Proteins, the machinery of life, perform a vast array of essential biochemical functions, evolving over time to acquire diverse roles within biological systems. This evolution, primarily driven by mutations within protein sequences, can profoundly impact protein function, potentially leading to various diseases. This thesis aims to dissect the intricate mechanisms through which genetic mutations influence protein functionality, focusing on the dynamic alterations induced by single and combined mutations. Employing a suite of computational tools, including molecular dynamics (MD) simulations and proven analysis metrics like the Dynamic Flexibility Index (DFI) and Dynamic Coupling Index (DCI), I analyze protein dynamics to uncover the common dynamic effects associated with disease causation and compensatory mechanisms. This analysis extends to exploring the concept of epistasis through the lens of protein dynamics, showing how combinations of mutations interact within the protein's 3D structure to either exacerbate or mitigate the functional impacts of individual mutations. The use of EpiScore, a computational tool designed to quantify the epistatic effects of mutations, provides insight on the combined dynamic effects two mutations might have. This is particularly evident in the analysis of rare alleles within human populations, where certain allele combinations, despite their individual rarity, frequently co-occur, suggesting a mechanism of dynamic compensation. This phenomenon is further investigated in the context of the SARS-CoV-2 spike protein, providing insights into viral evolution and the adaptive significance of specific mutations. Additionally, I delve into the role of Intrinsically Disordered Regions (IDRs) in protein function and mutation compensation, highlighting the need for sophisticated dynamics analysis tools to capture the full spectrum of mutation effects. By integrating these analyses, this thesis unveils a complex picture of how proteins' dynamic properties, shaped by mutations, underpin their functional evolution and disease outcomes.

Protein interactions with the environment are crucial for proper function, butinteraction mechanisms are not always understood. In G protein-coupled receptors (GPCRs), cholesterol modulates the function in some, but not all, GPCRs. Coarse grained molecular dynamics was used to determine a set of contact events for each residue and fit to a biexponential to determine the time scale of the long contacts observed in simulation. Several residues of interest were indicated in CCK1 R near Y140, which is known to render CCK1 R insensitive to cholesterol when mutated to alanine. A difference in the overall residence time between CCK1 R and its cholesterol insensitive homologue CCK2 R was also observed, indicating the ability to predict relative cholesterol binding for homologous proteins. Occasionally large errors and poor fits to the data were observed, so several improvements were made, including generalizing the model to include K exponential components. The sets of residence times in the improved method were analyzed using Bayesian nonparametrics, which allowed for error estimations and the classification of contact events to the individual components. Ten residues in three GPCRs bound to cholesterol in experimental structures had large tau. Slightly longer overall interaction time for the cholesterol sensitive CB1 R over its insensitive homologue CB2 R was also observed. The interactions between the cystic fibrosis transmembrane conductance regulator (CFTR) and GlyH-101, an open-channel blocker, were analyzed using molecular dynamics. The results showed the bromine in GlyH-101 was in constant contact with F337, which is just inside the extracellular gate. The simulations also showed an insertion of GlyH-101 between TM1 and TM6 deeper than the starting binding pose. Once inserted deeper between TMs 1 and 6, the number of persistent contacts also increased. This proposed binding pose may help in future investigations of CFTR and help determine an open-channel structure for the protein, which in turn may help in the development of treatments for various medical conditions. Overall, the use of molecular dynamics and state of the art analysis tools can be useful in the study of membrane proteins and eventuallyin the development of treatments for ailments stemming from their atypical function.

In this thesis, applications of sparsity, specifically sparse-tensors are motivated in physics.An algorithm is introduced to natively compute sparse-tensor's partial-traces, along with direct implementations in popular python libraries for immediate use. These applications include the infamous exponentially-scaling (with system size) Quantum-Many-Body problems (both Heisenberg/spin-chain-like and Chemical Hamiltonian models). This sparsity aspect is stressed as an important and essential feature in solving many real-world physical problems approximately-and-numerically. These include the original motivation of solving radiation-damage questions for ultrafast light and electron sources.

