Резюме
Щербик В. В., Бучацький Л. П. Будова центросоми і алгебра Кліффорда.
Розглянуто основні структурні елементи центросоми з використанням методів некомутативної геометрії. Головну роль в інтерпретації структурних особливостей центросоми виконують алгебра Кліффорда Cl(6) та її подалгебри. Зростання дочірніх процентріолей ортогонально центріолям центросоми обумовлено множиною еквівалентних алгебр Кліффорда Cl(3, 3) із різною знаковою сигнатурою. Триплет α–,β–,γ– тубуліну є різною реалізацією алгебри Кліффорда Cl(2, 1). Запропоновано функцію Q(p, q) = pq/q числа мікротрубочок, яка відповідає алгебрі Кліффорда Cl(p, q). Еволюція центросоми описується статистичним оператором фон Неймана, біфуркація якого визначає відмінність материнської та дочірньої центріолей.
Ключові слова: центросома, мікротрубочки, протофіламенти, алгебра Кліффорда.
Резюме
Щербик В. В., Бучацкий Л. П. Строение центросомы и алгебра Клиффорда.
Рассмотрены основные структурные элементы центросомы с использованием методов некоммутативной геометрии. Главную роль в интерпретации структурных особенностей центросомы выполняют алгебра Клиффорда Cl(6) и ее подалгебры. Рост дочерних процентриолей ортогонально центриолям центросомы обусловлен множеством эквивалентных алгебр Клиффорда Cl(3, 3) с разной знаковой сигнатурой. Триплет α–,β–,γ– тубулина является различной реализацией алгебры Клиффорда Cl(2, 1). Предложена функция Q(p, q) = pq/q числа микротрубочек, которая соответствует алгебре Клиффорда Cl(p, q). Эволюция центросомы описывается статистическим оператором фон Неймана, бифуркация которого определяет различие материнской и дочерней центриолей.
Ключевые слова: центросома, микротрубочки, протофиламенты, алгебра Клиффорда.
Summary
Stcherbic V. V., Buchatsky L. P. Centrosome structure and Clifford algebra.
The methods of non–commutative geometry was used to consider the main structural elements of the centrosome. Clifford algebra Cl(6) and its subalgebras perform major role in the interpretation of structural centrosome features. Growth of daughter procentrioles orthogonal to centrioles of centrosome is due to equivalent set of Clifford algebra Cl(3, 3) with a different sign signature.Triplet α–,β–,γ– tubulin is a different realization of the Clifford algebra Cl(2, 1). It was proposed the function Q(p, q) = pq/q for number of microtubules, which corresponds to the Clifford algebra Cl(p, q). Evolution of centrosome is described by von Neumann statistical operator which bifurcation defines the difference between mother and daughter centrioles.
Keywords: centrosome, microtubules, protofilaments, Clifford algebra.
Рецензент: д.біол.н., проф. Л.Т. Міщенко
УДК 576.311.346: 576.311.348.7
Киевский национальный университет имени Тараса Шевченко
Taras Shevchenko National University of Kyiv
stcherbic_v@bigmir.net, irido1@bigmir.net
Centrosomes are nonmembrane organelles present in most vertebrate cells and lower plants. Centrosome consists of two centrioles and pericentriolar material. One centriole, the mother, has appendages and pericentriolar satellites at the distal end. Another centriole, a daughter, has no additional structures. The centrosome is replicated in every cell cycle. After replication in a prophase cell cycle, centrosomes are located on opposite ends of metaphase spindle.
Our proposed explanation of the centrosome structure is based on the properties of noncommutative geometry. Noncommutative geometry is the triplet of operators (A, H, D), where
A - algebra, which determines the geometry of molecular structures;
H - set of functions of the algebra A;
D - operator containing all information about the geometry.
We use the Clifford algebras A = Cl(6) and their subalgebras.
Clifford algebras Cl(p, q) of different signature (p, q), where p + q = 6, are arranged as Hadamard matrix which maps in each string the signs of squares of the basis elements and signs of squares of various products of the basis elements.
Main Clifford algebra is the algebra Cl(3, 3). Growth of protofilaments is associated with a set of equivalent Clifford algebras Cl(3, 3) with a different sign signature. The replacing real on imaginary basis elements, imaginary on negative real basis elements in Clifford algebra Cl(3, 3) does not change the structure of the algebra subsets of vectors and corresponds to the left rotation of complex plane at an angle of 90°, which determines the location of centrioles perpendicular to each other and left rotation of the triplet microtubules relatively to proximal (initiating) end of centrioles. The dominant role of the nine real 2- and 4-vectors, which determine the ninth-order central symmetry of cartwheel, corresponds to base bundle of even vectors of the Clifford algebra Cl(3, 3). The shift of basis vectors of the Clifford algebra Cl(3, 3): e1,…, e6 ® e0′,…, e5′ leads to Clifford subalgebra Cl(3, 2) of signature (– – + + +). Shift algebras Cl(1, 0) Ì …Ì Cl(6, 5) are uniformly increase with step Cl(1, 1) and correspond to the building of six cartwheels inside procentrioles.
Representation of the number of microtubules is based on the assumption that sum of three numbers of the n-vectors states of the Clifford algebra Cl(3, 3) is equal to the number of protofilaments in each tube, and the product of n-vectors by n-vectors states is equal to positive part with minus sign of the signature algebra Cl(3, 3), i.e. –36, regardless of the number of microtubules.
For the Clifford algebra Cl(p, q) there is defined the map Q(p, q) = pq/q for the number of centrioles microtubules. The map Q(p, q) is the ratio of the set of functions q ® p to the set of initial elements q.
γ-TuRC is defined by Clifford algebras Cl(5, 1) @ Cl(1, 5), the first of which refers to the mother and the second to the daughter centrioles .
Triplet abg- tubulin is a representation of the Clifford algebra Cl(2, 1) with different sign signature. Interaction (ordering) between a-,b-,g-tubulin molecules occurs in four dimension space with noncommutative geometry.
Distal appendages of mother centriole are formed on all 9 triplets of centriole microtubules and in the Clifford algebra Cl(3, 3) defines a subset of vectors {[1], [3], [5], [6]}, which is in addition to a subset of vectors {[0], [2], [4]}of microtubules .
Subdistal appendages – pericentriolar satellites – are realized by set of proteins which are the representation of the sum by two Clifford algebras Cl(4, 2) Å Cl(2, 4), since in this case the 3-state vectors have the form 2[10+ 10–] and may be represented by globular proteins.
Centrosome evolution is described by the von Neumann statistical operator which bifurcation occurs because of a violation of equality Cl(p, q) @ Cl(q, p) and determines the difference in mother and daughter centrioles .
Conclusions. Clifford algebra Cl(3, 3) describes the structure of the centrosome within the noncommutative geometry. Clifford algebra Cl(6) performs a triple role: defines the structure of the centrosome, the microtubule structure and appendages, structure of proteins a-,b-,g- tubulin.
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