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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 Goldbeter–Koshland kinetics ： ウィキペディア英語版 Goldbeter–Koshland kinetics The Goldbeter–Koshland kinetics describe a steady-state solution for a 2-state biological system. In this system, the interconversion between these two states is performed by two enzymes with opposing effect. One example would be a protein Z that exists in a phosphorylated form ZP and in an unphosphorylated form ''Z''; the corresponding kinase ''Y'' and phosphatase ''X'' interconvert the two forms. In this case we would be interested in the equilibrium concentration of the protein Z (Goldbeter–Koshland kinetics only describe equilibrium properties, thus no dynamics can be modeled). It has many applications in the description of biological systems. The Goldbeter–Koshland kinetics is described by the Goldbeter–Koshland function: : $\backslash begin$ z = \frac = G(v_1, v_2, J_1, J_2) &= \frac with the constants : $\backslash begin$ v_1 = k_1 () ; \ v_2 &= k_2 () ; \ J_1 = \frac ; \ J_2 = \frac; \ B = v_2 - v_1 + J_1 v_2 + J_2 v_1 \end Graphically the function takes values between 0 and 1 and has a sigmoid behavior. The smaller the parameters ''J''1 and ''J''2 the steeper the function gets and the more of a ''switch-like'' behavior is observed. Goldbeter–Koshland kinetics is an example of ultrasensitivity. ==Derivation== Since we are looking at equilibrium properties we can write : $\backslash begin$ \frac \ \stackrel\ 0 \end From Michaelis–Menten kinetics we know that the rate at which ZP is dephosphorylated is $r\_1\; =\; \backslash frac$ and the rate at which ''Z'' is phosphorylated is $r\_2\; =\; \backslash frac$. Here the ''K''M stand for the Michaelis–Menten constant which describes how well the enzymes ''X'' and ''Y'' bind and catalyze the conversion whereas the kinetic parameters ''k''1 and ''k''2 denote the rate constants for the catalyzed reactions. Assuming that the total concentration of ''Z'' is constant we can additionally write that ()0 = () + () and we thus get: : $\backslash begin$ \frac = r_1 - r_2 = \frac &-\frac = 0 \\ \frac &= \frac \\ \frac)}+ (1 - \frac)} &= \frac}+ \frac} \\ \frac &= \frac \qquad \qquad (1) \end with the constants : $\backslash begin$ z = \frac ; \ v_1 = k_1 () ; \ v_2 &= k_2 () ; \ J_1 = \frac ; \ J_2 = \frac; \ \qquad \qquad (2) \end If we thus solve the quadratic equation (1) for ''z'' we get: : $\backslash begin$ \frac &= \frac \\ J_2 v_1+ z v_1 - J_2 v_1 z - z^2 v_1 &= z v_2 J_1+ v_2 z - z^2 v_2\\ z^2 (v_2 - v_1) - z \underbrace_ + v_1 J_2 &= 0\\ z = \frac &= \frac \cdot \frac}\\ z &= \frac \cdot \frac Thus (3) is a solution to the initial equilibrium problem and describes the equilibrium concentration of () and () as a function of the kinetic parameters of the phosphorylation and dephosphorylation reaction and the concentrations of the kinase and phosphatase. The solution is the Goldbeter–Koshland function with the constants from (2): : $\backslash begin$ z = \frac = G(v_1, v_2, J_1, J_2) &= \frac 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「Goldbeter–Koshland kinetics」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.