{"id":"https://openalex.org/W2169472594","doi":"https://doi.org/10.1186/1752-0509-6-99","title":"A variational approach to parameter estimation in ordinary differential equations","display_name":"A variational approach to parameter estimation in ordinary differential equations","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2169472594","doi":"https://doi.org/10.1186/1752-0509-6-99","mag":"2169472594","pmid":"https://pubmed.ncbi.nlm.nih.gov/22892133"},"language":"en","primary_location":{"id":"doi:10.1186/1752-0509-6-99","is_oa":true,"landing_page_url":"https://doi.org/10.1186/1752-0509-6-99","pdf_url":"https://bmcsystbiol.biomedcentral.com/counter/pdf/10.1186/1752-0509-6-99","source":{"id":"https://openalex.org/S92363890","display_name":"BMC Systems Biology","issn_l":"1752-0509","issn":["1752-0509"],"is_oa":true,"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":"BMC Systems Biology","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://bmcsystbiol.biomedcentral.com/counter/pdf/10.1186/1752-0509-6-99","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046026391","display_name":"Daniel Kaschek","orcid":"https://orcid.org/0000-0002-1953-7217"},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Daniel Kaschek","raw_affiliation_strings":["Institute of Physics, Freiburg University, Freiburg, Germany. daniel.kaschek@physik.uni-freiburg.de","Institute of Physics, Freiburg University, Freiburg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Physics, Freiburg University, Freiburg, Germany. daniel.kaschek@physik.uni-freiburg.de","institution_ids":["https://openalex.org/I161046081"]},{"raw_affiliation_string":"Institute of Physics, Freiburg University, Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5052831097","display_name":"Jens Timmer","orcid":"https://orcid.org/0000-0003-4517-1383"},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Jens Timmer","raw_affiliation_strings":["BIOSS Centre for Biological Signalling Studies, Freiburg University, Freiburg, Germany","Freiburg Center for Systems Biology (ZBSA), Freiburg University, Freiburg, Germany","Freiburg Institute for Advanced Studies (FRIAS), Freiburg University, Freiburg, Germany","Institute of Physics, Freiburg University, Freiburg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"BIOSS Centre for Biological Signalling Studies, Freiburg University, Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]},{"raw_affiliation_string":"Freiburg Center for Systems Biology (ZBSA), Freiburg University, Freiburg, Germany","institution_ids":[]},{"raw_affiliation_string":"Freiburg Institute for Advanced Studies (FRIAS), Freiburg University, Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]},{"raw_affiliation_string":"Institute of Physics, Freiburg University, Freiburg, Germany","institution_ids":["https://openalex.org/I161046081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5046026391"],"corresponding_institution_ids":["https://openalex.org/I161046081"],"apc_list":{"value":2290,"currency":"USD","value_usd":2290},"apc_paid":{"value":1677,"currency":"EUR","value_usd":1808},"fwci":0.0,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.11308623,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"6","issue":"1","first_page":"99","last_page":"99"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10621","display_name":"Gene Regulatory Network Analysis","score":0.7569000124931335,"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":0.7569000124931335,"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/T10791","display_name":"Advanced Control Systems Optimization","score":0.06390000134706497,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11053","display_name":"Process Optimization and Integration","score":0.02800000086426735,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.7697468996047974},{"id":"https://openalex.org/keywords/estimation-theory","display_name":"Estimation theory","score":0.6089097857475281},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5804574489593506},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5113039016723633},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4847838282585144},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.45250073075294495},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.41797104477882385},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.4133499562740326},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.41215670108795166},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3257352113723755},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.22411862015724182},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11284998059272766}],"concepts":[{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.7697468996047974},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.6089097857475281},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5804574489593506},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5113039016723633},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4847838282585144},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.45250073075294495},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.41797104477882385},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.4133499562740326},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.41215670108795166},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3257352113723755},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.22411862015724182},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11284998059272766},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D003627","descriptor_name":"Data Interpretation, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003627","descriptor_name":"Data Interpretation, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003627","descriptor_name":"Data