upon import by appending the sequential reaction number 5 Simulation details are not specified in SBML models but are required for simulation Therefore default values are set during the import process Sim ulation time is assumed to run from 0 1 by setting Time StartTime 0 and Time StopTime 1 The System solver is set to Runge Kutta RK4 by setting solver runge kutta and its time step 0 01 6 Finally simulation output is preconfigured with a graphical visualization of the time course data of all species using Analysis Logger plot Note that not all SBML concepts ca
31. ion 3 2 4 1 Homogeneous population The value specified in Celltype Property value serves as an initial condition for the whole population of this Celltype Therefore this method defines a cell population that is homogeneous with respect to this variable 3 2 4 2 Heterogeneous population A heterogeneous cell population in which initial conditions differ per cell may be configured using a mathematical expression in Population InitProperty This enables setting initial conditions according to cell ID cell id cell position cell center x or using random distributions e g rand_uni min max 3 2 4 3 Cell specific initial conditions Initial conditions may be specified for each cell separately in Population Cell PropertyData and likewise for vector and delay properties When using check pointing the state of each cell bound variable is stored here 3 3 Reaction diffusion models Solvers are available for initial boundary value problems on a one two or three dimensional domain of length L of the form 24 p24 f t y t with initial Ot Ox 17 18 Model formalisms conditions y tp 1 yo x and a set of boundary conditions such as y t 0 y t L 0 Reaction diffusion systems are solved using the sequential operator splitting method in which the original problem is split into two subproblems the reaction and diffusion steps that are solved sequentially both for the same time step
32. ith Reporters When the mapping between the two contexts is not unambiguous a statistic is required to define how a symbol should be mapped This requires the user to specify an appropriate way to transform the data from one sub model to the other sub model Defining such an explicit mapping is done using a Reporter The NeighborsReporter calculates the weighted average or sum of the cell bound Property in adjacent cells given by the input symbol and assigns this statistic to its output symbol The PDEReporter maps PDE variables for use in cell based models by reporting the average sum maximum or gradient of concentration of the input PDE Layer at the lattice sites occupied by the cell or optionally its discretized center of mass 5 2 Scheduling Integration of time continuous ODE and PDE models with time discrete CPM models and various auxiliary mathematical constructs requires a careful schedul ing of numerical updates While the general simulation schedule is executed in fixed order some processes must be scheduled according to the symbolic inter dependencies in order to guarantee correctness of simulation results The simula tion schedule is computed during intializatoin and printed to the standard output model xml out 5 2 1 Simulation schedule The general simulation cycle follows a fixed order in which temporal models are executed before the sequential processes and output processes as shown in figure 5 1 The scheduling
33. l for visualization of 2D simulations It is based on the graphing utility Gnuplot This 2 2 Simulation and results Table 2 1 Output files Description of common output and result files File Description model xml XML model generated by GUI upon model execution model xml out Standard output from simulator as in Output panel model xml err Error message generated by the simulator Title Time xml gz Compressed XML model with simulation state created if and only if checkpointing enabled Time SaveInterval logger_ Symbols log ASCII text output from the Analysis Logger plugin cells_ Time png Plot generated by the Analysis Gnuplotter plugin jpg gif pdf gnuplot_error log Gnuplot error messages if any analysis plugin can visualize both cells and PDE layers provides customizable color scales and can display output to screen wxt aqua x11 win or write to file in a number of image formats png jpg pdf svg For 3D simulations the TIFFPlotter provides writing image stacks that may contain 2D 5D data in the format X Y Z time channel that can be read by external image analysis software such as Fiji or ImageJ Alternatively 3D simulation data can be written to VTK format using VTKPlotter for post hoc visualization with ParaView The frequency of visualization is set by the interval attribute of the particular Analysis element 2 2 4 2 Logging and analysis Simulation data can be exported to log files eit
