Tangle Equations Of The Recombination Model Of DNA

This paper employs knot theory and the model of tangles to research how the circular DNA molecules change under the action of enzymes such as topoisomerases and recombinases.Meanwhile,this paper also reveals how the circular DNA molecules break the original structure and change the endpoints to reconnect them so that they can change the topological structure in the process of enzymatic reaction.In the whole process of recombination experiment of the circular DNA molecule,the role of the enzyme can be viewed as a tangle surgery which replaces the deleted tangles by new tangles.Therefore,it is very significant to study and analyze the abstracting model of tangles.Sometimes,n rounds of processive site-specific recombination will happen in the process.We use a set of?n+1?simultaneous tangle equations to construct a tangle system:N?S?=K₀?Substrate knot or 2-component link?N?S+iM?=K_i?1?i?n,i?n?N*??ith round product knot or 2-component link?.Where M denotes a rational tangle and S is either a rational tangle or the sum of two rational tangles,they are unknown.iM is the tangle sum of i copies of M and N denotes the numerator construction which is viewed as a map.In addition,K_i is known 2-bridge knots.There is a link between a rational tangle and a 2-bridge knot in this paper.And the system of four tangle equations N?S+iM?=K_i?i=0,1,2,3? solved by making use of K_i?i=0,1,2,3? crossing number and continued fraction of the tangle.This paper examines the condition of rational solution of tangle equations.When the tangle system includes four tangle equations,there is either a unique solution or no solution.Therefore,we give the general solution of tangle equations of N?S+iM?=K_i?1?i?n,i?n?N*?.Finally,we take full advantage of the achieved general solution to modify following tangle equations:N?S?=K₀=b?13,3?=b?13,9?=<4,2,1>N?S+M?=K₁=b?4,3?=b?4,3?=<1,2,1>N?S+M+M?=K₂=b?5,2?=b?5,3?=<2,1,1>N?S+M+M+M?=K₃=b?14,9?=b?14,11?=<1,1,1,3,1>We present a method of variable control,which change element of vector representative of some K_i?i=0,1,2,3? .And we research what variable meet so that tangle equations have a unique solution.We seek out anther K_i that is satisfied with the condition of tangle equations and make tangle equations have the only solution.This tangle equations system is put into use well.Analogously,we will get the change of topological structure of anther DNA molecule or some DNA molecules by studying original circular DNA molecule's variation in the process of enzymatic reaction when processive site-specific recombination?n iterations?happen.So we can reduce the amount of work time by avoiding research one by one.