Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
Bull Math Biol. 2010 Oct;72(7):1840-66. doi: 10.1007/s11538-010-9513-8. Epub 2010 Feb 19.
We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker-Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a "tug-of-war", and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.
我们提出了一种准稳态降阶方法，用于简化描述一个线性反应-双曲型主方程，该方程描述了一个由马达驱动的粒子在一维丝状轨迹上定向间歇性搜索隐藏目标的过程。粒子从轨迹的一端注入，并在固定搜索相和移动非搜索相之间随机切换，后者在顺向方向上存在偏差。由于轨迹的另一端存在吸收边界，粒子有可能无法找到目标，这种情况例如发生在囊泡货物被马达驱动运输到位于神经元轴突或树突上的突触目标的过程中。简化模型由一个标量福克-普朗克（FP）方程描述，该方程具有一个额外的非均匀衰减项，考虑了目标的吸收。FP 方程用于计算找到隐藏目标（命中概率）的概率和相应的条件平均首次通过时间（MFPT），这取决于随机搜索的有效漂移速度 V、扩散系数 D 和目标吸收率 λ。准稳态降阶方法根据基础马达运输模型的各种生物物理参数来确定 V、D 和 λ。我们首先将我们的分析应用于一个简单的 3 态模型，并表明我们的准稳态降阶方法在生理合理条件下，得到的结果与全系统的蒙特卡罗模拟非常吻合。然后，我们考虑了一个更复杂的双向运输多马达模型，其中两个相反的马达在“拔河”中竞争，并利用该模型探讨了 ATP 浓度如何调节货物向突触目标的输送。
