TimeNet's representation of the malaria example and the results of kinetic experiments, corresponding to this Petri Net (PN). Places are represented as circles and transitions as squares. The number 10 written inside the place P65 represents a marking of ten tokens that correspond to ten malaria parasites (merozoites). In this marking of the PN, only the transition MerozoiteInBlood is enabled because there are tokens in all its input places (in the case of this transition, there is only one input place: P65). When this transition fires, a token is passed from the input place P65 to all the output places. In this case, there is only one output place: P6. Tokens that reach place P1 may flow either through the transition Invasion or the transition ImmuneResponse. In the graph shown below the PN, the y-axis shows the values of the performance measures (average number of tokens at certain places) from time zero to a given time on the x-axis. These kinetic curves use a smoothing function, which does not measure the number of tokens at a given time but rather the average number of tokens over a time interval. R5 is a hyperbolic curve showing sink places that accumulate all the tokens in the system. The curve corresponds to place P66. R2 and R3 are parabolic curves showing transient stages. Tokens gradually enter these stages, stay there for a while, and then transition to the next stage. The curves correspond to places P5 and P7, respectively. R1 and R4 are curves showing stages that start with many tokens and lose them over time. The curves correspond to places P1 and P2, respectively.

Woflan's verification of the malaria Petri Net (PN). Woflan can be used to convert work flows in different work flow tool formats into PNs. The diagnosis option of Woflan verifies properties of the resulting PNs. Petri Nets can also be directly entered into Woflan, without the need to start from workflows. The properties that Woflan verifies are transition liveness and place boundedness (no infinite accumulation of tokens at places of the PN). Together, these two properties are termed soundness.

A Mobius model of malaria parasites invading a human host. Mobius uses composition to compose several atomic Petri Nets (PNs) into a composed model. Top: Shows a model composed of Rep1 copies of a Red_Blood_Cell PN (bottom left), Rep2 copies of the host's immune system cells PN (middle bottom), and one copy of the host's bloodstream PN (bottom right). The three atomic nets have a shared place (Place1). In the atomic PNs, places are depicted by circles and timed transitions as vertical bars. Small circles attached to a vertical bar depict transition cases (i.e., mutually exclusive processes that are all enabled by the same set of input places). Triangle depicts output gate. The output gate shown in the figure produces an integer number of tokens (burst size) in the output place to which it is linked, which allows the representation of several malaria parasites that are released from the host cell at the end of asexual development.

A Design/CPN model describing tRNA transcription and incorporation into proteins. The transition labeled as “Translation” is hierarchical and is expanded into a lower level Petri Net (not shown). Places are depicted as ovals and transitions as squares. Enabled transitions have a bold contour. Circles with a number in them indicate place marking. To the right of the circles, squares with text strings indicate the kind of tokens that mark a place. Each place has a color set (tRNA, mRNA, at the top left of each place), defined by a set of allowed token types (e.g., Initiator_tRNA, Asn). The places “Gene containing tRNA” and “mRNA” (bottom right) are initially marked, as specified in the expression below the place. The arcs are marked with 1'a and 1'm representing tokens belonging to the color sets tRNA and mRNA, respectively. The expressions in square brackets above transitions represent guarding conditions that must be true for a transition to be enabled. Hierarchical transitions (e.g., Translation) are labeled by two squares to their right. Transitions with bold contours indicate transitions that are enabled in the current marking. During the execution of the token game, one of the enabled transitions is randomly selected, and the tokens flow to create a new marking.

A Mobius model of the gemcitabine model. Gemcitabine is a chemotherapeutic drug that is an analog of the normal cytidine that participates in transcription and in DNA replication. The pathway starts from the drug's intake by the cell and ends in its effect on cancerous cells resulting in cell death. The pathway shows cellular reactions that inactivate the drug and negative feedback of the drug's processing by the cell, which is inhibited by normal deoxycytidine triphosphate nucleotide (dCTP). The part of the PN that is shown in the figure represents transport of the cytidine or the drug analog into the cytoplasm, followed by two phosphorylation steps or inactivation of these molecules. At the bottom, the Petri Net branches into two paths: one for cytidine and one for the drug analog. dCTP downregulates the drug effect by inhibiting the first phosphorylation step of the drug (and cytidine). This is represented in the model by the rate of the phosphorylation1 reaction, which is proportional to the concentration of the phosphorylation substrate (Place19) and inversely proportional to the concentration of dCTP. The reactions are modeled as activities with exponential time distribution function.