This thesis explores a diverse array of topics related to the role of dynamic allostery in regulating protein functions. Allostery is the phenomenon where a catalytic pocket responds to perturbations caused by binding at another distant site. This response often involves a conformational change resulting in a protein function alteration. However, it is essential to note the existence of dynamic allostery mechanisms that regulate protein function without relying on conformational changes but on dynamic motions. Within this thesis, position-specific equilibrium dynamics-based metrics like Dynamic Flexibility Index and Dynamic Coupling Index are employed to quantify the contributions of specific residues to protein dynamics. I investigated the role of dynamics in protein binding of the WW domain. In particular, I focused on how the mutations of distal positions modulate the binding site dynamics. By employing Dynamic Flexibility Index, I discovered that a residue, 10T, located distally from the binding pocket, plays a significant role in the observed dynamics difference between two variants: N21 (a native folded WW domain not binding Group I peptide) and CC16_N21 (an artificial WW domain binding Group I peptide). The T10H variant, created by exchanging the position 10 residue, enhances flexibility at positions 10 and 16. Consequently, this modification has led to an enhancement in the binding function of N21, enabling it to bind to Group I peptide effectively. Moreover, I investigated the influence of dynamic allostery on protein binding specificity, specifically in the PDZ domain PSD95. To gain insights into the binding process and accurately measure binding affinity, I employed two parallel computational approaches: Adaptive BP-docking and Steered Molecular Dynamics. These methods allowed me to model the binding interactions and quantify the binding strength robustly and comprehensively. The significance of allostery can serve as foundational knowledge in Deep Learning models, enabling the efficient mapping of protein sequences to their corresponding functionalities. One particular metric, Dynamic Coupling Index asymmetry, offers valuable insights into how the three-dimensional network of interactions facilitates communication within a protein structure. Leveraging these interactions, I developed a deep neural network architecture demonstrating enhanced capability in capturing epistatic interactions within Beta-lactamase and protein G function.

Natures hardworking machines, proteins, are dynamic beings. Comprehending the role of dynamics in mediating allosteric effects is paramount to unraveling the intricate mechanisms underlying protein function and devising effective protein design strategies. Thus, the essential objective of this thesis is to elucidate ways to use protein dynamics based tools integrated with evolution and docking techniques to investigate the effect of distal allosteric mutations on protein function and further rationally design proteins. To this end, I first employed molecular dynamics (MD) simulations, Dynamic Flexibility Index (DFI) and Dynamic Coupling Index (DCI) on PICK1 PDZ, Butyrylcholinesterase (BChE), and Dihydrofolate reductase (DHFR) to uncover how these proteins utilize allostery to tune activity. Moreover, a new classification technique (“Controller”/“Controlled”) based on asymmetry in dynamic coupling is developed and applied to DHFR to elucidate the effect of allosteric mutations on enzyme activity. Subsequently, an MD driven dynamics design approach is applied on TEM-1 β-lactamase to tailor its activity against β-lactam antibiotics. New variants were created, and using a novel analytical approach called "dynamic distance analysis" (DDA) the degree of dynamic similarity between these variants were quantified. The experimentally confirmed results of these studies showed that the implementation of MD driven dynamics design holds significant potential for generating variants that can effectively modulate activity and stability. Finally, I introduced an evolutionary guided molecular dynamics driven protein design approach, integrated co-evolution and dynamic coupling (ICDC), to identify distal residues that modulate binding site dynamics through allosteric mechanisms. After validating the accuracy of ICDC with a complete mutational data set of β-lactamase, I applied it to Cyanovirin-N (CV-N) to identify allosteric positions and mutations that can modulate binding affinity. To further investigate the impact of mutations on the identified allosteric sites, I subjected putative mutants to binding analysis using Adaptive BP-Dock. Experimental validation of the computational predictions demonstrated the efficacy of integrating MD, DFI, DCI, and evolution to guide protein design. Ultimately, the research presented in this thesis demonstrates the effectiveness of using evolutionary guided molecular dynamics driven design alongside protein dynamics based tools to examine the significance of allosteric interactions and their influence on protein function.

