RNA and protein molecules undergo a dynamic folding process that is important to their function. Computational methods are critical for studying this folding pro- cess because it is difficult to observe experimentally. In this work, we introduce new computational techniques to study RNA and protein energy landscapes, includ- ing a method to approximate an RNA energy landscape with a coarse graph (map) and new tools for analyzing graph-based approximations of RNA and protein energy landscapes. These analysis techniques can be used to study RNA and protein fold- ing kinetics such as population kinetics, folding rates, and the folding of particular subsequences. In particular, a map-based Master Equation (MME) method can be used to analyze the population kinetics of the maps, while another map analysis tool, map-based Monte Carlo (MMC) simulation, can extract stochastic folding pathways from the map. To validate the results, I compared our methods with other computational meth- ods and with experimental studies of RNA and protein. I first compared our MMC and MME methods for RNA with other computational methods working on the com- plete energy landscape and show that the approximate map captures the major fea- tures of a much larger (e.g., by orders of magnitude) complete energy landscape. Moreover, I show that the methods scale well to large molecules, e.g., RNA with 200+ nucleotides. Then, I correlate the computational results with experimental findings. I present comparisons with two experimental cases to show how I can pre- dict kinetics-based functional rates of ColE1 RNAII and MS2 phage RNA and their mutants using our MME and MMC tools respectively. I also show that the MME and MMC tools can be applied to map-based approximations of protein energy energy landscapes and present kinetics analysis results for several proteins.

Chemical reactions can be understood as transitions from basin to basin on a high dimensional potential energy landscape. Varying temperature only changes the average kinetic energy of the system. While applying voltages or external pressures directly tilts the landscape and drives the reactions in desired directions. In solids at relatively low temperature, where the entropy term is approximately invariant, the reaction spontaneity is determined by the energy difference between the reactant and product basins and the reaction rate can be calculated from the barriers in between. To achieve sufficient accuracy to explain experimental observations we are interested in, density functional theory (DFT) is usually employed to calculate energies. There are two types of reactions I have studied: the first type of reaction only involves a few number of individual atoms, corresponding to traveling in a small volume in the high dimensional configuration space; the other type involves a large amount of atoms moving in a concerted pattern, and the distance traveled in the configuration space is significantly longer. The scopes of these two in the energy landscapes are in different scales and thus proper metrics for distance measurements are required. In the first case, I have mainly studied Li/Na behaviors in the cathode materials of secondary batteries. Here resolving the energy landscape step by step with detailed information is possible and useful. By analyzing the energy landscapes with DFT plus the Hubbard U correction, I have explained several phenomena related to the degradation of lithium-rich layered oxides, rate performance of surface modified LiFePO₄, and capacity of vanadium-based fluorophosphates. Predictions on both thermodynamic and kinetic properties of materials are also made based on the calculation results and some are confirmed by experiments. In the second case, my focus is on solid-solid phase transitions. With a tremendous long reaction pathway, examining every possible atomic step is too expensive. By adopting periodic boundary conditions, a small supercell can represent the main feature of the energy landscape in a coarse grained way, where the connection between phases is easier to explore. After the big picture of a phase transition mechanism learned from this simplified model, details along the reaction pathway, like new phase nucleation and growth, could be resolved by using a larger supercell. In the above treatment, two types of variables, the cell vectors and atomic positions, span a generalized configuration space. Special consideration is required to balance these two to keep consistency under different supercells and avoid biases. A solid-state NEB (SSNEB) and a solid-state dimer (SSD) method are then developed to locate saddle points in the generalized configuration space. With the methodology well justified, we are able to efficiently find possible nucleation mechanisms, for examples the CdSe rock salt to wurtzite and Mo A15 to BCC phase transitions. SSNEB is also applied in studying phases transitions under pressures, including the graphite to diamond, and CaIrO₃ perovskite to post-perovskite transitions. Combined with the adaptive kinetic Monte Carlo (AKMC) algorithm, SSD shows the ability to find new polymorphs of CdSe and the connecting barriers between them. / text

