Affinity distribution models

In general, these distributions all share the intuitive property that the vast majority of ligands have low to moderate affinity, and fewer and fewer ligands have higher affinity. Though the Sips, log-normal and RAD distributions can be made visually quite similar, they will always differ in their moments and other mathematical properties. It should be stressed that these distributions are for affinities. For other properties, such as enzymatic activity, these distributions may not be relevant, though at least one model for catalytic activity is based on these distributions [19]. 

Throughout the following section, different search methods and different parameters for a particular search method are compared. Comparing different search methods requires a performance measure, a probabilistic measure of satisfying well-defined criteria for successful search. A common performance measure is the enrichment function [20,21], Enrichment can be defined as the ratio of either concentrations or mole fractions before and after selection as a function of affinity or as a function of ligand rank in the library for example, 

E(Ka) 1 means that ligands of affinity Ka are selected to a greater extent than the library on average, 0 E(Ka) 1 means that these ligands are selected to a lesser extent than the library on average, and E(Ka) = 0 means none of these ligands are selected. This definition of enrichment can also be expressed as 

Good performance measures will incorporate mole fraction information. For search without mutation, example measures are (i) the mole fraction of the highest affinity ligand (from the initial library) at a particular round (ii) the probability that a given number of ligands with affinities above a specified threshold appear in a sample taken after a particu- 

It is helpful to classify modeling approaches as to whether they are deterministic or 

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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. 

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).

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. 

While I have made a clear distinction between laboratory technique-based and landscape-based models, the distinction is more artifactual than representative of fundamental differences. The laboratory technique-based models do not include mutation or crossover, so the only landscape property they depend on is the affinity distribution p(Ka). Once mutation is included, some type of relationship between specific sequences and their affinities must be included. Landscapes are one means of including this relationship. Work with landscape-based models does not include laboratory techniques or parameters because the questions posed in this work do not require this added level of complexity and because of the paucity of experimental data to define actual affinity landscapes. If the landscape work is to solve actual laboratory protocol problems, the laboratory and chemistry details need to be included. Ideally, future work will include mathematically rigorous analyses of landscape-based models that incorporate chemical and experimental details. 

The application of the composite isotherms enables us to model the ion exchange on heterogeneous surfaces, such as rocks and soils. When the structure and composition of the sorbent is well known, we can choose the most probable site affinity distribution function. If not, it is desirable to fit the composite isotherm by different models. The just-described four isotherms provide an opportunity for this. In addition, when adsorption and ion exchange can take place simultaneously, adsorption and ion-exchange isotherms and site distribution functions can be combined (Cernik et al. 1996). 

For proton adsorption on well studied systems intrinsic affinity distributions can be obtained after conversion of the (T5(pH) curves into curves using the SGC model with a reasonable value for Ci,t. As indicated before the adequacy of the applied double layer model can be checked when adsorption data are available at a series of indifferent electrolyte concentrations. Replotting the surface charge as a function of pHs should lead to merging of the individual curves into a master curve [44, 45]. The master curve reflects the chemical heterogeneity [46-48]. 

Lutzenkirchen. J.. Influence of impurities on acid-base data for oxide minerals-analysis of observable surface charge and proton affinity distributions and model calculations for single crystal samples, Croat. Chem. Acta, 80, 333, 2007. 

Borkovec. M.. Rusch, U., and Westall, J.C.. Modeling of competitive ion binding to heterogeneous materials with affinity distributions, in Adsorption of Metals by Geomedia, Jenne, E., ed.. Academic Press, New York, 1998, p. 467. 

The simplest explanation is that there is a rubber-like network present and that this has a maximum extensibility due to the degree of entanglement, which is constant for a given grade of polymer and depends on its molar mass and method of polymerisation. This limiting extensibility is not to be confused with the limit of applicability of the affine rubber model for predicting orientation distributions discussed in section 11.2.1 because the limiting extension can involve non-affine deformation. 

When experimental methods to measure proton affinity distribution spectra (PADs) from potentiometric titration data became available [54a], it was found by Contescu et al. [58] that the apparent values of proton binding constants identified from PADs were in semiquantitative agreement with log K values for binding predicted by the MUSIC model (Table 1). The characteristic PADs for several oxides are shown in Fig. 2. 

Besides the application of specific models, an important amount of work has been devoted to extract affinity distributions directly from experimental data, attempting to invert Equations 11.31, 11.37, and/or 11.40 or similar. The extraction of affinity distribution from experimental data is an ill-posed problem (Provencher 1982a Cernik, Borkovec, and Westall 1995 Borkovec et al. 1996), because with experimental data one actually has, in the case of Equation 11.31,... 

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