Transition metal ions such as Zn2+, Mn2+, Co2+, and Fe2+ play crucial roles in organisms from all kingdoms of life. The homeostasis of these ions is mainly regulated by a group of secondary transporters from the cation diffusion facilitator (CDF) family. The mammalian zinc transporters (ZnTs), a subfamily of CDF, have been an important target for study as they are associated with several diseases, such as diabetes, delayed growth and osteopenia, Alzheimer’s disease, and Parkinsonism. The bacterial homolog of ZnTs, YiiP, is the first CDF transporter with a determined structure and is used as a model for studying the structural and mechanistic properties of CDF transporters. On the other hand, Molecular dynamics simulation has emerged as a valuable computational tool for exploring the physical basis of biological macromolecules' structure and function with atomic precision at femtosecond resolution. This work aims to elucidate the roles of the three Zn$2+ binding sites found on each YiiP protomer and the role of protons in the transport process of CDFs, which remain under debate despite previous thermodynamic and structural studies on YiiP. Cryo-EM, microscale thermophoresis (MST) and molecular dynamics (MD) simulations were used to address these questions. With a Zn2+ model that accurately reproduces experimental structures of the binding clusters, the dynamical influence of zinc binding on the transporter was accessed through MD simulations, which was consistent with the new cryo-EM structures. Zinc binding affinities obtained through MST were used to infer the stoichiometry of Zn2+/H+ antiport in combination with a microscopic thermodynamic model and constant pH simulations. The most likely microstates of H$^+$ and Zn2+ binding indicated a transport stoichiometry of 1 Zn2+ to 2-3 H+ depending on the external pH. A model describing the entire transport cycle of YiiP was finally built on these findings, providing insight into the structural and mechanistic properties of CDF transporters.

Secondary active transporters play significant roles in maintaining living cells' homeostasis by utilizing the electrochemical gradient in driving ions or protons as the source of free energy to transport substrate through biological membranes.A broadly recognized molecular framework, the alternating access model, describes the transport mechanism as the transporter undergoes conformational changes between different conformations and alternatingly exposes its binding site to intracellular and extracellular sides and, thus, exchanges ion and substrate in a cyclical manner. Recent progress in structural biology brought the first-ever structural insights into the mammalian Cation-Proton Antiporters (CPA) family of proteins. However, the dynamic atomic-level information about the interactions between the newly discovered structures and the bound ion or the corresponding substrate remains unknown. With Molecular Dynamics (MD), multiple spontaneous ion binding events were observed in the equilibrium simulations, revealing the binding site topology of Horse Sodium-Proton Exchanger 9 (NHE9) and Bison Sodium-Proton Antiporter 2 (NHA2) in their preferred protonation state. Further investigation into more CPA homologs compared various aspects, including sequence identity, binding site topology, and energetic properties, and obtained general insights into the similarities shared by the binding process of CPA members. The putative binding site and other conserved residues in their actively ion-bound poses were identified for each model, and their similarities were compared. The energetic properties accessed by the three-dimensional free energy profile, initially found to be binding unfavorable for the experimental structures, were recalculated based on the simulation data. The updated results show consistency with the correct binding affinity as indicated by the experimental methods. This work provided a general picture of the structures and the ion-protein interaction of CPA proteins and serves as comprehensive guidance for any related future structural and computational work.

Transportation of material across a cell membrane is a vital process for maintaininghomeostasis. Na+/H+ antiporters, for instance, help maintain cell volume and regulate intracellular sodium and proton concentrations. They are prime drug targets, since dysfunction of these crucial proteins in humans is linked to heart and neurodegenerative diseases. Due to their placement in a cell membrane, their study is particularly difficult compared to globular proteins, which is likely the reason the transport mechanisms for these proteins are not entirely known. This work focuses on the electrogenic bacterial homologs Thermus thermophilus NapA (TtNapA) and Echerichia coli NhaA (EcNhaA), each transporting one sodium from the interior of the cell for two protons on outside of the cell. Even though X-ray crystal structures for both of these systems have been resolved, their study through molecular dynamics (MD) simulations is limited. The dynamic protonation and deprotonation of the binding site residues is a fundamental process in the transport cycle, which currently cannot be explored intuitively with standard MD methodologies. Apart from this limitation, simulation performance is only a fraction of what is needed to understand the full transport process, particularly when it comes to global conformational changes. This work seeks to overcome these limitations through the development and application of a multiscale thermodynamic and kinetic framework for constructing models capable of predicting experimental observables, such as the dependence of transporter turnover on membrane voltage. These models allow interpretation of the effects of individual processes on the function as a whole. This procedure is demonstrated for TtNapA and the connection between structure and function is shown by computing cycle turnover across a range of non-equilibrium conditions.