Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi

İLKNUR KUŞBEYZİ AYBAR (YEDİTEPE Ü. FEN-EDEBİYAT F. MATEMATİK B.)
ORHAN ÖZGÜR AYBAR (PİRİ REİS Ü.)
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Proje Grubu: MFAG Sayfa Sayısı: 58 Proje No: 119F017 Proje Bitiş Tarihi: 01.03.2023 Metin Dili: Türkçe DOI: 119F017 İndeks Tarihi: 14-03-2024

Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi

Öz:
Biyokimyasal olayları temsil eden modellerin özellikleri, uygulamalarda ilgi çekici olmaya devam eden karmaşık kararlı durumlar ve sayısız durum geçişleri sergiler. Çatallanma teorisi ve hesaplamalı cebirin etkili yöntemlerini birleştirerek bu durumları incelemek, yakınında modelin niteliksel davranışının değiştiği çatallanma noktalarını ve belirli davranışı teşvik eden parametre aralıklarını elde etmek için son derece önemlidir. Bu proje, bazı biyokimyasal reaksiyon modellerinin daha önce tespit edilmemiş birkaç temel özelliğini ortaya koyan hesaplamalı cebirsel bir yaklaşım sunmaktadır. Bu yaklaşım sayesinde, önce, Lyapunov fonksiyonunu kullanarak, Brusselator modeli için Hopf çatallanmasını incelemek için ilk Lyapunov katsayısının genel formunu hesaplamaktayız. Daha sonra, en küçük üç boyutlu biyokimyasal reaksiyon modelindeki Hopf çatallanmasını incelemek için üçüncü dereceye kadar bir merkez manifold elde etmekteyiz. Bu iki model için elde ettiğimiz sonuçları sayısal benzetim ile görsel olarak sunmaktayız. Önde gelen metabolik yollardan biri olan glikoliz, bu temel süreci tanımlayan biyokimyasal modellerin pozitif kararlı durumlarında ortaya çıkan birçok farklı periyodik salınımı içermektedir. Ara ürünlerin moleküler difüzyonunu kullanan modellerden biri, sürekli salınımları açıklayabilmek için Higgins biyokimyasal modelidir. Projemizde, minimal Higgins modeli için hesaplamalı cebir yardımıyla model parametrelerinin genel değerleri için merkez odak problemini çalışmaktayız. İlk Lyapunov sayısının genel bir formunu bularak modelin her zaman kararlı bir odak noktasına sahip olduğunu göstermekteyiz. Daha sonra, model parametrelerinden ikisini değiştirerek, modelin kararlı odak noktası için periyot fonksiyonunun ilk üç katsayısını elde etmek için bir yaklaşım sunmaktayız. Bu sayede, tekil noktanın aslında [1,2] tipinde ikili-zayıf monodromik bir denge noktası olduğunu kanıtlamaktayız. Ek olarak, seçilen bir parametre için bir kritik periyodun küçük pertürbasyonlardan sonra bu tekil noktadan çatallanması için iki (küçük) aralık olduğunu göstermekteyiz. Son olarak, elde ettiğimiz tüm bulguları sayısal benzetimler ile görselleştirmekteyiz.
Anahtar Kelime: Limit çevrimi kararlılık biyokimyasal modeller kritik periyotlar çatallanmalar

Dynamical analysis of biochemical systems using the methods of computational algebra

Öz:
The characteristics of models representing biochemical phenomena exhibit complicated steady states and numerous state transitions that remain interesting in applications. Examining these states by combining the effective methods of bifurcation theory and computational algebra is profoundly appreciable to obtain bifurcation points near which the qualitative behavior of the model varies and parameter ranges that promote particular behavior. This study reveals several essential characteristics of two biochemical reaction models that have not been detected before. Utilizing the Lyapunov function, we compute the general form of the first Lyapunov coefficient to determine Hopf bifurcation for the Brusselator model. Then, for the smallest three dimensional biochemical reaction model, we obtain a center manifold up to third-degree to study Hopf bifurcation in this system. We demonstrate all results by numerical simulation. Glycolysis, one of the leading metabolic pathways, involves many different periodic oscillations emerging at positive steady states of the biochemical models describing this essential process. One of the models employing the molecular diffusion of intermediates is the Higgins biochemical model to explain sustained oscillations. In this paper, we investigate the center-focus problem for the minimal Higgins model for general values of the model parameters with the help of computational algebra. We demonstrate that the model always has a stable focus point by finding a general form of the first Lyapunov coefficient. Then, varying two of the model parameters, we obtain the first three coefficients of the period function for the stable focus point of the model and prove that the singular point is actually a bi-weak monodromic equilibrium point of type [1,2]. Additionally, we prove that there are two (small) intervals for a chosen parameter for which one critical period bifurcates from this singular point after small perturbations.
Anahtar Kelime: Limit çevrimi kararlılık biyokimyasal modeller kritik periyotlar çatallanmalar

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APA Kusbeyzi Aybar I, Aybar O (2023). Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. , 0 - 58. 119F017
Chicago Kusbeyzi Aybar Ilknur,Aybar Orhan Ozgur Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. (2023): 0 - 58. 119F017
MLA Kusbeyzi Aybar Ilknur,Aybar Orhan Ozgur Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. , 2023, ss.0 - 58. 119F017
AMA Kusbeyzi Aybar I,Aybar O Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. . 2023; 0 - 58. 119F017
Vancouver Kusbeyzi Aybar I,Aybar O Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. . 2023; 0 - 58. 119F017
IEEE Kusbeyzi Aybar I,Aybar O "Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi." , ss.0 - 58, 2023. 119F017
ISNAD Kusbeyzi Aybar, Ilknur - Aybar, Orhan Ozgur. "Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi". (2023), 0-58. https://doi.org/119F017
APA Kusbeyzi Aybar I, Aybar O (2023). Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. , 0 - 58. 119F017
Chicago Kusbeyzi Aybar Ilknur,Aybar Orhan Ozgur Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. (2023): 0 - 58. 119F017
MLA Kusbeyzi Aybar Ilknur,Aybar Orhan Ozgur Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. , 2023, ss.0 - 58. 119F017
AMA Kusbeyzi Aybar I,Aybar O Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. . 2023; 0 - 58. 119F017
Vancouver Kusbeyzi Aybar I,Aybar O Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi. . 2023; 0 - 58. 119F017
IEEE Kusbeyzi Aybar I,Aybar O "Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi." , ss.0 - 58, 2023. 119F017
ISNAD Kusbeyzi Aybar, Ilknur - Aybar, Orhan Ozgur. "Hesaplamalı Cebir Yöntemleri Kullanılarak Biyokimyasal Sistemlerin Dinamik Analizi". (2023), 0-58. https://doi.org/119F017
Sayfa Oluşturulma Tarihi
24-04-2024 12:55

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