Models studied, root mean squared

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

The challenge is then to achieve the same degree of accuracy in the derived values of the experimental electron density. Recent studies have shown that in some cases this is indeed within the reach of the present-day modelling techniques [3-5]. When the major sources of experimental error have been corrected for the typical root mean square electron density residual can reach values as low as 0.05 e A-3, with maxima below 0.20eA-3 in absolute value. The observed residuals are usually due to the... 

In all the studied systems addition of the surrounding protein in an ONIOM model clearly improves the calculated active-site geometries. This is clearly illustrated in Figure 2-13, which shows the root-mean-square deviation between calculated and experimental structures for four of the studied enzymes. 

Table 2. Scale factors for ab initio model vibrational frequencies adapted from (Scott and Radom 1996). Please note that these scale factors are determined by comparing model and measured frequencies on a set gas-phase molecules dominated by molecules containing low atomic-number elements (H-Cl). These scale factors may not be appropriate for dissolved species and molecules containing heavier elements, and it is always a good idea to directly compare calculated and measured frequencies for each molecule studied. The root-mean-squared (rms) deviation of scaled model frequencies relative to measured frequencies is also shown, giving an indication of how reliable each scale factor is.

Ellipsometry determines a certain average thickness th of the adsorbed layer. However, what is important for the evaluation of polymer conformations in this layer is the root-mean square thickness t. Hence, it is necessary to find a way of relating t to th. McCrackin and Colson66 studied this problem for several distributions of segments and found tnn, = th/l-5 for the exponential distribution and t - th/1.74 for the Gaussian distribution. Takahashi et al.67 showed that t = th/1.63 for the one-train and two-tail model (see Eqs. (B-110) and (B-lll)). 

From studies performed with well characterized substrates and polystyrene with a narrow M distribution, the measured values of T, p, and 0 at the theta point have been found to agree closely with the theories of Silberberg and Scheutjens and Fleer. Furthermore, it has been shown that the measured root-mean-square thickness of the adsorbed layer can be predicted semiquantitatively by the loop-train-tail conformation model. 

Table 4-2 reports the electrostatic and non-electrostatic components of AGsoi in water for the series of compounds included in the study computed from MST calculations. The deviation between experimental and calculated free energies of hydration is in general small, as noted in a mean signed errors (mse) close to zero and a root-mean square deviation around 0.9 kcal/mol, which compares with the statistical parameters obtained in the parametrization of the MST model [15]. 

Physiologic model-physiologically based pharmacokinetic model (PB/PK) A physiologically based model for Gl transit and absorption in humans is presented. The model can be used to study the dependency of the fraction dose absorbed (Fabs) of both neutral and ionizable compounds on the two main physico-chemical input parameters [the intestinal permeability coefficient (Pint) and the solubility in the intestinal fluids (Sint)] as well as the physiological parameters, such as the gastric emptying time and the intestinal transit time. For permeability-limited compounds, the model produces the established sigmoidal dependence between Fabs and Pnt. In case of solubility-limited absorption, the model enables calculation of the critical mass-solubility ratio, which defines the onset of nonlinearity in the response of fraction absorbed to dose. In addition, an analytical equation to calculate the intestinal permeability coefficient based on the compound s membrane affinity and MW was used successfully in combination with the PB-PK model to predict the human fraction dose absorbed of compounds with permeability-limited absorption. Cross-validation demonstrated a root-mean-square prediction error of 7% for passively absorbed compounds. 

In a more recent study, Hinchliffe and Willis (2003) model dynamic systems using genetic programming. The new approach is evaluated using two case studies, a test system with a time delay and an industrial cooking extruder. The objectives minimized are the root mean square error and the correlation and autocorrelations between residuals. The residuals of a model represent the difference between the predicted and actual values of the process output. In this work, two MOGPs are compared, one based on Pareto ranking but without preferences, and... 

Figure 6.4.7 shows the interpretation of two sets of relaxation data obtained from a specific site, Trp9, in the polypeptide backbone of gramicidin A [24]. From powder pattern studies it has been shown that the local motions occur about an axis consistent with the C, —C ,+i axis, and that the motion is a librational motion of about 20° [21], quite similar to the indole side-chain described earlier. The data in Fig. 6.4.7 has been interpreted in light of this experimentally defined motional model. However, the field-dependent relaxation data suggests that the amplitude is much less than 20°, in fact it is closer to a root mean square amplitude of 5°. However, this apparent... 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

An example from recent works is the study of the dynamics of unentangled PEO chains in 35% PEO/65% PMMA by quasielastic neutron scattering (Niedzwiedz et al., 2007), and in 25% PEO/75% PMMA by molecular dynamics simulations (Brodeck et al., 2010). The Rouse model has the mean-square displacement, (r (r)>, of a chain segment, which increases proportionally to the square root of time according to... 

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