Thermodynamics ensemble

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Eq. (1) would correspond to a constant energy, constant volume, or micro-canonical simulation scheme. There are various approaches to extend this to a canonical (constant temperature), or other thermodynamic ensembles. (A discussion of these approaches is beyond the scope of the present review.) However, in order to perform such a simulation there are several difficulties to overcome. First, the interactions have to be determined properly, which means that one needs a potential function which describes the system correctly. Second, one needs good initial conditions for the velocities and the positions of the individual particles since, as shown in Sec. II, simulations on this detailed level can only cover a fairly short period of time. Moreover, the overall conformation of the system should be in equilibrium. 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

It is important to emphasize that this lattice database is highly idealized compared to real databases. Unlike the lattice database, real databases cannot be treated as thermodynamic ensembles of protein-ligand complexes equilibrated at room temperature [33,34]. Two of the more straightforward reasons are mentioned here. First, real databases are inherently biased toward strong binders (K < 10 pM), because weak binders are difficult to crystallize and of lesser interest. Second, as mentioned above, real databases are not composed of a representative selection of proteins and ligands, and their compositions are biased toward peptide and peptidomimetic inhibitors and certain protein superfamilies. In contrast, because only one protein and four ligand types are used, the lattice database should have representative ligand compositions. 

Let A = yc, , 1 < / < Nc, be conformations generated for C using a computational method. Because the global free energy minimum conformation is expected to statistically dominate the thermodynamic ensemble, the predicted binding activity for C is determined by (C)=min F y. ) = F(yf ). 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

Kinetics deals with many-particle systems (thermodynamic ensembles). The properties measured as a function of time depend on the scale of observation, and this scale is chosen in relation to the question we wish to ask. The smaller the scale, the more inhomogeneous and fluctuating the homogeneous systems appear to be. For example, we describe the activated atomic jump frequency v as... 

The simplest method for controlling heat production is to rescale atomic velocities to yield the desired temperature. This approach was widely used in early MD simulations. Unfortunately, it has several disadvantages [49]. First, for typical system sizes, averaged quantities such as pressure do not correspond to any particular thermodynamics ensemble. Second, the dynamics produced are not time reversible, making results difficult to analyze. Finally, the rate and mode of heat dissipation are not determined by system properties, but instead depend on how often velocities are rescaled. This may influence the dynamics that are unique to a particular system. 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

Several of the chapters in this volume are concerned with the calculation of thermodynamic ensemble averages for systems of many particles. An introduction to this key application area is presented in the first chapter (by Siepmann), and advanced work is discussed in the last six chapters in this volume (by de Pablo and Escobedo, Valleau, Kofke, Siepmann, Johnson, and Barkema and Newmann). There are a large number of monographs and edited volumes with a major emphasis on techniques like those described above and their application to a wide variety of molecular simulations. Allen and Tildesley (1987), Heerman (1990), Binder and Heerman (1992), Binder (1995), and Frenkel and Smit (1996) may be consulted as a core library in this area. 

The potentials used to describe these interfaces are those used to describe bulk hquids. The differences stay in the choice of the thermodynamical ensemble (grand canonical ensembles are often necessary), in the boundary conditions to be used in calculations, and in the explicit introduction in the model of some properties and concepts not used for bulk liquids, like the surface tension. Much could be said in this preliminary presentation of liquid/gas interfaces, but we postpone the few aspects we have decided to mention, because they may be treated in comparison with the other kind of surfaces. 

Once again the potentials used for liquid/liquid surfaces are in general those used for bulk liquids eventually with the introduction of a change in the numerical values of the parameters." The difference from the bulk again regards the thermodynamical ensemble, the boundary conditions for the calculations and the use of some additional concepts. 

Chong S-H, Ham S Thermodynamic-ensemble independence of solvation free energy, J Chem Theoty Comput 11(2) 378—380, 2015. 

The central point to simulate decoherence is to generate independent MD simulations on the PESs of the two electronic states of interest. To have a nonzero overlap at time 1 = 0 (a coherent superposition of states), these MD simulations will share the same initial conditions (positions and momentum). In later times the classical nuclei will feel different forces on the two PESs and will diverge over time. Once the two trajectories have been carried out, the function can be calculated using the values of xy, X2j, Py, xy, and IsK 2 evaluated along the diverging trajectories. This operation is repeated with different initial conditions to sample the desired thermodynamics ensemble. 

Docking to multiple receptor conformations (sometimes abbreviated MRC [8]) can be applied on a set of conformations from any source. Experimentally different X-ray conformations or NMR solution stmctures can represent the same biomolecular system. NMR solution stmctures and conformations sampled during an MD simulation represent a valid thermodynamic ensemble, e.g., a canonical ensemble, for simulations with constant number of particles, volume, and temperature. Although, a collection of X-ray stmctures is not a valid thermodynamic ensemble, docking to multiple X-ray stmctures is sometimes also termed ensemble docking. 

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