34. logical and mathematical terms provides a powerful way to describe the relations and dynamics of biological processes 4 3 Encapsulation Models in Morpheus MDL describe both the model itself and specify details of its simulation including initial conditions see table 4 2 During a simulation the full simulation state including the position and state of all cells can be written in the same description language In this way a single model file contains the full specification of the model simulation The encapsulation in a single file significantly simplifies archiving checkpointing and restoring simulation models as well as the exchange of models between users 4 4 Two tiered architecture Morpheus MDL has a two tier architecture see figure 4 1 On the one hand the XML is used to store information about the model components or sub models in a hierarchically structured way On the other hand symbolic interdependencies rep resent the interactions and feedbacks between model components or sub models This combination provides a convenient way to express models of complex biolog ical processes and allows automation of model integration see chapter 5 4 4 1 XML The model description language is based on the eXtensible Markup Language XML This has a number of advantages It stores information in a well structured fashion that can be easily parsed and validated and it allows human readable domain specific terminology and can be extend
35. lore the effects of multiscale feedbacks between these pro cesses 1 1 1 Software Morpheus is a self contained application that covers the workflow from model construction to simulation and data analysis Internally it consists of two stand alone components a graphical user interface morpheus gui and a simulator morpheus The graphical user interface provides tools for model editing job execution and archiving and it generates XML based model descriptions based on user specified models The simulator takes these model description files as input to perform numerical simulation and generate data output and visualization The simulator can also be used from the command line interface 1 1 1 1 Download and installation Morpheus is available as binary package for all major operating systems These can be downloaded from the download page of the Morpheus website For Linux Morpheus is available as a Debian package deb that can be installed using the default package manager For other Linux distributions these packages can be converted using tools such as alien For Mac OSX 10 6 or higher Morpheus is distributed as an Apple disk image dmg containing the application bundle app Installing Morpheus app is done by dragging the app into Applications as usual It also contains a Gnuplot app required for data visualization to be installed in the same way For MS Windows XP or higher Morpheus uses an installer that guides the user t
36. lso stored in the model description file such as the simulation time spatial discretization initial conditions and the con figuration of visualization and data output For checkpointing the complete state of a simulation during execution can be stored in the same file format The fact that the complete simulation model including the description of its dynamics is encapsulated in single files render them suitable for archiving as well as model exchange between users 4 1 Declarative The Morpheus model description language MDL separates modeling from imple mentation by allowing the description of models in a declarative fashion Models describe what processes are to be simulated rather than how this should be ac complished This distinguishes declarative languages such as the Morpheus MDL from imperative programming language such as C or Python that focus on the description of algorithmic control flow 4 2 Domain specific The MDL uses a domain specific markup language using vocabulary that is derived from the application domain of multiscale and multicellular systems biology On the one hand it uses concepts such as cell types and populations and biological processes such as proliferation and chemotaxis table 4 2 On the other hand it defines a range of mathematical constructs such as constants functions equations 25 26 Model description language and systems of differential equations table 4 3 This combination of bio
37. lt List X KX 101530 TOS 10 10 10 20 20 20 10 10 10 20 20 20 square hexagonal square hexagonal Range Increment XXIX 0 2 10 0 2 6 8 10 Oxo AE 0 5 1 5 2 5 Intervals XIX XK 0 2 10 0 5 10 0 4 1 0 OO 23 OO ONS IMO Logarithmic x xlog x 1 210g 100 1 10 100 2 3 Parameter sweeps 2 3 2 Simulation and results Parameter sweeps are executed from the ParamSweep panel using the Start button in the toolbar Note that the ParamSweep panel must be opened in order to start a parameter sweep Upon execution a folder Name of Sweep is created that contains a file sweep_summary txt and subfolders for each simulation job The file contains the folder names for the individual jobs and their parameter sets and can be used for post hoc analysis 2 3 3 Restoring parameter sweeps Settings of previous parameter sweeps can be restored from the JobQueue provided that the selected model defined the same parameters and symbols 13 Model formalisms Morpheus supports the simulation of discrete cellular Potts models as well as contin uous ordinary differential equations ODEs and reaction diffusion systems These core model formalisms are implemented in a modular way such that they can be flexibly integrated into multiscale models Moreover the modularity enables them to be combined into auxiliary formalisms such as finite state machines cellular automata coupled ODE lattices or gradient based models 3 1 The System construct The System i
