Affinity distribution

Cemik, M., Borkovec, M. and Westall, J. C. (1995). Regularized least-squares methods for the calculation of discrete and continuous affinity distributions for heterogeneous sorbents, Environ. Sci. Technol., 29, 413-425. 

Altmann, R. S. and Buffle, J. (1988). The use of differential equilibrium functions for the interpretation of metal binding in complex ligand systems its relation to site occupation and site affinity distributions, Geochim. Cosmochim. Acta, 52, 1505-1519. 

Attention was paid early on to solution pH, and in particular, to a surface — bulk proton balance. Various models of hydroxyl chemistry have been developed in colloid science literature [21], Perhaps the simplest and most common model assumes a single type of OH group and amphoteric behavior (i.e., one set of Kx and K2 from Figure 6.1). More complicated models invoke multiple OH groups and proton affinity distributions [22]. It will be demonstrated below that the simpler type has worked well for the revised physical adsorption (RPA) model. 

We consider a simple reaction composed of two elementaiy steps in series with the affinity distribution ratio m - AgilAg and the stoichiometric niunbers vj and vj. The mean stoichiometric number v is then given by Eqn. 7-52 ... 

Fig. 7-12. Potential energy curves for a two-step reaction (a) near equilibrium indicating reaction affinity distributed to step 1 and (b) away from equilibrium indicating reaction affinity distributed to step 1 and step 2 - dG = affinity of overall reaction m = (dgi / dgj).

Rampey AM, Umpleby RJ, Rushton GT, Iseman JC, Shah RN, Shimizu KD. Characterization of the imprint effect and the influence of imprinting conditions on affinity, capacity, and heterogeneity in molecularly imprinted polymers using the Freundlich isotherm-affinity distribution analysis. Anal Chem 2004 76 1123-1133. 

Median value of affinity distribution for proton binding by carboxyl groups. 

Width of proton-affinity distribution of carboxyl groups. 

Median value of affinity distribution for proton binding by phenolic OH groups. AVidth of proton-affinity distribution of phenolic OH groups. 

Borkovec, M., Rusch, U., Cernik, M., Koper, G.J.M. and Westall, J.C. (1996) Affinity distributions and acid-base properties of homogeneous and heterogeneous sorbents exact results versus experimental data inversion. Colloid. Surf. A Physicochem. Eng. Aspects, 107, 285-296. 

Nederlof M.M., van Riemsdijk, W.H. and Koopal, L.K. (1990) Determination of adsorption affinity distributions a general framework for methods related to local isotherm approximations./. Coll. Interface Sci., 135, 410-426. 

Thakur, A.K., Munson, PJ., Hunston, D.L. and Rodbard, D. (1980) Characterization of ligand-binding systems by continuous affinity distributions of arbitrary shape. Anal. Biochem., 103, 240-254. 

At present, many popular applied molecular evolution protocols do not involve mutation or recombination. The laboratory technique-based models presented in this section are of this type. Incorporating mutation requires fitness landscape models or some other means of relating molecular properties to particular sequences. The more abstract models reviewed later allow for mutation and recombination and are based heavily on landscape structure. The models in the present section are based on affinity distribution p(Ka), the probability that a ligand chosen at random from the library has affinity Ka. 

The model consists of three stages setting the affinity distribution, equilibrium biopanning and dissociative biopanning (Fig. lb). 

Fig. 6. Effect of increasing stringency in the Levitan/Kauffman phage display model. Curves show the affinity distribution at screening rounds 0-4. The initial distribution, round O , is that given by the RAD model [16]. (a) Constant stringency (b) increasing stringency each generation in which T d, 0.25 [T mdH].

Fig. 11. Average highest affinity (average first-order statistic) in the initial library as a function of library size. The affinity distribution p(Ka) is log-normal with mean 3.2 x 106 M and standard deviation 107 (from Ref. 14). While the average affinity ofthe best ligand always increases with increasing library size, the incremental increase for one more library member decreases as the library is made larger. This diminishing return relates to the tradeoff in library size versus ligand copy number described in the Levitan/Kauffman model (see text).

Use of affinity distribution order statistics to assist in estimating experimental parameters. This approach is dependent on having estimates for p(Ka) and error bounds on the resulting order statistics. 

Random energy model The random energy model (REM) results from using a fitness distribution p(f) to assign fitnesses randomly to points in the landscape [ 14,59,60,70,71,81, 91,92], p(f) is the probability that a point in the sequence space has fitness fand is exactly analogous to affinity distribution p(Ka). Such landscapes have zero correlation (are very rugged), have many local fitness peaks, and result in very short adaptive walks. Very few of the local peaks are accessible by adaptive walks from any particular point. 

An order statistics approach could also be used to choose a library size. Given an affinity distribution p(Ka) and a minimum desired affinity, order statistics calculations can yield the number of ligands needed to include at least a specified number of ligands with this affinity or greater with a specified probability (Fig, 11). 

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