Hill’s coefficient

By using Hill s coefficient, it is possible to draw a conclusion about the character of the process and to determine ligand concentration in one cooperative unit. 

Figure 2. Dose-response curve of membrane depolarization as a function of PbTx-3 concentration (7). Data from a total of 22 axons were pooled each axon received only one dose. Data are plotted as means of depolarization amplitudes. TTie solid line is a theoretical 3rd order fit with an ED.q of 1.5 nM, maximum observed depolarization of 30 mV, and a Hill s coefficient of 2.

Table 11. Hill s coefficient for reaction of polymer-heme complexes with molecular oxygen...

System Solvent Additives Hill s coefficient (n) T/2a mmHg... 

This gives a linear relationship (plot) of logv/(n - v) or log 0/(1 - 0) versus log[A] with slope = h and intercept = log K. The interaction coefficient, h is also called the Hill s coefficient (uh) and is a measure of the degree of site-site interaction, i.e., n > h > 1. The closer the quantity, h (uh) approaches the number of sites n, the stronger the interaction, i.e. h = n for a biomacromolecule with strongly interacting multiple sites and h = 1 for a macromolecule with noninteracting multiple sites. 

Treatment of liquid drops is considerably more complex than bubbles, since the internal motion must be considered and internal boundary layers are difficult to handle. Early attempts to deal with boundary layers on liquid drops were made by Conkie and Savic (C8), McDonald (M9), and Chao (C4, W7). More useful results have been obtained by Harper and Moore (HIO) and Parlange (PI). The unperturbed internal flow field is given by Hill s spherical vortex (HI3) which, coupled with irrotational flow of the external fluid, leads to a first estimate of drag for a spherical droplet for Re 1 and Rep 1. The internal flow field is then modified to account for convection of vorticity by the internal fluid to the front of the drop from the rear. The drag coefficient. 

The Hill s interaction coefficient, h, is a measure of the strength of interaction among n-sites. If the n-equivalent sites are noninteracting, then h = 1, whereas h approaches n (number of equivalent sites) for the strongly interacting n-equivalent sites. 

The data below describe the binding of ligand L to the oligomeric receptor R (100 juM). Create a script hie and data hie to evaluate the association constant and the number of equivalent sites. Calculate Hill s interaction coefficient by perform regression analysis. 

T,RD = V /( u vX then a perturbation analysis of Hill s equation (11.11) shows that the uniform steady state of (11.1) with a temporally varying diffusion coeflBcient of the inhibitor is stable for sufficiently small oscillations, e < c 1 [436]. In other words, small oscillations in the diffusion coefficient have a stabilizing effect they delay the onset of the Turing instabihty. 

Figure 1.6 Effective volume of the interfadal film (normalized to the NaAOT volume) as a function of the bmimBF concentration (volume fraction) (closed circles) data from the lattice parameter of Hj phase obtained from SAXRD measurements, (closed stars) data from the selfdiffusion coefficients of AOT" and bmim measured in the Lj phase calculated from Equations 1.9 and 1.10 (see text), and (open stars) data obtained for the Lj phase doped with p-xylene (see text). The curve represents the best fit according to the Hill s cooperative binding (Eq. 1.13). Reproduced from Murgia et al. [26] with permission from American Chemical Society.

For the general case, the treatment suggested by Kern (Pmcc.s.s Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recommended. Because of the wide variations in fin-tube construction, it is convenient to convert all film coefficients to values based on the inside bare surface of the tube. Thus to convert the film coefficient based on outside area (finned side) to a value based on inside area Kern gives the following relationship ... 

The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direclion of motion) are given in Fig. 6-57 as a Function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton s law region is reported by Knudsen and Katz Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). 

Figure 10-170. Outside heat-transfer film coefficient as function of pipe temperature and O.D. (Used by permission Chapman, F. S., and Holland, F. A. Chemical Engineering, Dec. 20,1965, p. 79. McGraw-Hill, Inc. All rights reserved.)...

Coefficients of Cubical Expansion for Various Liquids and Aqueous Solutions, in Lange s Handbook of Chemistry, 12th ed., ed. John A. Dean (New York, McGraw-Hill 1979), Table 10-42. 

Saxena, S. C. Joshi, R. K., Thermal accommodation and adsorption coefficients of gases, In Vol. II 1 of McGraw Hill/CEMDAS Data Series on Material Properties Touloukian, Y. S. Ho, C. Y Eds. McGraw Hill, New York, NY, 1981... 

Figure 3.6 The relationship between swelling coefficient (S%) and WPG for a variety of linear chain anhydrides (from data of Hill and Jones, 1996a).

Clearly, the quantity = Xj/j) maps the region 0 < S < < into the interval 0 < S 2. The value of % = 2 is the maximum value of the Hill coefficient for the case m-l. One should be careful, however, to note that these particular methods are valid only for the case of two sites. When m > 2 there are various types of cooperativities and, in general, there is no single parameter that describes the cooperativity in the system. Even for the case m = 2 one could be misled in estimating the cooperativity of the system if one were to rely only on the/orm or the shape of the BI or any of its transformed functions, as will be demonstrated in Section 4.6 and again in Section 4.8 and Appendix F. 

Several other empirical relations for diffusion coefficients have been suggested Olson and Walton (01) have devised a means for estimating diffusion coefficients of organic liquids in water solution from surface-tension measurements. Hill (H5) has proposed a method based on Andrade s theory of liquids which allows for the concentration dependence of the diffusion coefficient in a binary liquid mixture. The formula of Arnold (A2, T6, p. 102) does not seem generally useful inasmuch as it contains two constants ( abnormality factors ) characteristic of the solute and of the solvent. 

K. S. Pitzer, Ion Interaction Approach Theory and Data Correlation , Chapter 3 of Activity Coefficients in Electrolyte Solutions, 2nd Edition, K. S. Pitzer, Editor, CRC Press, Boca Raton, 1991. Parameters for many electrolytes are summarized in this reference. The equations and parameters can also be found in K. S. Pitzer, Thermodynamics, Third Edition, McGraw-Hill, Inc., New York, 1995. 

Osmotic coefficients for KC1 and CaCh were obtained from Appendix 4 of G. N. Lewis, M. Randall, K. S. Pitzer, and L. Brewer, Thermodynamics, Second Edition, McGraw-Hill Book Company, New York, 1961. Values for LaCl3 were obtained from F. H. Spedding, H. O. Weber, V. W. Saeger, H. H. Petheram, J. A. Rard, and A. Habenschuss, Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25 °C 1. The Rare Earth Chlorides , J. Chem. Eng. Data, 21, 341-360 (1976). 

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