Helix operator

It should be remarked here that the Bloch-form of the one-electron orbitals [equation (2)] automatically implies translational symmetry. In the case of onedimensional polymers this symmetry operation can be combined with a simultaneous rotation around the polymer axis (helix operation). It can be shown that if the AO s xtt 0 are properly transformed in the translated and rotated elementary cells,8 7 the above described formalism can still be applied. 

It was easy to show that we can formulate the method also in the case of a combined symmetry operation (for instance helix operation = translation + rotation) instead of simple translation ( ). In this case k is defined on the combined symmetry operation and from going from one cell to the next one, one has (1) to put the nuclei in the positions required by the symmetry operation and (2) one has to rotate accordingly also the basis set. 

Structure Thus no symmetry relations, beyond the helix operation, are assumed. They are to be inferred from the resulting stable inking arrangements. However, this plan results in discovering the primitive unit cell. Very (Aen it is a convenience to impose some symmetry operations so that a symmetry-centered cell results directly. It is quite feasible to write a computer code that allows these operations to be imposed or not, according to the problem studied. 

There are several coordinate systems that have to be dealt with. Ultimately, in order to carry out the minimization process, the total energy is best expressed in terms of Cartesian coordinates. However, a general unit cell or lattice is characterized by non-orthogonal basis vectors. A cylindrical coordinate system is used to represent the molecular helix. The intramolecular energy is expressed in terms of valence coordinates. Thus transformations must be set up that relate the Cartesian coordinates to the helix parameters, the unit cell parameters and the valence coordinates. The helix operations and the unit cell parameters are considered first. 

We can apply the formalism developed in the preceding section also in the case of a combined symmetry operation. To show this let us consider a helix in which we pass from one unit to the next by a translation t and simultaneous rotation a. We can then introduce the helix operator... 

This means that by applying the helix operator to an AO we must... 

Finally, it is noteworthy that this derivations differs from those given by McCubbinS > and Ukrainski,S"> who start in the usual way with simple translational symmetry and analyze a posteriori the effects of other symmetry operations, such as the helix operation. 

Finally one should mention that with the help of simple group theoretical arguments one can show that the described formalism can be applied (in the ID case) also for combined symmetry operations (for instance to the helix operation which contains a translation and a rotation going from one unit to the next one) instead of a simple translation. As the detailed derivation shows in the case of a helix operation one has to put the nuclei in the next cell in the right position and rotate the basis functions which do not point in the direction of the polymer axis (say z axis) or are not spherically symmetric. Thus one has to rotate the and py functions and d, etc., functions. In this way in the case of a DNA helix (in DNA B) one can take a single nucleotide and not 10 of them as unit cell. 

Finally, it should be mentioned that the formalism described here is valid also for the case of a repeated combined symmetry operation (for instance helix operation). As group theoretical considerations show it in this case 1.) one has to put the nuclei into the right positions by moving from one cell to the next and 2.) one has to rotate correspondingly also the basis functions /14/. 

Approximately 10 base pairs are required to make one turn in B-DNA. The centers of the palindromic sequences in the DNA-binding regions of the operator are also separated by about 10 base pairs (see Table 8.1). Thus if one of the recognition a helices binds to one of the palindromic DNA sequences, the second recognition a helix of the protein dimer is poised to bind to the second palindromic DNA sequence. 

These genetic experiments clearly demonstrated that the proposed structural model for the binding of these proteins to the phage operators was essentially correct. The second a helix in the helix-turn-helix motif is involved in recognizing operator sites as well as in the differential selection of operators by P22 Cro and repressor proteins. However, a note of caution is needed many other early models of DNA-protein interactions proved to be misleading, if not wrong. Modeling techniques are more sophisticated today but are still not infallible and are certainly not replacements for experimental determinations of structure. 

Figure 8.15 Sequence-specific protein-DNA interactions provide a general recognition signal for operator regions in 434 bacteriophage, (a) In this complex between 434 repressor fragment and a synthetic DNA there are two glutamine residues (28 and 29) at the beginning of the recognition helix in the helix-turn-helix motif that provide such interactions with the first three base pairs of the operator region.

The elegant genetic studies by the group of Charles Yanofsky at Stanford University, conducted before the crystal structure was known, confirm this mechanism. The side chain of Ala 77, which is in the loop region of the helix-turn-helix motif, faces the cavity where tryptophan binds. When this side chain is replaced by the bulkier side chain of Val, the mutant repressor does not require tryptophan to be able to bind specifically to the operator DNA. The presence of a bulkier valine side chain at position 77 maintains the heads in an active conformation even in the absence of bound tryptophan. The crystal structure of this mutant repressor, in the absence of tryptophan, is basically the same as that of the wild-type repressor with tryptophan. This is an excellent example of how ligand-induced conformational changes can be mimicked by amino acid substitutions in the protein. 

The energy required to adjust the DNA to these receptor sites is given in Table VI. The DNA can kink equally well in both grooves with base pairs held at a distance sufficient for intercalation (Az = 6.76 A, ax = 0°) and for kinks (Az > 6.76 A, ax 0°). These receptor sites are constructed by operations on a pair of initially coincident base pairs. Each is rotated by +ax/2 and -ax/2 about a kink axis. This axis is perpendicular to the helix and dyad axes of the base, and parallel to the Cl (py)-Cl (pu) axis. It lies approximately along the C6(py)-C8(pu) axis. Then each base pair is rotated about the helix axis by +az/2 and -az/2 and separated by Az. The combinations of ax, az, and Az which permit the construction of a phosphate backbone defines families of receptor sites. With this approach, the base pairs adjacent to the BPDE are symmetrically... 

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