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Mathematical Modeling for Inherited Diseases Hindawi Computational and Mathematical Methods in Medicine Volume 2017, Article ID 1975780, 6 pages https://doi.org/10.1155/2017/1975780 Research Article Mathematical Modeling for Inherited Diseases Saima Anis,1 Madad Khan,1 and Saqib Khan2 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan Correspondence should be addressed to Madad Khan; [email protected] Received 22 December 2016; Revised 7 April 2017; Accepted 24 May 2017; Published 9 July 2017 Academic Editor: Delfim F. M. Torres Copyright © 2017 Saima Anis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra. 1. Introduction 23 pairs of chromosomes, with a total of 46 chromosomes. A gene is a unit of hereditary information and lies on chro- We introduced a new nonassociative and noncommutative mosomes. A gene can take different forms which are called algebra which has several properties similar to nonassociative alleles. In these 23 pairs, one represents the sexual character andcommutativealgebras.Therelationoftheleftalmost in males; a locus which occurs on 22 pairs of chromosomes algebra with other nonassociative algebras is useful and inter- is called autosomal, whereas a locus on one pair is called esting to be known; in this regard, we found some relations sex-linked. Diploid cells are those which carry a double set with known algebras, namely, flexible, alternative, and Jordan of chromosomes and haploid cells are those which carry a algebras. We discussed some characteristics of this algebra single set of chromosomes. Diploid cells produce sex cells which is similar to a commutative and associative algebra. We called gametes through a process called meiosis. Mitosis is discussed some of the genetic properties of this algebra. the process of reproduction in haploid cells. The gametes cells meet each other and give a zygote which is again a diploid cell. Some Basics from Genetics. Here, we discuss some simple ideas in genetics for nongeneticists and for those who are Basics of Genetic Algebra. After the theory of Charles Darwin, purely related to mathematics. Each cell of an organism it was Gregor Mendel who studied the natural laws of genetic contains long thread-like structures called chromosomes inheritance and tried to express them in a mathematical which are located inside the nucleus of animals and plants. language. In [1–3], Etherington introduced a new method Chromosomes are made of protein and a single molecule of nonassociative algebras to study genetics. Holgate stud- of deoxyribonucleic acid (DNA). After cell division, chro- ied these algebras with genetic realizations in [4–7]. Reed mosomes pass from parent cells to the newborn cells. The discussed the nonassociative algebraic structure of genetic particular intersections that the DNA carries make each inheritance in [8]. The genetic algebra with mutation has living creature unique from others. The molecules of the DNA been discussed in [9, 10]. Several other authors studied aretoolong,whichcanbefittedinsidethecellsonlyby nonassociative algebras with genetic realizations (for details, chromosomes. Moreover, chromosomes play an important see [10–14]). role in copying and distributing DNA accurately in the whole Let be a nonempty set together with two binary process of cell division. Problems in chromosomes in new operations “+” and “⋅”whichsatisfiesalltheaxiomsofan cells can create serious issues like leukemia and some types of associative ring (algebra) except an associative property with cancer. Males and females have different chromosomes; that respecttomultiplication;then,itisknownasanonassociative is, females have two chromosomes in their cells whereas ring (algebra). Lie ring was introduced by defining and males have one and one chromosome. Humans have replacing a new binary operation [, ] = ,forall − ,, 2 Computational and Mathematical Methods in Medicine with ordinary multiplication of an associative ring (algebra); the juxtaposition between 1 and 2 shows a binary operation obviously, it is a nonassociative ring (algebra). By defining a which is not associative because (12)2 ≠ 1(22) or new binary operation ⋅ = (1/2)( +) on an associative equivalently (12)2 =1(22) says that both alleles algebra over a field whose characteristic is not equal to 2, we 1 and 2 are the same. Therefore, the associative law obtain another important nonassociative algebra known as does not hold for the gametic algebra and it also does not Jordan algebra. It is worth mentioning here that the theory hold for the