38. mbers and radial even numbers neighborhoods as shown in figure 3 1 3 4 4 Monte Carlo Step MCS and time scaling Within the CPM a Monte Carlo step MCS is often interpreted as a discrete unit of time A single Monte Carlo step is defined as the number of random sampled updates equal to the number of lattice sites That is within one MCS on average each lattice site has been sampled for an update 3 4 Cellular Potts model The duration of a single MCS is scaled to the global simulation time as specified in CPM MCSDuration default 1 0 3 4 5 Cell shape interaction and motility 3 4 5 1 Hamiltonian Each cell occupies a set of lattice sites with its cell index o gt 0 whereas o 0 refers to the medium Changes in the spatial configuration of cells on the lattice are governed by a Hamiltonian H that describes the free energy of the lattice configuration In its simplest form ignoring intercellular interaction energies H Y y AV Ue V where vo is the actual volume i e number of lattice sites of cell o and V is the target volume Deviations from the target volume increase the free energy H according to the elasticity parameter Av Additionally a constraint on the perimeter of the cell is often used H Yeso Av vo Vi Ap po P where py is the actual perimeter i e number of interfaces between lattice sites of cell o and P is the target perimeter with Ap representing the elasticity
39. n be converted into Morpheus MDL In partic ular multiple compartments are not supported and conversion of delay equations is not possible since Morpheus only supports delay differential equations with con stant delays see section 3 2 3 Furthermore note that units and dimensions are discarded upon import After importing an SBML model into Morpheus it can be extended into a spatial multicellular or multiscale model Alternatively it can be copied as a module into an existing model using the Clipboard see section 2 1 7 2 3 Parameter sweeps The GUI provides a convenient interface for batch processing by generating multiple jobs with different parameter sets All parameters Attributes and Expressions can be selected for parameter sweep using the context menu item ParamSweep Upon selection the parameter appears in the ParamSweep view fig 2 3 This is opened when selecting the ParamSweep element in the Documents panel This shows the XML path of the parameter and its type The range of values for batch processing can be set be editing the values in the third column Values can be given explicitly as semicolon separated lists or using the list expansion syntax as shown in table 2 2 3 1 Multiple parameters By default parameter sets will be combined in a combinatorial fashion such that a job is created for each combination of parameter values Note that the number of simulation jobs can get prohibitively large for multiple param
40. ndary Center of mass of cell Cell length of major axis Orientation of major axis Cartesian vector coordinates Magnitude of vector Polar coordinates of vector 29 30 Table 4 2 Model description language Morpheus description language and their most important sub elements Model description language sub elements are printed in boldface Element Description Space Time CellTypes CellType CPM PDE CellPopulations Population Analysis Description Sets the name Title of the model used for naming the destination folder May include model annotation Details used for human readable annotation only Sets the size structure and boundary conditions of the lattice Lattice Optionally sets a symbols for the lattice size and current location Lattice Size symbol and SpaceSymbol Set the duration of a simulation StartTime and StopTime defining the global time Optionally sets a symbol for current time TimeSymbol May specify the interval to save the simulation state SaveInterval May set a random seed for stochastic simulations RandomSeed Allows multiple cell types to be defined Each cell type CellType sets a name and type i e biological or medium May define multiple properties Property for use in mathematical expressions Equation May contain reporters for spatial mapping Reporter May define systems of ordinary differential equations ODE System DiffEqn May specify a
41. nents colored boxes By defining and referring to symbolic identifiers model components can be linked together in a network arrows Systems rounded grey boxes provide an environment for tightly coupled differential equations in which self references and circular dependencies between symbols are allowed Similarly intracellular dynamics are modeled using a Systems of DiffEqn within a CellType while the PDE describing extracellular dynamics is defined in its own element outside of CellTypes The XML structure is convenient to represent the hierarchy between the compo nents of a model However it is not suitable to describe the network of interactions and feedbacks between these components which is done using symbolic identifiers 4 4 2 Symbols Model components can be linked using symbolic identifiers Symbolic identifiers and references establish interactions and feedbacks between sub models to repre sent the network like complexity in biological processes see fig 4 1 Symbolic identifiers or symbols can be specified to represent user defined model variables such as cell bound properties Property or concentrations of species in a reaction diffusion model Layer see table 4 1 Symbols can also 28 Model description language be specified for simulation related constants and variables such as lattice size and current time of simulation see table 4 1 Symbols can be used in mathematical expressions to define relations bet
