The hybrid model

From these considerations, one may finally conclude that the key role of water is to slow down the oxidation process of the Si(lOO) surface, but up to screening effects its influence upon the peptide binding process is rather limited. In particular, we do not expect that stable water layers form between adsorbate and substrate. These characteristic properties of etched Si(lOO) surfaces in de-ionized water effectively enter into the definition of the hybrid model which will then serve as the basis for the analysis and interpretation of the specificity of peptide adhesion on these substrates, 

Following the idea to treat the interaction between the peptide and the substrate on a coarsegrained, effective level, but to maintain atomic resolution for the peptide, the representation of the substrate is simplified and it is considered to consist only of atomic layers with 

Hybrid modei of a peptide-Si(IOO) interface, in this modei each atom of the peptide interacts with the surface iayer of the Si(IOO) crystai. The perpendicuiar distance of the /th atom from the surface is denoted by//. The image shows the iowest-energy conformation of the peptide SI. Because this peptide sustains its cc-heiicai structure that it aiso adopts in soiution, the binding to the substrate is supposed to be weak, it shouid aiso be noted that the popuiation of this conformation is very iow at room temperature. 

z = (zi, Z2. z r) is the perpendicular distance vector of all N peptide atoms from the surface layer of the substrate (see Fig. 14.8). The effect of the surrounding solvent is implicitly contained in the force field parameters. The all-atom peptide model [223,271, 272] is the same that has been used in the study of the peptides in solution in Section 14.4. 

The interaction of the peptide with the substrate is modeled in a simplified way, i.e., each peptide atom feels the mean field of the atomic substrate layers. The atomic density of these layers is dependent on the surface characteristics, i.e., it depends on the crystal orientation (the Miller index hkl) of the substrate at the surface. We make the following assumptions for setting up the model. According to our considerations about Si(lOO) surface properties in de-ionized water, the Si(lOO) surface can be considered to be hydrophobic. This has the effect that it is not favorable for water molecules to reside between the adsorbed peptide and the substrate. Furthermore, polarization effects between side chains and substrate are not expected. 

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Simulation of RF Discharges in Silane-Hydrogen Mixtures with the Hybrid Model... 

For one specific set of discharge parameters, in a comparison between the hybrid approach and a full PIC/MC method, the spectra and the ion densities of the hybrid model showed some deviations from those of the full particle simulation. Nevertheless, due to its computational advantages, the hybrid model is appropri-... 

Two more sets of observables are Introduced Into the hybrid models the emissions factors and the dispersion factors. It Is the difficulty of quantifying these that led to the use of a receptor model over the source model In the first place, so It would seem there Is little advantage In reintroducing them. The advantage of the hybridization Is that the number of Individual emission and dispersion factors can be considerably reduced and that the relative values rather than the absolute values are used. These relative values are more accurate In most cases. Still, the uncertainties of emission and dispersion factors need to be evaluated and Incorporated Into any source/receptor hybrid model. 

Expecting the EPA to lower the IVD value in the future, we were asked by Lubrizol to design a fuel additive for an intake valve deposit of 10 milligrams. We used a genetic algorithm in the hybrid model (see Figure 9.2) to predict the properties of some designed molecules. 

One structure we discovered that came close to meeting our needs (99.3 percent fitness, 12 milligrams IVD) had been already discovered by the Lubrizol scientists through their intuitive guess-and-test approach. However, the hybrid model discovered two other better structures. The best of the three had completely novel chemistry and was a combination of molecules that had never been thought of. The hybrid model used out of the box thinking that opened up possibilities of new chemistry for generating leads in a much shorter time frame. 

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

Employing the additivity approximation, we find dielectric response of a reorienting single dipole (of a water molecule) in an intermolecular potential well. The corresponding complex permittivity jip is found in terms of the hybrid model described in Section IV. The ionic complex permittivity A on is calculated for the above-mentioned types of one-dimensional and spatial motions of the charged particles. The effect of ions is found for low concentrated NaCl and KC1 aqueous solutions in terms of the resulting complex permittivity e p + Ae on. The calculations are made for long (Tjon x) and rather short (xion = x) ionic lifetimes. 

In the third period, which ended in 1999 after the book VIG was published, various fluids had been studied strongly polar nonassociated liquids, liquid water, aqueous solutions of electrolytes, and a solution of a nonelectrolyte (dimethyl sulfoxide). Dielectric behavior of water bound by proteins was also studied. The latter studies concern hemoglobin in aqueous solution and humidified collagen, which could also serve as a model of human skin. In these investigations a simplified but effective approach was used, in which the susceptibility % (m) of a complex system was represented as a superposition of the contributions due to several quasi-independent subensembles of molecules moving in different potential wells (VIG, p. 210). (The same approximation is used also in this chapter.) On the basis of a small-amplitude libration approximation used in terms of the cone-confined rotator model (GT, p. 238), the hybrid model was suggested in Refs. 32-34 and in VIG, p. 305. This model was successfully employed in most of our interpretations of the experimental results. Many citations of our works appeared in the literature. 

We shall show below that the hybrid model gives a satisfactory description of the usual Debye relaxation at the microwave region and explains a quasiresonance absorption band in the FIR region. 

We should note that in many articles and in the book VIG other formulas for the summands of spectral function (121) of the hybrid model are employed ... 

These formulas were previously used for all possible ranges of the cone angles— that is, for p [0,7t/2], but without theoretical justification. Because of the simplicity of Eq. (126a) in comparison with the formulas for the spectral function pertinent to the rigorous hat-plane model (see Section IV.C.3), the hybrid model was often applied for calculation of dielectric properties of various polar fluids. 