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A basse température, c'est à dire dans le régime 1K, les verres présentent des propriétés remarquablement différentes de celles des cristaux de même composition. Par exemple, la dépendance en température de la capacité thermique est presque linéaire et celle de la conductivité thermique est approximativement quadratique (dans les cristaux, on trouve une dépendance cubique pour les deux propriétés).Beaucoup de ces observations peuvent être expliquée par le “Standard Tunneling Model” (STM), dont l’hypothèse de base est l'existence de potentiels locaux à double puits, ou “systèmes à deux niveaux” dans la surface d'énergie potentielle, où des excitations localisées (une particule ou plutôt un cluster de particules) subissent l'effet tunnel à travers la barrière. Récemment les systèmes de tunnels (TSs) ont attiré une attention considérable pour la fabrication de qubits pour les ordinateurs quantiques impliquant jonctions Josephson supraconductrices amorphes.Toutefois, malgré le succès du STM, de nombreuses caractéristiques du modèle sont encore peu claires, par example, la nature microscopique des TS reste inconnue. En outre, des effets magnétiques inattendus ont été découverts dans des verres multi-composants non-magnétiques, comme par exemple des variations non-monotones de la constante diélectrique et de la chaleur spécifique en présence de champs magnétiques faibles.Une explication possible de ces observations est donnée dans l’ “Extended Tunneling Model” (ETM) dans lequel on suppose la présence de régions mieux ordonnées, avec les TSs dans leur interstices, qui doivent être décrites par des potentiels à trois puits (TWPs), dans la structure à moyen distance des verres. En conséquence, les particules concernées par l’effet tunnel peuvent se coupler avec le champ magnétique sous l'effet Aharonov-Bohm.La présente thèse se compose de deux parties: Dans la première nous réalisons des calculs analytiques pour généraliser le modèle ETM au cas d'un potentiel de quatre puits tétraédriques dans un champ magnétique. Nos calculs montrent qu’en fait, le TWP peut être considéré comme le modèle de travail le plus simple pour décrire les verres réels. Nous dérivons également la contribution à l'aimantation des TWPs de notre modèle, et nous obtenons des ajustements qui sont en bon accord avec les données expérimentales. Nous montrons qu’en tenant compte une telle géométrie de tunneling nous obtenons un bon accord entre les concentrations d'impuretés extraites de l'aimantation et celles extraites à partir de mesures de la capacité thermique à basse température. Ceci est une autre preuve de la présence d'inhomogénéités structurales dans les verres à basse température.L'objectif de la deuxième partie est d'élucider la nature des TSs via des simulations numériques. Pour cela nous avons d'abord développé de nouveaux algorithmes pour étudier la géométrie des minima et les barrières d'un modèle simple bidimensionnel de surface d'énergie potentielle. Cette étude est le point de départ d'une nouvelle méthode, nommée "Effective Isopotential Method”, que nous introduisons pour effectuer une analyse locale et systématique de la surface d'énergie en proximité du fond des minima locaux.Nous testons ce méthode sur un cristal CFC Lennard-Jones, puis sur un mélange binaire Lennard-Jones du verre à basse température. Nous constatons que la forme géométrique du minimum local à basse température n’est pas régulière, mais caractérisée par des vallées internes. Dans le cas du cristal on observe la présence de six vallées symétriques associés à une particule donnée, tandis que dans le cas du verre on constate que la plupart des particules présentent deux vallées.Parmi elles, nous trouvons des géométries dont les caractéristiques sont en accord semi-quantitatif avec celles de modèles phénoménologiques décrivant les TSs. Nous avons maintenant une representation de la géométrie de ces TSs. / At low temperature, i.e. in the 1K regime, glasses show properties that are remarkably different from those of the corresponding crystalline counterparts, e.g., the heat capacity depends approximately linearly and the thermal conductivity almost quadratically on temperature (in crystals one finds a cubic dependence for both properties).Many of these observations can be rationalized by the so-called “Standard Tunneling Model” (STM) whose basic assumption is the existence of local double-well potentials, or two level systems, in the potential energy landscape, where localized excitations (a particle or rather a cluster of particles) undergo quantum tunneling through the barrier. In recent times the tunneling systems (TSs) have attracted considerable attention for the fabrication of coherent qubits for quantum computers involving amorphous superconducting Josephson junctions.However, despite the success of the STM, many features of the model are still unclear in that, e.g., the microscopic nature of the TSs remains unknown. In addition, unexpected magnetic effects have been discovered in non-magnetic multi-component glasses, like e.g. the non-monotonous changes of the dielectric constant and the specific heat in the presence of weak magnetic fields.A possible explanation of these observations is the so-called “Extended Tunneling Model” (ETM) in which one assumes the presence of better ordered regions, hosting TSs in the interstices, that have to be described by three-well potentials (TWPs), in the intermediate range structure of glasses; as a result