Estimation and Regulation of Spatially Distributed Signals
Detection and estimation of unknown parameters in random systems, and regulation of signal sensing, are both important in the study of a wide range of systems including communication, ecological, biological, and manufacturing systems. For instance, the rate of photons hitting at the array of photon detectors, the number of cells of interest in a fluorescence cell sorting experiment, and identifying faulty products through random inspection in manufacturing, are all examples of randomly occurring phenomena that are analogous to detection of random objects using a randomly distributed set of detectors or detection attempts. Additionally, an efficient and cost-effective detection system for random targets requires precise knowledge of the number of detectors, position of the detectors, or the rate of detection attempts that can detect the targets with sufficient accuracy. Moreover, many systems, such as biochemical systems, require signal sensing with remarkable resilience in response to various forms of intrinsic and extrinsic variations. One such system is pattern formation by morphogen during different stages of species development. A morphogen gradient transmits positional information to a homogeneous field of cells and differentiates them into distinct patterns. Many factors, such as modulators, regulate morphogen signaling to ensure reproducibility of patterns. During signal transduction, morphogen always degrades, and an important question that needs unraveling is how different degradation mechanisms of morphogen affect system's ability to scale. To study these problems, we use point processes to approximate the detection of randomly arriving targets by randomly distributed detection attempts, and the analysis of the impact of degradation schemes on scaling is done using a partial differential equation (PDE) model. Specifically, we propose a Maximum Likelihood Estimator (MLE) framework to estimate the rate of random targets by detecting the targets using randomly distributed set of detectors. In this analysis, both the random targets, and the detectors, are modeled as Poisson processes. With the assumption that detection is certain when targets and detection attempts are within a fixed detection window, the proposed MLE successfully estimates the underlying rate of the targets, and the performance of the MLE is shown to be largely dependent on the detection window considered. In addition, research reveals that with a sufficiently large rate of detection attempts, bias of the estimation reduces to a negligibly low value. It is worth mentioning that this MLE is applicable to estimate rate of random arrival in any given continuum, and the MLE formulation can be used for non-Poisson processes as well. Furthermore, to study the role of degradation schemes on morphogen signaling and its ability to scale, we use a partial differential equation (PDE) model for a two component system (TCS) comprising of a morphogen and a modulator. Using the model, we compare the role of two different degradation mechanisms-- i) linear degradation of morphogen, ii) linear + quadratic degradation of morphogen. A large-scale parametric screen reveals that both decay mechanisms are able to achieve scaling, and a quadratic decay of morphogen is not mandatory to achieve scaling in patterning systems. These findings increase our understanding of how morphogen gradient scales in many species.
Umulis, Purdue University.
Biomedical engineering|Electrical engineering
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