Nonmonotonic Models

In Section IV we provide illustrations of the modeling concepts presented in Section II and how the strategy of nonmonotonic planning has been used to synthesize the switchover operating strategy for a chemical process. 

In this section we will offer several illustrations of the various aspects of nonmonotonic operations planning (discussed in earlier sections) including the following (1) development of hierarchical models for the process and its operations, (2) conversion of constraints to temporal orderings of primitive operations, and (3) synthesis of complete plans. 

Chandra and his coworkers have developed analytical theories to predict and explain the interfacial solvation dynamics. For example, Chandra et al. [61] have developed a time-dependent density functional theory to predict polarization relaxation at the solid-liquid interface. They find that the interfacial molecules relax more slowly than does the bulk and that the rate of relaxation changes nonmonotonically with distance from the interface They attribute the changing relaxation rate to the presence of distinct solvent layers at the interface. Senapati and Chandra have applied theories of solvents at interfaces to a range of model systems [62-64]. 

For the g2SC phase, the typical results for the default choice of parameters H = 400 MeV and r/ = 0.75 are shown in Figure 4. Both the values of the diquark gap (solid line) and the mismatch parameter 5/j, = /i,./2 (dashed line) are plotted. One very unusual property of the shown temperature dependence of the gap is a nonmonotonic behavior. Only at sufficiently high temperatures, the gap is a decreasing function. In the low temperature region, T < 10 MeV, however, it increases with temperature. For comparison, in the same figure, the diquark gap in the model with /je = 0 and /./, = 0 is also shown (dash-dotted line). This latter has the standard BCS shape. 

Pore size and dielectric constant s of water in pores exhibit a strong effect on proton distributions, as studied in Eikerling. Model variants that take into account the effect of strongly reduced s near pore walls ° and the phenomenon of dielectric saturation ° 2° lead to nonmonotonous profiles in proton concentration with a maximum in the vicinity of the pore wall. 

A schematic presentation is given in Figure 4.51 for the simplified diffusion-reaction two-enzymes/two-compartments model. From an enzyme kinetics point of view, we consider the most general case, in which both enzymes have nonmonotonic dependence on the substrate and hydrogen-ion concentrations. 

In the opposite case, /3A<1, the Raman process is dominant, the probability of which js proportional to n(w,)[n(w ) +1)] under the condition - temperature dependences of both processes for various A are shown in Figure 6.19. The nonmonotonic dependence of t 1 on A predicted by this model was not observed experimentally because this effect is pertinent to inaccessibly low temperatures. However, the T4 dependence was verified experimentally for benzonic acid crystals by Oppenlander et al. [1989]. 

Nonmonotonic mass variation (Fig. 14-14a), well predicted by the kinetic model (Rychly et al., 1997). 

Considering the highly processive mechanism of the protein degradation by the proteasome, a question naturally arises what is a mechanism behind such translocation rates Let us discuss one of the possible translocation mechanisms. In [52] we assume that the proteasome has a fluctuationally driven transport mechanism and we show that such a mechanism generally results in a nonmonotonous translocation rate. Since the proteasome has a symmetric structure, three ingredients are required for fluctuationally driven translocation the anisotropy of the proteasome-protein interaction potential, thermal noise in the interaction centers, and the energy input. Under the assumption that the protein potential is asymmetric and periodic, and that the energy input is modeled with a periodic force or colored noise, one can even obtain nonmonotonous translocation rates analytically [52]. Here we... 

Recent molecular dynamics simulations of water between two surfactant (sodium dodecyl sulfate) layers, reported by Faraudo and Bresme,14 revealed oscillatory behaviors for both the polarization and the electric fields near a surface and that the two fields are not proportional to each other. While the nonmonotonic behavior again invalidated the Gruen—Marcelja model for the polarization, the nonproportionality suggested that a more complex dielectric response of water might, be at the origin of the hydration force. The latter conclusion was also supported by recent molecular dynamics simulations of Far audo and Bresme, who reported interactions between surfactant surfaces with a nonmonotonic dependence on distance.15... 

This type of oscillatory hydration was previously observed experimentally in interactions between mica surfaces in water,16 and has been associated with the layering of water in the vicinity of a surface.16 17 The discrete nature of the water molecules, considered hard spheres, was suggested to be responsible for these nonmonotonic interactions 18 19 however, the high fluidity of the water confined in molecularly thin films20 seems to be inconsistent with the crystallization of water predicted by the hard-sphere model.18... 

It was recently shown via molecular dynamics simulations14 that, in the close vicinity of a surface, water molecules exhibit an anomalous dielectric response, in which the local polarization is not proportional to the local electric field. The recent findings are also in agreement with earlier molecular dynamics simulations, which showed that the polarization of water oscillates in the vicinity of a dipolar surface,11,14 leading therefore to a nonmonotonic hydration force.15 Previous models for oscillatory hydration forces, based either on volume-excluded effects,18,19 or on a nonlocal dielectric constant,f4 predicted many oscillations with a periodicity of 2 A, which is inconsistent with these molecular dynamics simulations,11,18,14 in which the polarization exhibits only a few oscillations in the vicinity of the surface, with a larger periodicity. 

However, the use of (5.24) should not be considered as a panacea for modeling nonmonotonic dissolution curves. Obvious drawbacks of the model (5.24) are ... 

Dichotomous responses. Some endpoints (e.g., behavioral parameters) can lead to nonmonotonous concentration-response curves, which are more difficult to model than the monotonous concentration-response curves that are observed, for example, in survival studies. Also in case of hormesis, nonmonotonous concentration-response curves are found. Nonmonotonous curves pose inherent problems for the application of both mixture toxicity concepts (Backhaus et al. 2004 Belz et al. 2008). 

Beside monotonous relaxation kinetics, which is usually treated using one of the above models, there is experimental evidence for nonmonotonous relaxation kinetics [78], Some of these experimental examples can be described by the model... 

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