Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Abstract

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.

Exploiting stochastic Petri nets with fuzzy parameters to predict efficient drug combinations for Spinal Muscular Atrophy

“…A question of practical interest may be the issue of determining precise fuzzy numbers to be assigned to kinetic parameters with uncertain or unknown experimental values. In [16], the authors adopted the following scheme for fuzzy parameter estimation: A fuzzy number is initially represented as a union of its α -cuts. The α -cut for each output is obtained by decomposing all fuzzy parameters into their α -cuts and then running stochastic simulations at each α level.…”

“…Lamprecht et al [13] used SPNs to create model of Ca 2+ release sites composed of a number of intracellular channels that have stochastic behavior, and Marwan et al [14] investigated enteric bacteria phosphate regulation by using SPNs, while Castaldi et al [15] developed SPN model of the tissue factor-induced coagulation cascade. Liu et al [16] used fuzzy SPNs to create a yeast polarization model, and Bashirov et al [17] presented stochastic simulation-based validation and analysis of the p16-mediated pathway, the disruption of which is among major causes of human cancers. Software tools used to conduct the above research include Snoopy [18], Möbius [19] and GreatSPN [20], while https://www.informatik.unihamburg.de/cgi-bin/TGI/tools/ collects links to 23 Petri net tools and software supporting SPNs.…”

“…Mehraei [24] exploited fuzzy stochastic hybrid Petri nets in modelling of mood disorder treatment. Liu et al [16] combined SPNs and fuzzy logic in order to create a quantitative model of biological systems in which reaction rates are associated with fuzzy numbers. Bordon et al [25] demonstrated how fuzzy logic and Petri nets can be used to deal with unknown or imprecise data arising from gene regulatory processes, and Liu et al [26] represented interactions among genes with fuzzy rules implementing colored fuzzy Petri nets.…”

“…So far, there is no widespread evidence of the applications of both SPNs and fuzzy logic in biological models. To the best of our knowledge, the present paper is one of the first attempts [16,25] to associate fuzzy numbers with reaction rates and concentrations in an SPN model of a biological network. Moreover, this work is the first account on the use of SPNs with fuzzy parameters in a target-based drug combination discovery.…”

Advantages of Petri-Net Modeling and Simulation for Biological Networks

“…Fuzzy Petri-Nets [22], learning Petri-Nets [23] and object oriented Petri-Nets [7] represent only a snapshot of this wide collection of higher Petri-Nets. …”