1. Matlab Toolbox Download
2. Symbolic Math Toolbox Download

Symbolic Math Toolbox™ provides functions for solving, plotting, and manipulating symbolic math equations. You can create, run, and share symbolic math code using the MATLAB ® Live Editor. The toolbox provides functions in common mathematical areas such as calculus, linear algebra, algebraic and ordinary differential equations, equation.

1 INTRODUCTION The expanding field of Systems Biology has stimulated the formalization of an increasing number of biological/biochemical models. The Systems Biology Markup Language (SBML), an XML-based format for computational models of biochemical networks, is becoming accepted as a de facto standard for the representation of such models and thus facilitates their systematic exchange. In addition to promoting the creation of models, Systems Biology has also motivated the development of a range of software packages that can interact with these models, perform simulations and analyses on them, produce graphical representations of models and facilitate the creation of new models. However, the commercially available software package MATLAB provides a wide spectrum of this type of functionality combined with the facility to easily develop user-specific functions.

Thus an alternative approach to that of developing new software exists in the form of developing a toolbox that provides users with an interface between basic MATLAB data structures and a format such as SBML. This not only enables users to leverage their existing skills in using the environment to work with a new format such as SBML, but it also provides a substrate enabling other analysis tools in the environment to be applied to data represented in SBML. MATLAB is a particularly attractive environment in this regard because it is already popular worldwide in both engineering and scientific research, and as the field of Systems Biology continues to attract researchers with an engineering or physical science background, the use of MATLAB within the field is likely to proliferate. Also, there currently exist many tools, both commercial and freely available (, ), that apply the computational and analytical capabilities of MATLAB to models and data in a variety of formats. SBMLToolbox was initially developed specifically to meet two separate needs: (1) those of existing MATLAB users wishing to import SBML models and apply functionality appropriate to their goals, whether built into the environment or purpose-written and (2) those of users experienced with SBML wishing to apply the computational power of MATLAB to their models.

Thus, in addition to importing SBML, the toolbox includes functionality serving as an example of using MATLAB in the manipulation and analysis of models. However, the actual analytical functionality is limited and while it is possible to simulate a range of models with the toolbox, it should not be considered a simulation tool but rather a facilitator for the development of other functions and toolboxes. To date we are aware of at least two freely available toolboxes that use SBMLToolbox for precisely this purpose; namely SBToolbox and SBMLSim (, ). 2 FEATURES The core functions of SBMLToolbox are for importing and exporting SBML models. The toolbox also contains a range of functions that allow a user to manipulate the content of the resulting MATLAB data structures (that is, the model being considered). A set of validation functions allow the structure to be checked for appropriate fields and the values assigned to these fields. 2.1 Import and export SBMLToolbox offers two import functions that take an SBML model and return a MATLAB structure that replicates the Unified Modelling Language (UML) definition of an SBML model.

Although the structure returned by both import functions is identical, two versions of this function are included within the toolbox: one in which the SBML file is verified against the XML Schema (, ) for SBML (see ) and a number of consistency checks are performed prior to import, and a second version in which the verification stage is skipped. In the toolbox, the returned structure is called a MATLABSBML structure. It is populated with values drawn from the model being imported. Each of the SBML ‘listOf’ elements is reproduced as an array of structures defining the particular element, whereas each element is itself another structure. Illustrates a portion of MATLABSBML corresponding to a reaction and its relationship to the definition of SBML's Reaction. MATLABSBML structure representing a ‘reaction’ and the UML definition of a ‘reaction’ within an SBML model. For export, SBMLToolbox provides a function for translating a MATLABSBML structure back to an XML file in SBML format.

As a convenience to users, the toolbox also provides additional functions allowing a MATLABSBML model structure to be written as a MATLAB data file known as a MAT file. This allows faster access to stored models. The use of a MATLAB MAT file also facilitates using MATLAB external library functions for the manipulation of these data files. SBMLToolbox includes functions for browsing models stored in MAT files. 2.2 Manipulation functions One set of functions act directly on the structures and mimic the functions provided by libSBML, the library developed and maintained by the SBML Team as part of the software infrastructure supporting SBML development (, ). Many developers of SBML compliant software are familiar with libSBML and thus this set of functions gives them an easy transition into the MATLAB environment.

The functions include those for creating objects, e.g. Modelcreate; functions for setting field values, e.g. CompartmentsetId; functions for returning field values, e.g. ReactiongetId; functions for querying whether a field value has been set, e.g.

SpeciesisSetId and functions for adding structures to arrays within the given structure, e.g. A second set of manipulation functions in SBMLToolbox drive a set of graphical user interfaces (GUI) that display the field names and corresponding values for each element. For example, the ViewSpecies function produces a GUI that shows the name/id of the species, the compartment containing it, the initial concentration and units and so on. The GUI does not provide editing functionality; it is intended principally as a simple tool for displaying model structures, and as a starting point for users' development of more elaborate capabilities.

In order to interact with the MATLABSBML model structure it is necessary to extract the required information. The use of the standard MATLAB structure allows easy access to field names and indexing into arrays of structures within structures, e.g. The name of the second species in ‘model’ can be accessed as model.species(2).name. However the interaction with a model may be more complex and SBMLToolbox provides a third set of designed functions to return more detailed information from the model; e.g. GetRateLawsFromReactions returns two arrays, one of the names of each species in the model and the other containing the character representation of the rate law used to determine each species in the same order as the species, thus allowing the two to be paired.

