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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.

The free energy landscape of a protein-like chain is constructed from exhaustive simulation studies using a combination of discontinuous molecular dynamics and parallel tempering methods. The protein model is a repeating sequence of four kinds of monomers, in which hydrogen bond attraction, electrostatic repulsion, and covalent bond vibrations are modeled by step, shoulder and square-well potentials, respectively. These protein-like chains exhibit a helical structure in their folded states. The model allows a natural definition of a configuration by considering which beads are bonded. In the absence of a solvent, the relative free energy of dominant structures is determined from the relative populations, and the probabilities predicted from the calculated free energies are found to be in excellent agreement with the observed probabilities at different temperatures. The free energy landscape of the protein-like chain is analyzed and confirmed to have funnel-like characteristics, confirmed by the fact that the probability of observing the most common configuration approaches unity at low enough temperatures for chains with fewer than 30 beads. The effect on the free energy landscape of an explicit square-well solvent, where the beads that can form intra-chain bonds can also form (weaker) bonds with solvent molecules while other beads are insoluble, is also examined. Simulations for chains of 15, 20 and 25 beads show that at low temperatures, the most likely structures are collapsed helical structures. The temperature at which collapsed helical structures become dominant is higher than in the absence of a solvent. Finally, the dynamics of the protein-like chain immersed in an implicit hard sphere solvent is studied using a simple model in which the implicit solvent interacts on a fast time scale with the chain beads and provides sufficient friction so that the motion of monomers is governed by the Smoluchowski equation. Using a Markovian model of the kinetics of transitions between conformations, the equilibration process from an ensemble of initially extended configurations to mainly folded configurations is investigated at low effective temperatures for a number of different chain lengths. It was observed that folding profiles appear to be single exponentials and independent of temperature at low temperatures.

The free energy landscape of a protein-like chain is constructed from exhaustive simulation studies using a combination of discontinuous molecular dynamics and parallel tempering methods. The protein model is a repeating sequence of four kinds of monomers, in which hydrogen bond attraction, electrostatic repulsion, and covalent bond vibrations are modeled by step, shoulder and square-well potentials, respectively. These protein-like chains exhibit a helical structure in their folded states. The model allows a natural definition of a configuration by considering which beads are bonded. In the absence of a solvent, the relative free energy of dominant structures is determined from the relative populations, and the probabilities predicted from the calculated free energies are found to be in excellent agreement with the observed probabilities at different temperatures. The free energy landscape of the protein-like chain is analyzed and confirmed to have funnel-like characteristics, confirmed by the fact that the probability of observing the most common configuration approaches unity at low enough temperatures for chains with fewer than 30 beads. The effect on the free energy landscape of an explicit square-well solvent, where the beads that can form intra-chain bonds can also form (weaker) bonds with solvent molecules while other beads are insoluble, is also examined. Simulations for chains of 15, 20 and 25 beads show that at low temperatures, the most likely structures are collapsed helical structures. The temperature at which collapsed helical structures become dominant is higher than in the absence of a solvent. Finally, the dynamics of the protein-like chain immersed in an implicit hard sphere solvent is studied using a simple model in which the implicit solvent interacts on a fast time scale with the chain beads and provides sufficient friction so that the motion of monomers is governed by the Smoluchowski equation. Using a Markovian model of the kinetics of transitions between conformations, the equilibration process from an ensemble of initially extended configurations to mainly folded configurations is investigated at low effective temperatures for a number of different chain lengths. It was observed that folding profiles appear to be single exponentials and independent of temperature at low temperatures.

