Solutions to stochastic Boolean models are usually estimated by Monte Carlo simulations, but as the state space of these models can be enormous, there is an inherent uncertainty about the accuracy of Monte Carlo estimates and whether simulations have reached all attractors. Moreover, these models have timescale parameters (transition rates) that the probability values of stationary solutions depend on in complex ways, raising the necessity of parameter sensitivity analysis. We address these two issues by an exact calculation method for this class of models. We show that the stationary probability values of the attractors of stochastic (asynchronous) continuous time Boolean models can be exactly calculated. The calculation does not require Monte Carlo simulations, instead it uses graph theoretical and matrix calculation methods previously applied in the context of chemical kinetics. In this version of the asynchronous updating framework the states of a logical model define a continuous time Markov chain and for a given initial condition the stationary solution is fully defined by the right and left nullspace of the master equation’s kinetic matrix. We use topological sorting of the state transition graph and the dependencies between the nullspaces and the kinetic matrix to derive the stationary solution without simulations. We apply this calculation to several published Boolean models to analyze the under-explored question of the effect of transition rates on the stationary solutions and show they can be sensitive to parameter changes. The analysis distinguishes processes robust or, alternatively, sensitive to parameter values, providing both methodological and biological insights. Up to an intermediate size (the biggest model analyzed is 23 nodes) stochastic Boolean models can be efficiently solved by an exact matrix method, without using Monte Carlo simulations. Sensitivity analysis with respect to the model’s timescale parameters often reveals a small subset of all parameters that primarily determine the stationary probability of attractor states.

中文翻译：

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随机连续时间布尔模型的精确求解和敏感性分析。

随机布尔模型的解通常由蒙特卡洛模拟来估计，但是由于这些模型的状态空间可能很大，因此蒙特卡洛估计的准确性以及模拟是否已达到所有吸引子都存在固有的不确定性。此外，这些模型具有时标参数（转换率），固定解的概率值以复杂的方式依赖于此，因此增加了参数敏感性分析的必要性。我们通过针对此类模型的精确计算方法解决了这两个问题。我们表明，可以精确地计算出随机（异步）连续时间布尔模型吸引子的平稳概率值。该计算不需要蒙特卡洛模拟，取而代之的是，它使用先前在化学动力学中应用的图形理论和矩阵计算方法。在此版本的异步更新框架中，逻辑模型的状态定义了一个连续时间马尔可夫链，对于给定的初始条件，固定解由主方程动力学矩阵的左右零空间完全定义。我们使用状态转移图的拓扑排序以及零空间与动力学矩阵之间的依存关系，无需进行仿真即可得出平稳解。我们将此计算应用于几个已发布的布尔模型，以分析过渡速率对平稳解的影响的未充分研究的问题，并表明它们对参数变化敏感。通过分析可以区分出流程是否健壮，或者 对参数值敏感，提供了方法和生物学方面的见识。在不使用蒙特卡洛模拟的情况下，可以通过精确的矩阵方法有效地求解最大中等大小（分析的最大模型为23个节点）的随机布尔模型。关于模型的时标参数的敏感性分析通常会发现所有参数的一小部分，这些参数主要决定吸引子状态的平稳概率。