Cleavage center

Next let us show how one can compute the proteasome output if the transport rates are given. In our model we assume that the proteasome has a single channel for the entry of the substrate with two cleavage centers present at the same distance from the ends, yielding in a symmetric structure as confirmed by experimental studies of its structure. In reality a proteasome has six cleavage sites spatially distributed around its central channel. However, due to the geometry of its locations, we believe that a translocated protein meets only two of them. Whether the strand is indeed transported or cleaved at a particular position is a stochastic process with certain probabilities (see Fig. 14.5). 

The protein strand can be cleaved if it lies close to the cleavage center or it could be transported forward by one amino acid. We assume that the probability of transport depends only on the length of the strand inside the proteasome. The probability of transport is, therefore, given by a translocation rate function, v(x + D) where x + D is the length of the strand inside the proteasome, in terms of amino acids. The probability of cleavage is assumed to be a constant, denoted by y. We also assume that the degradation of proteins by the proteasome is a highly processive mechanism [2], i.e., in other words, the protein is not released by the proteasome until it is completely processed. This leads to the possibility of the proteasome... 

Fig. 14.5 Schematic diagram of the protein degradation by the proteasome. The protein strand (denoted by 0-0-0-0 ) enters the proteasome from the left to be cleaved by the cleavage centers (denoted by the scissors). The length of the strand after crossing the cleavage center is denoted byx.

D Distance between a cleavage center and the proteasome end Amino adds 15... 

We take the offset of the coordinate x (measured in amino acids) along the proteasome at the first cleavage center (see Fig. 14.5). During the time interval dt, the protein strand can move by one amino acid with the probability v(x + D)dt and can be cut with the probability ydt. 

In this case, the length distribution does not coincide with the distribution of protein coordinates. Let us start with the distribution of protein coordinates w(x). Beyond the cleavage centers it obeys the equation which is similar to Eq. (12) ... 

As shown below, the presence of a second cleavage centre does not significantly change the results. Hence, to find the conditions for a maximum in the peptide length distribution, we use the analytical expressions for the case of one cleavage center. This peptide length distribution has extrema at the points where ... 

This equation should be fulfilled at least in one point for x + D > 0. To note y is here the rate of cleavage. Hence, there are no limitations for its value. Eq. (22) shows that the condition for obtaining a maximum when we have a single cleavage center is independent of the actual form of the transport rate function, but is rather dependent on its slope. This would suggest that it is possible to obtain a peak in the length distribution even when the transport rate function is monotonically decreasing, and indeed that is what we find from our numerical simulations. 

As predicted by theory (solid line) the numerically computed peptide length distribution (denoted by boxes) has a maximum for one cleavage center (b), two cleavage centers with immediate disappearance of cleavage products (c), and in the case where the cleavage products do not disappear (d). 

The mechanism of anionic polymerization of cyclosiloxanes has been the subject of several studies (96,97). The first kinetic analysis in this area was carried out in the early 1950s (98). In the general scheme of this process, the propagation/depropagation step involves the nucleophilic attack of the silanolate anion on the sUicon, which results in the cleavage of the siloxane bond and formation of the new silanolate active center (eq. 17). 

As chemists proceeded to synthesize more complicated stmctures, they developed more satisfactory protective groups and more effective methods for the formation and cleavage of protected compounds. At first a tetrahydropyranyl acetal was prepared, by an acid-catalyzed reaction with dihydropyran, to protect a hydroxyl group. The acetal is readily cleaved by mild acid hydrolysis, but formation of this acetal introduces a new stereogenic center. Formation of the 4-methoxytetrahy-dropyranyl ketal eliminates this problem. 

In contrast to the extensive body of work on the preparation of these zinc carbenoids, few investigations are on record concerning the mechanism of the Furu-kawa method for carbenoid formation. Two limiting mechanisms can be envisioned - a concerted metathesis via a four-centered transition structure or a stepwise radical cleavage-recombination (Scheme 3.11). 

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