Fig 1.
Schematic diagram of the entities and pathways linking feeding-fasting signals to Circadian Clock and Oscillators.
The core circadian clock comprises of the entities CLOCK, BMAL1, CLOCK-BMAL1, RORs, PERIOD, CRYPTOCHROME, and PER-CRY. The metabolic arm of the circadian system comprises of the enzymes NAMPT, NMNAT, and SIRT1 with the metabolites Nic, NMN, and NAD+. Finally, the entities PARP1, HSF1, CREB, and ChREBP link the feeding and fasting signals to the rest of the circadian system via their interactions.
Fig 2.
Work Flow Diagram presenting the structure and organisation of the study.
The procedure follows after the literature search when a BRN is constructed from the pathway details, upon which the wet-lab experimental observations are applied via CTL Model Checking to generate the set of parameters. The parameters are then modelled as a discrete (Standard) Petri Net for property analysis, after which the system is converted to a Hybrid model for simulations.
Fig 3.
Graph showing the natural continuous, and modelled discrete changes of concentration of an entity X.
After the activation or deactivation signal, the protein concentration starts changing immediately while the discrete level remains the same, until a respective threshold is reached.
Fig 4.
Example of a standard Petri Net.
(A) A simple Petri Net model using the standard Petri Net framework. The set P = {p1, p2} is the set of places, T = {t1, t2} is the set of transitions, f = {p1t1, p2t2, t1p2, t2p1} is the set of directed arcs all of which have an arc weight of 1, and m0 = (2, 0) being the initial marking for the ordered tuple (p1, p2). (B) The Reachability Graph obtained from the PN from the initial marking m0. The graph shows three cycles: (2, 0) → (1, 1) → (0, 2) → (1, 1) → (2, 0), (2, 0) → (1, 1) → (2, 0), and(1, 1) → (0, 2) → (1, 1); and contains no deadlocks.
Fig 5.
The BRN generated by abstracting the entities and pathways of the schematic diagram shown in Fig. 1.
The proteins CLOCK-BMAL1 and HSF1 are showing activatory interactions, while PER-CRY and PARP1 are showing inhibitory interactions only. SIRT1 is showing both activatory and inhibitory interactions. The Feeding signal, being an abstracted environmental cue, is represented as an oval.
Table 1.
Table showing the parameters, their respective resource sets, and the values that were allowed in SMBioNet, generated by SMBioNet in the parameter sets, and finally selected for modelling in Petri Nets.
Fig 6.
Partial discrete Petri Net model of the BRN.
This part represents the complete Place and Transition sets of the entity PC (P = {PC, cPB}, T = {Kpc_, Kpc_cb, Kpc_sirt1, Kpc_cbsirt1}), where the place PC models the presence of PC in the system, and cPC models its absence. PC has four parameters, KPC{CB}, KPC{CB, SIRT1} ∈ °p, and KPC{}, KPC{SIRT1} ∈ p° for p = PC. Highly abstracted sets of the influencers of PC (CB and SIRT1) are also represented simply as the respective presence and absence places, along with generic source and sink transitions. As CB is the activator of PC, the transitions where it is in the resource set of PC link to the place CB, and the transitions where it is absent from the resource set link to the place cCB. On the other hand, this pattern is inverted in the case of SIRT1 as it is an inhibitor of PC, and thus the transitions where it is in the resource set of PC link to the place cSIRT1, while the transitions where it is absent from the resource set link to the place SIRT1 instead.
Fig 7.
The generated reachability graph of the complete discrete Petri Net model.
The order of the tuple in each marking is ⟨CB, SIRT1, PC, Feed, HSF1, PARP1⟩. The graph was generated from the initial marking m0 = (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1), shown at the top. It consists of 64 unique markings and 224 marking transitions, and is itself a single strongly connected component.
Table 2.
Table showing the rates used for the continuous transitions of each entity. The concentration increase rate specifies the rate for the transitons t ∈ °p, and the concentration decrease rate specifies for t ∈ p°, for place p representing the entity in the system.
Fig 8.
Simulation results of the 3 meals/day entrainment base scenario.
Each grid box shows 1 complete day on the x-axis, and a complete level on the y-axis. The graph shows the oscillations of the entities in accordance with the 0800 hrs breakfast, 1400 hrs lunch, and 2000 hrs dinner, with an overnight fast of 12 hours, for a duration of 10 days. The entities were able to entrain on the third day, and continued the periodic behaviour from then on. All entities are utilising their respective rate values given in Table 2, except for the Feeding signal which is modelled as a periodic discrete signal, shown as the vertical black pillars. The number of meals, and their timings was assumed to be the prevalent regimen, and were thus used as the base scenario to model other differing regimen scenarios.
Fig 9.
Simulation results of the 2 meals/day entrainment explained in Scenario 1.
(A) The graph shows the oscillations of the entities in accordance with the 0800 hrs breakfast, and 2000 hrs dinner, with an overnight fast of 12 hours. (B) The graph shows the oscillation beyond the 456th hour, in which the PER-CRY (PC) protein complex converges into its rhythm.
Fig 10.
Simulation results of a single 0800 hrs breakfast/day entrainment explained in Scenario 2.
The system was able to entrain but showed high levels of stress via higher expressions of SIRT1 and HSF1.
Fig 11.
Simulation results depicting the over feeding scenario with 5 meals/day.
The meals were simulated on 0800 hrs, 1100 hrs, 1400 hrs, 1700 hrs, and 2000 hrs, along with the 12 hour overnight fast. The results indicate an over expression of PARP1, with suppressed expressions of almost all entities, other than HSF1.