These functions are also replicated to return symbolic representations of the model information which facilitates the use of MATLAB's Symbolic Toolbox—an additional commercial toolbox from The Mathworks that incorporates symbolic computation into the MATLAB environment. Finally, the MATLABSBML structure stores any formula/MathML in the original model as a string representation of the formula. In order to support interaction with the model, SBMLToolbox contains a set of ‘convenience’ functions that facilitate the manipulation of the string representations of formula; e.g. PairBrackets that determines the order in which brackets within a formula should be considered and SubstituteFunction which allows arguments to be passed to a formula defined by a function definition.

2.3 Simulation SBMLToolbox is not intended to be a simulation tool; however, it does contain a set of example functions that allow the user to simulate an SBML model using MATLAB's built-in ODE solvers. The functionality provided includes writing out a script for use with the ODE solvers, running the solvers, and displaying the data, as well as a number of supporting functions. These functions deal with all aspects of an SBML model. The simulation output passes all the tests in the test-suite produced by the SBML Team (, ).

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The one exception is that the package does not currently handle the ‘delay’ feature in SBML. 3 IMPLEMENTATION SBMLToolbox works with all versions of MATLAB from version 6.5 onwards and depends on the Xerces-C version of libSBML. The import/export functions do require compilation; however, scripts are included within the toolbox to facilitate this. There are two downloads available: a Windows installer that pre-packages the necessary libraries and precompiled functions, and a ‘source’ download that can be used to install the toolbox on Linux, Unix or Mac OS platforms. The functions providing interaction between a model and the symbolic toolbox obviously require the presence of the MATLAB Symbolic Toolbox; however, these functions are self-contained and the full functionality of the toolbox can be achieved without using symbols. In addition, the source download contains a number of test files, thus facilitating further development of the toolbox functions. Full documentation is included with SBMLToolbox and each function has extensive MATLAB help.

SBMLToolbox is distributed under the GNU Lesser General Public License. This work was supported by the BBSRC e-Science Initiative UK and grant number R01 GM70923 from the NIH National Institute of General Medical Sciences USA. Conflict of Interest: none declared.

X,Y = meshgrid(-1.01:1); Z1 = -abs(X) - abs(Y); Z2 = -1 - sqrt(1 - X.^2 - Y.^2); Z2 = real(Z2); W1 = Z1; W2 = Z2; W1(Z1 Z2 W2(Z1 Z2 hand = figure;% handle to the figure, since we'll plot more later set(gcf,'Color','w')% white background surf(X,Y,W1,'LineStyle','none'); hold on surf(X,Y,W2,'LineStyle','none'); view(-44,18) There is a slight gap between the upper and lower surfaces of the figure. This is an artifact of the general plotting routine used to create the figure.

Matlab Toolbox Download

This routine erases any rectangular patch on one surface that touches the other surface. The syntax and structures of the two sets of toolbox functions differ. In particular, symbolic variables are real or complex scalars, but Optimization Toolbox™ functions pass vector arguments. So there are several steps to take to generate symbolically the objective function, constraints, and all their requisite derivatives, in a form suitable for the interior-point algorithm of fmincon. Energy = sym(0); for i = 1:3:25 for j = i+3:3:28 dist = x(i:i+2) - x(j:j+2); energy = energy + 1/sqrt(dist.' .dist); end end gradenergy = jacobian(energy,x).'

; hessenergy = jacobian(gradenergy,x); Create the Objective Function File The objective function should have two outputs, energy and gradenergy. Put both functions in one vector when calling matlabFunction to reduce the number of subexpressions that matlabFunction generates, and to return the gradient only when the calling function ( fmincon in this case) requests both outputs. This example shows placing the resulting files in your current folder. Of course, you can place them anywhere you like, as long as the folder is on the MATLAB path. Function c,ceq,gradc,gradceq = democonstr(in1)%DEMOCONSTR% C,CEQ,GRADC,GRADCEQ = DEMOCONSTR(IN1). X101 = in1(28,:). C = t417.^2 +.; if nargout 1 ceq = ; end if nargout 2 gradc = 2.x11.; end if nargout 3 gradceq = ; end This function has the correct form for a constraint function with a gradient; see.

Symbolic Math Toolbox Download

Generate the Hessian Files To generate the Hessian of the Lagrangian for the problem, first generate files for the energy Hessian and for the constraint Hessians. The Hessian of the objective function, hessenergy, is a very large symbolic expression, containing over 150,000 symbols, as evaluating size(char(hessenergy)) shows. So it takes a substantial amount of time to run matlabFunction(hessenergy). To generate a file hessenergy.m, run the following two lines. Xfinal2,fval2,exitflag2,output2.iterations,output2.funcCount xfinal2 = 0.0000 1.0000 -1.0000 0.6689 -0.6644 -1.3334 -0.6644 0.6689 -1.3334 0.0000 -1.0000 -1.0000 0.6357 0.6357 -1.4380 -0.0317 -0.0317 -1.9990 1.0000 0.0000 -1.0000 -1.0000 0.0000 -1.0000 0.0000 0.0000 -0.0000 -0.6667 -0.6667 -1.3334 fval2 = 34.1365 exitflag2 = 1 ans = 77 ans = 2434 In this run the number of function evaluations (in output2.funcCount) is 2434, compared to 28 (in output.funcCount) when using gradients and Hessian. Clear the Symbolic Variable Assumptions The symbolic variables in this example have the assumption, in the symbolic engine workspace, that they are real. To clear this assumption from the symbolic engine workspace, it is not sufficient to delete the variables.

You must clear the variables using the syntax.