Average correlation

Pressure drops typically can match conventional tubular exchangers. Again from the APV handbook an average correlation is as follows ... 

Now consider a pair of reservoirs in equilibrium with respect to the extensive parameters Xj and Xk, with instantaneous values of Xj and Xk. Let 6Xj denote a fluctuation from the instantaneous value. The average value of 6Xj is zero, but the average of its square (SXj)2 = ((6Xj)2) 0. Likewise, the average correlation moment (5Xj5Xk) / 0. 

The physical dimensions and dynamics of calmodulin have also been investigated by tyrosine fluorescence. To learn about the internal mobility of calmodulin, Lambooy et al 1 and Steiner et al measured the steady-state fluorescence anisotropy of the tyrosine. Since the average correlation... 

One can easily adjust the values of the dielectric constants D(, and Dj to obtain the experimental values of W, as in Table 4.4. With a choice of = 19.6 and Dj. = 51.0 for water, and D. = 12.5 and Dj. = 31.8 for 50% water-ethanol, we obtain the experimental values of W. We now compute the total correlation function for the two-state model for succinic acid. Here the correlation cannot be computed as an average correlation of the two configurations (see Section 4.5). The total correlation of the equilibrated two-state model is... 

In two-site systems, there is only one correlation function which characterizes the cooperativity of the system. In systems with more than two identical sites, for which additivity of the higher-order correlations is valid, it is also true that the pair correlation does characterize the cooperativity of the system. This is no longer valid when we have different sites or nonadditivity effects. In these cases there exists no single correlation that can be used to characterize the system, hence the need for a quantity that measures the average correlation between ligands in a general binding system. There have been several attempts to define such a quantity in the past. Unfortunately, these are valid only for additive systems, as will be shown below. 

Note that each correlation has been defined as the X 0 limit and hence independent of X or C. Here we show that the average correlation, constructed from the X. — 0 limit of the correlations, is concentration-dependent. 

It is clear that neither Wyman s nor Minton and Saroff s measures do justice to aU types of cooperativities in the system, and certainly cannot account for variation in cooperativity at different stages of the binding process. In the next subsection we define a new measure of the average correlation in any binding system, and show how to extract this quantity from experimental data. 

For all the above reasons we have defined g(C) without reference to any hypothetical, independent-site system. One simply extracts both 1(C) and all from the experimental data, and then constructs the quantity g(C). When the sites are identical in a weak sense, i.e., all k = k, some of the correlations for a given / might differ. For example, four identical subunits arranged in a square will have only one intrinsic binding constant k, but two different pair correlation functions. For this particular example we have four nearest-neighbor pair correlations g (2), and two second-nearest-neighbor pair correlations gJJ)- The average correlation for this case is... 

We compare here the average correlation in the three models of the four-site system. In the case of direct interactions only, it is intuitively clear and easily proven that the average correlation depends only on the sign of 5(2) - 1 (assuming the subunits are identical, that direct interactions are pairwise additive, and neglecting long-range interactions). Hence, when 5(2) > 1 (positive direct correlation), we always have... 

Figure 6.2. Binding isotherms and the average correlation, g(C) - 1 for the tetrahedral (T), square (S), and linear (L) models. The sites are identical and all correlations are due to direct ligand-ligand pairwise additive interactions, (a) Curves for positive cooperativity, S(2) = 10 (b) curves for negative coopera-tivity, S(2) = 0.1. Note that in these systems the cooperativity increases in absolute magnitude from L to S to T.

The average correlation, plotted as f(C)- 1, shows that the square model starts initially with a small positive value and increases monotonously to the very large value of 37,348 at C -> < . On the other hand, the g(C) - 1 curve for the tetrahedral model starts from a very small value and reaches the value of about 21,058 at very high concentrations. Clearly, both of the Bis appear as positive cooperative, but with much stronger cooperativity for the square model, in apparent defiance of the density of interaction argument. 

Figure 6.3. The binding isotherm and the average correlation [as f(G) -1] for the square and tetrahedral models discussed in Section 6.6. No direct ligand-ligand interactions are assumed. The parameters for the indirect correlations are A = 0.01, AT = 1, = 4Cu, and t = 0.01. The lower two...

Figure 6.7 shows the average correlation g(PQ) as a function of oxygen partial pressure. At the low-pressure limit (determined by the pair correlation only) there is no clear-cut monotonic dependence on temperature. At the high-pressure limit (as determined by the quadruplet correlations, see Section 5.8), we see that the... 

Figure 6.7. Average correlation s(Pq ) as a function of the partial pressure of oxygen Pq (in torrs), for the same system as in Fig. 6.5. (a) Low-pressure limit (b) high-pressure limit. The temperatures are indicated next to each curve.

It is clearly seen that within each group of cinves the overall cooperativity became weaker as the temperature increases. This can be judged qualitatively from the steepness of the Bis, as well as quantitatively from the average correlations (shovm on the lower panel of Fig. 6.12, in the range of 30-100 torr). The utility function, computed for P2 = 100 torr and Pj = 30 torr, is nearly zero for the system with no added solutes (besides the buffer solution). It is small for the system with 2 mM of BPG added and becomes relatively large for the system with 2 mM of IHP added. (Note, however, that the utility values do not increase monotonically with temperature this is clear for the BI on the Ihs of Fig. 6.12.) Thus, the change in utility values upon addition of BPG and IHP is much more dramatic than the change due to temperature. 

Figure 6.12. BI (upper panel) and average correlation g(P) (lower panel), for human adult Hb at temperatures of 10, 15, 20, 25, 30, 35 °C. The upper curves correspond to 10 °C and the lowest to 35 °C. All systems were in a fixed buffer solution [for details, see Imai and Yonetani (1975)]. (a) As in Fig. 6.5, no added BPG or IHP (b) 2 mM of BPG added (c) 2 mM of IHP added. The Bis were drawn in the same range of 0-100 torr, but the average correlations are drawn between 30-100 torr. The two vertical lines drawn at 30 and 100 torr are used to estimate the utility value for each system.

Figure 6.13. BI and average correlation for Hb at three different pH values. The data are the same as in Fig. 6.8, but here the BI is drawn in the pressure range of 0-30 torr. The utility function is computed for the arbitrarily selected pair of pressures = 5 and P2= 10 torr. The average correlations are drawn...

The different behavior of the open and closed system is due to the difference in the pattern of correlations in the two systems hence the average correlation differs for the two systems with the same m. 

For the open linear system the pattern of pair correlations is different. We have again m m - l)/2 total number of pairs, but now only (m - 1) of these are nn pairs, contributing S each, and the remaining (m - l)(m - 2)/2 are non-nn pairs, contributing unity to the average correlation g(C 0) see Table 7.1. 

In the C °o limit, all the sites are bound the average correlation g(C is determined by the mth-order correlation function, which is 5 for the cyclic and 5 for the open linear system. This is true within the pairwise additive approximation for direct interaction, and neglecting long-range correlations. 

In terms of these average binding constants we define the average correlations, in the same formal manner as we define correlations between specific sites ... 

So in this case each average correlation is the arithmetic average of all the different correlations of the same order. For strictly identical sites Eqs. (J.l 1) reduce to... 

The other quantity is a sort of average correlation time defined as... 

D. Axelson Carbon-13 NMR allows for the measurement of the average correlation time for each individual carbon atom. For the glass temperature problem we are obviously only concerned with the correlation time of the backbone carbons. 

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