A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this articl...A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this article, concerning this conjecture, a topological model of prion proteins （PrPc, PrPsc） called a prion-tangle is introduced to discuss a question of whether or not the prion proteins are easily entangled by an approach from the mathematical knot theory. It is noted that any prion-string with trivial loop which is a topological model of a prion protein can not be entangled topologically unless a certain restriction such as ＂Rotaxsane Property＂ is imposed on it. Nevertheless, it is shown that any two split prion-tangles can be changed by a one-crossing change into a non-split prion-tangle with the given prion-tangles contained while some attentions are paid to the loop systems. The proof is made by a mathematical argument on knot theory of spatial graphs. This means that the question above is answered affirmatively in this topological model of prion-tangles. Next, a question of what is the simplest topological situation of the non-split prion-tangles is considered. By a mathematical argument, it is determined for every n 〉 1 that the minimal crossing number of n-string non-split prion-tangles is 2n or 2n-2, respectively, according to whether or not the assumption that the loop system is a trivial link is counted.展开更多
文摘A conformal structure of a prion protein is thought to cause a prion disease by S.B. Prusiner＇s theory. Knot theory in mathematics is useful in studying a topological difference of topological objects. In this article, concerning this conjecture, a topological model of prion proteins （PrPc, PrPsc） called a prion-tangle is introduced to discuss a question of whether or not the prion proteins are easily entangled by an approach from the mathematical knot theory. It is noted that any prion-string with trivial loop which is a topological model of a prion protein can not be entangled topologically unless a certain restriction such as ＂Rotaxsane Property＂ is imposed on it. Nevertheless, it is shown that any two split prion-tangles can be changed by a one-crossing change into a non-split prion-tangle with the given prion-tangles contained while some attentions are paid to the loop systems. The proof is made by a mathematical argument on knot theory of spatial graphs. This means that the question above is answered affirmatively in this topological model of prion-tangles. Next, a question of what is the simplest topological situation of the non-split prion-tangles is considered. By a mathematical argument, it is determined for every n 〉 1 that the minimal crossing number of n-string non-split prion-tangles is 2n or 2n-2, respectively, according to whether or not the assumption that the loop system is a trivial link is counted.