42. nitRectangle A CellType with a System of synchronously updated Rules describes the state transitions based on the states of cells in the local neighborhood reported using a NeighborReporter Gradient based models Gradient based models can be used to describe pattern ing of tissues under influence of a morphogen gradient such as Wolpert s classical French flag model Gradient based models are built using a non diffusive PDE Layer initialized by a mathematical expression using an InitPDEExpression A regular lat tice of cells is configured by an InitRectangle in which each cell measures the morphogen concentration at its location using a PDEReporter based on which an Equation defines the cellular identity 23 Model description language Morpheus simulation models are specified in a custom domain specific model de scription language MDL The XML based language uses biological and math ematical terminology to declaratively describe multiscale multicellular simulation models It is composed of human readable tags to represent the components of biological processes table 4 2 and a number of mathematical constructs to define their dynamics and relations table 4 3 Morpheus simulation models are fully specified by single model description files These include the definition configuration and and parameterization of sub models as well as the specification how these sub models are interlinked Details on the numerical simulations are a
43. ns the audience from computational experts to more biologically trained students and researchers within this multi disciplinary field and it can accelerate research by facilitating rapid development of new hypotheses into computational models and simulation results Ultimately it is required to streamline the scientific workflow in the face of the growing complexity of computational models in systems biology The usability and workflow management in Morpheus is most tangible in the graphical user interface with its tools for model construction simulation batch processing and archiving However it also extends into the modular design of the model formalisms generating the flexibility to construct a range of auxiliary model formalisms Moreover the domain specific language that Morpheus uses for model description separates the process of modeling from its numerical implementation and allows users to construct complex models in a transparent fashion 1 1 2 1 Target audience Morpheus is aimed at researchers and students interested in mathematical modeling of multicellular systems Its use does not presuppose knowledge of computational modeling or programming It is suitable for anyone with basic knowledge of math ematical modeling of biological systems including researchers and students with a background in biology mathematics physics and related fields 1 1 2 2 Limitations Morpheus is not an advanced solver for ordinary differential equation
44. nsions act to prevent updates altogether based on some criterion These include the Freezer that disables all updates for a particular cell based on a user specified condition lt also includes the ConnectivityConstraint that ensures the cell is simply connected by preventing updates that would break this topological constraint 3 4 6 Initial condition of spatial configuration Simulations of cellular Potts models require the specification of at least one popula tion of cells in CellPopulations Population By default the specified number of cells is distributed randomly in space using a uniform distribution each cell occupying a single lattice site Spatially structured initial populations can be configured using InitRectangle and InitCircle These initializers attempt to fit the specified number of cells in a regular fashion with each cell occupying a single lattice site Note that artefacts may occur due to spatial discretization Initializing a population of cells with geometrical shapes can be configured using the Cell Objects initializer This allows for the specification of spheres cylinders boxes and ellipsoids that can be arranged along the orthogonal axes Cell populations can also be initialized from images using the TIFFReader initializer This provides an interface to configure models from microscopy images TIFF images may be in 8 16 and 32 bit format and may contain multiple z slices image stacks By convention all pixels
