Browsing by Subject "Markov Chains"

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    Advancements in Analytical Methods and Opto-Mechanical Techniques to Study Molecular Motors and Mutations
    (2018-11) Bhaban, Shreyas
    Transport of important cargo inside the cell occurs through ‘motor-proteins': nanoscale biological machines that carry cargo over tracks formed by filaments called microtubules. Defects in intracellular transport mechanisms are linked to numerous neuro-degenerative disorders such as Alzheimer's and Huntington’s disease. Understanding motor-protein functionality and related diseases at the cellular level requires detailed investigation of motor protein behavior. My thesis enables the study through the following key contributions: 1. Innovation of new experimental and computational methods to analyze transport inside cells, with unprecedented resolution and accuracy. 2. Deployment of the new methods to study motor-protein behavior and how mutations in proteins affect intracellular transport, leading to neuro-degenerative diseases. My research makes these contributions through analytical and experimental approaches. In the first part, I have developed analytical tools that are capable of probing how the motor proteins function while carrying cargoes inside the cell, with a particular focus on the transport of cargos by teams of motors. Utilizing the properties of Markov processes, I have developed a computational engine that can provide exact probabilities of the various configurations adopted by motors in the team containing single or multiple species. This methodology helps in answering previously unanswered questions on the mechanisms adopted by motors in response to external events, such as obstructions or impediments along the path of cargo travel, changing load conditions and varying ATP concentrations. The analysis predicts that motors develop a cooperative mechanism and cluster together while carrying progressively heavier loads; but tend to adopt a non-clustered configuration in response to sudden spikes in loads (which is akin to sudden obstacles along the path of travel). A further extension of the computational tool enabled the investigation of the impact of disease causing mutations on team based transport, where the mutations are known to impair single motor behavior. I analyzed the consequence of having a minority of mutated motors in a team, with the mutation that has been implicated in Huntington's disease. Under certain conditions, the teams with even a few mutated motors are seen to gain a significant competitive advantage, potentially contributing to the dysregulation of cargo transport, impairment of neuronal function and the onset of neuro-degeneration. A separate contribution of this work is the underlying analytical technique, which facilitates exact computations, detection of rare events, ease of usage and a reduction in computational resources needed for its functionality, each of which is a notable improvement over existing approaches. The above analytical conclusions need to be supported by experimental evidence, which is not possible without commensurate improvements in the available platforms of experimental interrogation. Existing instruments, in particular those using optical forces for probing such as 'Optical Tweezers', lack of utilization of a modern controls paradigm and a systems engineering based approach leading to a constrained performance. In the second part of my thesis, I developed a platform for disturbance estimation based on multi-objective synthesis. This facilitated the analysis of multi-input and multi-output systems where the regulation of a certain variable is the objective, in the presence of an external disturbance while simultaneously providing estimates of the disturbances in real time. The platform is designed to function in situations where the disturbance is altered by the noise, whose impact needs to be mitigated. I tested the efficacy of the platform on an 'Optical Tweezer' instrument using live samples of the protein Kinesin-I and demonstrated over 50 % improvement over previous studies, while working in extremely low force ranges of femto-newtons and offering a unique ability to regulate load force on the motor-protein itself. This platform can be utilized to experimentally probe the conclusions borne out of the previous analytical studies involving multiple motors and mutations and can also be extended to study faster motors in their native environment, something which was not possible using the earlier approaches. While numerous techniques exist that can study motor proteins through the dynamics of the cargo they carry, there are far fewer experimental techniques that can effectively probe the motors themselves. This is mainly due to the restrictions imposed by the extremely small dimensions of the motors and the limited probing capacity of optical instruments such as fluorescent microscopes. To tackle this issue, I have developed a unique and low cost instrument called 'TIRF-Tweezer', that combines the capabilities of total internal reflection fluorescence (TIRF) microscopy and optical tweezers. This greatly enhances the probing capacity of optical tweezers and facilitates the investigation of, amongst other things, the previously drawn conclusions on the relative configurations adopted by motors while transporting cargo under varying conditions. To further bolster such investigations, I have also developed robust biological protocols that can recreate the required cellular environments in an in-vitro environment and help provide experimental evidence to the analytical conclusions. With these analytical and experimental tools, I intend to equip biologists and biophysists with useful tools that can assist in the task of analysis of molecular motor behavior.
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    The Ergodic Theorem and Markov Chain Strong Laws
    (2015-08) Huang, Yan
    The purpose of this paper is to explain the pointwise Ergodic Theorem and then to apply it to stationary Markov Chains. The Ergodic Theorem is a theorem which shows that the time-averages of a stationary sequence of random variables converge almost surely, and also gives a way to evaluate the limit of these averages. In the setting of Markov chains, the Ergodic Theorem can be used to obtain an important convergence fact about Markov chains.
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    Markov Chains Meet Molecular Motors
    (2022-11) Shrivastava, Rachit
    Transportation of important cargoes inside the cells of living organism is critical for several cellular functions. Most form of cargo transportation inside the cells is accomplished by teams of nanometer sized proteins called molecular motors. These proteins walk on filamentous tracks inside the cells while carrying a common cargo from its source to destination. Malfunctions in this process could lead to several life-threatening maladies ranging from neurodegenerative diseases to cancers. Thus, fundamental understanding of how molecular motors coordinate the transport of a common cargo is of immense scientific importance. Due to small sizes and stochastic nature of these motor proteins, experiments often lack the spatial and temporal resolutions to investigate the intracellular cargo transport process with molecular details. Mathematical modeling provides a helping hand and can not only add to the information obtained experimentally, but also guide future experiment design, thereby helping us understand the genesis of diseases and aid to the discovery of cures. In this thesis, we have utilized the theory of Markov chains to mathematically model intracellular cargo transport process by teams of molecular motors. Backed by the mathematical theory, we developed a numerical framework which improves upon the existing methodologies and enables us to numerically compute the biologically important statistics of the cargo transport by teams of motor proteins in a more realistic setting on a regular desktop PC. In contrast, previous methodologies regularly employed supercomputing clusters to obtain these statistics. Thus our methods are more accessible to biophysicists studying molecular motors but lacking the access to appropriate computational infrastructure. Further, we develop toy models to mathematically analyze the cargo transport process by two molecular motors replicating the well known "tug of war" and "coordinated movement" type scenarios during multi-motor cargo transport. Our results show that the cargo transport velocities could display non-trivial characteristics and phase transitions depending on certain experimental parameters which is an unexpected phenomenon. The methods developed here are not only limited to modeling cargo transport by molecular motors but also serve as a stepping stone for modeling more general class of processes where a group of stochastic agents accomplishes a common goal.