Entropy and stochastic properties in catalysis at nanoscale

This work approaches the Michaelis-Menten model for enzymatic reactions at a nanoscale, where we focus on the quasi-stationary state of the process. The entropy and the kinetics of the stochastic fluctuations are studied to obtain new understanding about the catalytic reaction. The treatment of this problem begins with a state space describing an initial amount of substrate and enzyme-substrate complex molecules. Using the van Kampen expansion, this state space is split into a deterministic one for the mean concentrations involved, and a stochastic one for the fluctuations of these concentrations. The probability density in the fluctuation space displays a behavior that can be described as a rotation, which can be better understood using the formalism of stochastic velocities. The key idea is to consider an ensemble of physical systems that can be handled as if they were a purely conceptual gas in the fluctuation space. The entropy of the system increases when the reaction starts and slightly diminishes once it is over, suggesting: 1. The existence of a rearrangement process during the reaction. 2. According to the second law of thermodynamics, the presence of an external energy source that causes the vibrations of the structure of the enzyme to vibrate, helping the catalytic process. For the sake of completeness and for a uniform notation throughout this work and the ones referenced, the initial sections are dedicated to a short examination of the master equation and the van Kampen method for the separation of the problem into a deterministic and stochastic parts. A Fokker-Planck equation (FPE) is obtained in the latter part, which is then used as grounds to discuss the formalism of stochastic velocities and the entropy of the system. The results are discussed based on the references cited in this work.