45. of sequential processes such as Reporters and Equations involves the determination of the order of updates as well as the setting of intervals in which the various model components should be computed 5 2 Scheduling Termination Initialization Simulation cycle Lattices Analysis Celltypes Save state CPM Systems Diffusion PDE Cell populations Equations Symbols Reporters Events Q A E p Y p U StopTime Reporters Equations Analysis Analysis Save state Save state Figure 5 1 Overview of the general simulation schedule Between initialization and termination the simulation cycle is iteratively executed until time StopTime A single simulation cycle proceeds in three phases First the temporal models such as CPM Systems and Diffusion are updated according to user defined time steps Second the sequential processes such as Equations and Reporters are updated in an order and interval that is determined by dependencies between symbols Third output is generated and the simulation state is written to file at user defined intervals Initialization First the lattice is created and initialized cell types are created the CPM is initial ized and PDE layers are created and initialized PDEInitExpression Afterwards populations of cells are created and put in the lattice randomly or according to a user specified initializer e g InitRectangle Then symbols are registered and set to initial values
46. or on the combination of properties between neighboring cells to represent binding between heterophilic HeterophilicAdhesion or homophilic adhesion molecules HomophilicAdhesion These plugins can be specified in CPM Interaction Contact 22 Model formalisms 3 4 5 3 Kinetic terms The classical CPM has been extended by non Hamiltonian terms Since these terms directly affect the change in energy AH and may change the energy of the system they are called kinetic terms A widely used kinetic term biases motility of a cell o in the direction of a local concentration gradient of a species w in a reaction diffusion model in order to represent chemotactic migration AH y w we We Kinetic terms that bias motility Chemotaxis Haptotaxis DirectedMotion Persistence can be specified in CellTypes CellType 3 4 5 4 Event based terms Other CPM extensions are based on the cell state at the end of a Monte Carlo step and are evaluated only once per Monte Carlo step instead of every update trial This includes extensions that model cell division Proliferation and cell death Apoptosis Proliferation triggers the division of a cell into two daughter cells based on a condition Apoptosis triggers the immediate removal of a cell lysis or setting its target volume to zero shrinking and removing the cell from the population only after it does not occupy any lattice sites 3 4 5 5 Update preventing terms Some CPM exte
47. rin Plk1 v E Example CellCycle men 2 Job14222 EE amp Job 14221 BTZms dCDK1 dt ES Job 14220 Tm amp Job 14219 555 827m5 a 2 1 5 amp Job 14218 545 074ms oda 7 aus amp Job 14217 K Job 14212 aaa er UGC amp Job 14025 0519 a2 Ae Fa amp Job 14024 APC Anaphase promotin E se a CDK1 Cyclin dependent ki ES Job 14018 B _ K Michaelis constant Job 7 0 pos Plk1 Polo like kinase 1 Job 1401 i vt Target volume amp Job 14016 at c division timeout amp Job 14015 am cc cellcount amp Job 14014 em gt cell amp Job 14013 v d divisions A gt ES Example CellCycl n Hill coefficient we gt a Example Dictyost Documentation gt E a pa Sc gt Example Gameof i Expression to be evaluated during run gt ES Example LateralSi A time gt E Examole Pancreab RG lt gt Operators i a nee nn Starting Job 14218 a a A z if condition then e 2 1 6 8 Starting Job 14221 x v sin cos tan asin acos i Starting Job 14222 tanh asinh acosh atanh loq2 log10 Tm S Model loaded successfully Figure 2 1 Editor view This view is opened when selecting an element in the Documents panel Numbers correspond to sections in the text Graphical user interface 2 1 1 Examples A number of example models is included in the application accessible through the Examples menu in main toolbar These examples demonstrate key features of Morpheus show diverse use cases and
48. rs are scheduled according to their symbolic interdependencies Sequential processes are ordered according to the dependencies in their input and output symbol based on the following rule e Before updating a process its input symbols must all be updated This is achieved by scheduling all processes that have these symbols as an output before Note that this is only possible if no algebraic loops or circular dependencies exist between these processes Therefore such loops are only allowed within the System environment The order of execution of data output and visualization in Analysis is arbitrary and does not affect the simulation itself Therefore these are executed in the order in which they are given in the model description file 5 2 3 Update interval The simulation cycle is iterated for the period from StartTime to StopTime called the global simulation time The number of iterations that are executed during this period depends on the temporal process with the smallest time step During each iteration only the processes that require updating are executed The intervals at which each process requires updating are determined during initialization based on user specified time steps or symbolic interdependencies The frequency in which CPM and Systems are updated is based on user specified information The update interval of CPM models is the duration of a single Monte Carlo step specified in CPM MCSDuration see 3 4 4 The update int
