Lattice models contact energies

The Lattice Model Contact Energy iv Approximates Intermolecular Interactions... 

Despite their simplicity, certainly compared to the all-atom potentials used in molecular dynamics studies, these contact energy functions enable the exploration of different interaction scenarios. This diversity is achieved by changing the heterogeneity of the sequence, by altering the number N of different types of residues that are being used. The most elementary lattice model involves only two types of monomers hydrophobic... 

The formulation of a proper surface boundary condition is a delicate matter, as noted by DiMarzio (1965) and de Gennes (1969). Lattice models simply require that P(i, s) = 0 for layers i < 0, a form proven correct by DiMarzio (1965). In continuum models, chains intersecting the surface undergo both reflection and adsorption, the relative amount of each depending on the energy of contact at the surface. The result is a mixed boundary condition expressed by de Gennes (1969) as... 

Fig. 5. Free energy landscape of a lattice model protein (see Sect. 2.2), as a function of two order parameters, the number of contacts C and the number of native contacts Qo (see Sect. 2.3). Unlike the energy landscape funnel picture, the free energy shows two stable states separated by a barrier (the transition state). Extended unfolded conformers quickly collapse to the molten globule, and have to overcome a barrier to folding to the native state. The funnel picture is thus reconciled with the two-state concept of a free energy barrier. Reprinted from Dinner et ah. Trends Biochem. Sci. 25, 331, (2000) with permission from Elsevier...

Most lattice methods rely on an extremely simple potential function, either a two state interresidue contact energy corresponding to native/nonnative contacts, or a three state model, corresponding to hydrophobic-hydrophobic, hydrophilic-hydrophilic, and hydrophobic-hydrophilic interactions. The interaction of the twenty naturally-occurring amino acids in real proteins are obviously more complex. 

The gas-lattice model considers liquids to be a mixture of randomly distributed occupied and vacant sites. P and T can change the concentration of holes, but not their size. A molecule may occupy m sites. Binary liquid mixtures are treated as ternary systems of two liquids (subscripts 1 and 2 ) with holes (subscript 0 ). The derived equations were used to describe file vapor-Uquid equilibrium of n-alkanes they also predicted well the phase behavior of -alkanes/PE systems. The gas-lattice model gives the non-combinatorial Helmotz free energy of mixing expressed in terms of composition and binary interaction parameters, quantified through interaction energies per unit contact area (Kleintjens 1983 Nies et aL 1983) ... 

The classical models of adsorption processes like Langmuir, BET, DR or Kelvin treatments and their numerous variations and extensions, contain several uncontrolled approximations. However, the classical theories are convenient and their usage is very widespread. On the other hand, the aforementioned classical theories do not start from a well - defined molecular model, and the result is that the link between the molecular behaviour and the macroscopic properties of the systems studied are blurred. The more developed and notable descriptions of the condensed systems include lattice models [408] which are solved by means of the mean - field or other non-classical techniques [409]. The virial formalism of low -pressure adsorption discussed above, integral equation method and perturbation theory are also useful approaches. However, the state of the art technique is the density functional theory (DFT) introduced by Evans [410] and Tarazona [411]. The DFT method enables calculating the equilibrium density profile, p (r), of the fluid which is in contact with the solid phase. The main idea of the DFT approach is that the free energy of inhomogeneous fluid which is a function of p (r), can be... 

Shahknovich and Gutin [88] have introduced a 3-D lattice model of protein folding in which amino acids are placed inside a cube in which protein is depicted as a compact self-avoiding chain structure, like one shown in Figure 3.11, which was considered by Sali, Shahknovich, and Karplus [101], As outlined by Sali et al, for such protein models, one can calculate the total energy E of conformation, which is given by the sum of the contact energies between the non-bonded adjacent amino acids within the 3 x 3 x 3 lattice ... 

In practice, few solutions are truly ideal. They involve energies of mixing. In the lattice model, the total energy of mixing is the sum of the contact interactions of noncovalent bonds of all the pairs of nearest neighbors in the mixture. For a lattice solution of A and B particles. Figure 15.4 shows the three possible types of contact an AA bond, a BB bond, or an AB bond. There are no other options, because the lattice is completely filled by A s and B s. 

The boundary between two condensed phases is an interface. The interfacial tension yab is the free energy cost of increasing the interfacial area between phases A and B. If yab is large, the two media will tend to minimize their interfacial contact. Let s determine yab by using the lattice model for molecules of types A and B that are identical in size. 

Figure 16.10 iJ° for the transfer of hydrocarbons from aqueous solution to pure liquid hydrocarbon at 25 °C, based on solubility measurements of C McAuliffe, J Phys Chein 70, 1267 (1 966). The transfer free energy is linearly proportional to the number of solute-solvent contacts, represented by z in the lattice model. Source ... 

In the lattice model, the driving force for partitioning depends on the lattice coordination number z, which serves as a measure of the surface area of contact between the solute molecule and surrounding medium. It follows that Pj (A) -p j (B) should be a linear function of the sizes of solutes that partition between different media. Figure 16.10 shows examples in which the partitioning free energy increases linearly with the solute size. For the same reason, the Henry s law constant should increase with solute size. Figure 16.12 shows this result. 

Lattice model mixture. You have a solution with molecules of type A mixed with molecules of type B at T = 300 K. Suppose that every two AB molecular contacts are less stable than one AA plus one BB contact, by an energy kT, and that every molecule has an average of six neighbor contacts. For x,4 = xb = 1/2, to w hat temperature do you have to heat the system to get A to dissolve fully in B ... 

Fig. 10 Interaction between cellular force dipoles, (a) Possible transition of a dipole to the neighboring unoccupied edges on a triangular lattice (with probability P (A H ) is shown by the lightly shaded ellipses surrounding the original dipole (dark ellipses). Two dipoles are not allowed occupy a common node i.e. direct cell-cell contact is forbidden in this model, (b) Energies of a two-dipole system for various configurations. Simulation data suggests that cells preferably position themselves next to each other to minimize the total energy.

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