{"id":"https://openalex.org/W2114989670","doi":"https://doi.org/10.1007/s11047-015-9531-4","title":"Modeling biological gradient formation: combining partial differential equations and Petri nets","display_name":"Modeling biological gradient formation: combining partial differential equations and Petri nets","publication_year":2015,"publication_date":"2015-10-31","ids":{"openalex":"https://openalex.org/W2114989670","doi":"https://doi.org/10.1007/s11047-015-9531-4","mag":"2114989670","pmid":"https://pubmed.ncbi.nlm.nih.gov/27881934"},"language":"en","primary_location":{"id":"doi:10.1007/s11047-015-9531-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11047-015-9531-4","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11047-015-9531-4.pdf","source":{"id":"https://openalex.org/S149828480","display_name":"Natural Computing","issn_l":"1567-7818","issn":["1567-7818","1572-9796"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Natural Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007%2Fs11047-015-9531-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036637876","display_name":"Laura M.F. Bertens","orcid":"https://orcid.org/0000-0001-9974-9887"},"institutions":[{"id":"https://openalex.org/I121797337","display_name":"Leiden University","ror":"https://ror.org/027bh9e22","country_code":"NL","type":"education","lineage":["https://openalex.org/I121797337"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Laura M. F. Bertens","raw_affiliation_strings":["LIACS, Leiden University, Leiden, The Netherlands","[LIACS, Leiden University Leiden, The Netherlands]"],"affiliations":[{"raw_affiliation_string":"LIACS, Leiden University, Leiden, The Netherlands","institution_ids":["https://openalex.org/I121797337"]},{"raw_affiliation_string":"[LIACS, Leiden University Leiden, The Netherlands]","institution_ids":["https://openalex.org/I121797337"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014564304","display_name":"Jetty Kleijn","orcid":"https://orcid.org/0000-0001-9506-4071"},"institutions":[{"id":"https://openalex.org/I121797337","display_name":"Leiden University","ror":"https://ror.org/027bh9e22","country_code":"NL","type":"education","lineage":["https://openalex.org/I121797337"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Jetty Kleijn","raw_affiliation_strings":["LIACS, Leiden University, Leiden, The Netherlands","[LIACS, Leiden University Leiden, The Netherlands]"],"affiliations":[{"raw_affiliation_string":"LIACS, Leiden University, Leiden, The Netherlands","institution_ids":["https://openalex.org/I121797337"]},{"raw_affiliation_string":"[LIACS, Leiden University Leiden, The Netherlands]","institution_ids":["https://openalex.org/I121797337"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079511393","display_name":"Sander C. Hille","orcid":"https://orcid.org/0000-0003-0437-6745"},"institutions":[{"id":"https://openalex.org/I121797337","display_name":"Leiden University","ror":"https://ror.org/027bh9e22","country_code":"NL","type":"education","lineage":["https://openalex.org/I121797337"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Sander C. Hille","raw_affiliation_strings":["Mathematical Institute, Leiden University, Leiden, The Netherlands","Mathematical Institute Leiden University, Leiden, the Netherlands"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, Leiden University, Leiden, The Netherlands","institution_ids":["https://openalex.org/I121797337"]},{"raw_affiliation_string":"Mathematical Institute Leiden University, Leiden, the Netherlands","institution_ids":["https://openalex.org/I121797337"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002026696","display_name":"Monika Heiner","orcid":"https://orcid.org/0000-0003-1815-1205"},"institutions":[{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Monika Heiner","raw_affiliation_strings":["Department of Computer Science, Brandenburg Technical University Cottbus-Senftenberg, Cottbus, Germany","Department of Computer Science, Brandenburg Technical University Cottbus-Senftenberg, Cottbus, Germany#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Brandenburg Technical University Cottbus-Senftenberg, Cottbus, Germany","institution_ids":["https://openalex.org/I51783024"]},{"raw_affiliation_string":"Department of Computer Science, Brandenburg Technical University Cottbus-Senftenberg, Cottbus, Germany#TAB#","institution_ids":["https://openalex.org/I51783024"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042490356","display_name":"Maciej Koutny","orcid":"https://orcid.org/0000-0003-4563-1378"},"institutions":[{"id":"https://openalex.org/I84884186","display_name":"Newcastle University","ror":"https://ror.org/01kj2bm70","country_code":"GB","type":"education","lineage":["https://openalex.org/I84884186"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Maciej Koutny","raw_affiliation_strings":["School of Computing Science, Newcastle University, Newcastle Upon Tyne, UK","School of Computing Science; Newcastle University; Newcastle upon Tyne