Milne R K, Yeo G F, Edeson R O, Madsen B W
Department of Mathematics, University of Western Australia, Nedlands.
Proc R Soc Lond B Biol Sci. 1988 Apr 22;233(1272):247-92. doi: 10.1098/rspb.1988.0022.
Stochastic models of ion channels have been based largely on Markov theory where individual states and transition rates must be specified, and sojourn-time densities for each state are constrained to be exponential. This study presents an approach based on random-sum methods and alternating-renewal theory, allowing individual states to be grouped into classes provided the successive sojourn times in a given class are independent and identically distributed. Under these conditions Markov models form a special case. The utility of the approach is illustrated by considering the effects of limited time resolution (modelled by using a discrete detection limit, xi) on the properties of observable events, with emphasis on the observed open-time (xi-open-time). The cumulants and Laplace transform for a xi-open-time are derived for a range of Markov and non-Markov models; several useful approximations to the xi-open-time density function are presented. Numerical studies show that the effects of limited time resolution can be extreme, and also highlight the relative importance of the various model parameters. The theory could form a basis for future inferential studies in which parameter estimation takes account of limited time resolution in single channel records. Appendixes include relevant results concerning random sums and a discussion of the role of exponential distributions in Markov models.
离子通道的随机模型很大程度上基于马尔可夫理论，在该理论中必须指定各个状态和转换速率，并且每个状态的逗留时间密度被约束为指数分布。本研究提出了一种基于随机和方法与交替更新理论的方法，只要在给定类别中的连续逗留时间是独立且同分布的，就允许将各个状态分组为类别。在这些条件下，马尔可夫模型构成一种特殊情况。通过考虑有限时间分辨率（通过使用离散检测极限ξ建模）对可观测事件属性的影响，特别是对观测到的开放时间（ξ - 开放时间）的影响，来说明该方法的实用性。针对一系列马尔可夫和非马尔可夫模型，推导了ξ - 开放时间的累积量和拉普拉斯变换；给出了几个对ξ - 开放时间密度函数有用的近似。数值研究表明，有限时间分辨率的影响可能非常大，并且还突出了各种模型参数的相对重要性。该理论可以为未来的推断研究奠定基础，在这些研究中参数估计会考虑单通道记录中的有限时间分辨率。附录包括关于随机和的相关结果以及对指数分布在马尔可夫模型中的作用的讨论。