Part of the Genomic Object Net's Petri Net of the gemcitabine example shown in and some of the graphs produced that follow the number of tokens at some places as a function of time. These graphs were produced by the Cell Illustrator tool, version 1.02, which is a new version of Genomic Object Net, which has better capabilities but is not freely available.

The results of experiments performed by the Mobius tool (plotted using Microsoft Excel). Whereas in TimeNET and Genomic Object Net, a single experiment can examine individual places (or sets of places in TimeNET) as a function of time, a Mobius experiment measures the number of tokens in a place or the number of times that a transition was fired at a single point in time or the total or average number over an interval of time. To generate a kinetic profile in Mobius, several experiments need to be executed. On the other hand, in a single experiment in Mobius, the user can vary many structural parameters. However, plots are not generated automatically by Mobius. A: The number of transitions fired as a function of immune_probability with burst size 50. This curve plots the number of two transitions that were fired: Immune_Response and Asexual_Development as a function of the probability of going in the path of an immune response from Place1 of the composed model shown in , rather than executing the other transition that uses Place1 as an input place (MerozoiteInBlood, left-hand SAN in ). The parameter burst size indicates the number of tokens that are output by the output gate shown on the left-hand side of . B: Drug effect (cell death as a function of drug dose). C: Inactive drug/drug effect as a function of drug dose. B, C: Results that were generated from simulation of the Gemcitabine model shown in . The plots show token markings as a function of time: the number of tokens at a place that indicates drug effect (CellDeathPlace + Place13) (B) and the ratio of the sum of tokens in the places that indicate inactivated drug (DeactivatedC, DeactivationDrug, InactiveDrug, marked by arrows in ) divided by the number of tokens at the place that represents drug effect (CellDeath + Place13) (C). Because the places that indicate inactivated drug include two places that store both drug tokens and cytidine tokens (DeactivatedC, DeactivationDrug), we multiplied the number of tokens stored in those places by the ratio of drug concentration to the concentration of cytidine + drug.

The algorithm is described below. Steps 3–8 are illustrated by the drawing. Convert every PN Place to a SAN Place. Copy the name and number of tokens.Convert every PN transition to a timed activity with exponential distribution. The rate of the transition = 1/(min_transition_time + (max_transition_time–min_transition_time)/2) if these parameters have values. Otherwise, the rate is 1.Convert a PN arc a that connects a transition to a place and has no weight or a weight of 1 into the corresponding straight line connector from transition to place (see Case 1 below).Convert a PN arc that connects a transition to a place p and has weight w into the corresponding straight line connector from transition to output gate. The gate function will add w to the number of tokens at place p. Link the gate to p (see Case 2 below).Convert a PN arc a that connects a place to a transition and has no weight or a weight of 1, and there is no other arc whose :TO slot equals the :TO slot of a into the corresponding straight line connector from place to transition (see Case 3 below).Convert a PN arc a that connects a place to a transition and has weight w, and there is no other arc whose :TO slot equals the :TO slot of a into the corresponding straight line connector from place to input gate and set the function of the gate to consume w tokens from :TO. Then link the gate to the transition (see Case 4 below).Convert a PN arc a that connects a place to a transition and has no weight or a weight of 1, and there is another arc whose :FROM slot equals the :FROM slot of a into the corresponding straight line connector from place to transition. This transition should have n cases, where n equals the number of arcs on that PN whose :TO slot equals the :TO of a. Link each case to a place. Give the new transition and the new places dummy names. Each case probability will be determined from the probability of each arc. Link the new place that corresponds to the case whose probability was determined from the arc's probability to the transition that is specified in that arc's :FROM slot. Do the same to all the other arcs that share the same input place, remembering not to process them again during the PN → SAN conversion (see Case 5 below).Modification of the previous step for arcs that have weight w: instead of simply linking a new place to the corresponding transition, place an input gate between the two and update its function according to w (see Case 6 below).