Department of Physics and Astronomy, University of Denver, Denver, Colorado 80208, USA.
J Chem Phys. 2013 Sep 28;139(12):121915. doi: 10.1063/1.4816527.
We study stochastic dynamics of two competing complexation reactions (i) A + B↔AB and (ii) A + C↔AC. Such reactions are common in biology where different reactants compete for common resources--examples range from binding enzyme kinetics to gene expression. On the other hand, stochasticity is inherent in biological systems due to small copy numbers. We investigate the complex interplay between competition and stochasticity, using coupled complexation reactions as the model system. Within the master equation formalism, we compute the exact distribution of the number of complexes to analyze equilibrium fluctuations of several observables. Our study reveals that the presence of competition offered by one reaction (say A + C↔AC) can significantly enhance the fluctuation in the other (A + B↔AB). We provide detailed quantitative estimates of this enhanced fluctuation for different combinations of rate constants and numbers of reactant molecules that are typical in biology. We notice that fluctuations can be significant even when two of the reactant molecules (say B and C) are infinite in number, maintaining a fixed stoichiometry, while the other reactant (A) is finite. This is purely due to the coupling mediated via resource sharing and is in stark contrast to the single reaction scenario, where large numbers of one of the components ensure zero fluctuation. Our detailed analysis further highlights regions where numerical estimates of mass action solutions can differ from the actual averages. These observations indicate that averages can be a poor representation of the system, hence analysis that is purely based on averages such as mass action laws can be potentially misleading in such noisy biological systems. We believe that the exhaustive study presented here will provide qualitative and quantitative insights into the role of noise and its enhancement in the presence of competition that will be relevant in many biological settings.
我们研究了两个竞争复合反应（i）A + B↔AB 和（ii）A + C↔AC 的随机动力学。这些反应在生物学中很常见，不同的反应物竞争共同的资源——从酶动力学结合到基因表达的例子不胜枚举。另一方面，由于拷贝数小，随机性是生物系统固有的。我们使用耦合复合反应作为模型系统，研究竞争和随机性之间的复杂相互作用。在主方程形式主义中，我们计算了复合物数量的确切分布，以分析几个可观察量的平衡波动。我们的研究表明，一种反应（例如 A + C↔AC）提供的竞争的存在可以显著增强另一种反应（A + B↔AB）的波动。我们为生物学中典型的不同速率常数和反应物分子数组合提供了这种增强波动的详细定量估计。我们注意到，即使两个反应物分子（例如 B 和 C）数量无穷大，保持固定的化学计量，而另一个反应物（A）是有限的，波动也可能很显著。这纯粹是由于通过资源共享介导的耦合，与单反应情况形成鲜明对比，在单反应情况下，一个组件的大量存在确保了零波动。我们的详细分析进一步突出了数值估计的质量作用解与实际平均值可能存在差异的区域。这些观察结果表明，平均值可能是系统的一个较差表示，因此仅基于平均值（例如质量作用定律）的分析在这种嘈杂的生物系统中可能会产生误导。我们相信，这里提出的详尽研究将为噪声及其在竞争存在下的增强作用提供定性和定量的见解，这将在许多生物学环境中具有重要意义。
