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SBML2Murphi: a Translator from a Biology Markup Language to Murphy Andrea Romei Ciclo di Seminari su Model Checking Dipartimento di Informatica Università di Pisa Pisa, April 2009

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2 Targets of this talk Provide a quick look to Murphi descriptive language Show how a biological system can be modelled as a Murhpi Finite State System in an (semi-)automatic way.

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Outline I. Introduction II. The Murphi Language III. The “sort” Example IV. The Systems Biology Markup Language V. SBML2Murphi Translator VI. An example of application VII. Final remarks 3

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4 Introduction Model checking is the process of checking whether a given structure is a model of a given logical formula. Murphi is a formal verifier based on explicit state enumeration. Originally developed at Stanford, now maintained at University of Utah. It works on various versions of Unix/Linux.

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5 Murphi in brief Murphi takes a Finite State System S and checks that a given invariant property for S is satisfied An execution is a finite sequence of states s 0,s 1,…: s 0 is one of the start states s i+1 is obtained by applying one transition rule whose condition is true in s i and whose action transforms s i in s i+1 The invariants are applied whenever a state is explored s i can satisfy several conditions, the verifier must cover all the possibilities

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6 The Murphi Program A Murphi program has the following structure constant, type, and variable declarations procedure and function declarations rules, start states, and invariants A Murphi program implicitly determines a state graph (i.e. an assignment of a value to each global variable)

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7 Constants, types and variables Murphi supports the specification of constants, variables and types The simple types are boolean, enumerations, finite sub-ranges of integers The compound types are arrays or records of compound or simple types Sub-ranges and enumerations are important to avoid memory consumption

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8 Constants, types and variables

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9 Functions and procedures Functions and procedures can have side effects Formal parameters declared “var” are passed by reference. Formals that are not declared “var” are passed by reference, but the function or procedure is not allowed to modify them.

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10 Rule A rule determines a transition from one state of the finite automaton to another It consists of a body and a condition A program must have at least one rule

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11 Start State A start state is a special type of rule It is only executed at the beginning of an execution A start state must assign a value to every global variable

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12 Invariant An invariant is a boolean expression that references the defined variables Its value doesn’t change during the program execution An invariant is a special rule

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13 Example: sorting by swapping Array of integers 0..N-1 and pointers for swapping Swaps the positions “i” and “j” of the array Increases the input pointer mod N

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14 Example: sorting by swapping Increments the “i” pointerIncrements the “j” pointer Swaps the elements pointed by “i” and “j” if they are not in the right order

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15 Example: sorting by swapping The program violates invariant when the data is sorted

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16 Example: sorting by swapping Murphi usage: Write sort.m Run the Murphi compiler over it by typing “./mu sort.m”; this yields a file “sort.C” Run the C++ compiler over “sort.C” together with the verifier code Run “sort.exe –ta”

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17 Example: sorting by swapping

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18 SBML (from wikipedia) The Systems Biology Markup Language (SBML) is a machine-readable language for representing models of biochemical reaction networks It is based on the XML technology SBML can represent metabolic networks, cell- signaling, pathways, regulatory networks and other kinds of systems studied in systems biology

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19 SBML purposes SBML has three main purposes enabling the use of multiple software tools without rewriting models for each tool enabling models to be shared and published ensuring the survival of models beyond the lifetime of the software used to create them SBML serves as a “lingua franca”

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20 SBML Hierarchy

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21 Species and Reactions A specie is a substance or entity that takes part in a reaction (e.g. the glucose molecule) A reaction is a statement describing some transformation that can change the amount of one or more species It describes how certain entities (reactants) are transformed into certain other entities (products) It has associated a kinetic rate expression describing how quickly it takes place

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22 SBML Example The reactions of human CO hemoglobin An equation used in enzyme characterizations formulated by A. Hill in 1910 to describe the sigmoidal binding curve of hemoglobin Five species: S 0, S 1, S 2, S 3, S 4 Four reactions:

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23 Why an SBML2Murphi translator? Reactions are driven by empirical laws Species in a model change their values along the time The target may be to verify some basic properties of a biological model E.g. verify the consistency of the model when the initial concentrations of the species varies in a computed range

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24 The Law of Mass Action It states that the rate of a chemical reaction is proportional to the probability that the reacting molecules will be found together in a small volume Produces a differential equation for each distinct specie belonging to a reaction The system of differential equations can be solved to determine how species change values against the time

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The Law of Mass Action: Example Reactants Products Kinetic Rate

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26 The law of Mass Action: Example

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Reactions in SBML 27

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The Role of the Translator 28 MurphiSBMLHow Variables & Constants SpeciesDirectly by parsing the SBML model State transitionsReactionsComputed by means of a procedure that implements the computation of the law of mass action Start states?? Invariants??

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29 The “Hill” model in Murphi

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30 The “Hill” model in Murphi

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31 The “Hill” model in Murphi The start states in the Hill model indicate what happens when the initial concentrations vary. I used about 500 assignments to S1, S2 and S3 by varying three parameters. I got about 500 starting states each of them corresponds to a different experiment (i.e. an initial concentration).

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32 The “Hill” model in Murphi The invariant is used to test the consistency of the model by varying the initial concentrations I assigned to “Hill.m” ranges in which the species must fall into during the experiment 0 <= S 1 < 16 0 <= S 2 < <= S 3 < 0.5

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Conclusion 33 MurphiSBMLHow Variables & Constants SpeciesDirectly by parsing the SBML model State transitionsReactionsComputed by means of a procedure that implements the computation of the law of mass action Start states-Empirical assignments Invariants-Ranges in which the species fall into

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34 References

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35 References

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