Longbotham, Abraham

Radulescu, Anca R

English

VCU Scholars Compass

Virginia Commonwealth University VCU Scholars Compass Biology and Medicine Through Mathematics Conference 2022 May 18th, 5:00 PM - 7:00 PM Effects of local mutations in quadratic iterations Anca R. Radulescu State University of New York at New Paltz, radulesa@newpaltz.edu Abraham Longbotham SUNY New Paltz, longbotha1@newpaltz.edu Follow this and additional works at: https://scholarscompass.vcu.edu/bamm  Part of the Dynamic Systems Commons, Medicine and Health Sciences Commons, and the Non-linear Dynamics Commons https://scholarscompass.vcu.edu/bamm/2022/wed/50 This Event is brought to you for free and open access by the Dept. of Mathematics and Applied Mathematics at VCU Scholars Compass. It has been accepted for inclusion in Biology and Medicine Through Mathematics Conference by an authorized administrator of VCU Scholars Compass. For more information, please contact libcompass@vcu.edu. Effects of local mutations in quadratic iterationsAnca Rǎdulescu, SUNY New Paltz, radulesa@newpaltz.eduAbraham Longbotham, SUNY New Paltz, longbota1@newpaltz.eduThe study of copying mechanisms is of great importance to genetics. We study in a theoretical system how a mutation(replication error) affects the temporal evolution of the system, on both a local and global scale (from tumor formationto overall systemic unsustainability).We introduce a new mathematical framework for studying replication mechanisms, in the form of discrete iterationsof complex quadratic maps. This approach builds upon a century of existing knowledge of iterated maps towards ob-taining results with potential impact on applications. More specifically, our modeling framework considers a “correct”function acting on the complex plane (representing the space of genes to be copied) and a “mutation,” acting at aspecific focal point, of a given size r, and moves radially toward an outer radius R. We use the Julia set of the systemto quantify simultaneously the long-term behavior of the entire space under such transformations. We analyze howthe position, timing and size of the mutation can alter the topology of the Julia set, hence the system’s long-termevolution, its progression into disease, but also its ability to recover or heal.In the context of genetics, such results may help shed some light on aspects such as the importance of location,size and type of mutation when evaluating a system’s prognosis, and of customizing timing of treatment to addresseach specific situation. Our current work is a proof of principle. Once these aspects are understood theoretically, theycan be further applied to empirically driven genetic models, validated with data and used for predictions.1

Effects of local mutations in quadratic iterations

https://scholarscompass.vcu.edu/cgi/viewcontent.cgi?article=1463&context=bamm

Effects of local mutations in quadratic iterations

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