Biological spin Hamiltonians

In this chapter we continue our journey into the quantum mechanics of paramagnetic molecules, while increasing our focus on aspects of relevance to biological systems. For each and every system of whatever complexity and symmetry (or the lack of it) we can, in principle, write out the appropriate spin Hamiltonian and the associated (simple or compounded) spin wavefunctions. Subsequently, we can always deduce the full energy matrix, and we can numerically diagonalize this matrix to obtain the stable energy levels of the system (and therefore all the resonance conditions), and also the coefficients of the new basis set (linear combinations of the original spin wavefunctions), which in turn can be used to calculate the transition probability, and thus the EPR amplitude of all transitions. 

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The EPR spectrum is a reflection of the electronic structure of the paramagnet. The latter may be complicated (especially in low-symmetry biological systems), and the precise relation between the two may be very difficult to establish. As an intermediate level of interpretation, the concept of the spin Hamiltonian was developed, which will be dealt with later in Part 2 on theory. For the time being it suffices to know that in this approach the EPR spectrum is described by means of a small number of parameters, the spin-Hamiltonian parameters, such as g-values, A-values, and )-values. This approach has the advantage that spectral data can be easily tabulated, while a demanding interpretation of the parameters in terms of the electronic structure can be deferred to a later date, for example, by the time we have developed a sufficiently adequate theory to describe electronic structure. In the meantime we can use the spin-Hamiltonian parameters for less demanding, but not necessarily less relevant applications, for example, spin counting. We can also try to establish... 

Although the same nuclear spin interactions are present in solid-state as in solution-state NMR, the manifestations of these effects are different because, in the solid, the anisotropic contribution to the spin interactions contributes large time-independent terms to the Hamiltonian that are absent in the liquid phase. Therefore, the experimental methods employed in solids differ from the ones in the liquid state. The spin Hamiltonian for organic or biological solids can be described in the usual rotating frame as the sum of the following interactions ... 

The solution structures, dynamics, and interactions of and between biological macromolecules are topics of widespread interest in biochemistry. The chapter on electron spin labels, by Millhauser et al. illustrates how the EPR spectra of the stable nitroxide free radical can be used to address such problems. The chemistry of the nitroxides and their modes of attachment to the host molecules are discussed first. The details of the EPR spectra and of the spin Hamiltonian are then presented, showing how the intrinsic tensorial nature of the EPR spectrum of the reporter group is affected by motion. Such dynamic information is then extracted from some small peptides. The interaction between pairs of nitroxides is used to extract structural information. Finally, an example of Fourier transform EPR is introduced. ... 

Direct observation of very small dipolar couplings between distant protons in the overcrowded spectra of biological molecules represents a challenging task. Now Bax and co-workers have presented a method which, by the appropriate manipulation of the nuclear spin Hamiltonian, allows effective decoupling of all protons outside a spectral region of interest and facilitates observation of interactions between remote protons separated by distances of up to 12 A. The method has been applied to measure remote dipolar couplings in an unlabelled nucleic acid. 

Mn(II) EPR spectra in biological systems are very much like those in glasses — e.g., that in lithium-borate glass (Griscom Griscom, 1967) matches closely that in kinase oxalate ternary complex (Reed Markham, 1984 referred to hereafter as RM, and references therein). Table 1 lists the measured values of the spin-Hamiltonian parameters parameters (g, D, E) in some proteins, as taken from RM. 

Electron Spin Echo Envelope Modulation (ESEEM) and pulse Electron Nuclear Double Resonance (ENDOR) experiments are considered to be two cornerstones of pulse EPR spectroscopy. These techniques are typically used to obtain the static spin Hamiltonian parameters of powders, frozen solutions, and single crystals. The development of new methods based on these two effects is mainly driven by the need for higher resolution, and therefore, a more accurate estimation of the magnetic parameters. In this chapter, we describe the inner workings of ESEEM and pulse ENDOR experiments as well as the latest developments aimed at resolution and sensitivity enhancement. The advantages and limitations of these techniques are demonstrated through examples found in the literature, with an emphasis on systems of biological relevance. 

In practice, the EPR spectroscopist will thus be using a variety of multifrequency cw-EPR and hyperfine spectroscopy techniques to determine the different spin Hamiltonian parameters. This has led to new insights in biological systems as well as in material sciences and it would be impossible to give an exhaustive account of all these applications. Here, we merely give a few examples to give the reader an idea... 

This Hamiltonian leads to dephasing of the S -spin signal recorded as function of time (increasing number of rotor periods Nc in the REDOR experiment) as illustrated in Fig. lb. REDOR has been a key experiment in biological solid-state NMR, as for example used recently for determination of statherin binding to biomineral surfaces as illustrated in Fig. lc, with numerous REDOR determined intemuclear distances high-lighted in Fig. Id [79]. 

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