We derive approximate expressions under various conditions of validity over both pre- and post-steady state regimes of the velocity-substrate-inhibitor spaces of the Michaelis-Menten enzyme kinetic schemes with fully and partial competitive inhibition. Our refinement over the currently available standard quasi steady state approximations (sQSSA) seems to be valid over wide range of enzyme to substrate and enzyme to inhibitor ratios. Further, we show that under certain conditions the enzyme-inhibitor-substrate system can exhibit temporally well-separated two different steady states with respect to both enzyme-substrate and enzyme-inhibitor complexes. We define the ratios f_S = v_max⁄(K_MS + e₀) and f_I = u_max⁄(K_MI + e₀) as the acceleration factors with respect to the catalytic conversion of substrate and inhibitor into their respective products. Here K_MS and K_MI are the Michaelis-Menten parameters associated with the binding of substrate and inhibitor with the enzyme, v_max and u_max are the respective maximum reaction velocities and e₀, s₀, and i₀ are total enzyme, substrate and inhibitor levels. When (f_S ⁄ f_I) < 1, then enzyme-substrate complex will show multiple steady states subsequently reaches the full-fledged steady state only after the depletion of enzyme-inhibitor complex. When (f_S ⁄ f_I) > 1, then the enzyme-inhibitor complex will show multiple steady states and subsequently reaches the full-fledged steady state only after the depletion of enzyme-substrate complex. This complex behavior exclusively when (f_S ⁄ f_I) ≠ 1 is the root cause of large amount of error in the estimation of various kinetic parameters both in the cases of fully and partial competitive inhibition schemes using the sQSSA methods. Remarkably, we show that our refined expressions for the reaction velocities over enzyme-substrate-inhibitor space can control this error more significantly than the currently available sQSSA velocity expressions.

中文翻译：

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具有竞争性抑制的Michaelis-Menten型酶动力学速率方程理论

我们推导了具有完全和部分竞争性抑制的 Michaelis-Menten 酶动力学方案的速度-底物-抑制剂空间的稳态前和稳态后的各种有效性条件下的近似表达式。我们对当前可用的标准准稳态近似（sQSSA）的改进似乎在广泛的酶与底物以及酶与抑制剂的比率范围内是有效的。此外，我们表明，在某些条件下，酶-抑制剂-底物系统可以表现出就酶-底物和酶-抑制剂复合物而言暂时良好分离的两种不同稳态。我们将比率f _S = v _max ⁄( K _MS + e ₀ ) 和f _I = u _max ⁄( K _MI + e ₀ ) 定义为底物和抑制剂催化转化为其各自产物的加速因子。这里K _MS和K _{MI是与底物和抑制剂与酶的结合}相关的米氏参数，v _max和u _max是各自的最大反应速度，e ₀、s ₀和 i ₀是总酶，底物和抑制剂水平。当 ( f _S ⁄ f _I ) < 1 时，酶-底物复合物将呈现多个稳态，只有在酶-抑制剂复合物耗尽后才达到完全稳态。当 ( f _S ⁄ f _I ) > 1 时，酶-抑制剂复合物将显示多个稳态，并且仅在酶-底物复合物耗尽后才达到完全稳态。仅当 ( f _S ⁄ f _I ) ≠ 1 时的这种复杂行为是在使用 sQSSA 方法的完全和部分竞争性抑制方案的情况下估计各种动力学参数时产生大量误差的根本原因。值得注意的是，我们表明，与目前可用的 sQSSA 速度表达式相比，我们对酶-底物-抑制剂空间上的反应速度的精确表达式可以更显着地控制此误差。