The simplest way to account for the influence of diffusion on the kinetics of multisite phosphorylation is to modify the rate constants in the conventional rate equations of chemical kinetics. We have previously shown that this is not enough and new transitions between the reactants must also be introduced. Here we extend our results by considering enzymes that are inactive after modifying the substrate and need time to become active again. This generalization leads to a surprising result. The introduction of enzyme reactivation results in a diffusion-modified kinetic scheme with a new transition that has a negative rate constant. The reason for this is that mapping non-Markovian rate equations onto Markovian ones with time-independent rate constants is not a good approximation at short times. We then developed a non-Markovian theory that involves memory kernels instead of rate constants. This theory is now valid at short times, but is more challenging to use. As an example, the diffusion-modified kinetic scheme with new connections was used to calculate kinetics of double phosphorylation and steady-state response in a phosphorylation-dephosphorylation cycle. We have reproduced the loss of bistability in the phosphorylation-dephosphorylation cycle when the enzyme reactivation time decreases, which was obtained by particle-based computer simulations.

中文翻译：

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扩散影响的多位点磷酸化与酶再激活的动力学

解释扩散对多位点磷酸化动力学影响的最简单方法是修改化学动力学常规速率方程中的速率常数。我们之前已经表明，这还不够，还必须引入反应物之间的新转变。在这里，我们通过考虑修饰底物后失去活性并需要时间再次变得活跃的酶来扩展我们的结果。这种概括导致了令人惊讶的结果。酶再激活的引入导致扩散修改的动力学方案，具有负速率常数的新转变。其原因是，将非马尔可夫速率方程映射到具有与时间无关的速率常数的马尔可夫方程在短时间内并不是一个好的近似。然后我们开发了一种非马尔可夫理论，涉及内存内核而不是速率常数。该理论现在在短时间内有效，但使用起来更具挑战性。例如，使用具有新连接的扩散修改动力学方案来计算磷酸化-去磷酸化循环中的双磷酸化和稳态响应的动力学。我们重现了当酶重新激活时间缩短时磷酸化-去磷酸化循环中双稳定性的丧失，这是通过基于粒子的计算机模拟获得的。