Bridging Across Scales to Model Cone Phototransduction

视网膜中的视杆细胞和视锥细胞调节视力。杆状细胞检测白光，而视锥细胞则负责色觉。它们的作用在高敏性视力中是至关重要的，因为它们在与年龄相关的黄斑变性导致失明的过程中丧失。这些细胞的一个共同特征是它们复杂的几何结构，由大约1000个薄煎饼状的褶皱（圆盘）组成，通过这些褶皱，光被转换成电脉冲（光导），供大脑“看到”。光被位于这些褶皱上的受体/色素捕获，并通过一系列生化步骤转化为细胞内外之间的电流。这些步骤包括光信号的放大器（换能器和效应器）和信号的载体（第二信使）在这些褶皱中扩散。由于这些褶皱的数量（大约1000个）和厚度（几纳米），对这些过程的数学理解变得复杂。均质化的数学理论试图用更简单的圆柱形结构取代复杂的几何结构，同时保留原始系统的所有生化和生物物理功能。为了视觉感知的一致性，转导级联必须是可靠/稳定的，并且可以通过数学建模来分解和分析级联中不稳定的来源和稳定机制。这项跨学科研究涉及数学（均质化）、计算科学（有限元代码编写）、生物化学（激活/失活级联）和生理学（第二信使的扩散）。该项目涉及学生和博士后学员，并将通过培训课程和研讨会进行传播。本研究由数学科学部数学生物学项目和综合有机体系统生理机制和生物力学项目共同资助。视锥细胞捕捉红色、蓝色和绿色波长的光，因此，除了它们的几何形状外，它们的生化和生物物理功能也不同于杆状细胞。特别是它们从不饱和，它们有更快更小的响应，它们对昏暗的光不敏感，并且有更快的失活。光反应是由第二信使钙和CGMP在脂质盘内的细胞质中扩散产生的。为了克服圆盘的复杂几何形状（大约一千个，每个几纳米厚），我们提出了一种均质化过程，通过该过程，圆盘的数量趋于无穷大，厚度趋于零，而可用于扩散的体积保持不变，生化和生物物理功能保持在极限内。极限“均质”锥变成圆柱状，没有圆盘，扩散过程在均质域上分为内部扩散和边界扩散。激活/失活过程由跟踪级联各个步骤的连续时间马尔可夫链来建模。所得到的模型是混合的，因为它包含一个确定性部分（第二信使的均匀扩散）和一个随机输入（激活/失活级联的随机步骤）。这允许人们分析随机失活事件转变为稳定光响应的机制，并确定可变性和可变性抑制的原因。将创建一个参数数据库来填充模型并影响数值模拟，以与实验数据进行比较，包括缺乏饱和度，低灵敏度和更快的失活。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Rods and cones in the retina mediate vision. While rods detect white light, cones are designed for color vision. Their role is crucial in high acuity vision as underscored by their loss in age-related macular degeneration leading to blindness. A common feature of these cells is their intricate geometrical structure, consisting of about one thousand thin pancake-like folds (discs) through which light is transformed into electrical pulses (phototransduction) for the brain to "see." Light is captured by receptors/pigments residing on these folds, and transformed into a current between interior and exterior of these cells, by a cascade of biochemical steps. These steps involve amplifiers of the light signal (transducers and effectors) and carriers of the signal (second messengers) diffusing within these folds. A mathematical understanding of these processes is complicated by the numbers of these folds (about one thousand) and their thickness (a few nanometers). The mathematical theory of homogenization seeks to replace the complex geometry with a simpler, cylindrical structure while preserving all the biochemical and biophysical functions of the original system. The transduction cascade must be reliable/stable for consistency of visual perception, and sources of instability and stabilizing mechanisms in the cascade can be broken down and analyzed via mathematical modeling. This interdisciplinary investigation involves mathematics (homogenization), computational sciences (finite element code writing), biochemistry (activation/deactivation cascades), and physiology (diffusion of second messengers). The project involves students and postdoctoral trainees and will be disseminated through training courses and seminars. This research is funded jointly by the Division of Mathematical Sciences Mathematical Biology Program and the Division of Integrative Organismal Systems Physiological Mechanisms and Biomechanics Program. Cones capture light in the red, blue and green wavelength, and as such, besides their geometrical shape, their biochemical and biophysical functions are different than rods. In particular they never saturate, they have a faster and smaller response, they are little sensitive to dim light and have a faster deactivation. The photoresponse is generated by diffusion of the second messengers calcium and CGMP in the cytoplasm within the lipidic discs. To overcame the intricate geometry of the discs (about a thousand, each a few nanometers thick) we propose an homogenization process, by which the number of discs goes to infinity and their thickness goes to zero, while the volume available for diffusion remains unchanged, and the biochemical and biophysical functions are preserved in the limit. The limiting "homogenized" cone becomes cylinder-like, with no discs, and the diffusion process is separated into interior and boundary diffusion on the homogenized domain. The activation/deactivation process is modeled by a continuous-time Markov chain tracking the various steps of the cascade. The resulting model is hybrid as it contains a deterministic part (homogenized diffusion of the second messengers) with a random input (stochastic steps of the activation/deactivation cascade). This permits one to analyze the mechanism by which random deactivation events turn into a stable photoresponse, and identify the causes of variability and variability suppression. A database of parameters will be created to populate the model and effect numerical simulations to be compared with experimental data, including lack of saturation, low sensitivity and faster deactivation.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.