{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:59:55Z","timestamp":1760230795078,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T00:00:00Z","timestamp":1660608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We revisit our earlier paper, with two of the coauthors, in which we proposed an unbiased and consistent estimator \u03bc^n for an unknown mutation rate \u03bc of microorganisms. Previously, we proved that the associated sequence of estimators \u03bc^n converges to \u03bc almost surely pointwise on a nonextinct set \u03a90. Here, we show that this sequence converges also in the mean square with respect to conditional probability measure P0\u00b7=P\u00b7\u2229\u03a90\/P\u03a90 and that, with respect to P0, the estimator is asymptotically unbiased. We further assume that a microorganism can mutate or turn to a different variant of one of the two types. In particular, it can mean that bacteria under attack by a virus or chemical agent are either perishing or surviving, turning them to stronger variant. We propose estimators for their respective types and show that they are a.s. pointwise and L2-consistent and asymptotically unbiased with respect to measure P0.<\/jats:p>","DOI":"10.3390\/sym14081701","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T03:15:27Z","timestamp":1660706127000},"page":"1701","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Determination of Mutation Rates with Two Symmetric and Asymmetric Mutation Types"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6718-1526","authenticated-orcid":false,"given":"Jewgeni H.","family":"Dshalalow","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, College of Engineering and Science, Melbourne, FL 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Van Minh","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, College of Engineering and Science, Melbourne, FL 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0270-6907","authenticated-orcid":false,"given":"Richard R.","family":"Sinden","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Biology, and Health Sciences, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5524-629X","authenticated-orcid":false,"given":"Ryan T.","family":"White","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Florida Institute of Technology, College of Engineering and Science, Melbourne, FL 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1146\/annurev.genet.33.1.57","article-title":"Mechanisms of Stationary Phase Mutation: A Decade of Adaptive Mutation","volume":"33","author":"Foster","year":"1999","journal-title":"Annu. Rev. Genet."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1534\/genetics.108.095588","article-title":"Reinventing the Ames Test as a Quantitative Lab That Connects Classical and Molecular Genetics","volume":"181","year":"2009","journal-title":"Genetics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1093\/genetics\/28.6.491","article-title":"Mutations of Bacteria from Virus Sensitivity to Virus Resistance","volume":"28","author":"Luria","year":"1943","journal-title":"Genetics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"7160","DOI":"10.1073\/pnas.88.16.7160","article-title":"A Constant Rate of Spontaneous Mutation in DNA-based Microbes","volume":"88","author":"Drake","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S0076-6879(05)09012-9","article-title":"Methods for determining spontaneous mutation rates","volume":"Volume 409","author":"Foster","year":"2006","journal-title":"Methods in Enzymology"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1093\/genetics\/136.3.1209","article-title":"Luria-Delbruck Fluctuation Experiments: Design and Analysis","volume":"136","author":"Jones","year":"1994","journal-title":"Genetics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"255","DOI":"10.2307\/3214564","article-title":"Analysis of the Luria-Delbr\u00fcck Distribution Using Discrete Convolution Powers","volume":"29","author":"Ma","year":"1992","journal-title":"J. Appl. Probab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1006\/tpbi.2000.1504","article-title":"Improved Inference of Mutation Rates: II. Generalization of the Luria-Delbr\u00fcck Distribution for Realistic Cell-Cycle Time Distributions","volume":"59","author":"Oprea","year":"2001","journal-title":"Theor. Popul. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1006\/meth.1999.0901","article-title":"Determining Mutation Rates in Bacterial Populations","volume":"20","author":"Rosche","year":"2000","journal-title":"Methods"},{"key":"ref_10","first-page":"173","article-title":"On Fluctuation Analysis: A New, Simple and Efficient Method for Computing the Expected Number of Mutants","volume":"85","author":"Sarkar","year":"1992","journal-title":"Genetics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.mrfmmm.2008.11.015","article-title":"A Robust Estimator of Mutation Rates","volume":"661","author":"Wu","year":"2009","journal-title":"Mutat. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jtbi.2014.11.009","article-title":"Fast Maximum Likelihood Estimation of Mutation Rates Using a Birth-Death Process","volume":"366","author":"Wu","year":"2015","journal-title":"J. Theor. Biol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s00285-008-0225-8","article-title":"A Stochastic Model for Estimation of Mutation Rates in Multiple-replication Proliferation Processes","volume":"59","author":"Xiong","year":"2009","journal-title":"J. Math. Biol."},{"key":"ref_14","unstructured":"Drake, J.W. (1970). The Molecular Basis of Mutation, Holden-Day."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1093\/genetics\/137.4.1139","article-title":"Fluctuation Tests: How Reliable Are the Estimates of Mutation Rates?","volume":"137","author":"Stewart","year":"1994","journal-title":"Genetics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0025-5564(99)00045-0","article-title":"Progress of a Half Century in the Study of the Luria\u2013Delbr\u00fcck Distribution","volume":"162","author":"Zheng","year":"1999","journal-title":"Math. Biosci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.jtbi.2015.09.003","article-title":"The Luria-Delbr\u00fcck Protocol Is Still the Most Practical","volume":"386","author":"Zheng","year":"2015","journal-title":"J. Theor. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.jtbi.2012.01.029","article-title":"On an Unbiased and Consistent Estimator for Mutation Rates","volume":"300","author":"Niccum","year":"2012","journal-title":"J. Theor. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1128\/br.21.4.263-272.1957","article-title":"Synchronization of Cell Division","volume":"21","author":"Campbell","year":"1957","journal-title":"Bacteriol. