Modeling schemes

Much use has been, and continues to be, made of simplified model schemes representative of general classes of chemical or themial feedback. The oregonator and Lengyel-Epstein models for the BZ and CDIMA systems have been given earlier. Pre-eminent among the more abstracted caricature models is the bnisselator introduced by Prigogine and Lefever [47] which has the following fomi ... 

In the classical diffusion control model it is assumed that propagation occurs according to the terminal model (Scheme 7.1). The rate of the termination step is limited only by the rates of diffusion of the polymer chains. This rate may be dependent on the overall polymer chain composition. However, it does not depend solely on the chain end.166,16... 

Figure 11.1. A flow-model scheme for treating the protein routing question. Labels refer to flow rates of carbon. The total carbon flux, into and out of the body, is 1, divided into F (for protein) and 1 - F for the remainder. The significant relevant internal fluxes are between the amino acid pool (coupled to the body protein pool), and the energy metabolism pool . The extent to which protein routing is observable in the body protein composition depends on the value ofX (See Fig. 11.2). Numbers in refer to suggested isotopic fractionations associated with a metabolic path, which are consistent with the data of the Ambrose and Norr (1993) and Tieszen and Fagre (1993) data set (see Section 4.1).

Elgiire 11.3. A flow-model scheme intended to represent relevant nitrogen flows, especially with regard to which flows are reversihle. The labeled reactions 1, 11, 111 IV are all potentially iso-topically fractionating. Because reaction 11 is not reversible, subsequent fractionations in the excretory pathway should not influence the isotopic composition of the body protein pool. 

Figure 11.4. (a) Flow-model scheme for a simple food chain with one predator-prey relationship. See text for discussion, (b) The steps involved whereby atoms from prey collagen (i.e., the diet) may be transferred to a predator s collagen (i.e., the consumer tissue). Each arrow represents a potential change in carbon isotopic composition, complicating the relationship between prey collagen 5 C and predator 5 C. 

The product selectivities in propane metathesis can also be explained by using the same model in which [1,3]- and [1,2]-interactions determine the ratio of products. For instance, the butane/pentane ratios are 6.2 and 4.8 for [(= SiO)Ta(= CHfBu)(CH2tBu)2] and [(= SiO)2Ta - H], respectively (Table 5). A similar trend is observed for the isobutane/isopentane ratio, which are 4.1 and 3.0, respectively. The higher selectivity in butanes (the transfer of one carbon via metallacyclobutanes involving [l,3]-interactions) than that of pentanes (the transfer of two carbons via metallacyclobutanes involving [1,2]-interactions) is consistent with this model (Scheme 28). 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

Diastereoselectivity of the reactions of the cation 26 and the anion 29 derived from 25 (R2 = Me) was modeled by self-consistent reaction field solvation models obtained from ab initio SCF-MO calculations. The experimentally found cis/trans ratios confirmed the model (Scheme 1) <2002JOC2013>. 

Diastereoselective addition of a wide range of Grignard reagents to C -alkyl and C-aryl-A-[a.-phenyl- or u-methyl-j3-(benzyloxy)ethyl nitrones is determined by the presence of a stereogenic A -substituent (136, 137). High diastereoselectiv-ity in the addition of organometalic compounds to A-(( i-methoxyalkyl) nitrones can be explained by a simple chelation model (Scheme 2.132) (136). 

Figure 6. The model scheme of heavy metal behavior in the atmosphere (Ilyin et al., 2004).

The many sulphur dyes synthesised via quinonimine intermediates are polymeric products containing numerous disulphide crosslinkages that can be broken by reduction in aqueous sodium sulphide thioether groups survive the reduction process. The smaller thiolate-containing molecules formed are substantive to cellulose. Although the actual structures of such dyes are complex, their essential features can be illustrated in an idealised model (Scheme 6.23), in which X = S, NH or O, and R indicates substituents or annelated rings. 

