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Our discussion of multi-dimensional NMR has thus far focused almost exclusively on the determination of threedimensional structures of proteins (Example 6.2). However, this group of techniques is also useful as a general probe for protein structure which can also shed light on aspects of protein dynamics such as enzyme catalysis, protein protein complex formation, ligand binding and protein folding (Example 6.3). In principle, any biomolecule containing atoms possessing spin can give an NMR spectrum and resonances due to individual atoms can be followed by multi-dimensional methods. Although, high-resolution structural information may not always be obtainable from such spectra, they can nonetheless yield important clues to shape or function. The technique is therefore generally applicable to other classes of biomolecules such as DNA and carbohydrates and a similar approach can be taken to analysis of structure and function in these molecules as for proteins (Table 6.2). In the case of DNA, multidimensional NMR studies have tended to concentrate on short oligonucleotides (10 20 bp) rather than large structures. This can yield structural information on functionally important parts of DNA such as promoters and operators. Since there are only four bases commonly found in DNA (compared to twenty amino acid residues in proteins), proton NMR spin systems are more easily assigned to bases than to amino acids. The major interproton distances measured from NOESY in DNA samples are between sequentially adjacent bases rather than the more long-distance interactions which we have come across in protein structure. NOEs are also measurable between protons on separate, individual strands of double-stranded DNA which happen to be brought close together by hydrogen

we see that if there is an integer m such that 3x = m 4 , then the condition for a resonance is satisfied. To examine the phase space near resonance, we go to rotating coordinates as we did in the more elementary treatment. Now, of course, this requires yet another canonical transformation. We take

Figure 13.10 Oesophageal complications of obesity: (a) Barrett s oesophagus, with ascending gastric-type mucosa (appearing darker); (b) carcinoma of the oesophagus with Barrett s oesophagus. Images reproduced by courtesy of Dr. N Sarwar, Royal Liverpool University Hospital.

First of all, the script creates two custom exception classes derived from the built-in Exception class: InputException and FuelException. An InputException object is to be thrown if the calling code has supplied an invalid warp factor (outside the range 1 through 9). The InputException class adds an $_invalidWarpFactor private property to store the supplied warp factor, and extends the Exception constructor to also allow the supplied warp factor to be stored in the InputException object when it s created. Finally, it provides a getInvalidWarpFactor() method to retrieve the invalid warp factor that was supplied:

12.2 64

Whitfield Diffie and Martin Hellman developed public key cryptography in 1976 It was also developed earlier by secret government agencies that don t get published (A few years later, Whit Diffie also developed the earliest seeds of the presentation software that became PowerPoint I m convinced that PowerPoint is one of the roots of evil, and yet public key cryptography is good The mind boggles) Public key encryption isn t an encryption algorithm for turning text into gobbledygook; it s a system for generating and managing the keys those algorithms use Cryptography works by combining plaintext with a key to result in gobbledygook Only someone with the key can recover the plaintext Before public key encryption, there was secret key encryption: the person who encrypted the message (traditionally called Bob) uses the same key as the person who decrypts the message (Alice).

Figure 31-9: Calculating how the delta-x and delta-y affect a hatch pattern. Finally, you can add additional definition lines. One of the definition lines should start at 0,0, but the others may start anywhere. Here is the definition for the pattern in Figure 31-10. It actually creates the shape of the trailers. Although you see the rectangular shape, the hatch pattern is created from four separate lines, two at 0 degrees and two at 90 degrees. Note that the two 0-degree lines are the same, except that they start at different base points. The same is true for the two 90-degree lines.