The Master Molecule

The genome in the nucleus of every human cell (except red blood cells) consists of 6 billion deoxyribonucleic acid (DNA) nucleotides packaged in 48 chromosomes. The life of every human being b jins with DNA, a polymer of a long series of nucleotides, with a backbone of five-carbon sugars. Ribonucleic acid (RNA) contains ribose in the place of deoxyribose. Human DNA contains 22,000 genes, containing 750 MB of data (the Bible contains 5 MB). 

Humans and chimps evolved separately from a common ancestor about 6 million years ago. Nearly 99% of the gene sequences in chimps are identical to those of humans. Macaque monkeys branched off the family tree of a common ancestor about 25 million years ago. Even in 93% of macaque DNA is the same as that of human beings. Ninety-nine percent of each person s DNA is identical to that of all other human beings. 

the evolution of genes, programmed cell death (apoptosis), and the action of messenger RNA (mRNA) are three major targets of research. mRNA contains the blueprint for every protein in the body. It is transcribed from a DNA template, and carries information to ribosomes, the sites of protein synthesis. The sequences of nucleic acid polymers are translated by transfer RNA (tRNA) into amino acid polymers. tRNA recognizes the three-nucleotide sequences that encode each amino acid. Ribosomal RNA directs the ribosome s production of proteins. Codons carry the messages that terminate protein synthesis. 

The DNA of genes is made of four nucleotide bases two purines, adenine (A) and guanine (G) and two pyrimidines, thymine (T) and cytosine (C). The genetic code is based on these four letters, AGCT, that encode the amino acids making up the body s peptides and proteins. The genetic code is the same in all living creatures. 

Nobel Prize winner Sydney Brenner showed that a sequence of three nucleotide bases in a row encode each specific amino acid. RNA strands complementary to a DNA strand are responsible for the process called transcription, which is brought about by the action of the enzyme, RNA polymerase. 

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Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

RNApolymerase molecules are involved in the process. If the system is well stirred so that spatial degrees of freedom play no role, birth-death master equation approaches have been used to describe such reacting systems [33, 34]. The master equation can be simulated efficiently using Gillespie s algorithm [35]. However, if spatial degrees of freedom must be taken into account, then the construction of algorithms is still a matter of active research [36-38]. 

The Montroll-Shuler equation can also predict how fast a molecule which is created in a highly excited vibrational state will decay to the equilibrium state. This is of interest in connection with chemiluminescence phenomena. In certain cases one finds experimentally that this relaxation is much faster than what one would expect from the master equation of Montroll and Shuler and improved versions of this equation. One possible mechanism for this fast relaxation is that although most of the collisions in which the diatomic molecule participates are between the diatomic molecule and an inert gas atom, there will also be some collisions between diatomic molecules. In the latter case we have the situation where two diatomic molecules in quantum state n collide producing, with fairly high probability, molecules in quantum states n I and n + 1, respectively. The number of such collisions is, of course quite small compared to the number of collisions of the first kind, but since they are so extremely efficient they may still be of importance. This mechanism, we believe, was first suggested in connection with chemiluminescence by Norrish in a Faraday Society discussion.5 The equations describing this relaxation had, however, been discussed several years earlier by Shuler6 and Osipov.7... 

The same assumptions on which the macroscopic equation (2.2) is based lead to a definite form of the master equation for P, apart from a small but essential modification.In (2.1) the probability for a collision involving Sj molecules Xj is taken proportional to ri-j more precisely this factor should be... 

This is not the only possible way of adding terms to the master equation that give rise to the macroscopic terms (4.1). The molecules might be injected, e.g., in clusters. Such a different choice for the mesoscopic description would affect the fluctuations in n. In general, whenever a system is subject to an external force or agency, one cannot compute the fluctuations if that force is merely known macroscopically, one must also know its stochastic properties. ... 

The probability (6.3) of the gross state changes whenever one of the molecules jumps from one level n to another level n. The probability that one of the Nn> molecules in level n makes such a jump during At is Wnn Nn At. The probability that two or more jump is of order (At)2. Hence P obeys the master equation... 

This master equation is uniquely determined by the one-molecule master equation (6.1), because the physics of both is the same. We shall now show how the solutions of (6.4) can be expressed in terms of those of (6.1). 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

Conclusions, (i) In the literature the / in the Boltzmann equation is variously defined as the number, the average number, the probable number or the most probable number (of molecules in a unit phase cell). In our treatment, based on the master equation, it appears as the macroscopic value of that number obtained in the limit of zero fluctuations. Moreover, to order A 1 the fluctuations are symmetric, so that / is also the average. It cannot be called the most probable number because it is not an integer. 

J. Nyg rd, Quantum Chaos of the NO2 Molecule in High Magnetic Fields, Master Thesis, Copenhagen, 1996. 

Since S values tor any desired pair of S-aT/AO hybrids can. readily be obtained using Eqs. (77/ (79) and the master tables, no extensive hybrid tables will be given here. Nevertheless, two sets of Slater-AO hybrid 5 tables (Tables XXIV-XXVIII) have been computed explicitly in order to aid the reader in appreciating quanritarivciy the rather remarkable checvs ot hybridization on A values. Further examples of hybrid 5 values for specific molecules, and a discussion. wili be given in a following paper.nti... 

The crucial step in the development of unimolecular theory was the postulate of a time lag between the activation and reaction steps in the master mechanism for all elementary reactions given in Chapter 1. During this time an activated molecule can either be deactivated in a deactivating energy transfer collision, or it can alter configuration to reach the critical configuration and react. All elementary reactions involve three steps, two energy transfer steps and one reaction step, and for unimolecular reactions... 

The essential progress in calculation of transport properties in the strong electron-vibron interaction limit has been made with the help of the master equation approach [104-112]. This method, however, is valid only in the limit of very weak molecule-to-lead coupling and neglects all spectral effects, which are the most important at finite coupling to the leads. 

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