Figure 13. Dimensionless absorption versus normalized frequency calculated rigorously (solid lines), from the PL-RP approximation (dashed lines), and for the hybrid model (dashed-and-dotted lines). The cone angle P = tt/8 and the reduced collision frequency y = 0.2. The reduced well depth u = 3.5 (a) and 5.5 (b). Left and righ vertical lines mark the frequency peaks estimated, respectively, in the rotational and librational ranges.

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... 

We employ the following equations Eq. (142) for the complex susceptibility X, Eq. (141) for the complex permittivity , and Eq. (136) for the absorption coefficient a. In (142) we substitute the spectral functions (132) for the PL-RP approximation and (133) for the hybrid model, respectively. In Table IIIB and IIIC the following fitted parameters and estimated quantities are listed the proportion r of rotators, Eqs. (112) and (127) the mean number m of reflections of a dipole from the walls of the rectangular well during its lifetime x, Eqs. (118)... 

Figures 15a and 15b show the wideband absorption and loss spectra, respectively, calculated for the PL-RP version. Figures 15c and 15d demonstrate similar spectra pertinent to the hybrid model. It is seen that the spectra calculated for both versions of the rectangular-well model agree in their main features with the experimental dielectric/FIR spectra recorded in the region 0-1000 cm-1.

The last formula obtained by using the hybrid-model version refers to the quantity, slightly differing from this number, namely, to the number of reorientation cycles, averaged over the total ensemble. 

In Table V the fitted free and estimated statistical parameters are presented. For calculation of the spectral function we use rigorous formulas (130) and Eqs. (132) for the hybrid model. For calculation of the susceptibility %, complex permittivity , and absorption coefficient a we use the same formulas as those employed in Section IV.G.2 for water.29... 

Figure 18. The absorption (a) and loss (b) spectrum of CH3F at T = 133 K. Solid and dashed lines refer, respectively, to rigorous and the hybrid-model versions of the hat-flat model, circles do to experiment. The vertical line marks the frequency of relaxational loss peak.

Rigorous and simplified variants of the hat-flat model agree satisfactorily with the experiment at low temperature. The hybrid model yields worse result for the temperature T = 293 K near the critical one The theoretical loss-peak position is noticeably shifted to lower frequencies in comparison with the experimental position. This important result shows that rigorous30 consideration is preferable in the case of a nonassociated liquid. 

We remind the Reader that we have no theoretical justification (see the end of Section IV.E) for application of the hybrid model at high temperatures, at which the libration angle p is rather large. 

A simplified version of this model, termed the hybrid model (VIG, p. 305) [32-34, 39] (see also Section IV.E) was proposed for the case of a small cone angle p. In this model the rotators move freely over the barrier U0 as if they do not notice the conical surface the librators move in the diametric sections of a cone—that is, they librate. The hybrid model was widely used for investigation of dielectric relaxation in a number of nonassociated and associated liquids, including aqueous electrolyte solutions (VIG, p. 553) [53, 54]. The hat model was recently applied to a nonassociated liquid [3] and to water [7, 12c]. 

More success was gained in the calculations [32] based on application of the cosine squared (CS) potential applied to nonassociated liquids (CH3F and CHF3). However, the results obtained by using the CS model are poor if compared with those given by the hybrid model, since the CS model yields... 

The main purpose of this section is consideration of the FIR spectra due to the second dipole-moment component, p(f). However, for comparison with the experimental spectra [17, 42, 51] we should also calculate the effect of a total dipole moment ptot. In Refs. 6 and 8 the modified hybrid model44 was used, where reorientation of the dipoles in the rectangular potential well was considered. In this section the effect of the p(f) electric moment will be found for the hat-curved, potential, which is more adequate than the rectangular potential pertinent to the hybrid model. In Section VI.B we present the formula for the spectral function of the hat-curved model modified by taking into account the p(f) term (derivation of the relevant formula is given in Section VI.E). The results of the calculations and discussion are presented, respectively, in Sections VI.C and VI.D. 

A brief description of the hybrid model based on application of the rectangular well potential was given in Section V.E. 

The main advantage of the hat-curved potential is that it is possible to narrow the width Avor of the librational absorption band by decreasing the form factor /. Indeed, Avor attains its maximum value when/ = 1. Note that / = 1 is just the case of the hat flat or its simplified variant, the hybrid model, both of which were described in Section IV. The latter was often applied before (VIG) and is characterized by a rather wide absorption band, especially in the case of heavy water. In another extreme case, / — 0, the linewidth Avor becomes very low. When / = 0, we have the case of the parabolic potential well, whose dielectric response was described, for example, in GT and VIG. Thus, when the form factor/of the hat-curved well decreases from 1 to 0, the width Avor decreases from its maximum to some minimum value. 

The second (ionic) term in Eq. (387) is assumed to vanish in the limit co —> oo, just as does the term As (v) in Eq. (278) stipulated by oscillating charges of a nonrigid dipole. The first term in Eq. (387) will be calculated below in terms of the hybrid model, which was briefly described in Section IV.E. For the limit co —> oo we set this term to be equal to optical permittivity n2, the same as in pure water. 

If the electrolyte concentration Cm varies, the wideband spectra are controlled only by one parameter (x) of the hybrid model. Other parameters of this model—the normalized well depth u, the libration amplitude ft, and the p-correcting coefficient —can be set independent of Cm and therefore could be fit by comparison of the calculated and recorded [70, 71] spectra of water (see Table XVI). 

Further elaboration of the hybrid models stipulated by the necessity to model chemical processes in polar solvents or in the protein environment of enzymes, or in oxide-based matrices of zeolites, requires the polarization of the QM subsystem by the charges residing on the MM atoms of the classically treated solvent, or protein, or oxide matrix. This polarization is described by renormalizing the one-electron part of the effective Hamiltonian for the QM subsystem ... 

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