{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T11:07:03Z","timestamp":1768820823888,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T00:00:00Z","timestamp":1768176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471038"],"award-info":[{"award-number":["12471038"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct left-symmetric algebras arising from DNA insertion operations. That is, we define a modified insertion operation by weighting the simplified insertion. It generalizes the left-symmetric algebra constructed from the simplified DNA insertion operation. We prove that the algebra F(R) (over a field F of characteristic 0, with R being an infinite free semigroup generated by DNA nucleotides {A,G,C,T}) forms a left-symmetric algebra if and only if the function f satisfies a certain multiplicative condition for all positive integers m, n, and p. A key example of such a function is f(m,n)=exp{g(m,n)}, where g(m,n)=k\u00b7mn, and k is a fixed positive number, which effectively models length-dependent DNA insertion dynamics. This work contributes an algebraic framework that may be useful for quantitative modeling of DNA recombination processes.<\/jats:p>","DOI":"10.3390\/axioms15010055","type":"journal-article","created":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T14:57:25Z","timestamp":1768229845000},"page":"55","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Left-Symmetric Algebras Arising from Modified DNA Insertion Operations"],"prefix":"10.3390","volume":"15","author":[{"given":"Chen","family":"Yuan","sequence":"first","affiliation":[{"name":"School of Science, Beijing Forestry University, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhixiang","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang University, Hangzhou 310027, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Beijing Forestry University, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,12]]},"reference":[{"key":"ref_1","unstructured":"Sverchkov, S.R. (2010, January 24\u201326). Algebraic Theory of DNA Recombination. Proceedings of the ISCA 2nd International Conference on Bioinformatics and Computational Biology, BICoB-2010, Sheraton Waikiki Hotel, Honolulu, HI, USA."},{"key":"ref_2","unstructured":"Bremner, M. (2005, January 15\u201317). DNA computing, insertion of words and left-symmetric algebras. Proceedings of the 2005 Maple Conference, Waterloo, ON, Canada."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1090\/S0273-0979-97-00712-X","article-title":"Algebraic structure of genetic inheritance","volume":"34","author":"Reed","year":"1997","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1090\/conm\/672\/13470","article-title":"Invitation to research of new mathematics from biology: Evolution algebras","volume":"672","author":"Tian","year":"2016","journal-title":"J. Top. Funct. Anal. Algebr. Contemp. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"053302","DOI":"10.1063\/5.0215165","article-title":"Infinite-dimensional genetic and evolution algebras generated by Gibbs measures","volume":"66","author":"Coletti","year":"2025","journal-title":"J. Math. Phys."},{"key":"ref_6","first-page":"340","article-title":"Convex homogeneous cones","volume":"12","author":"Vinberg","year":"1963","journal-title":"Transl. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"448","DOI":"10.2991\/jnmp.1999.6.4.5","article-title":"What a classical r-matrix really is","volume":"6","author":"Kupershmidt","year":"1999","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s002200050779","article-title":"Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem","volume":"210","author":"Connes","year":"2000","journal-title":"Commun. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"105280","DOI":"10.1016\/j.geomphys.2024.105280","article-title":"Cohomologies, non-abelian extensions and wells exact sequences of \u03bb-differential pre-lie algebras","volume":"204","author":"Sun","year":"2024","journal-title":"J. Geom. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1155\/S1073792801000198","article-title":"Pre-lie algebras and the rooted trees operad","volume":"2001","author":"Chapoton","year":"2001","journal-title":"Int. Math. Res. Not."},{"key":"ref_11","first-page":"242","article-title":"On the theory of the analytical forms called trees","volume":"3","author":"Cayley","year":"1890","journal-title":"Collect. Math. Pap. Arthur Cayley"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"515","DOI":"10.24033\/bsmf.1572","article-title":"Domaines born\u00e9s homogenes et orbites de groupes de transformations affines","volume":"89","author":"Koszul","year":"1961","journal-title":"Bull. Soc. Math. Fr."},{"key":"ref_13","first-page":"215","article-title":"Affine structures on complex manifolds","volume":"5","author":"Matsushima","year":"1968","journal-title":"Osaka J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"267","DOI":"10.2307\/1970343","article-title":"The cohomology structure of an associative ring","volume":"78","author":"Gerstenhaber","year":"1963","journal-title":"Ann. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/aima.2001.1993","article-title":"Theory of finite pseudoalgebras","volume":"162","author":"Bakalov","year":"2001","journal-title":"Adv. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1142\/S1005386723000354","article-title":"Left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra","volume":"30","author":"Chen","year":"2023","journal-title":"Algebra Colloq."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"104343","DOI":"10.1016\/j.jpaa.2025.108086","article-title":"Left-symmetric superalgebras and Lagrangian extensions of Lie superalgebras in characteristic 2","volume":"229","author":"Benayadi","year":"2025","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.1002\/mana.201300339","article-title":"Left-symmetric algebroids","volume":"289","author":"Liu","year":"2016","journal-title":"Math. Nachrichten"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.difgeo.2018.04.003","article-title":"Left-symmetric bialgebroids and their corresponding Manin triples","volume":"59","author":"Liu","year":"2018","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s11005-017-0973-8","article-title":"Pre-symplectic algebroids and their applications","volume":"108","author":"Liu","year":"2018","journal-title":"Lett. Math. Phys."},{"key":"ref_21","first-page":"323","article-title":"Left-symmetric algebras, or pre-Lie algebras in geometry and physics","volume":"4","author":"Burde","year":"2006","journal-title":"Cent. Eur. J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5381","DOI":"10.1080\/00927872.2018.1468903","article-title":"Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a,b)","volume":"46","author":"Liu","year":"2018","journal-title":"Commun. Algebra"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2250086","DOI":"10.1142\/S0219498822500864","article-title":"Compatible left-symmetric algebraic structures on high rank Witt and Virasoro algebras","volume":"21","author":"Xu","year":"2022","journal-title":"J. Algebra Its Appl."},{"key":"ref_24","first-page":"201","article-title":"Ordered constraint semigroups: A novel class of algebraic systems","volume":"37","author":"Lekkoksung","year":"2025","journal-title":"J. Math. Comput. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.tcs.2004.03.043","article-title":"Families of languages defined by ciliate bio-operations","volume":"320","author":"Daley","year":"2004","journal-title":"Theor. Comput. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"482","DOI":"10.22436\/jmcs.035.04.07","article-title":"The stability of bi-derivations and bihomomorphisms in Banach algebras","volume":"35","author":"Khan","year":"2024","journal-title":"J. Math. Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"373904","DOI":"10.1155\/2012\/373904","article-title":"Approximation of Homomorphisms and Derivations on non-Archimedean Lie C*-algebras via Fixed Point Method","volume":"2012","author":"Cho","year":"2012","journal-title":"Discret. Dyn. Nat. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T09:17:09Z","timestamp":1768468629000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,12]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010055"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010055","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,12]]}}}