Folding and Unfolding

To define the problems of folding and unfolding, consider the functions 

Folding the function 5(x) with the function R x,x ) to obtain the function Mix) means to perform the integration 

Unfolding means to obtain the function Six), knowing Mix) and Rix, x ). Thus, folding is an integration, as shown by Eq. 11.33. Unfolding, on the other hand, entails solving the integral equation, Eq. 11.33—known as the Fredholm equation—for the unknown function Six). 

In the field of radiation measurements, folding and (especially) imfolding are very important operations that have to be applied to the experimental data. In most radiation measurements, the variable x is the energy of the particle, and for this reason the discussion in this section will be based on that variable. The reader should be aware, however, that x may represent other quantities, such as time, velocity, or space variables. If x is the energy of the particle, the functions Six), Mix), and Rix, x ) have the following meanings (also given in Sec. 9.7)  

SiE) dE = source spectrum = number of particles emitted by the source with energy between E and E + dE 

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Measuring Protein Sta.bihty, Protein stabihty is usually measured quantitatively as the difference in free energy between the folded and unfolded states of the protein. These states are most commonly measured using spectroscopic techniques, such as circular dichroic spectroscopy, fluorescence (generally tryptophan fluorescence) spectroscopy, nmr spectroscopy, and absorbance spectroscopy (10). For most monomeric proteins, the two-state model of protein folding can be invoked. This model states that under equihbrium conditions, the vast majority of the protein molecules in a solution exist in either the folded (native) or unfolded (denatured) state. Any kinetic intermediates that might exist on the pathway between folded and unfolded states do not accumulate to any significant extent under equihbrium conditions (39). In other words, under any set of solution conditions, at equihbrium the entire population of protein molecules can be accounted for by the mole fraction of denatured protein, and the mole fraction of native protein,, ie. 

Solving the master equation for the minimally frustrated random energy model showed that the kinetics depend on the connectivity [23]. Eor the globally connected model it was found that the resulting kinetics vary as a function of the energy gap between the folded and unfolded states and the roughness of the energy landscape. The model... 

M Karplus, A Sail. Theoretical studies of protein folding and unfolding. Curr Opin Struct Biol 5 58-73, 1995. 

A Caflisch, M Karplus. Molecular dynamics studies of protein and peptide folding and unfolding. In K Merz Jr, S Le Grand, eds. The Protein Eoldmg Problem and Tertiary Structure Prediction. Boston Birkhauser, 1994, pp 193-230. 

We will discuss three different approaches to engineer a more thermostable protein than wild-type T4 lysozyme, namely (1) reducing the difference in entropy between folded and unfolded protein, which in practice means reducing the number of conformations in the unfolded state, (2) stabilizing tbe a helices, and (3) increasing the number of bydropbobic interactions in tbe interior core. 

Hermans, Jr., J., Lohr, D. and Ferro, D. Treatment of the Folding and Unfolding of Protein Molecules in Solution According to a Lattic Model. Vol. 9, pp. 229-283. 

S. Kelly and N. Price, The application of circular di-chroism to studies of protein folding and unfolding, Biochim. Biophys. Acta, 1338, 161 (1997). 

Zhang O, Kay LE, Olivier JP, Forman-Kay JD. Backbone 11 and 15N resonance assignments of the N-terminal SFI3 domain of drk in folded and unfolded states using enhanced sensitivity pulsed field gradient NMR techniques. J Biol NMR 1994 4 845-858. 

Three theory papers are also included. Determinants of the Polyproline II Helix from Modeling Studies by Creamer and Campbell reexamines and extends an earlier hypothesis about Pn and its determinants. Hydration Theory for Molecular Biophysics by Paulaitis and Pratt discusses the crucial role of water in both folded and unfolded proteins. Unfolded State of Peptides by Daura et al. focuses on the unfolded state of peptides studied primarily by molecular dynamics. 

Tanford (1968) reviewed early studies of protein denaturation and concluded that high concentrations of Gdm-HCl and, in some cases, urea are capable of unfolding proteins that lack disulfide cross-links to random coils. This conclusion was largely based on intrinsic viscosity data, but optical rotation and optical rotatory dispersion (ORD) [reviewed by Urnes and Doty (1961) ] were also cited as providing supporting evidence. By these same lines of evidence, heat- and acid-unfolded proteins were held to be less completely unfolded, with some residual secondary and tertiary structure. As noted in Section II, a polypeptide chain can behave hydrodynamically as random coil and yet possess local order. Similarly, the optical rotation and ORD criteria used for a random coil by Tanford and others are not capable of excluding local order in largely unfolded polypeptides and proteins. The ability to measure the ORD, and especially the CD spectra, of unfolded polypeptides and proteins in the far UV provides much more incisive information about the conformation of proteins, folded and unfolded. The CD spectra of many unfolded proteins have been reported, but there have been few systematic studies. 

We present a molecular theory of hydration that now makes possible a unification of these diverse views of the role of water in protein stabilization. The central element in our development is the potential distribution theorem. We discuss both its physical basis and statistical thermodynamic framework with applications to protein solution thermodynamics and protein folding in mind. To this end, we also derive an extension of the potential distribution theorem, the quasi-chemical theory, and propose its implementation to the hydration of folded and unfolded proteins. Our perspective and current optimism are justified by the understanding we have gained from successful applications of the potential distribution theorem to the hydration of simple solutes. A few examples are given to illustrate this point. 

Folded and unfolded proteins in solution are dense materials characterized in large part by different degrees of conformational flexibility and solvent exposure. Thus, packing is a foundational issue for their solution thermodynamic properties. Although the developments above... 

Shea, J. E., and Brooks, C. L. (2001). From folding theories to folding proteins A review and assessment of simulation studies of protein folding and unfolding. Annu. Rev. Phys. Chem. 52, 499-535. 

The characteristics that discourage the use of RPLC for preparative isolation of bioactive proteins favor its use as an analytical tool for studying protein conformation. Chromatographic profiles can provide information on conformational stability of a protein and the kinetics of folding and unfolding processes. Information about solvent exposure of certain amino acid residues (e.g., tryptophan) as a function of the folding state can be obtained by on-line spectral analysis using diode array UV-vis detection or fluorescence detection. 

During the cell cycle, chromosome structures shuttle between de-condensed interphase and condensed mitosis states. Dynamic changes also occur at the lower levels of architectures, i.e., at the chromatin and nucleosome levels. Upon gene activation and inactivation, folding and unfolding of the nucleosome structure and the chromatin fibers occur at limited loci of the genome. Namely, the structures of the chromosome are dynamic and mobile. Nevertheless, there are basic structural units that remain stable and constitute the fundamental chromosome architecture. 

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