49. s a mathematical construct in Morpheus MDL that plays a central role in modeling of temporal dynamics including ordinary differential equation reaction diffusion systems as well as rule based models such as cellular automata A System is an environment for tightly coupled sets of differential equations Rules or differential equations in a System environment are updated synchronously such that the states of variables at time are calculated on the basis of the state of the variables at the previous time t At Algebraic loops within or between rules or differential equations are allowed only within a System Therefore re currence equations Rule or tightly coupled differential equations DiffEqn can be modeled within this environment A System is associated with a numerical solver and a time step as explained in section 3 2 1 The time step sets the integration time step used by the solver and should be set by the user on the basis of performance large time step and accuracy small time step considerations In addition the System has an optional attribute time scaling default 1 0 which scales the time within a System relative to the global time This automatically scales the System time step such that the numerical accuracy is preserved The time scaling attribute provides a convenient way to scale the dynamics of sub models to each other 3 2 Differential equations A number of numerical solvers are available for a subset of
50. s or reaction diffusion systems The use of explicit forward solver methods precludes the sim 1 2 About this manual ulation of stiff systems and the use of fixed time stepping methods requires the specification of reasonable time steps by the user to ensure numerical stability un der acceptable computational performance The current version does not support solving partial differential equations with spatial derivatives other than the most commonly used second order and thereby excludes the simulation of e g advective terms 1 1 3 Developers Morpheus has been developed by J rn Starru Walter de Back and others in the group of Prof Andreas Deutsch at the Center for Information Services and High Performance Computing ZIH at the Technische Universit t Dresden 1 2 About this manual This manual provides information on the main functionalities of the graphical user interface chapter 2 the core model formalisms it implements chapter 3 the domain specific language in which Morpheus models are expressed chapter 4 and the methods for model integration of the various formalisms chapter 5 The manual does not provide an extensive list of descriptions of features or model components For documentation on individual model elements users are refered to the Documentation panel in the graphical user interface see section 2 1 6 and figure 2 1 It intends to concisely describe Morpheus to a degree that is relevant for con stru
51. stepping adaptive time stepping solvers are not yet implemented 3 2 2 Stochastic differential equations For stochastic differential equations the Euler Maruyama or Heun Maruyama method is used see table 3 1 These methods scale the noise amplitude to the time step h that is used Morpheus automatically switches to Maruyama solvers 3 3 Reaction diffusion models when a DiffEqn contains a normally distributed random number i e rand_ norm mean stdev Note that other random number distributions e g uniform or gamma distribu tions are not allowed in this context Moreover stochastic differential equations cannot be solved using the Runge Kutta method 3 2 3 Delay differential equations Morpheus supports delay differential equations through the use of a property with a delay aDelayProperty This property has an attribute delay that specifies the length of history i e the lag between assignment of a value and the time at which it is returned Note that only constant time delays are supported Further note that the delay must be a multiple of the solver time step time scaling Also note that a delay smaller than this time step has no effect 3 2 4 Initial condition Initial conditions for cell bound variables i e Cel ltype Property can be specified in various ways depending on whether the initial condition should be specified in a homogeneous or heterogeneous way or to set an initial conditions in a cell specific fash
