Simultaneous binding of divalent ligands to two identical molecules is a widespread phenomenon in biology and chemistry. Here, we describe this binding event as a divalent ligand AA that can bind to two identical monovalent molecules B to form the complex AA·B₂. Cases where the total concentration [AA]_T is either much larger or much smaller than the total concentration [B]_T have been studied earlier, but a description of intermediate concentrations is missing. In this paper, we describe the general case of any ratio of ξ ≡ [B]_T/[AA]_T. We show that the concentration of the intermediate complex AA&·B is governed by a cubic equation and discuss several scenarios in which this cubic equation simplifies. Our numerical results, which cover the entire range of 0 < ξ <∞, are relevant to processes wherein the concentrations of free ligands and proteins both decrease upon binding. Such ligand and protein depletion is expected to be important in cellular contexts, e.g., in antigen detection and in coincidence detection of proteins or lipids.

中文翻译：

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二价配体一价分子结合

二价配体与两个相同分子的同时结合是生物学和化学领域的普遍现象。在这里，我们将这种结合事件描述为可以与两个相同的单价分子B结合形成复合物A·B ₂的二价配体AA 。总浓度[AA] _T比总浓度[B] _T大得多或小得多的情况已经在较早的研究中，但是缺少中间浓度的描述。在本文中，我们描述了任意比率ξ≡[B] _T / [AA] _T的一般情况。我们表明，中间配合物AA＆·B的浓度受三次方程控制，并讨论了简化该三次方程的几种情况。我们的数值结果涵盖了0 <ξ<∞的整个范围，与其中结合时游离配体和蛋白质的浓度均降低的过程有关。预期这种配体和蛋白质的消耗在细胞环境中很重要，例如，在抗原检测和蛋白质或脂质的同时检测中。