zygotic algebra. Generally, algebras associated of nonassociative algebras is a fruitful branch of algebra. with genetics are commutative but nonassociative. In [3], Most importantly, the class of nonassociative algebras has Etherington proved that the zygotic algebra can be obtained closed connections with other branches of mathematics; from the gametic algebra through a commutative duplication also, it has closed connections with quantum mechanics, process. Now, it is interesting to note that there is a class of physics, biology, and other sciences. The crucial part of this algebras which is nonassociative and noncommutative but theory is the theory of nearly associative rings and algebras: possesses many characteristics similar to commutative and Lie, alternative, and Jordan algebras. In short, considering associativealgebrasandhascloserelationswithcommutative nonassociative algebras over real number fields has several algebras. Using notions from these algebras, we give a more applications in biology especially in genetics. Moreover, there general definition of gametic and zygotic algebras. Moreover, are some other classes of nonassociative algebras closely this nonassociative and noncommutative algebra has closed related to genetics which are popular among mathematicians connections with Jordan and flexible algebras. In addition and geneticists for their usefulness in genetics. Generally, to this, if it contains a left identity, then it satisfies the thesetypesofalgebrasarecommutativeandnonassociative. famous Jordan identity and the generalized Jordan identity. In fact, one can study the properties of genetics by making This algebra works mostly like a commutative algebra; for 2 2 2 2 the mathematical models using nonassociative, commutative instance, = ,forall, ,holdsinacommutative algebras.Tovisualizetheconceptofsuchalgebras,letuspay algebra while this equation also holds for a left almost attention to some specific classes of algebras like gametic, algebra, and if a left almost algebra contains a left identity zygotic, and copular algebras. ,then = (),forany,.Infact,thestructureis Here, we will discuss some simple algebras with genetic nonassociative and noncommutative but it possesses many realizations. In order to understand the algebraic properties properties which usually hold in associative and commutative in genetics, simple Mendelian inheritance has been consid- algebraic structures. Also, defining a new operation on this ered in [10]. If we consider a single gene with two alleles 1 algebra gives a commutative and associative algebra. and 2,aftermatingofalleles1 and 2, we get the zygotes In this paper, we will discuss those algebras which are 11, 12, 21,and22,where11,22 are known as nonassociative and noncommutative but have close connec- homozygotes while 12,21 are called heterozygotes. In tions with nonassociative and commutative algebras. We particular, 12 =21,whichmeansthatthecommutative generalized the definition available in [8] and introduced a law holds in this case, which genetically represents the notion new generalized definition for gamete algebras. Moreover, we that mating of allele 1 with allele 2 is the same as mating of introduce a new class of a nonassociative algebra called left 2 with 1. Let us consider the equation 12 = (1/2)(1 + almost algebra. 2). This linear equation represents the notion that 11 = Werestrictourselvesbyconsideringageneonaparticular 1 and 22 =2. The algebra with the multiplication locus on a chromosome. Here, we begin with the definition tableaboveisknownasgametealgebra.Moreover,the of gametic algebra [8]. Consider gametes 1,2,..., zygotic algebra can be obtained from the gametic algebra by asbasiselementsofan-dimensional real vector space. commutative duplication. From mathematical perspectives, Multiplication is defined by = ∑ , =1 (1) such that 0≤ ≤1, ,,=1,2,...,, ∑ =1, ,,=1,2,...,, =, ,,=1,2,..., =1 where represent the relative gene frequencies. Let , ,,where ∈ is a left almost semigroup with left The resulting algebra S is called an -dimensional identity .Then, gametic algebra. () =() , ∀, , ∈. (2) Some Basic Notions of Nonassociative Algebras.Agroupoid is called a left almost semigroup if it satisfies the following left But the converse of the above statement is not true. invertive law, () = ,forall() , ,.Holgatecalled ∈ If a left almost semigroup contains a right identity, then it italeftinvertivegroupoid[4].Anelement of a groupoid becomes a commutative semigroup. A left almost semigroup is called left (right)
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