<div> The folding and unfolding processes of poly-protein has been tremendously studied recently. The poly-protein dynamics under an external force can play an important role in addressing the issue of the mechanics of muscle tissue. In this research, we use a single-molecule technique: magnetic tweezers to observe the dynamics of 8-mer poly-protein L under different loads applied and then in different Tris-buffered salines. Our result shows that more protein domains unfold as the force load becomes larger. At 6, 7 and 8 pN loads, the poly-protein is most likely to stay in state 1, 3 and 6 with 1, 3 and 6 domains unfolded, respectively according to the probability distribution. This can be well explained by our constructed free energy-related model. The fit results give protein L parameters of persistence length of 0.4 nm, contour length of 18.8 nm and the unfolding energy of 6.5 kT, all in reasonable ranges based on previously reported literature.</div><div> Besides, we also find the dependency of transition rate on force load and salt. The poly-protein has lower transition rate at high force than at low force due to the free energy tilting effect since high force extremely decreases the possibility of protein unfolding that results in a huge drop in the total number of folding and unfolding events. This inverse proportion effect can also be seen in different TRIS-buffered salines (TRIS-150mM NaCl, TRIS-1M NaCl, and TRIS-1M KCl,). We explore the effect of salt concentration, when the concentration of NaCl is increased, the transition rate increases while the probability distribution remains almost the same, indicating the protein unfolding barrier is lowered without altering the overall energy landscape. We attribute this to, first, the charge shielding effect that more interactions between ions and water molecules occur, causing fewer water molecules available to interact with the charged part of protein than before, and, second, more direct interactions of ions with protein that might affect the electrostatic-related transition rate. Considering the effect of salt type, the two 1M alkali metal-chloride salines are compared. We conclude that ions with larger size have less effect on transition rate because ions with smaller size (Na+) can create stronger bonds with water that increase the interference on the protein interaction with water and can easier penetrate into protein to directly interact with the protein.</div>

Comprendre la dynamique locale des protéines dans leur état natif (structure repliée etfonctionnelle) est essentiel pour comprendre leur dynamique globale et leur fonction biologique. Aucours de cette thèse, nous avons étudié la dynamique locale de plusieurs petites protéines enmesurant les fluctuations de sondes locales le long de la séquence d’acide aminé de ces protéines.Nous avons essayé de comprendre la dynamique de ces sondes locales, comment celles-ci serelaxaient entre leurs différentes conformations, comment leurs fluctuations étaient corrélées lesunes aux autres et comment peuvent-elles être reliées à la fonction biologique des protéines.Dans les trois premiers chapitres, nous introduisons les concepts du mouvement Browniende rotation libre, de la spectroscopie par Résonance Magnétique Nucléaire (RMN) et de ladynamique moléculaire (DM). Dans les chapitres 4 et 5, nous avons étudié la dynamique desliaisons amides de la chaine principale (backbone) des protéines sur leurs paysages d’énergie libre.Dans le chapitre 4, nous avons démontré que les fluctuations des liaisons amide dubackbone de la protéine VA3 sont décrites par une diffusion rotationnelle anormale plutôt que parune diffusion rotationnelle libre généralement utilisée pour interpréter les données RMN enrelaxation de spins et en couplage résiduel dipolaire. [...] Dans le chapitre 5, nous avons démontré la diffusion rotationnelle anormale de ces liaisons jusqu’à une échelle de temps de 100 ns en utilisant dix simulations de DM de 1 μs de la protéineUbiquitine. Nous avons aussi étudié la convergence des paramètres RMN extraits des trajectoiresde DM en fonction de leur durée. [...] Dans le chapitre 6, nous avons réalisé une l’étude de la corrélation entre les mouvements du backbone et des chaines latérales des protéines. [...] Dans la première partie du dernier chapitre de cette thèse, nous avons étudié l’évolution de la corrélation dynamique entre les chaines latérales et la chaine principale d’une protéine durant des évènements de dépliement/repliement. Pour cette étude préliminaire, nous avons utilisé unesimulation de DM d’un « ultra-fast folder » nommé Trp-cage réalisée à 380K. Nous avons confirméles résultats précédemment trouvés pour les protéines dans leur état natif. Nous avons observél’augmentation de la corrélation entre les séries temporelles yn(t) and δn(t) pendant un évènementde dépliement caractérisé par la sortie du tryptophane de sa cage. Un paramètre stérique s aégalement été défini afin de quantifier les intéractions des chaines latérales avec leurenvironnement. Dans une seconde partie de ce dernier chapitre, nous présentons une étudepréliminaire du dépliement d’un « downhill folder » nommé gpW sous contrainte d’une force. Pourcaractériser le dépliement de la protéine gpW, nous avons calculé les chemical shifts des atomes Cª et Hⁿ du backbone le long de sa séquence en fonction d’une coordonnée réactionnelle choisie comme étant la distance entre les Cª de résidus C- et N- terminaux. Nous avons démontré qu’il était difficile de discerner un comportement particulier à partir des tous les chemical shifts en fonction de la distance. Cependant, en moyennant la valeur des chemical shifts en sur tous lesrésidus de la protéine nous trouvons que l’évolution de cette valeur moyenne en fonction de ladistance permettait de décrire les évènements du dépliement de la protéine en fonction de lacoordonnée de réaction durant la simulation de DM / Understand the local dynamics of proteins in their native state, i.e. in their folded functionalstructure, is a prerequisite to understand their global dynamics and their biological function. In thepresent thesis, we investigated the local dynamics of several small proteins by recording thefluctuations of local probes along the amino-acid sequence of those proteins. We tried tounderstand the dynamics of the local probe, i.e. how they relax between their differentconformations, how their fluctuations are correlated to each other, how their fluctuations arerelated to the function of the proteins. In the first three chapters, we introduced the concepts of the free rotational Brownian motion, of the Nuclear Magnetic Resonance spectroscopy and of the Molecular Dynamics (MD)simulations. In chapters 4 and 5, we studied the dynamics of the backbone amide bonds of theproteins on their free-energy landscape. In chapter 4, we demonstrated that the fluctuations of the backbone amide bonds of the protein VA3 are described by a rotational anomalous diffusion rather than by a free rotationaldiffusion, as often assumed in the interpretation of the raw NMR-measured data (Spin relaxation(SR) data and Residual Dipolar Coupling (RDC) data. [...] In chapter 5, we demonstrated the anomalous diffusion of backbone amide bonds up to 100 ns by using ten MD trajectories of 1 μs of duration for the protein ubiquitin. We also studied the convergence of the NMR-derived parameters extracted from the MD trajectories in function of their duration. [...] In chapter 6, we addressed the question of the correlation between the motions of the side chains and main chain of a protein. [...] In the first part of the final chapter of the present thesis, we investigated the evolution of the correlation between the side-chain and the main-chain motions of a protein during unfolding/folding events. In this preliminary work, we used a single MD simulation of the ultrafast folder Trp-cage performed at 380 K. We confirmed the results found for proteins in theirnative state. We observed an increase of the correlation between the two time series yn(t) and δn(t) during an unfolding event characterized, here, by the exit of the TRP residue of its “cage”.A steric parameter s was also defined in order to quantify interactions of the amino-acid side chainwith its environment. In a second part of the last chapter, we present a preliminary study of theunfolding of the downhill folder gpW under a mechanical force. To characterized the unfolding ofgpW, we computed the chemical shift of the Cª and of the Hⁿ atoms along the amino-acidsequence of the protein in function of a reaction coordinate: the distance, rCªCª , between the Cª atoms of the N and C terminal residues. We demonstrated that it is hard to distinguish a typical behavior of all the chemical shift of all the residues along the amino-acid sequence in function of the distance rCªCª . However, by averaging the chemical shift over all the residues of the protein we found that the evolution of the average value of the chemical shift described the unfolding eventsof the protein during the MD simulations

