Boie Sebastian, Kirk Vivien, Sneyd James, Wechselberger Martin
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand.
School of Mathematics and Statistics, University of Sydney, Camperdown, NSW 2006, Australia.
J Theor Biol. 2016 Mar 21;393:16-31. doi: 10.1016/j.jtbi.2015.12.011. Epub 2016 Jan 8.
Many biophysical models have the property that some variables in the model evolve much faster than others. A common step in the analysis of such systems is to simplify the model by assuming that the fastest variables equilibrate instantaneously, an approach that is known as quasi-steady state reduction (QSSR). QSSR is intuitively satisfying but is not always mathematically justified, with problems known to arise, for instance, in some cases in which the full model has oscillatory solutions; in this case, the simplified version of the model may have significantly different dynamics to the full model. This paper focusses on the effect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We first illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then categorize Hopf bifurcations according to whether they involve fast variables, slow variables or a mixture of both, and show that Hopf bifurcations that involve only slow variables are not affected by QSSR, Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simplified system, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model.
许多生物物理模型具有这样的特性，即模型中的某些变量比其他变量演化得快得多。分析此类系统的一个常见步骤是通过假设最快的变量瞬间达到平衡来简化模型，这种方法被称为准稳态约简（QSSR）。QSSR在直观上令人满意，但在数学上并不总是合理的，已知会出现一些问题，例如，在某些完整模型具有振荡解的情况下；在这种情况下，模型的简化版本可能与完整模型具有显著不同的动力学。本文重点关注QSSR对通过一个或多个霍普夫分岔产生振荡解的模型的影响。我们首先通过将QSSR应用于一些著名模型来说明可能出现的问题。然后，我们根据霍普夫分岔是否涉及快速变量、慢速变量或两者的混合对其进行分类，并表明仅涉及慢速变量的霍普夫分岔不受QSSR影响，涉及快速和慢速变量的霍普夫分岔（即奇异霍普夫分岔）在简化系统中只要保留一个快速变量通常在QSSR下得以保留，而主要涉及快速变量的霍普夫分岔可能会被QSSR消除。最后，如果希望在尽量减少无意中破坏原始模型基本特征风险的同时使用该方法，我们给出了一些应用QSSR的指导原则。
