On Symmetry and Function

As Monod states in his treatise. On Symmetry and Function in Biological Systems, One may set aside the simple problem of fibrous proteins. Being used as scaffolding, shrouds or halyards, they fulfill these requirements by adopting relatively simple types of translational symmetries. Therefore, it was not anticipated that positive cooperativity, the effect Monod thought to be the second secret of life, second only to the structure of DNA, would be most beautifully demonstrated by designed variations of a repeating sequence of the mammalian elastic fiber based on translational symmetry. 

J. Monod, On Symmetry and Function in Biological Systems. In Nobel Symposium 11 Symmetry and Function of Biological Systems at the Macromolecular Level, A. Engstrom and B. Strandberg, Eds. Almqvist Wiksell Forlag AB, Stockholm, 1968, p. 1527. Also reprinted in Selected Papers in Molecular Biology by Jacques Monod, A. Lwoff, and A. Ullmann, Eds., Academic Press, New York, 1978, p. 708. 

The authors feel that compounds need to be sufficiently complex in order to provide a significant level of information content. On the other hand, compounds that are too complex may be viewed as undesirable. A number of measures for characterizing molecular complexity have been developed over the years. Complexity has been defined in terms of size, elemental composition, topology, symmetry, and functional groups present in a molecule (17,18). In the present work, we have applied a simple, chemically intuitive measure developed by Barone and Channon (18). This approach is based on a simple additive model, where values are assigned to specific classes of molecular features such as the types of atoms and bonding patterns, and the number and types of rings. This is illustrated in Eq. 2.1 ... 

Changeux, J. P. Remarks on the symmetry and cooperative properties of biological membranes. In Symmetry and Function of Biological Systems at the Macromolecular Level (Engstrom, A., Strandberg, B., Eds), Almqvist and Wiksell Stockholm, 1969, pp. 235—256. 

We can think of a symmetry operator as a mathematical operator that can operate both on points and functions, or we can think of a symmetry operation as a means of moving a point in three-dimensional space. The effect of the symmetry operator on a point is the same as the effect of the symmetry operation. A symmetry operation will be denoted by the same symbol as a symmetry operator without the caret ( ). When operating on the nuclei of a molecule we will sometimes use the terms symmetry operators and symmetry operations interchangeably. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

L°° Tl) depend only on x and possess the usual properties of symmetry and positive definiteness. As a matter of convenience we choose arbitrary fixed functions G and M - G L Q) satisfying the conditions... 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

The precedings sections concentrated on single determinant variational functions. One may wonder whether going to multieonfigurational SCF funetions will restore symmetry and when. 

Based on the same two step proeedure as presented above for C2H4 (MCSCF ealeulations followed by Schmidt orthogonalization of Rydberg functions), a systematic search was conducted by progressively incorporating groups of orbitals in the active space. Two types of wave functions proved well adapted to the problem, one for in-plane excitations, the other for out-of-plane excitations from the carbene orbital. The case of the Ai states will serve as an illustration of the general approach done for all symmetries and wave functions. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Another ligand including a thiophene moiety but lacking the C2-symmetry and thus bearing electronically different phosphorus atoms was prepared by these authors, in 2001. The electrochemical oxidative potential was obtained by cyclic voltammetry. The oxidation potential of the phosphine group located on the phenyl ring was found to be 0.74 V (vs. Ag/Ag" ) and the authors attributed a value of 0.91 V to the phosphine attached to the thiophene moiety. This second functionality is a rather electron-poor phosphine. As shown... 

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