The modelling of single-molecule experiments is of vital interest to gain new insights into processes which were hitherto not accessible by measurements performed on bulk systems. In the first part of this thesis, the kinetics of a triple-branch DNA molecule with four conformational states is investigated by employing pulling experiments with optical tweezers and theoretical modelling. Probability distributions of first rupture forces, which are calculated by applying transition rate theory to a free energy model, show good agreement with experimental findings. Permanently frayed molecules could be identified by analysing the number of opening base pairs in force jumps. In the second part of the thesis, DNA hairpin molecules with periodic base sequences are studied. Their unfolding kinetics allows an analytical treatment, because they exhibit a regular coarse-grained free energy landscape as a function of the number of opened base pairs. A procedure is developed for determining all relevant parameters of the landscape, which relies on probabilities that can be easily sampled from the unfolding trajectories. By means of Monte Carlo simulations it is shown that already 300 trajectories, as typically measured in single-molecule experiments, provide faithful results for the energetic parameters. The approach in particular opens a new access to improve loop contributions in the free energy landscape. In the third part of the thesis, a simulation method is developed for modelling the unfolding kinetics of DNA molecules with arbitrary base sequences. The method is validated against experimental data for five DNA hairpin molecules with different length of the end-loop. Applications of the method enable one, among others, to improve the parameter determination in functional forms suggested for the tail behaviour of work distributions. Such work distributions enter detailed and integral fluctuation theorems, which are useful for estimating free energy differences between folded and unfolded states from nonequilibrium measurements.