Time-dependent structural transformation analysis to high-level Petri net model with active state transition diagram

Abstract

Background: With an accumulation of in silico data obtained by simulating large-scale biological networks, a new interest of research is emerging for elucidating how living organism functions over time in cells. Investigating the dynamic features of current computational models promises a deeper understanding of complex cellular processes. This leads us to develop a method that utilizes structural properties of the model over all simulation time steps. Further, user-friendly overviews of dynamic behaviors can be considered to provide a great help in understanding the variations of system mechanisms.

Results: We propose a novel method for constructing and analyzing a so-called active state transition diagram (ASTD) by using time-course simulation data of a high-level Petri net. Our method includes two new algorithms. The first algorithm extracts a series of subnets (called temporal subnets) reflecting biological components contributing to the dynamics, while retaining positive mathematical qualities. The second one creates an ASTD composed of unique temporal subnets. ASTD provides users with concise information allowing them to grasp and trace how a key regulatory subnet and/or a network changes with time. The applicability of our method is demonstrated by the analysis of the underlying model for circadian rhythms in Drosophila.

Conclusions: Building ASTD is a useful means to convert a hybrid model dealing with discrete, continuous and more complicated events to finite time-dependent states. Based on ASTD, various analytical approaches can be applied to obtain new insights into not only systematic mechanisms but also dynamics.

Figure 1

The symbols of HFPNe.

Figure 2

Schematic overview of our method. The method includes Algorithm 1 and Algorithm 2 for constructing an ASTD (active state transition diagram) from simulation data (EDF file).

Figure 3

An example illustrating the process of performing Algorithm 1. For a given transition t, the processes of extracting a temporal subnet H'(x) from H at time x by applying Algorithm 1. Block (b) is obtained from (a) by performing the subroutine RMVA(H') (step 2 in TSN(H, x)). Block (c) is obtained from (b) by performing the subroutine RMVP(H') (step 4 in TSN(H, x)). Note that t is connected by an enabled inhibitory arc, therefore t cannot be removed by RMVT(H') (step 3 in TSN(H, x)) in Algorithm 1.

Figure 4

HFPNe model and its simulation result of circadian rhythm in Drosophila. (a) HFPNe model of circadian rhythm in Drosophila. The accompanying variable m_x at a place represents the concentration of the corresponding mRNA, protein or the compound. For example, the variable m₁ indicates the concentration of dClk mRNA. Reaction speed (the rate of transcription, translation, complex formation or degradation) is expressed by a simple formula at each transition. For example, the formula m₁/5 indicates the translation rate of dCLK protein that depends on the variable m₁ for the dClk Mrna concentration. The real number over an arc is the threshold for the content of the place attached to this arc. For example, the translation of tim mRNA occurs during the period that the place value of tim mRNA exceeds 1.0. (b) Oscillations of tim, per, dClk mRNAs, and the proteins TIM, PER, dCLK, PER/TIM, PER/DBT, dCLK/CYC (left y-axis) and DBT, CYC (right y-axis). The unit of x-axis is [pt] ([pt] is the virtual time unit of the HFPNe model), while that of y-axis is [vc] ([vc] is the virtual concentration unit).

Figure 5

Schematic representation of ASTD^Wand ASTD^L. The four bold-line blocks of z₁₉, z₂₀, z₁₀ and z₁₁ on the left side show the corresponding temporal subnets that are the minimal element sets at respective time points. Dashed-line block shows the state transitions of "z₁₉→z₂₀→z₁₀↔z₁₁" in the ASTD and corresponding detailed regulation variations of HFPNe elements. For example, in the structural transformation from z₁₉ to z₂₀, two inhibitory arcs a(p₁₃, t₁₀) and a(p₁₃, t₁₄) are enabled due to the increase in the concentration of PER/TIM, resulting in the deletion of four arcs a(p₅, t₁₀), a(p₅, t₁₄), a'(t₁₀, p₆) and a'(t₁₄, p₈). Note that the transformation from z₁₁ to z₁₀ is omitted.

Figure 6

Three characteristic overviews of the ASTD for the circadian rhythm model. (a) ASTD^W and ASTD^L characterized with respect to duration (upper) and out-degree (lower). (b) ASTD^W characterized with duration and the total concentration difference for per and dClk mRNAs.