Proliferation exponential

If the hyperbolic dynamical system is moreover chaotic, the periodic orbits are known to proliferate exponentially as [19]... 

The concatenation is performed without constraint on the successive symbols so that the motion on the repeller corresponds to a Bernoulli random process. The regions around the fundamental periodic orbits are successively visited in a random fashion without memory of the previous fundamental periodic orbit visited. As a consequence, the periodic orbits proliferate exponentially with their period, as described by (2.17). The topological entropy per symbol is equal to htop = In M. 

The approximation in (2.2.30) is acceptable for large n. This formula yields an important result The number of periodic orbits of the shift map increases roughly exponentially with the length of the period. In other words, periodic orbits in the shift map proliferate exponentially. 

For classically chaotic quantum systems the forward application of the trace formula is difiicult because of the following reasons, (i) In a chaotic system the number of periodic orbits proliferates exponentially (ii) the orbits have to be computed numerically and (iii) there are convergence problems with (4.1.72) that have to be circumvented with appropriate resummation prescriptions such as, e.g., cycle expansions (Cvitanovic and Eckhardt (1989), Artuso et al. (1990a,b)). Nevertheless, sometimes valuable information on the structure of atomic states can be obtained by retaining only the shortest orbits in the expansion (4.1.72). This was... 

The nucleotide pools of chick fibroblasts proliferating exponentially in culture have been determined by an isotopic method as seen in Table l-III, in these cells the pool of ATP is about 10 times that of any other ribonucleoside triphosphate and more than 100 times that of dATP or dTTP (14)- The small size of the deoxyribonucleotide pool in the exponentially proliferating cells is consistent with observations by others and suggests that the deoxyribonucleotides are synthesized as needed for DNA synthesis and do not accumulate. Skoog and Nordenskjold (5) have estimated that in cultures of mouse embryo cells undergoing DNA synthesis, the pool of dGTP was sufficient to support only about 30 seconds of DNA... 

The process known as SPREAD (Surface Promoted Replication and Exponential Amplification of DNA Analogues) attempts to reach the target, striven for by many researchers, of an exponential proliferation of biomolecules in model systems. As already mentioned, product inhibition (e.g., by dimerisation of the new matrices to give C2) only allowed parabolic growth. In the SPREAD process, both solid phase chemistry and feeding have a positive effect on the synthesis. Thus, no separation processes are required, as excess reagents can be removed just by washing. The synthetic process consists of four steps ... 

Fig. 8.3 The proliferation curves of RNA strands (the Q beta system) for decreasing concentrations of added matrix molecules. If the number of matrix molecules is larger than that of the enzymes, a linear proliferation is observed (first curve). This slows down at high concentrations, due to product inhibition. RNA proliferation is exponential if the amount of enzyme is larger than that of the matrix. If no matrix is added, the system goes through an incubation phase and then forms an RNA sequence which is related to certain Q beta fragments (Eigen et al., 1982)...

L. T. Vassilev, W. C. Burhans, and M. L. DePamphilis, Mapping an origin of DNA replication at a single-copy locus in exponentially proliferating mammalian cells. Mol. Cell. Biol. 10,4685-4689 (1990). 

Note that this equation will only apply during the initially exponential growth phase. As the number of cells becomes much greater, the daughter cells compete for limiting quantities of one or more nutrients. This will reduce the efficiency of cell division, and the proliferation curve will reach a plateau that is often called the stationary growth phase. 

The rate of growth of human and experimental cancers is initially quite rapid (exponential) and then slows until a plateau is reached. The decrease in growth rate with increasing tumor size is related both to a decrease in the proportion of cancer cells actively proliferating (termed the growth fraction) and to an increase in the rate of cell loss due to hypoxic necrosis, poor nutrient supply, immunological defense mechanisms, and other processes. 

The two models of cell proliferation differ largely in the predici-ton of the Smith and Martin model that the proportion of cells remaining in interphase (a) should decline exponentially with age, beginning at time TB after mitosis. Thus a semilog plot of a against age after mitosis should show a linear decay after a lag period equal toTB. As shown in Fig. 10.4 this is borne out by the data. In fact there is an initial downward curvature before linearity is attained caused by the variability in TB. 

Time sequenced, step growth of branch cells and dendrimer structure Self replication of branch cells throughout dendrimer construction Structural proliferation of dendrimers with exponential amplification of branch cells" and surface functionality as a function of generation... 

More recently an interesting dendron construction method was reported independently by Frechet et al. [102] as well as by Neenan and Miller [103] which has been referred to as the 1(B) Convergent Method. This method differs from the divergent dendron approach in that exponential proliferation of surface groups is accomplished by use of a limited number of growth reactions per sequence. Its advantages and limitation will be discussed later. With this method a variety of dendron families have been successfully synthesized and subsequently coupled to various point-like as well as hyper-branched cores, to produce dendrimers. This... 

Alternatively, dendrimers may be synthesized directly by our original II Divergent Core Proliferation method. This method may involve the exponential covalent assembly of monomer units around a multi-valent core to produce branch cells in situ or it may involve the direct use of pre-formed branch cell reagents. In either case the resulting covalent structure consists of precise numbers of dendrons organized around the initiator core. 

If there is a change in the physiology of the cells, can one conclude that there are biochemical events, pathways, which are more sensitive than others to the influence of copper, ie., is a cell in a non-proliferating (lag phase) stage more susceptible to copper stress than exponentially growing cells ... 

Cell viability. Exponentially growing HL-60 cells, 15 x 106 in 75.0 mL of RPMI 1640, were incubated with either ATRA, or UDMA or BDDMA for five days. The total cell number was determined every day using the trypan blue dye exclusion test. Both the cellular proliferation - as area under curves (AUC) - and cellular mortality were calculated. 

R17. Rocha, B., Lembezat, M. P., Freitas, A., and Bandeira, A. Interleukin-2 receptor expression and interleukin-2 production in exponentially growing T cells Major differences between in vivo and in vitro proliferating T lymphocytes. Eur. J. Immunol. 19, 1137-1145 (1989). 

This result reflects both the common decay of potential as the distance from the center of a charged sphere (r/R term) increases, and a more rapid exponential decay due to the presence of the diffuse layer (exponential term). Consequently, the potential of a sphere surrounded by diffuse layer decays with distance faster than the potential near the particle in a dielectric medium, or the potential in the vicinity of a flat interface with a diffuse layer. One can say that the distant regions with low potentials are mostly proliferated around the charged particle, while those with high potentials occupy a small volume in direct vicinity of the particle surface. For potentials at large distances from the surface of strongly charged particles one can use an expression similar to eq. (III. 19) ... 

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