Mechano–chemical study of rotatory molecular motors

Cells are the minimum unit of life. They are born, they eat, the may grow, they may move, and, eventually, they die. By contrast, from a physicist point of view, cells are systems out of equilibrium continuously transducing between matter, energy and information. This transduction is what grants the cell their active properties. In order to perform such tasks, cells have a set of macromolecules, a machinery, which are called, Molecular Motors or Molecular Machines. The operation of molecular motors is multiple. For instance, kinesins are molecular motors able to transport cargoes along the cell, or the Bacterial Flagellar Motor works as a nanometric ionic turbine transmitting its rotation to bacterial flagella propelling the cell. The energy input of such nanometric devices have two primary sources. On one hand the hydrolysis of nucleotide derivatives, such as ATP. On the other hand, molecular motors can also be found in biological membranes obtaining energy from the natural flux of ions crossing the membrane due to mechano-chemical energetic differences at each side. The recycling of ATP molecules takes place in another molecular machine, the F0F1 ATP synthase. F0F1 is made up of two subunits that can be separated themselves in two different molecular machines. This way, the F1 motor can couple a rotatory motor with the synthesis/hydrolysis of ATP. Understanding the working of molecular motors is not straightforward. The transduction processes result from a complex set of interactions of all the molecules conforming the motor plus all the interactions with the surrounding molecules. Thus, different approaches with different levels of abstraction are necessary. In the current thesis, molecular motors are studied through the identification of the energetic transduction cycles out of the trajectory of the motor. Trajectories allow to identify the different mechanical and chemical processes driving the motor and allow to propose a spatio-temporal potential for the motor that give information of the energetic performance of the motor such as power and efficiency. This analysis is performed on the F1 motor (in its hydrolysis regime). Such analysis allowed to identify the origin of two well differentiated mechanical and chemical processes that were quantified by means of the reaction kinetics theory and the overdamped dynamics associated with the nanometric biological scale. From this analysis resulted a prediction for the average velocity of the motor with the experimental control parameters. The resulting velocity matches experimental measures of the average velocity without fitting any parameter since all the parameters needed can be extracted from alternative experimental assays. The appealing results of the average velocity lead to a proposal of motor potential for the F1 motor consisting on two linear piece-wise potentials flashing between them. Each potential presenting the experimental characteristics observed when the catalytic site of the motor is empty or occupied. The potential also hold the substepping mechanism observed experimentally. Thus, the resulting potential can be tested, together with the overdamped dynamics of the potential and the thermal fluctuations characteristic of the biological cellular scale. This results in a Langevin equation leading the dynamics of the motor. Again, the stochastic dynamics proposed are able to reproduce the velocity of the motor returning a better approximation than the deterministic approach. As happened in the previous case, there is no fitting in the parameters to test the validity of the velocity expression. Actually, the model is able to predict the measured substep angle from optimisation arguments. The mismatch between the deterministic and the stochastic results was identified as a result of a loss of ATP hydrolysis events due to thermal fluctuations that has been also properly quantified through the Fokker-Planck formalism of the corresponding Langevin equation. The motor potential proposed was also used to study experimental assays of the F1 motor working against conservative forces. The effect of a conservative torque in the working of the motor contains contributions both mechanical and chemical. Altogether, this contributions were successfully addressed presenting again an analytical and stochastic prediction for the velocity of the motor that matches the experimental observations without the need of any parameter fitting. This analysis also entailed a study of the energetic performance of the motor which is unavailable experimentally. The results show a complete divergence between the stochastic and deterministic predictions. The divergence is specially dramatical near the stall force of the motor where the determenistic analysis predicts an efficiency maximum and the stochastic analysis returns a null efficiency. This points out that the stochastic effects are very relevant to the energetic performance of the motor and can not be missed in a proper energetic study of a molecular machine. Besides the study of the F1 motor, also a rotatory device working with an ionic flux was analised. The aim of the analysis was the devise of a minimal mechanistic turbine and the study of its main working features. Such a machine is composed by a mobile piston with periodic boundary conditions at both ends of a nanometric channel separating two particle reservoirs. Hence, the turbine is able to transduce energy between the flux of ions and an external force hindering the natural motion of the piston. Again, thermal fluctuations provide a stochastic dynamic that must be studied through a Langevin equation that can be tackled analytically. This study revealed that the velocity and the flux are not coupled. Specially, two different stall forces appear for the motor. One for the velocity and one for the flux. This results in an intermediate zone where there is a continuous leakage of ions that does not allow any energetic output. This effect is originated from thermal fluctuations. Thus, when the energetic performance is evaluated, a similar behaviour than the one obtained for the F1 motor is recuperated. This minimal model was extended with more complex turbines that take into account more thoroughly the biophysics of molecular machines. All of them result in the same energetic landscape where a minimum of efficiency is obtained near the stall of the motor. Additionally, a new formalism has been developed to simplify the resulting Langevin equations (Fokker-planck white noise limit) and a new algorithm has been devised able to integrate Langevin equations with non-continuous multiplicative noise