Closed association model, concentration

The general principles of self-assembly of amphiphilic molecules into finite-sized aggregates (micelles) are described in a number of classic books [34-36]. In our analysis of micelle formation we apply the equilibrium close association model. That is, we assume first that only one population of micelles, with an aggregation number p (number of copolymers in one aggregate), is present in the system at any given concentration of amphiphiles in the solution, or that there are no micelles at all and second, that the free energy per molecule in a micelle, Fp, exhibits a minimum at a certain value of the aggregation number, p = po. 

The closed association model can accoimt for the observation of a critical micelle concentration. It is also known as the mass action model. It is assumed that there is a dynamic equilibrium between molecules and micelles containing p molecules. In practice, micelles are not monodisperse (Section 1.8), i.e. there is a range of values of association number. Usually, the dispersity in p amounts to about 20-30 % of its value, which is not large enough to change the behaviour captured by models for monodisperse micelles. In the following, we consider the equilibrium between nonionic surfactant molecules and monodisperse micelles in dilute solution ... 

The closed association model of micellization can be used to obtain the fraction of molecules in micelles as a function of total amphiphile concentration. Using concepts from Section 6.4.5 derive expressions for the fractions of associated and unassociated molecules and hence obtain plots similar to those in Fig. 4.18. 

As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a true model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. 

Comparison of the depositional fluxes shows that diatoms were the most important particle component transporting P to the sediment surface, accounting for 50-55% of the flux (Table II). Terrigenous material and calcite were also important transport vectors. Deposition varied markedly with season, as shown by the time series plot of the major particle components (Figure 13). The total P flux calculated by using the particle components model agreed with the flux measured by sediment traps (157-227 versus 185 mg/m2). The close agreement indicated that the major particle vectors were represented and associated P concentrations were accurately quantified. 

Micelles are formed by association of molecules in a selective solvent above a critical micelle concentration (one). Since micelles are a thermodynamically stable system at equilibrium, it has been suggested (Chu and Zhou 1996) that association is a more appropriate term than aggregation, which usually refers to the non-equilibrium growth of colloidal particles into clusters. There are two possible models for the association of molecules into micelles (Elias 1972,1973 Tuzar and Kratochvil 1976). In the first, termed open association, there is a continuous distribution of micelles containing 1,2,3,..., n molecules, with an associated continuous series of equilibrium constants. However, the model of open association does not lead to a cmc. Since a cmc is observed for block copolymer micelles, the model of closed association is applicable. However, as pointed out by Elias (1973), the cmc does not correspond to a thermodynamic property of the system, it can simply be defined phenomenologically as the concentration at which a sufficient number of micelles is formed to be detected by a given method. Thermodynamically, closed association corresponds to an equilibrium between molecules (unimers), A, and micelles, Ap, containingp molecules ... 

Figure 9-24. A plot of the concentration dependence of r/sp/c for various polymer-solvent and polymer-polymer interactions. Left diagram nonassociating polymers right diagram associating polymers. (I) Good solvent, (II) poor solvent, (o) Open assocation, (c) closed association. The multiplicity of curves derives from the number of possible models as well as from the relative influences of the equilibrium association constant, the association number, the molar mass, the size and shape of the molecules and associates, and the solvent interaction.

Hasal et al. (1997) used similar techniques to model the effects of external electric fields on calcium waves like those shown in Figure 13.3. The phenomena observed resemble those found experimentally in the BZ system. Calcium waves play key roles in many biological phenomena and in neuronal-glial networks, for example, they are closely associated with local electrical field gradients arising from spatiotemporal patterns of ionic concentrations (Cooper, 1995). 

Hunter (60) reported a self-assembled open polymer formed by a zinc porphyrin bearing one para-aniline substituent at the meso position. The ortho- and mela-analogs discussed above form closed dimers, but the geometry of the para-derivative precludes this, and polymerization is the only alternative (76, Fig. 31). Although the dilution experiments could be fitted to a non-cooperative polymerization model with a pairwise association constant (K = 190 M 1) practically identical to that found for simple aniline-zinc porphyrin complexes (K = 130 M 1), broadening of the 4H NMR spectrum at high concentrations is characteristic of oligomerization. 

The bioavailability of selenium to a benthic deposit-feeding bivalve, Macoma balthica from particulate and dissolved phases was determined from AE data. The selenium concentration in the animals collected from San Francisco Bay was very close to that predicted by a model based on the laboratory AE studies of radiolabelled selenium from both particulate and solute sources. Uptake was found to be largely derived from particulate material [93]. The selenium occurs as selenite in the dissolved phase, and is taken up linearly with concentration. However, the particle-associated selenium as organoselenium and even elemental selenium is accumulated at much higher levels. The efficiency of uptake from the sediment of particulate radiolabelled selenium was 22%. This contrasts with an absorption efficiency of ca. 86% of organoselenium when this was fed as diatoms - the major food source of the clam. The experiments demonstrated the importance of particles in the uptake of pollutants and their transfer through the food web to molluscs, but the mode of assimilation was not discussed. 

In the proposed criteria, the linear model was used to calculate the concentrations associated with incremental lifetime risks of 10-5. However, in response to public comment, the USEPA ultimately decided to adopt the linearized multistage model to make full use of all available data. Comparison of the values reported in the box indicates that, for most cases, the concentrations calculated by either model for a given nominal risk are very close. 

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