52. tep May contain algebraic loop and self references Mathematical expression that defines a differential equation Only allowed in System environment May contain algebraic loop and self references Reporters Explicit data mappings Computes a statistic average mean etc of the input data and assigns this to the output symbol Updates are scheduled depending on its symbol dependencies Environments Environment for tightly coupled sets of differential equations and rules that are synchronously updated see section 3 1 Scheduled according to user specified System time step and time scaling Environment for conditional or timed events Triggered periodically or if Condition is specified whenever the condition changes from false to true Updates are scheduled according to time step if specified or depending on its symbol dependencies otherwise Symbol graph Constant Global Property Layer Function Equation JE 31 32 Model description language Table 4 4 Operators and predefined functions available in mathematical expressions Class Operators Logical operators Comparison Functions Random number Condition Description Addition Subtraction Multiplication Division Power Logical and Logical or Exclusive or Equal Not equal Smaller Greater Smaller or equal Greater or equal Sine Cosine Tangens Arc sine Arc cosine Arc tangens Hyperbolic sine Hyp
53. tries are validated by regular expressions As long as an entry is invalid it is marked by a red background 2 1 5 Expression editor The Expression panel is a multiline editor to enter or edit the expression that is currently selected in the Editor panel The editor enables users to specify math ematical expressions as text in a familiar infix notation using common operators and functions listed in table 4 4 Expressions are formulated using predefined and user defined symbols see table 4 1 For each model the available symbols are shown in a list below the editor 2 2 Simulation and results 2 1 6 Documentation The Documentation panel provides a context sensitive documentation of the ele ment or attribute that is currently selected in the Editor or Attribute editor 2 1 7 Clipboard The Clipboard provides the ability to cut copy and paste model elements Because the clipboard is shared between documents elements can be copied and pasted between models 2 1 8 Fixboard When loading documents with outdated or broken yet well formed models Mor pheus attempts to repair the model by e g adding required elements or removing elements that are not allowed The Fixboard provides a list of changes that have been made Items in this list link to the element to which the changes have been applied 2 2 Simulation and results The GUI provides tools to execute multiple simulations called jobs view simu lation results browse
54. ust be in 8 bit format in which zero pixels are interpreted as background outside domain non zero pixels are set as foreground inside domain Irregular domains can have constant or noflux boundaries 3 3 3 Initial condition Initial conditions for each species in the reaction diffusion system can be set us ing mathematical expressions in PDE Layer Initial InitPDEExpression This enables setting up random initial conditions or spatial gradients By specifying symbols for space and lattice size using user defined symbols see table 4 1 het erogenous initial conditions can be specified independent of spatial discretization 3 4 Cellular Potts model 19 Table 3 2 Cellular Potts model parameters and their location in the model description language Symbol Metropolis kinetics T Y N Interaction energy J Shape constraints Vi Av P AP Non Hamiltonian u Description Temperature Yield Neighborhood Sampling stepper algorithm Interface energy Target volume Strength of volume constraint Target perimeter Strength of perimeter constraint Chemotactic sensitivity Morpheus MDL CPM MetropolisKinetics temperature yield Neighborhood stepper CPM Interaction Contact CellTypes CellType VolumeConstraint target VolumeConstraint strength SurfaceConstraint target SurfaceConstraint strength CellType CellType Chemotaxis strength 3 4 Cellular Potts model The cellular Potts model CPM is a cell based time
55. ween model components Additionally symbols provide a convenient way to integrate different sub models by defining symbolic identifiers in one sub model and using them in another sub model 4 5 Mathematical constructs The model description language provides a number of mathematical constructs to define constants and variables and to express functions equations and condi tional events as well as the specification of tightly coupled systems of differential equations An overview of the available constructs is given in table 4 3 Mathematical expressions may be specified in terms of user defined or prede fined symbols see table 4 1 Expressions are entered in plain text using standard infix notation An overview of the available operators functions and random num ber generators and their syntax is given in table 4 4 Expressions are parsed during initialization using the fast math parser muparser muparser beltoforion de Muparser converts expressions to reverse Polish notation represented in byte code that is used to evaluate the mathematical ex pressions at run time 4 6 XML schema The rules constraints and contents of the Morpheus model description language are laid down in an XML schema The XML schema description XSD is em bedded in the GUI and provides the information to edit validate and repair model descriptions Although the XML schema validates the XML structure the GUI does not check the correctness of symbolic linking
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