The Chain Population

Z]/[Q]), so that R ranges from 0 to v as the molecules making up the system become more condensed. In systems in which finite chain and ring structures coexist, it is advisable to calculate an Rehains value for the chain population alone by subtracting the concentrations [T], [Z] and [Q] associated with cyclic molecules from the total concentrations of the T s, Z s and Q s respectively. 

The families of a,o)-dimethoxyl poly(methylphosphonates) for which p 2, alkyl polyphosphates for which p = 2, sodium polyphosphates for which p = 3 and aliphatic hydrocarbons for which p % 2 are examples of systems for which more than (v — 1) equilibrium constants are needed to represent the scrambling equilibria involving the chain population (since p > 1). 

The nuclear chain reaction can be modeled mathematically by considering the probable fates of a typical fast neutron released in the system. This neutron may make one or more coUisions, which result in scattering or absorption, either in fuel or nonfuel materials. If the neutron is absorbed in fuel and fission occurs, new neutrons are produced. A neutron may also escape from the core in free flight, a process called leakage. The state of the reactor can be defined by the multiplication factor, k, the net number of neutrons produced in one cycle. If k is exactly 1, the reactor is said to be critical if / < 1, it is subcritical if / > 1, it is supercritical. The neutron population and the reactor power depend on the difference between k and 1, ie, bk = k — K closely related quantity is the reactivity, p = bk jk. i the reactivity is negative, the number of neutrons declines with time if p = 0, the number remains constant if p is positive, there is a growth in population. 

Finally, an alchemical free energy simulation is needed to obtain the free energy difference between any one substate of system A and any one substate of system B, e.g., Ai- In practice, one chooses two substates that resemble each other as much as possible. In the alchemical simulation, it is necessary to restrain appropriate parts of the system to remain in the chosen substate. Thus, for the present hybrid Asp/Asn molecule, the Asp side chain should be confined to the Asp substate I and the Asn side chain confined to its substate I. Flat-bottomed dihedral restraints can achieve this very conveniently [38], in such a way that the most populated configurations (near the energy minimum) are hardly perturbed by the restraints. Note that if the substates AI and BI differ substantially, the transfomnation will be difficult to perform with a single-topology approach. 

Figure 12.5. (a) Lattice model showing a polymer chain of 200 beads , originally in a random configuration, after 10,000 Monte Carlo steps. The full model has 90% of lattice sites occupied by chains and 10% vacant, (b) Half of a lattice model eontaining two similar chain populations placed in contact. The left-hand side population is shown after 50,0000 Monte Carlo steps the short lines show the loeation of the original polymer interface (courtesy K. Anderson). 

Because dideoxynucleotides lack 3 -OH groups, these nucleotides cannot serve as acceptors for 5 -nucleotide addition in the polymerization reaction, and thus the chain is terminated where they become incorporated. The concentrations of the four deoxynucleotides and the single dideoxynucleotide in each reaction mixture are adjusted so that the dideoxynucleotide is incorporated infrequently. Therefore, base-specific premature chain termination is only a random, occasional event, and a population of new strands of varying length is synthesized. Four reactions are run, one for each dideoxynucleotide, so that termination, although random, can occur everywhere in the sequence. In each mixture, each newly synthesized strand has a dideoxynucleotide at its 3 -end, and its presence at that position demonstrates that a base of that particular kind was specified by the template. A radioactively labeled dNTP is included in each reaction mixture to provide a tracer for the products of the polymerization process. 

In conventional radical polymerization, the chain length distribution of propagating species is broad and new short chains are formed continually by initiation. As has been stated above, the population balance means that, termination, most frequently, involves the reaction of a shorter, more mobile, chain with a longer, less mobile, chain. In living radical polymerizations, the chain lengths of most propagating species are similar (i.e. i j) and increase with conversion. Ideally, in ATRP and NMP no new chains are fonned. In practice,... 

Figure 1. Conceptual model illustrating examples of major anthropogenic contaminant sources and contaminants, their distribution within the abiotic environmental media, their movement into biota with potential food chain contamination, and potential effects at the organismal, population, conmiunity and ecosystem level of organization.

Using this procedure, the classic RIS model for the atomistic chain has been adapted so that it influences the conformations populated by the coarsegrained chain on the high coordination lattice. 

Figure 10-5. Representative conformations of the (5 amyloid peptide (10-42) under different pH conditions. The conformations were obtained as centroids of the most populated clusters from the replica-exchange CPHMD folding simulations [43, 44]. The N-terminal residues 10-28 are shown in blue the C-terminal residues 29-42 are shown in red. In the most aggregation-prone state (pH 6), the side chains of the central hydrophobic cluster Leu-17, Val-18, Phe-19, Phe-20 and Ala-21 are shown as van der Waals spheres in pink, grey, cyan, purple and green, respectively...

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