Matematičko modeliranje kinetike reakcija

master's thesis

• Title (english): Mathematical modeling of reaction kinetics
• Author: Benjamin Vinković
• Mentor: Marko Radulović (mentor)
• Committee member: Marko Radulović (predsjednik povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu smo se bavili enzimskim reakcijama i njihovim modelima. Pokazali smo kako napraviti model iz enzimske reakcije i ujedno ga analizirali. U prvom poglavlju smo definirali sustave diferencijalnih jednadžbi, prezentirali uvod i najosnovnije pojmove vezane uz teoriju stabilnosti rješenja sustava diferencijalnih jednadžbi te definirali i opisali metodu asimptotskog sparivanja. U drugom poglavlju smo se bavili načinom kako izraditi matematički model neke enzimske reakcije, čime dobijemo sustav diferencijalnih jednadžbi te obavili analizu samog tog modela. Analizu smo prvo proveli na najjednostavnijem modelu enzimske reakcije - Michaelis-Mentenovoj kinetici. Kvantitativnu analizu smo napravili metodom asimptotskog sparivanja čime smo dodatno pojasnili primjenu te metode. Potom smo analizirali i dva modela za drugačije tipove reakcija, model kinetike kooperativnih i inhibicijskih reakcija. Pokazalo se da je metoda asimptotskog sparivanja dobra aproksimativna metoda za traženje rješenja sustava diferencijalnih jednadžbi za modele kinetike reakcija. U zadnjem poglavlju smo se kratko osvrnuli na stabilnost rješenja sustava, tj. kritične točke i moguća ponašanja kritičnih točki.

Abstract (english)

In this work we deal with enzyme reactions and models of said reactions. We show how to make a model out of the reaction and also how to perform the mathematical analysis. In Chapter 1, we defined basic mathematical definitions and terms necessary for the work in further chapters. We define systems of differential equations, define the necessary terms of stability theory and describe method of matched asymptotic expansions. Chapter 2 introduces how to make a model of a reaction which gives us a system of differential equations and how to analyze that model. We first perform an analysis of the easiest model of enzyme reactions - Michaelis-Menten kinetics. Quantitative analysis was performed using method of matched asymptotic expansions which furthered our understanding of the method. After that, we analyze two more complex reactions and their models, cooperative and inhibition model. We show that the method of matched asymptotic expansions is a good approximate method for finding solutions to systems of differential equations. In the last Chapter, we briefly turn our attention to stability of solutions to systems of differential equations, that is steady states.

• Keywords: enzimske reakcije; diferencijalne jednadžbe; metoda asimptotskog sparivanja; matematički model enzimske reakcije; Michaelis-Mentenovoj kinetika
• Keywords (english): enzyme reactions; differential equations; method of matched asymptotic expansions; Michaelis-Menten kinetics
• Language: croatian
• URN:NBN: urn:nbn:hr:217:765009
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2023-08-25 07:50:54