UK"],"affiliations":[{"raw_affiliation_string":"School of Computing Science, Newcastle University, Newcastle Upon Tyne, UK","institution_ids":["https://openalex.org/I84884186"]},{"raw_affiliation_string":"School of Computing Science; Newcastle University; Newcastle upon Tyne UK","institution_ids":["https://openalex.org/I84884186"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022631255","display_name":"Fons J. Verbeek","orcid":"https://orcid.org/0000-0003-2445-8158"},"institutions":[{"id":"https://openalex.org/I121797337","display_name":"Leiden University","ror":"https://ror.org/027bh9e22","country_code":"NL","type":"education","lineage":["https://openalex.org/I121797337"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Fons J. Verbeek","raw_affiliation_strings":["LIACS, Leiden University, Leiden, The Netherlands","[LIACS, Leiden University Leiden, The Netherlands]"],"affiliations":[{"raw_affiliation_string":"LIACS, Leiden University, Leiden, The Netherlands","institution_ids":["https://openalex.org/I121797337"]},{"raw_affiliation_string":"[LIACS, Leiden University Leiden, The Netherlands]","institution_ids":["https://openalex.org/I121797337"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5036637876"],"corresponding_institution_ids":["https://openalex.org/I121797337"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.3019,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.62234791,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"15","issue":"4","first_page":"665","last_page":"675"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10621","display_name":"Gene Regulatory Network Analysis","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10492","display_name":"Microtubule and mitosis dynamics","score":0.9846000075340271,"subfield":{"id":"https://openalex.org/subfields/1307","display_name":"Cell Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10497","display_name":"Fungal and yeast genetics research","score":0.9799000024795532,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/petri-net","display_name":"Petri net","score":0.8063159584999084},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6327009201049805},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6007999181747437},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.5779811143875122},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5267942547798157},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.4892404079437256},{"id":"https://openalex.org/keywords/stochastic-petri-net","display_name":"Stochastic Petri net","score":0.4650844931602478},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.44414493441581726},{"id":"https://openalex.org/keywords/differential","display_name":"Differential (mechanical device)","score":0.4225032329559326},{"id":"https://openalex.org/keywords/biological-system","display_name":"Biological system","score":0.4222835898399353},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.296200156211853},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.25363293290138245},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1310926079750061},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12464675307273865},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.07129362225532532},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.06499066948890686}],"concepts":[{"id":"https://openalex.org/C38677869","wikidata":"https://www.wikidata.org/wiki/Q724168","display_name":"Petri net","level":2,"score":0.8063159584999084},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6327009201049805},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6007999181747437},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.5779811143875122},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5267942547798157},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.4892404079437256},{"id":"https://openalex.org/C197551870","wikidata":"https://www.wikidata.org/wiki/Q7617809","display_name":"Stochastic Petri net","level":3,"score":0.4650844931602478},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44414493441581726},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.4225032329559326},{"id":"https://openalex.org/C186060115","wikidata":"https://www.wikidata.org/wiki/Q30336093","display_name":"Biological system","level":1,"score":0.4222835898399353},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.296200156211853},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.25363293290138245},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1310926079750061},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12464675307273865},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.07129362225532532},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.06499066948890686},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s11047-015-9531-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11047-015-9531-4","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11047-015-9531-4.pdf","source":{"id":"https://openalex.org/S149828480","display_name":"Natural