Similarly, the proton transfer on the hydroxo oxo complex is illustrated by Eqs. (13)-(15), based on the Eigen model (Scheme 4), and can again be due to protolysis, hydrolysis, or direct proton exchange... 

The exchange mechanism of proton transfer in these systems is explained in terms of the Eigen model (Scheme 4) as discussed above by either hydrolysis (Mo(IV) and W(IV)) or protolysis (Tc(V) and Re(V)) pathways, coupled with direct proton transfer in the intermediate pH... 

An exception to the general model arises for systems bearing a substituent directly adjacent to the alkene radical cation. Here, the syn diastereomer cyclizes with a high degree of stereocontrol, as predicted by the model (Scheme 37) [131,144,145]. 

Conversely, the anti isomer cyclizes with much lower selectivity in the opposite sense to that predicted by the model (Scheme 38). This situation arises because cyclization on the initial contact ion pair, on the opposite face to... 

The degree of asymmetric induction with 119 increased in the order CF3 — Me < Et < Bu < Pr As with the alkoxyaluminum dichloride reagent (Sect. II1-A-2) the cyclic transition state model (Scheme 20) would predict a continuous... 

If the response of the organism is produced by a combination of the two compounds, then they are said to exert joint action. This joint action can be further classified into simply additive, more than additive (i.e., synergistic), and less than additive (i.e., antagonistic). When this scheme is applied to multicomponent mixtures present in leachates of solid wastes, the analysis becomes more complex because the joint actions of different compound pairs may fall into different types of joint action. In the next section, three different modeling schemes are presented. 

The occurrence of a persistent absorbance change (AA ) for the [M, Fe] hybrids is a slight complication that requires an extended kinetic model (Scheme II) in which (MP) is reduced not only by Fe P (regenerating the [MP, Fe P] state), but also by an as yet unidentified amino acid residue X and/or solution impurities (Eq. (3)) leading to [M, Fe " ] ... 

A study of debrominations of vtc-dibromides promoted by diaryl tellurides and din-hexyl telluride has established several key features of the elimination process the highly stereoselective reactions of e/7f/tro-dibromides are much more rapid than for fhreo-dibromides, to form trans- and cw-alkenes, respectively the reaction is accelerated in a more polar solvent, and by electron-donating substituents on the diaryl telluride or carbocation stabilizing substituents on the carbons bearing bromine. Alternative mechanistic interpretations of the reaction, which is of first-order dependence on both telluride and vtc-dibromide, have been considered. These have included involvement of TeAr2 in nucleophilic attack on carbon (with displacement of Br and formation of a telluronium intermediate), nucleophilic attack on bromine (concerted E2- k debromination) and abstraction of Br+ from an intermediate carbocation. These alternatives have been discounted in favour of a bromonium ion model (Scheme 9) in which the role of TeArs is to abstract Br+ in competition with reversal of the preequilibrium bromonium ion formation. The insensitivity of reaction rate to added LiBr suggests that the bromonium ion is tightly paired with Br. ... 

CHEMK, A Computer Modeling Scheme for Chemical Kinetics (undated). G. Z. Whitten and J. P. Meyer. Systems Applications, Inc., 950 Northgate Drive, San Rafael, CA 94903. 

The presence of a stereogenic center on the aldehyde can strongly inlinence the diastereoselectivity in allylboration reactions, especially if this center is in the a-position. Predictive rules for nucleophilic addition on snch a-snbstitnted carbonyl substrates such as the Felkin model are not always snitable for closed transition structures.For a-substituted aldehydes devoid of a polar substituent, Roush has established that the minimization of ganche-ganche ( syn-pentane ) interactions can overrule the influence of stereoelectronic effects. This model is valid for any 3-monosubstituted allylic boron reagent. For example, althongh crotylboronate (E)-7 adds to aldehyde 39 to afford as the major prodnct the diastereomer predicted by the Felkin model (Scheme 2), " it is proposed that the dominant factor is rather the minimization of syn-pentane interactions between the Y-snbstitnents of the allyl unit and the a-carbon of the aldehyde. With this... 

---