Outline Introduction Petri Nets Suitable models Bibliography Biological Pathways Representation by Petri Nets and extensions Andrea Marin December 6, 2006 Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Suitable models Bibliography 1 Introduction The cell Pathways 2 Petri Nets Definitions 3 Suitable models Introduction Representing Metabolic Pathways Representing Message Passing Pathways Representing Gene Regulation Processes 4 Bibliography Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography The cell (1) The cell is the structural and functional unit of all living organisms. It is the building block of life. Two types of cells: Prokaryotic cells: small, lack of nuclear membrane, usually singletons; Eukaryotic cells: 10 time bigger than Prokaryotic ( 1000 in volume), contain membrane-bound compartments in which specific metabolic activities take place, usually found in multi-cellular organisms; Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography The cell (2) Eukaryotic Cell Prokaryotic Cell Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography Cell Activities All cells share the following activities: Reproduction by cell division Use and production of enzymes and other proteins coded by DNA genes Response to external and internal stimuli such as changes in temperature, pH or nutrient levels Metabolism: take in raw materials, building cell components, converting energy, molecules. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography Biological Pathways Biological Pathways A Biological Pathway is a molecular interaction network in biological processes. We consider three types of biological pathways: Metabolic pathways Message Passing Regulatory pathways Gene Regulatory Pathways Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography Metabolic Pathways Metabolic Pathways A metabolic pathway is a series of chemical reactions occurring within a cell, catalyzed by enzymes, resulting in either the formation of a metabolic product to be used or stored by the cell, or the initiation of another metabolic pathway. The functioning of a cell depends upon its ability to extract and use chemical energy stored in organic molecules. This energy is derived from metabolic pathways to ATP. Example of metabolic pathways are: glycolysis which generates high-energy molecules, ATP and NADH Pentose phosphate pathway ... Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction The cell Petri Nets Pathways Suitable models Bibliography Regulatory Pathways Regulatory Pathways Regulatory Pathways can be classified in: Genetic information processing: Transcription, Translation, Sorting and Degradation. Replication and Repair Environmental information processing: Membrane transport, Signal transduction, Ligand receptor interaction Cellular processes: Cell motility, Cell growth and death, Cell communication, Development, Behaviour Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography Standard Petri Net (definition) A Petri Net is a 5-tuple = PN {P , T , A , W , m 0 } where: P is the set of places. E.g. P = { P 1 , P 2 , P 3 , P 4 , P 5 } T is the set of transitions. E.g. T = { T 1 , T 2 } A ⊆ T × P ∪ P × T is the set of arcs. E.g. A = { ( P 1 , T 1 ) , ( P 2 , T 1 ) , ( P 2 , T 2 ) , ( P 3 , T 2 ) } ∪ { ( T 1 , P 4 ) , ( T 2 , P 5 ) } W : A → N is a function which assigns to each arc a weight m 0 is the initial marking vector. E.g. m 0 = (2 , 2 , 1 , 0 , 0). Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography Enabled Transitions, Input and Output Vector Definition A transition T i is enabled by the marking m when ∀ ( P j , T i ) ∈ T , W ( P j , T i ) ≤ m j . It is disabled otherwise. In the previous example T 1 is enabled and T 2 is disabled. Definition The Input vector I ( T i ) is a vector whose k -th component is W ( P k , T i ) if ( P k , T i ) ∈ A , 0 otherwise. The Output vector O ( T i ) is a vector whose k -th component is W ( T i , P k ) if ( T i , P k ) ∈ A , 0 otherwise. In the previous example I ( T 1 ) = (2 , 1 , 0 , 0 , 0) and O ( T 1 ) = (0 , 0 , 0 , 1 , 0). Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography Incidence Matrix Definition The Incidence Matrix of a PN is a matrix with a row for each transition and a column for each place. The k -th matrix row is the vector O ( T k ) − I ( T k ) and it represents the marking change when T k fires. The incidence matrix of the previous example is: � − 2 � − 1 0 1 0 A = 0 − 2 − 2 0 1 Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography Dynamic Definition When a transition is enabled it can fire. The firing of the transition T i changes the marking of the net from m to m − I ( T i ) + O ( T i ). Definition The reachability set RS ( m 0 ) of a Petri Net is the set of all possbile markings reachable from the initial marking m 0 . Note that: In the general case, the number of marking of the RS grows exponentially with the number of places of the net and the number of tokens of the initial marking, The problem of deciding if a marking is reachable is NP-hard. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography S-invariant Definition A S-Invariant for a PN with A as incidence matrix is a vector such that: A · S = 0 . The existence of a no-zero components S-invariant tells us that the weight sum of the tokens in the net is costant. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Petri Nets Definitions Suitable models Bibliography T-invariant Definition A T-Invariant for a PN with A as incidence matrix is a vector such that: A T · T = 0 . Suppose that a PN has 4 transitions T = { T 1 , . . . , T 4 } and let T = (2 , 0 , 1 , 0) be a T-invariant. Then we know that, starting from any marking m , if T 1 fires twice and T 3 fires once (in any order), and no other transition fires, the final marking is again m . Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Introduction Representing Metabolic Pathways Petri Nets Representing Message Passing Pathways Suitable models Representing Gene Regulation Processes Bibliography What should a suitable model allow? 1 readability = ⇒ supports understanding both for computer scientists and biologists 2 executability = ⇒ allows even no-experts to get familiar with the model 3 validations techniques = ⇒ consistency checks, does the model respect natural laws? 4 analysis techniques = ⇒ qualitative and/or quantitative analysis. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Introduction Representing Metabolic Pathways Petri Nets Representing Message Passing Pathways Suitable models Representing Gene Regulation Processes Bibliography A possible path for models The animation step gives the idea on the possible simplifications, e.g. ignoring reactions which seem not important The qualitative model is a simplified model which allows one to compare pathways, or point out cycles or steady states The quantitative model is an extended model which consider compound concentrations and reaction kinetic. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Introduction Representing Metabolic Pathways Petri Nets Representing Message Passing Pathways Suitable models Representing Gene Regulation Processes Bibliography Catalyzed chemical reaction A chemical reaction takes its substrate and give a product. For example the glycosulfatase reaction has as substrate D-glucose 6-sulfate + H20 and for product D-glucose and sulfate: D-glucose 6-sulfate + H2O = D-glucose + sulfate Most of the reactions in cell are catalyzed by enzymes. Note that: enzymes are not consumed by the reaction enzymes speed up the reaction In the glycolysis the glycosulfatase is catalyzed by the EC 3.1.6.3 enzyme. Andrea Marin Biological Pathways Representation by Petri Nets and extension
Outline Introduction Introduction Representing Metabolic Pathways Petri Nets Representing Message Passing Pathways Suitable models Representing Gene Regulation Processes Bibliography Petri Net for a simple chemical reaction The basic idea is to associate to each substrate or product compound a place, to the enzyme another place and set the arcs to destroy the substrate and return the product: EASY! Andrea Marin Biological Pathways Representation by Petri Nets and extension
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