Unfolding

The so-called Tucker3 model is defined by the decomposition of a three-way table X into a three-way core matrix Z and three two-way loading matrices A, B, C (one for each mode)  

In this way, an nxpxq table X is decomposed into an rxsxt core matrix Z and the nxr, pxs, qxt loading matrices A, B, C for the row-, column- and layer-items of X. The loading matrices are column-wise orthonormal, which means that  

The number of factors r, s and t, assigned to each mode, is generally different. They are chosen such as to be less than the dimensions of the original three-way table in order to achieve a considerable amount of data reduction. The elements of Z represent the magnitude of the factors and the extent of their interaction [57]. Computationally, the core matrix Z and the loadings matrices A, B and C are derived such as to minimize the sum of squared residuals. 

By analogy with singular vector decomposition (SVD) and ordinary PCA, one can also define the TuckerS model in an extended matrix notation  

The extended matrix notation is represented here by the three dots that surround the core matrix. A graphical example of the TuckerS model is rendered in Fig. 31.19. 

This comparatively simple approach is sometimes sufficient but the PCA calculation neglects to take into account the relationships between the variables. For example, the 

Problem 4.1 Grouping of Elements from Fundamental Properties Using PCA 

The table below lists 27 elements, divided into six groups according to their position in the periodic table together with five physical properties. 

Element Group Melting point (K) Boiling point (K) Density Oxidation number Electronegativity 

Standardise the five variables, using the population (rather titan sample) standard deviation. Why is this preprocessing necessary to obtain sensible results in this case  

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Consider that at low temperatures, a lubricant is a poor solvent for polymer chains. When the temperature increases, interactions between polymer chains decrease the space occupied by the polymer ball takes on greater volume and consequently, the viscosity decrease due to the lubricant temperature increase is compensated by the unfolding of the polymer chain and the result is a reduction of the difference between the viscosities at low and high temperature, and therefore an Increase in viscosity index. 

Proteins often have the same high-affinity isotherms as do synthetic polymers and are also slow to equilibrate, due to many contacts with the surface. Proteins, however, have the additional complication that they can partially or completely unfold at the solid-liquid interface to expose their hydrophobic core units to a hydrophobic surface... 

Plenary 2. S A Asher et al, e-mail address asher ,vms.cis.pitt.edu/asher+ (RRS, TRRRS). UV RRS is used to probe methodically the secondary structure of proteins and to follow unfolding dynamics. Developing a library based approach to generalize the mediod to any protein. 

Rief M, Gautel M, Oesterhelt F, Fernandez J M and Gaub FI E 1997 Reversible unfolding of individual titin immunoglobulin domains by AFM Science 276 1109... 

Siska P E 1973 Iterative unfolding of intensity data, with application to molecular beam scattering J. Chem. Rhys. 59 6052-60... 

Given that a sequence folds to a known native stmcture, what are the mechanisms in the transition from the unfolded confonnation to the folded state This is a kinetics problem, the solution of which requires elucidation of the pathways and transition states in the folding process. 

For these sequences the value of Gj, is less than a certain small value g. For such sequences the folding occurs directly from the ensemble of unfolded states to the NBA. The free energy surface is dominated by the NBA (or a funnel) and the volume associated with NBA is very large. The partition factor <6 is near unify so that these sequences reach the native state by two-state kinetics. The amplitudes in (C2.5.7) are nearly zero. There are no intennediates in the pathways from the denatured state to the native state. Fast folders reach the native state by a nucleation-collapse mechanism which means that once a certain number of contacts (folding nuclei) are fonned then the native state is reached very rapidly [25, 26]. The time scale for reaching the native state for fast folders (which are nonnally associated with those sequences for which topological fmstration is minimal) is found to be... 

Taketomi H, Ueda Y and Go N 1975 Studies on protein folding, unfolding, and fluctuations by computer simulation Int. J. Pept. Protein Res. 7 445-59... 

Philips C M, Mizutani Y and Hochstrasser R M 1995 Ultrafast thermally induced unfolding of RNase A Proc. Natl Acad. Sol. USA 92 7292-6... 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

One should also note that the unfolding procedure makes the average density of levels unifomi over the entire energy range. Thus, the difference... 

Creveld, L., Amadei, A., Van Schaik, C., Pepermans, R., De Vlieg, J., Berendsen, H.J.C. Identification of functional and unfolding motions of cutinase as obtained from molecular dynamics computer simulations. Submitted (1998). 

These events marked the beginning of the Ig domain unfolding, after which the strands unraveled one at a time, accompanied by a large reduction in the recorded force. After an extension of 260 A, the domain was completely unfolded further stretching of the already extended polypeptide chain caused the force to increase dramatically. 

Kellermayer et aJ., 1997] Kellermayer, M., Smith, S., Granzier, H., and Bustamante, C. Folding-unfolding transition in single titin modules characterized with laser tweezers. Science. 276 (1997) 1112-1116... 

Tskhovrebova et al., 1997] Tskhovrebova, L., Trinick, J., Sleep, J., and Simmons, R. Elasticity and unfolding of single molecules of the giant protein titin. Nature. 387 (1997) 308-312... 

To facilitate conformational transitions in the before-mentioned adenylate kinase, Elamrani and co-workers scaled all atomic masses by a large factor thus allowing the use of a high effective simulation temperature of 2000K ([Elamrani et al. 1996]). To prevent protein unfolding, elements of secondary structure had to be constrained. 

M. H. Hao, M. R. Pincus, S. Rackovsky, and H. A. Scheraga. Unfolding and refolding of the native structure of bovine pancreatic trypsin inhibitor studied by computer simulations. Biochemistry, 32 9614-9631, 1993. 

Figure 8-8. a) A non-mirror-symmetric pair of points b) folding the pair of points shown in (a) results in a non-coincident cluster of two points c) the noncoincident cluster is averaged d) the averaged point in c) is unfolded to a mirror-symmetric pair. 

In order to calculate S, one has to find the nearest configuration of Pi points that is G-symmetric. In the majority of cases, S is the distance of a chiral object from a reflection mirror. The fblding/unfolding procedure was developed for finding that configuration. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

The biologiccJ function of a protein or peptide is often intimately dependent upon the conformation(s) that the molecule can adopt. In contrast to most synthetic polymers where the individual molecules can adopt very different conformations, a protein usually exists in a single native state. These native states are found rmder conditions typically found in Uving cells (aqueous solvents near neutred pH at 20-40°C). Proteins can be unfolded (or denatured) using high-temperature, acidic or basic pH or certain non-aqueous solvents. However, this unfolding is often reversible cind so proteins can be folded back to their native structure in the laboratory. 

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