Molecules complex boundary condition

Molecular dynamics simulations are capable of addressing the self-assembly process at a rudimentary, but often impressive, level. These calculations can be used to study the secondary structure (and some tertiary structure) of large complex molecules. Present computers and codes can handle massive calculations but cannot eliminate concerns that boundary conditions may affect the result. Eventually, continued improvements in computer hardware will provide this added capacity in serial computers development of parallel computer codes is likely to accomplish the goal more quickly. In addition, the development of realistic, time-efficient potentials will accelerate the useful application of dynamic simulation to the self-assembly process. In addition, principles are needed to guide the selec-... 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

To solve eqn. (114) with the boundary condition above, even for the steady state, is a task of considerable complexity and tedium. Sole and Stockmayer [256] considered the case of axially symmetric molecules and fixed the laboratory framework with 0 = 0. Hence, only 0, 0A and 0B appear in eqn. (115) and thence in the diffusion eqn. (114). By analogy to eqn. (19), the rate coefficient is... 

The relatively small system size feasible in molecular simulation can have a large effect on the results [21,23]. To minimize such finite size effects, an infinite system is mimicked by appl3dng periodic boundary conditions. These conditions effectively create an infinite number of copies of the system in each direction. The cubic box is simplest to implement, but more complex tilings of space such as the truncated octahedron, or the dodecahedron results in smaller system sizes because they require fewer solvent molecules, and hence are more efficient [21,22]. 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

Exterior complex transformation of the coordinate z. In our numerical approach we transform the real coordinate z into a curved path in the complex plane z. This transformation leaves intact the Hamiltonian in the internal part of the system, but supplies the complex rotation of z (and the possibility to use the zero asymptotic boundary conditions for the wavefunction) in the external part of the system. The transformation can be applied both for atoms and molecules and provides precise results for fields from weak up to superstrong with some decrease of the numerical precision in the regime Re.E << Im [39]. 

Remember that this reaction-rate constant was derived for a single molecule of A. A similar procedure can be used to estimate the reverse-rate constant, except that the boundary conditions on the diffusion equation must be modified instead of Cg = 0 at the complex surface and Cg = cb,oo far from that surface (as was used to find Equation 4-64), the reverse reaction starts with a single B molecule bound in the encounter complex (of volume 4na /3), which must subsequently diffuse into an unbounded fluid in which its concentration is negligible. The rate of dissociation of the complex is equal to the rate of diffusion of B molecules away from the complex surface ... 

Periodic boundary conditions are not always used in computer simulations. Some systems, such as liquid droplets or van der Waals clusters, inherently contain a boundary. Periodic botmdary conditions may also cause difficulties when simulating inhomogeneous systems or systems that are not at equilibrium. In other cases the use of periodic boundary conditions would require a prohibitive number of atoms to be included in the simulation. This particularly arises in the study of the structural and conformational behaviour of macromolecules such as proteins and protein-ligand complexes. The first simulations of such s) tems ignored all solvent molecules due to the limited computational resources then available. This corresponds to the unrealistic situation of simulating an isolated protein in vacuo and then comparing the results with experimental data obtained in solution. Vacuum calculations can lead to significant problems. A vacuum boundary tends to minimise the surface area and so may distort the shape of the system if it is non-spherical. Small molecules may adopt more compact conformations when simulated in vacuo due to favourable intramolecular electrostatic and van der Waals interactions, which would be dampened in the presence of a solvent. 

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