{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:20:46Z","timestamp":1760242846199,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,12,31]],"date-time":"2016-12-31T00:00:00Z","timestamp":1483142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"KAKENHI","award":["16H02896"],"award-info":[{"award-number":["16H02896"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Despite the recent rapid progress in high throughput measurements of biological data, it is still difficult to gather all of the reaction speed data in biological pathways. This paper presents a Petri net-based algorithm that can derive estimated values for non-valid reaction speeds in a signaling pathway from biologically-valid data. In fact, these reaction speeds are reflected based on the delay times in the timed Petri net model of the signaling pathway. We introduce the concept of a \u201cdependency relation\u201d over a transition set of a Petri net and derive the properties of the dependency relation through a structural analysis. Based on the theoretical results, the proposed algorithm can efficiently shrink the transitions with two elementary structures into a single transition repeatedly to reduce the Petri net size in order to eventually discover all transition sets with a dependency relation. Finally, to show the usefulness of our algorithm, we apply our algorithm to the IL-3 Petri net model.<\/jats:p>","DOI":"10.3390\/a10010004","type":"journal-article","created":{"date-parts":[[2017,1,2]],"date-time":"2017-01-02T09:54:18Z","timestamp":1483350858000},"page":"4","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dependent Shrink of Transitions for Calculating Firing Frequencies in Signaling Pathway Petri Net Model"],"prefix":"10.3390","volume":"10","author":[{"given":"Atsushi","family":"Mizuta","sequence":"first","affiliation":[{"name":"Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi 7538512, Japan"}]},{"given":"Qi-Wei","family":"Ge","sequence":"additional","affiliation":[{"name":"Faculty of Education, Yamaguchi University, Yamaguchi 7538513, Japan"}]},{"given":"Hiroshi","family":"Matsuno","sequence":"additional","affiliation":[{"name":"Graduate School of Science and Technology for Innovation, Yamaguchi University, Yamaguchi 7538512, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,31]]},"reference":[{"key":"ref_1","unstructured":"Reisig, W. 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