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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク ideal chain ： ウィキペディア英語版 ideal chain An ideal chain (or freely-jointed chain) is the simplest model to describe polymers, such as nucleic acids and proteins. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers. Although it is simple, its generality gives insight about the physics of polymers. In this model, monomers are rigid rods of a fixed length ''l'', and their orientation is completely independent of the orientations and positions of neighbouring monomers, to the extent that two monomers can co-exist at the same place. In some cases, the monomer has a physical interpretation, such as an amino acid in a polypeptide. In other cases, a monomer is simply a segment of the polymer that can be modeled as behaving as a discrete, freely jointed unit. If so, ''l'' is the Kuhn length. For example, chromatin is modeled as a polymer in which each monomer is a segment approximately 14-46 kbp in length. == The model == ''N'' monomers form the polymer, whose total unfolded length is: :$L\; =\; N\backslash ,l$, where ''N'' is the number of monomers. In this very simple approach where no interactions between monomers are considered, the energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell–Boltzmann distribution. Let us call $\backslash vec\; R$ the total end to end vector of an ideal chain and $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ the vectors corresponding to individual monomers. Those random vectors have components in the three directions of space. Most of the expressions given in this article assume that the number of monomers ''N'' is large, so that the central limit theorem applies. The figure below shows a sketch of a (short) ideal chain. The two ends of the chain are not coincident, but they fluctuate around each other, so that of course: :$\backslash langle\; \backslash vec\; \backslash rangle\; =\; \backslash Sigma\_^N\; \backslash langle\; \backslash vec\; r\_i\backslash rangle\; =\; \backslash vec\; 0~$ Throughout the article the $\backslash langle\; \backslash rangle$ brackets will be used to denote the mean (of values taken over time) of a random variable or a random vector, as above. Since $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ are independent, it follows from the Central limit theorem that $\backslash vec\; R$ is distributed according to a normal distribution (or gaussian distribution): precisely, in 3D, $R\_x,\; R\_y,$ and $R\_z$ are distributed according to a normal distribution of mean ''0'' and of variance: :$\backslash sigma\; ^2\; =\; \backslash langle\; R\_x^2\; \backslash rangle\; -\; \backslash langle\; R\_x\; \backslash rangle\; ^2\; =\backslash langle\; R\_x^2\; \backslash rangle\; -0$ :$\backslash langle\; R\_x^2\backslash rangle\; =\backslash langle\; R\_y^2\; \backslash rangle\; =\; \backslash langle\; R\_z^2\backslash rangle\; =\; N\backslash ,\backslash frac$ So that $\backslash langle\; \backslash vec\; \backslash rangle\; =\; N\backslash ,\; l^2\; =\; L\; \backslash ,\; l~$. The end to end vector of the chain is distributed according to the following probability density function: :$P(\backslash vec\; R\; )\; =\; \backslash left\; (\; \backslash frac\; \backslash right\; )^e^\}$ The average end-to-end distance of the polymer is: :$\backslash sqrt\; =\; \backslash sqrt\; N\; \backslash ,\; l\; =\; \backslash sqrt~$ A quantity frequently used in polymer physics is the radius of gyration: :$\backslash mathit\_G\; =\; \backslash frac$ It is worth noting that the above average end-to-end distance, which in the case of this simple model is also the typical amplitude of the system's fluctuations, becomes negligible compared to the total unfolded length of the polymer $N\backslash ,l$ at the thermodynamic limit. This result is a general property of statistical systems. Mathematical remark: the rigorous demonstration of the expression of the density of probability $P(\backslash vec\; R)$ is not as direct as it appears above: from the application of the usual (1D) central limit theorem one can deduce that $R\_x$, $R\_y$ and $R\_z$ are distributed according to a centered normal distribution of variance $N\backslash ,\; l^2/3$. Then, the expression given above for $P(\backslash vec\; R)$ is not the only one that is compatible with such distribution for $R\_x$, $R\_y$ and $R\_z$. However, since the components of the vectors $\backslash vec\; r\_1,\backslash ldots\; ,\backslash vec\; r\_N$ are uncorrelated for the random walk we are considering, it follows that $R\_x$, $R\_y$ and $R\_z$ are also uncorrelated. This additional condition can only be fulfilled if $\backslash vec\; R$ is distributed according to $P(\backslash vec\; R)$. Alternatively, this result can also be demonstrated by applying a multidimensional generalization of the central limit theorem, or through symmetry arguments. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ideal chain」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.