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1021\/acssynbio.9b00058","article-title":"Microfluidic Synchronizer Using a Synthetic Nanoparticle-Capped Bacterium","volume":"8","author":"Chang","year":"2019","journal-title":"ACS Synth. Biol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1073\/pnas.50.4.767","article-title":"Bacterial Synchronization by Selection of Cells at Division","volume":"50","author":"Helmstetter","year":"1963","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"238","DOI":"10.3389\/fmicb.2015.00238","article-title":"A Ten-Year Search for Synchronous Cells: Obstacles, Solutions, and Practical Applications","volume":"6","author":"Helmstetter","year":"2015","journal-title":"Front. Microbiol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1126\/science.131.3407.1098.a","article-title":"Synchronization of Division in Escherichia Coli","volume":"131","author":"Anderson","year":"1960","journal-title":"Science"},{"key":"ref_24","first-page":"3","article-title":"Automatic synchronization of growth of \u201cEscherichia coli\u201d (author\u2019s transl)","volume":"131","author":"Kepes","year":"1980","journal-title":"Ann. Microbiol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1288","DOI":"10.1002\/bit.260261105","article-title":"Freeze Preservation of Synchrony in Cultures of Enterobacteriaceae Synchronized by Continuous Phasing in Phosphate-Limited Media","volume":"26","author":"Kepes","year":"1984","journal-title":"Biotechnol. Bioeng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1016\/S0006-3495(68)86521-X","article-title":"Linear Cell Growth in Escherichia Coli","volume":"8","author":"Kubitschek","year":"1968","journal-title":"Biophys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.synbio.2019.06.001","article-title":"A Novel Synchronization Approach Using Synthetic Magnetic Escherichia Coli","volume":"4","author":"Ling","year":"2019","journal-title":"Synth. Syst. Biotechnol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s00449-007-0180-6","article-title":"Modeling Synchronous Growth of Bacterial Populations in Phased Cultivation","volume":"31","author":"Noack","year":"2008","journal-title":"Bioprocess Biosyst. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1128\/jb.103.3.789-792.1970","article-title":"Synchronous Growth of Enteric Bacteria","volume":"103","author":"Shehata","year":"1970","journal-title":"J. Bacteriol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.cell.2016.06.052","article-title":"The Synchronization of Replication and Division Cycles in Individual E. Coli Cells","volume":"166","author":"Wallden","year":"2016","journal-title":"Cell"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/S0027-5107(00)00076-2","article-title":"The Bacterial Tryptophan Reverse Mutation Assay with Escherichia Coli WP2","volume":"455","author":"Mortelmans","year":"2000","journal-title":"Mutat. Res."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hamel, A., Roy, M., and Proudlock, R. (2016). The bacterial reverse mutation test. Genetic Toxicology Testing, Elsevier.","DOI":"10.1016\/B978-0-12-800764-8.00004-5"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"E2774","DOI":"10.1073\/pnas.1210309109","article-title":"Rate and Molecular Spectrum of Spontaneous Mutations in the Bacterium Escherichia Coli as Determined by Whole-Genome Sequencing","volume":"109","author":"Lee","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Szafra\u0144ska, A.K., Junker, V., Steglich, M., and N\u00fcbel, U. (2019). Rapid Cell Division of Staphylococcus Aureus during Colonization of the Human Nose. BMC Genom., 20.","DOI":"10.1186\/s12864-019-5604-6"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3540","DOI":"10.1128\/AAC.46.11.3540-3548.2002","article-title":"An Elevated Mutation Frequency Favors Development of Vancomycin Resistance in Staphylococcus Aureus","volume":"46","author":"Schaaff","year":"2002","journal-title":"Antimicrob. Agents Chemother."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"16033","DOI":"10.1038\/srep16033","article-title":"Natural Transformation Occurs Independently of the Essential Actin-like MreB Cytoskeleton in Legionella Pneumophila","volume":"5","author":"Juan","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/S0306-9877(03)00205-6","article-title":"Facilitation of Horizontal Transfer of Antimicrobial Resistance by Transformation of Antibiotic-Induced Cell-Wall-Deficient Bacteria","volume":"61","author":"Woo","year":"2003","journal-title":"Med. Hypotheses"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1093\/jac\/dkv391","article-title":"Horizontal Gene Transmission of the Cfr Gene MRSA Enterococcus: Role of Staphylococcus epidermidis as a reservoir and alternative pathway for the spread of linezolid resistance","volume":"71","author":"Cafini","year":"2016","journal-title":"J. Antimicrob. Chemother."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1624","DOI":"10.1093\/jac\/dkx056","article-title":"Resistance Gene Transfer: Induction of Transducing Phage by Sub-Inhibitory Concentrations of Antimicrobials Is Not Correlated to Induction of Lytic Phage","volume":"72","author":"Laing","year":"2017","journal-title":"J. Antimicrob. Chemother."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Dshalalow, J.H. (2001). Real Analysis: An Introduction to the Theory of Real Functions and Integration, Chapman & Hall. Studies in Advanced Mathematics.","DOI":"10.1201\/9781420036893"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Dshalalow, J.H. (2014). Foundations of Abstract Analysis, Springer-Verlag. [2nd ed.].","DOI":"10.1007\/978-1-4614-5962-0"},{"key":"ref_42","unstructured":"Number, B. (1972). Branching Processes, Springer-Verlag. 196 in Die Grundlehren Der Mathematischen Wissenschaften in Einzeldarstellungen Mit Besonderer Ber\u00fccksichtigung Der Anwendungsgebiete."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kimmel, M., and Axelrod, D. (2015). Branching Processes in Biology, Springer-Verlag. [2nd ed.]. Interdisciplinary Applied Mathematics.","DOI":"10.1007\/978-1-4939-1559-0"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1701\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:09:18Z","timestamp":1760141358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1701"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,16]]},"references-count":43,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081701"],"URL":"https:\/\/doi.org\/10.3390\/sym14081701","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,16]]}}}