the effective tunneling particles can couple to the magnetic field via the Aharonov-Bohm effect.This work consists of two parts: In the first one we carry out analytical calculations to generalize the ETM model to the case of a four-well tetrahedral trapping potential in a magnetic field. Our calculations show that in fact the TWP can be considered as the simplest working model to describe real glasses. We also derive the contribution to the magnetization from the TWPs of our model, and obtain fits that are in good agreement with the experimental data. We show that only taking into account such tunneling geometry we get a good agreement between the impurity concentrations extracted from the magnetization and those extracted from low-temperature heat capacity measurements. This is thus evidence for the presence of structural inhomogeneities in glasses at low temperature.The goal of the second part is to elucidate the nature of the TSs via computer simulations. For this we first develop new algorithms to study the geometry of the minima and barriers of a simple two-dimensional model of a potential energy surface. This study is the starting point for a novel method, the so-called “Effective Isopotential Method”, that we introduce to perform a local and systematic analysis of the energy landscape close to the bottom of the local minimum.We apply this method to a test case, a Lennard-Jones FCC crystal, and then to a binary mixture Lennard-Jones glass at low temperature. We find that the geometric shape of the IS at low temperature is not smooth but is characterized by internal valleys, i.e. points of the configuration space where the potential energy is lower than the immediate neighborhood. In the case of the crystal we observe the presence of six symmetric valleys associated with a given particle, while in the glass case we find that most of the particles show only two valleys. Amongst them we find the geometries with the right semi-quantitative features (in agreement with the phenomenological models) to be considered as TSs, so that we finally know how they look like in reality.

The dynamics of colloidal particles driven over optical potential energy landscapes is studied. Experiments are conducted using colloids driven by solvent flow or piezo-stage, optical tweezers, magnetic fields, and video-microscopy. Firstly, the properties of optical traps and potential energy landscapes are determined using driven colloidal particles and clusters. The trap stiffness and potential depth of single Gaussian traps are measured directly. It is shown that the nature of optical potential energy landscapes may be fully engineered and predicted using a sum of single Gaussian potentials. Next, the motion of colloidal particles driven by a constant force over a periodic optical potential energy landscape is considered. The average particle velocity is found as a function of the driving velocity, and the wavelength of the optical potential energy landscape, which is found to be sinusoidal at small trap spacings. The critical driving velocity required for a particle to move across the landscape is determined as a function of the wavelength. Brownian motion is found to have a significant effect on the critical driving velocity, but a negligible effect at high driving velocity. Subsequently, the dynamic mode locking caused by adding a modulation to the driving force is studied. This synchronisation manifests as a `Devil's staircase' in the average particle velocity as a function of driving velocity. The amplitude and frequency dependence of the mode locked steps are studied. Furthermore, particle trajectories are examined, and phase portraits show locked (unlocked) states as closed (open) loops in phase space. A state diagram of mode locked steps is constructed. Finally, driven systems of magnetically interacting colloidal particles are examined in potential energy landscapes. The critical driving velocity of a chain of coupled particles driven by a constant force is found to depend strongly on the chain length and the magnetic field. Secondly, a mobile density wave (kink) in an optically pinned chain of coupled particles is exposed to a constant and modulated drive. The kink is found to behave as a quasi-particle, exhibiting analogous dynamic mode locking behaviour to the single particle case. Finally, the mode locking of a finite mobile chain is considered, and found to be affected by the chain flexibility, which is a function of the magnetic field.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.

Nonequilibrium work theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, allow one to use nonequilibrium measurements to determine equilibrium free energies. For example, it has been demonstrated that the Crooks fluctuation theorem can be used to determine RNA folding energies. We used single-molecule manipulation with an atomic force microscope to measure the work done on poly(dA) as it was stretched and relaxed. This single-stranded nucleic acid exhibits unique base-stacking transitions in its force-extension curve due to the strong interactions among A bases, as well as multiple pathways. Here we showed that free energy curves can be determined by using the Crooks fluctuation theorem. The nonequilibrium work theorem can be used to determine free energy curves even when there are multiple pathways.