Computing","issn_l":"1567-7818","issn":["1567-7818","1572-9796"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Natural Computing","raw_type":"journal-article"},{"id":"pmid:27881934","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/27881934","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Natural computing","raw_type":null},{"id":"pmh:oai:eprint.ncl.ac.uk:191705","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402485","display_name":"Newcastle University ePrints (Newcastle Univesity)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I84884186","host_organization_name":"Newcastle University","host_organization_lineage":["https://openalex.org/I84884186"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""},{"id":"pmh:oai:pubmedcentral.nih.gov:5101295","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/5101295","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Nat Comput","raw_type":"Text"},{"id":"pmh:oai:scholarlypublications.universiteitleiden.nl:item_2868829","is_oa":false,"landing_page_url":"https://hdl.handle.net/1887/47984","pdf_url":null,"source":{"id":"https://openalex.org/S4306400850","display_name":"Leiden Repository (Leiden University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I121797337","host_organization_name":"Leiden University","host_organization_lineage":["https://openalex.org/I121797337"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Natural computing","raw_type":"Article / Letter to editor"},{"id":"pmh:ul:oai:scholarlypublications.universiteitleiden.nl:item_2868829","is_oa":true,"landing_page_url":"http://hdl.handle.net/1887/47984","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Natural computing, 154(4), 665 - 675","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1007/s11047-015-9531-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11047-015-9531-4","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11047-015-9531-4.pdf","source":{"id":"https://openalex.org/S149828480","display_name":"Natural Computing","issn_l":"1567-7818","issn":["1567-7818","1572-9796"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Natural Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2114989670.pdf","grobid_xml":"https://content.openalex.org/works/W2114989670.grobid-xml"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W14540139","https://openalex.org/W46848391","https://openalex.org/W67242188","https://openalex.org/W130113173","https://openalex.org/W139832837","https://openalex.org/W172473584","https://openalex.org/W644441850","https://openalex.org/W653950636","https://openalex.org/W858225545","https://openalex.org/W1469014383","https://openalex.org/W1487395029","https://openalex.org/W1510249251","https://openalex.org/W1561267729","https://openalex.org/W1587747784","https://openalex.org/W1720193968","https://openalex.org/W1756322562","https://openalex.org/W1786236181","https://openalex.org/W1904811091","https://openalex.org/W1965558704","https://openalex.org/W1968706614","https://openalex.org/W1981747964","https://openalex.org/W2011835017","https://openalex.org/W2033543895","https://openalex.org/W2036614629","https://openalex.org/W2048987027","https://openalex.org/W2058943257","https://openalex.org/W2078692950","https://openalex.org/W2092608858","https://openalex.org/W2122792487","https://openalex.org/W2128584428","https://openalex.org/W2133552211","https://openalex.org/W2135648313","https://openalex.org/W2142216488","https://openalex.org/W2148357359","https://openalex.org/W2148895036","https://openalex.org/W2156375169","https://openalex.org/W2168259679","https://openalex.org/W2581610748","https://openalex.org/W2913376619","https://openalex.org/W3021856877","https://openalex.org/W4256259255","https://openalex.org/W4404342747"],"related_works":["https://openalex.org/W1543378523","https://openalex.org/W1491709828","https://openalex.org/W2916743419","https://openalex.org/W2381491984","https://openalex.org/W2257573499","https://openalex.org/W2148229336","https://openalex.org/W1980200158","https://openalex.org/W1513546254","https://openalex.org/W2091203523","https://openalex.org/W2125164140"],"abstract_inverted_index":{"Both":[0],"Petri":[1,48],"nets":[2],"and":[3],"differential":[4,35],"equations":[5],"are":[6,43,58],"important":[7],"modeling":[8,21],"tools":[9],"for":[10],"biological":[11,28],"processes.":[12],"In":[13],"this":[14],"paper":[15],"we":[16],"demonstrate":[17],"how":[18],"these":[19],"two":[20],"techniques":[22],"can":[23],"be":[24],"combined":[25],"to":[26],"describe":[27],"gradient":[29,41,65],"formation.":[30],"Parameters":[31],"derived":[32],"from":[33],"partial":[34],"equation":[36],"describing":[37],"the":[38,55,68],"process":[39],"of":[40,54,64],"formation":[42,66],"incorporated":[44],"in":[45,67],"an":[46],"abstract":[47],"net":[49],"model.":[50],"The":[51],"quantitative":[52],"aspects":[53],"resulting":[56],"model":[57],"validated":[59],"through":[60],"a":[61],"case":[62],"study":[63],"fruit":[69],"fly.":[70]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
