Stochastic Shielding for Dimension Reduction in Models of Biological Systems

该项目将开发数学工具，使科学家能够理解生物系统在只有部分系统可以直接观察到的情况下如何工作。在该项目的第一部分，PI将与研究囊性纤维化的基本生物学的科学家合作。当某人拥有制造某种蛋白质的基因变体时，就会发生这种疾病。这种蛋白质的行为不能直接观察到。PI将帮助科学家根据现有的部分测量结果更好地了解囊性纤维化蛋白的功能。在该项目的第二部分，PI将研究被称为离子通道的分子，这种分子位于大脑中的神经细胞中，使脑细胞能够相互通信。离子通道可以打开和关闭。虽然离子通道的随机开启和关闭不能直接观察到，但通过它们对单个神经细胞电活动的影响，可以间接观察到离子通道的波动。PI将提供了解由于离子通道的随机打开和关闭而导致的某些神经细胞的电活动的不规则程度的方法。最后，PI将工作在某些生命阶段的动物可以直接观察，而其他动物则不能。生态学家仔细研究了某些青蛙的数量是如何随着时间的推移而变化的。青蛙和它们的卵可以在它们生活的池塘中计数，但未成熟的青蛙在池塘附近的陆地上生活一年，在那里它们很难观察到。PI将提供数学工具，我们的生态学家合作者可以用来理解只能部分观察到的种群的波动。离散状态，连续时间马尔可夫过程发生在细胞生物学、神经科学和生态学中，代表过程在多个位置或状态之间转换的随机动力学。这类模型的复杂性降低旨在以最小的精度损失，通过更简单的表示来捕捉基本的动力学和随机属性。经典的方法，如节点聚合和快速变量的消除，导致简化的模型不再是马尔可夫模型。随机屏蔽提供了一种替代方法，它简化了对驱动过程的噪声的描述，同时通过从模型中删除对过程的可观察特征影响最小的那些波动来保持马尔可夫性质。PI将以他们之前的工作为基础，通过三种方式开发随机屏蔽框架。1.PI将为随机屏蔽近似建立严格的数学基础，通过建立一个非线性主方程描述未观察到的节点的概率的演变，条件是该过程的可观测轨迹。2.PI将建立随机屏蔽对非平稳混合(条件确定性)过程的适用性。以前对随机屏蔽近似的严格分析局限于平稳过程，例如电压钳下基于随机混合电导的离子通道模型。在当前的钳制条件下，每条边对Path属性的波动有明显的贡献，例如尖峰间隔方差。3.PI将把随机屏蔽从适合于噪声离子通道的恒定种群的情况扩展到种群可以增长或减少的情况，如在生态模型中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will develop mathematical tools to enable scientists to understand how biological systems function when only parts of those systems can be directly observed. In the first part of the project the PI will collaborate with scientists who study the basic biology of cystic fibrosis. This disease arises when someone has a variant of the gene making a certain protein. The behavior of this protein cannot be observed directly. The PI will help scientists better understand what the cystic fibrosis protein is doing based on the partial measurements available. In the second part of the project the PI will study molecules called ion channels, located in nerve cells in the brain that allow brain cells to communicate with each other. Ion channels can open and close. Although the random opening and closing of ion channels cannot be observed directly, their fluctuations can be observed indirectly through their effects on the electrical activity of individual nerve cells. The PI will provide ways to understand how irregular the electrical activity of certain nerve cells will be, due to the random opening and closing of ion channels. Finally, the PI will work on certain life stages of animals can be directly observed while others cannot. Ecologists have carefully studied how populations of certain frogs change over time. The frogs and their eggs can be counted in the ponds where they live, but immature frogs spend a year living on land near the ponds, where they are difficult to observe. The PI will provide mathematical tools that our ecologist collaborators can use to understand fluctuations in populations that can only be partially observed.Discrete state, continuous time Markov processes occur throughout cell biology, neuroscience, and ecology, representing the random dynamics of processes transitioning among multiple locations or states. Complexity reduction for such models aims to capture the essential dynamics and stochastic properties via simpler representations, with minimal loss of accuracy. Classical approaches, such as aggregation of nodes and elimination of fast variables, lead to reduced models that are no longer Markovian. Stochastic shielding provides an alternative approach by simplifying the description of the noise driving the process, while preserving the Markov property, by removing from the model those fluctuations that have the least impact on observable features of the process. The PI will build on their prior work developing the stochastic shielding framework in three ways. 1. The PI will establish rigorous mathematical foundations for the stochastic shielding approximation by developing a nonlinear master equation description for the evolution of the probability of the unobserved nodes, conditioned on the observable trajectory of the process. 2. The PI will establish the applicability of stochastic shielding to nonstationary hybrid (conditionally deterministic) processes. Prior rigorous analysis of the stochastic shielding approximation was confined to stationary processes, such as stochastic hybrid conductance-based ion channel models under voltage clamp. Under current clamp conditions, each edge makes a distinct contribution to fluctuations in pathwise properties such as interspike interval variance. 3. The PI will extend stochastic shielding from the constant population case, appropriate for noisy ion channels, to the case where populations can grow or decline, as in ecological models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.