Interpretation, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008433","descriptor_name":"Mathematics","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D008962","descriptor_name":"Models, Theoretical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D049490","descriptor_name":"Systems Biology","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D049490","descriptor_name":"Systems Biology","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D049490","descriptor_name":"Systems Biology","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false}],"locations_count":5,"locations":[{"id":"doi:10.1186/1752-0509-6-99","is_oa":true,"landing_page_url":"https://doi.org/10.1186/1752-0509-6-99","pdf_url":"https://bmcsystbiol.biomedcentral.com/counter/pdf/10.1186/1752-0509-6-99","source":{"id":"https://openalex.org/S92363890","display_name":"BMC Systems Biology","issn_l":"1752-0509","issn":["1752-0509"],"is_oa":true,"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":"BMC Systems Biology","raw_type":"journal-article"},{"id":"pmid:22892133","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/22892133","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":"BMC systems biology","raw_type":null},{"id":"pmh:oai:europepmc.org:2537054","is_oa":true,"landing_page_url":"http://europepmc.org/articles/PMC3534504","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"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":"","raw_type":"Text"},{"id":"pmh:oai:freidok.uni-freiburg.de:11673","is_oa":false,"landing_page_url":"https://freidok.uni-freiburg.de/data/11673","pdf_url":null,"source":{"id":"https://openalex.org/S4306401057","display_name":"FreiDok plus (Universit\u00e4tsbibliothek Freiburg)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I161046081","host_organization_name":"University of Freiburg","host_organization_lineage":["https://openalex.org/I161046081"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"BMC Systems Biology. 6 (2012), 99, ISSN: 1752-0509","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:3534504","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/3534504","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":"BMC Syst Biol","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1186/1752-0509-6-99","is_oa":true,"landing_page_url":"https://doi.org/10.1186/1752-0509-6-99","pdf_url":"https://bmcsystbiol.biomedcentral.com/counter/pdf/10.1186/1752-0509-6-99","source":{"id":"https://openalex.org/S92363890","display_name":"BMC Systems Biology","issn_l":"1752-0509","issn":["1752-0509"],"is_oa":true,"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":"BMC Systems Biology","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2169472594.pdf","grobid_xml":"https://content.openalex.org/works/W2169472594.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W622112888","https://openalex.org/W1574514837","https://openalex.org/W2012500654","https://openalex.org/W2100486339","https://openalex.org/W2131448225","https://openalex.org/W2143894094","https://openalex.org/W2168648059","https://openalex.org/W3102649939"],"related_works":["https://openalex.org/W1631776469","https://openalex.org/W2950866767","https://openalex.org/W4299610791","https://openalex.org/W3019662020","https://openalex.org/W3106300234","https://openalex.org/W4286900056","https://openalex.org/W1963826769","https://openalex.org/W4297571715","https://openalex.org/W3134934667","https://openalex.org/W2043671984"],"abstract_inverted_index":{"BACKGROUND:":[0],"Ordinary":[1],"differential":[2,111],"equations":[3,112],"are":[4,194],"widely-used":[5],"in":[6,67,132,148],"the":[7,48,89,114,128,146,180,183,203],"field":[8],"of":[9,46,50,53,61,110,136,171,179,185,200],"systems":[10],"biology":[11],"and":[12,138,160,192],"chemical":[13,17],"engineering":[14],"to":[15,25,42,57,72,105,127,156],"model":[16],"reaction":[18,90,173],"networks.":[19],"Numerous":[20],"techniques":[21],"have":[22],"been":[23],"developed":[24],"estimate":[26],"parameters":[27],"like":[28],"rate":[29],"constants,":[30],"initial":[31],"conditions":[32],"or":[33,81],"steady":[34],"state":[35,119],"concentrations":[36],"from":[37,189],"time-resolved":[38],"data.":[39],"In":[40,164],"contrast":[41],"this":[43,68,166],"countable":[44],"set":[45,60],"parameters,":[47],"estimation":[49,76,124,135,143],"entire":[51],"courses":[52,137,150],"network":[54,91],"components":[55,116],"corresponds":[56],"an":[58,107],"innumerable":[59],"parameters.":[62,139],"RESULTS:":[63],"The":[64,141],"approach":[65,144],"presented":[66],"work":[69],"is":[70,95,101,125],"able":[71],"deal":[73],"with":[74],"course":[75],"for":[77,197],"extrinsic":[78],"system":[79,109,130],"inputs":[80],"intrinsic":[82],"reactants,":[83],"both":[84],"not":[85],"being":[86],"constrained":[87],"by":[88],"itself.":[92],"Our":[93],"method":[94],"based":[96],"on":[97],"variational":[98,186],"calculus":[99],"which":[100],"carried":[102],"out":[103],"analytically":[104],"derive":[106],"augmented":[108,129],"including":[113],"unconstrained":[115],"as":[117],"ordinary":[118],"variables.":[120],"Finally,":[121],"conventional":[122],"parameter":[123,158],"applied":[126],"resulting":[131],"a":[133],"combined":[134,142,195],"CONCLUSIONS:":[140],"takes":[145],"uncertainty":[147],"input":[149],"correctly":[151],"into":[152],"account.":[153],"This":[154],"leads":[155],"precise":[157],"estimates":[159],"correct":[161],"confidence":[162],"intervals.":[163],"particular":[165],"implies":[167],"that":[168],"small":[169],"motifs":[170],"large":[172],"networks":[174],"can":[175],"be":[176],"analysed":[177],"independently":[178],"rest.":[181],"By":[182],"use":[184],"methods,":[187],"elements":[188],"control":[190],"theory":[191],"statistics":[193],"allowing":[196],"future":[198],"transfer":[199],"methods":[201],"between":[202],"two":[204],"fields.":[205]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}
