Stochastic Shielding for Dimension Reduction in Models of Biological Systems

该项目将开发数学工具，使科学家能够了解生物系统如何在只能直接观察到这些系统的一部分时发挥作用。 在项目的第一部分，PI将与研究囊性纤维化基础生物学的科学家合作。这种疾病发生时，有人有一个变异的基因，使某种蛋白质。 这种蛋白质的行为不能直接观察到。 PI将帮助科学家更好地了解囊性纤维化蛋白在现有部分测量的基础上做什么。 在该项目的第二部分，PI将研究称为离子通道的分子，位于大脑神经细胞中，允许脑细胞相互交流。离子通道可以打开和关闭。 虽然不能直接观察到离子通道的随机打开和关闭，但可以通过它们对单个神经细胞电活动的影响间接观察到它们的波动。 PI将提供一些方法来了解由于离子通道的随机打开和关闭，某些神经细胞的电活动将是多么不规则。最后，PI将工作在动物的某些生命阶段，可以直接观察，而其他人不能。 生态学家仔细研究了某些青蛙的种群如何随着时间的推移而变化。 青蛙和它们的卵可以在它们生活的池塘中计数，但是未成熟的青蛙在池塘附近的陆地上生活了一年，在那里它们很难被观察到。PI将提供数学工具，我们的生态学家合作者可以使用它来理解只能部分观察到的种群波动。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，代表多个位置或状态之间过渡过程的随机动态。这种模型的复杂性降低旨在通过更简单的表示来捕获基本的动力学和随机特性，并以最小的精度损失。经典的方法，如节点的聚合和快速变量的消除，导致减少模型，不再是马尔可夫。随机屏蔽提供了一种替代方法，通过简化驱动过程的噪声的描述，同时保留马尔可夫属性，通过从模型中删除那些对过程的可观察特征影响最小的波动。PI将以他们先前的工作为基础，以三种方式开发随机屏蔽框架。1. PI将建立严格的数学基础的随机屏蔽近似，通过开发一个非线性主方程描述的不可观察节点的概率的演变，条件是可观察的轨迹的过程。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。先前严格分析的随机屏蔽近似被限制到静态过程，如随机混合电导为基础的离子通道模型电压钳。在当前钳位条件下，每个边缘作出了独特的贡献，如峰间间隔方差的路径特性的波动。3. PI将扩展随机屏蔽从恒定人口的情况下，适合嘈杂的离子通道，人口可以增长或下降的情况下，如在生态models.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.