The Single-Population Model

In Section 4, competition between two populations is analyzed. Again, the equations can be reduced to a system that can be directly compared to the systems derived in Chapters I and 2. Section 5 explores the evolution in time of the population average length, surface area, and volume in Section 6 we formulate the conservation principle, which played such a crucial role in earlier chapters. The steady-state size distribution of a population is determined in Section 7. Our findings are summarized in a discussion section, where a comparison is made between the conclusions derived from the size-structured model and the unstructured models considered in Chapters 1 and 2. 

An individual in the population is characterized by its length /. It is assumed that all individuals have the same length (/ ) at birth and that they do not shrink, so that /. If nutrient is available at concentration S in the chemostat, then the rate of nutrient uptake by an individual of length / is given by 

Finally, the washout rate D in the chemostat and the population death rate d are assumed to be constant, independent of /. Therefore, the removal rate of the organism is given hy D — D + d. 

Having described the behavior of individuals, we now focus attention at the population level. Let p(t, 1) be the density of individuals of size / at time t, so that 

it is assumed that the nutrient S(/) is given for t tg. The number of individuals in this cohort can change only owing to mortality and washout, so 

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The mandibular salivary glands (ca. 0.25 g) were isolated and perfused arterially with a modified Krebs solution. Na NMR spectra were collected at 8.4, 4.7 and 2.34 T by using MSL-100, AM-200wb and AMX-360wb, respectively. The T2 double-quantum filter and Tj double-quantum filter were used. Results are summarized in brief 1) The transverse relaxation rate constants showed a slight Bq dependency, which promises a poor fitting by the single population model. 2) When temperature was decreased (37 - 5°C), the ratio of the relaxation... 

Population PK analysis following a single dose of 30 mg microencapsulated octreotide acetate intramuscularly the PK profile of microencapsulated octreotide acetate was effectively described by the derived population model. The relationship between IGF-1 and drug concentration could be used to guide optimization of therapeutic octreotide dosage regimens... 

Simulating the model allows investigation of the single populations x3 to x4 of STAT-5. The in silico results are given in Fig. 17.1-6. 

The single-cell model Is limited In Its usefulness for engineering calculations unless a population model can be constructed from the Information In the single-cell model. It Is possible to build a population model using an ensemble of singlecell models to mimic the response of a large population of cells (24, 46, 68). Since computer capacity Is finite, the question Is really, "how few cell models can be Included In a population and still allow reasonable predictions of the behavior of a natural population of cells "... 

We can immediately see that a necessary condition for coexistence is that 0)2 t The values of o) for each species are determined by the same equation as in the single population case, unaffected by the presence of the other species. We can, therefore, move directly to a graphical description of the steady-state behavior for our competition model. Figure 11 shows isoclines of w in the plane of (k,X) values. If we specify values Xi and ki for population 1, this yields a value for wi. We can then immediately discover the permissible steady-states for any species 2 with parameter values X2 and <2 (see Figure 12). If <2 < only species 1 can survive, unless X2 xs such that 0)2 > Wj — which would allow coexistence. If Kj > Kj, only species 2 can survive, unless X2 is such that 0)2 < o)i, which again allows coexistence. 

Implicit in the use of Equation (10) is the assumption that desorption occurs from the surface populated by one type of adsorbed species. This single-state model will be shown to be adequate to describe the dynamic behavior of CO desorption at the high temperature (723 K) considered here. The analysis of the more general case of multi-state desorption can be found in Donnelly et al. (32) and Winterbottom (3). 

The chiral sites which are able to rationalize the isospecific polymerization of 1-alkenes are also able, in the framework of the mechanism of the chiral orientation of the growing polymer chain, to account for the stereoselective behavior observed for chiral alkenes in the presence of isospecific heterogeneous catalysts.104 In particular, the model proved able to explain the experimental results relative to the first insertion of a chiral alkene into an initial Ti-methyl bond,105 that is, the absence of discrimination between si and re monomer enantiofaces and the presence of diastereoselectivity [preference for S(R) enantiomer upon si (re) insertion]. Upon si (re) coordination of the two enantiomers of 3-methyl-l-pentene to the octahedral model site, it was calculated that low-energy minima only occur when the conformation relative to the single C-C bond adjacent to the double bond, referred to the hydrogen atom bonded to the tertiary carbon atom, is nearly anticlinal minus, A- (anticlinal plus, A+). Thus one can postulate the reactivity only of the A- conformations upon si coordination and of the A+ conformations upon re coordination (Figure 1.16). In other words, upon si coordination, only the synperiplanar methyl conformation would be accessible to the S enantiomer and only the (less populated) synperiplanar ethyl conformation to the R enantiomer this would favor the si attack of the S enantiomer with respect to the same attack of the R enantiomer, independent of the chirality of the catalytic site. This result is in agreement with a previous hypothesis of Zambelli and co-workers based only on the experimental reactivity ratios of the different faces of C-3-branched 1-alkenes.105... 

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