Derivation of antibody distribution from experimental binding data.

It can be demonstrated that antibody populations cannot be suitably described in terms of a Sips distribution. Use of the Stieltjes transform allows instead derivation, from experimental binding data, of the most probable distribution of the association constants. From graphical interpolation of the experimental data four parameters can be obtained, which characterize a satisfactory bimodal distribution. By this procedure, analysis of data taken at different times of a humoral immune response, indicates that the relative abundance of the two sub-populations, rather than their mean association constant, is mainly affected by time or by variations of antigen dose. By use of the same procedure it is also possible to show that, at least as far as haptens are concerned, the slope of the 50% antigen-binding plot, often taken as a measure of the “avidity” of antibodies, is instead a function of the relative weight of the two sub-populations and of the spread of each of them.     

Ionization of clusters. III. Evaluation of interaction parameters from binding data

Interactions between ionizable groups on the same molecule modulate the binding of protons to an extent where the binding constants may be shifted by orders of magnitude. The first two papers of this series discussed the family of carboxylic acids, pairwise isotropic interactions, and evaluation of single site binding data. This paper presents an extended group of hypothetical binding isotherms. Procedures are illustrated for deriving interaction parameters from binding data. The interaction parameters for about 25 representative compounds with two and three interacting ionizable groups are evaluated and tabulated.     

Co-operativity and the methods of plotting binding and steady-state kinetic data.

The features that distinguish positive from negative co-operativity in double-reciprocal Eadie-Hofstee-Scatchard and Hanes plots, often incorrectly stated to be the sign of curvature or second derivatives, are explained. It is shown how to determine the 'Hill exponent' and interaciton free energies from curves in these plots, and in the simple plot of ligand binding or velocity against free ligand or substrate concentration. New types of plots, where the kind of co-operative behaviour is more obvious than in the traditional ones, are proposed.     

Analysis of derivative binding isotherms. Theoretical considerations

The basic theoretical groundwork for the use of derivative binding isotherms in the analysis of ligand binding is presented. The derivative binding isotherm is defined as Γ (Y) = df/dy where f = fractional degree of saturation and y = natural logarithm of the free ligand concentration. Since Γ (y) is a positive function, which goes to zero as y → ±∞, the mean value of y, 〈y〉, and the second and third moments, μ₂ and μ₃ about 〈y〉 are well defined. For a macromolecular system consisting of N equivalent and independent binding sites, Γ (y) is a symmetrical bell-shaped function with one maximum. The maximum occurs when y = ?ln K_assoc; μ₂ = π²/3, and μ₃ = 0. For multiple sets of independent binding sites, Γ (y) is a superposition of Γ-type functions. If the sets are sufficiently well separated in binding free energy, multiple extrema may be seen at positions corresponding to the logarithms of the dissociation constants for the individual sets. In any case, 〈y〉 is equal to the mean value of the logarithms of the dissociation constants for the sets; μ₂ > π²/3 and equal to π²/3 plus the variance of the logarithms of the dissociation constants about their mean value; and μ₃ is, except by coincidence, not equal to zero and equals the third moment of the distribution of logarithms of the dissociation constants about their mean value. Analysis of Γ(y) for the case of cooperative interactions within a set of binding sites was investigated by examining (1) the Hill model (whose mathematical representation is equivalent to that used to describe antibody heterogeneity except that in the latter case the parameter a, the Sips, constant, is constrained (0 < a ≤1);(2) a common model for cooperativity in which the cooperative free energy is a linear function of the fraction bound; and (3) a general representation of cooperative interactions within a set of sites in terms of ?(f), a smooth function that gives the interaction free energy in units of RT. For the Hill model (or Sips model) Γ(y) is a symmetrical function with one maximum at y = (?1)/a)lnK, μ₂ = π²/3a²; and μ₃ = 0. For the case in which the cooperative free energy is a linear function of f [?(f) = cf], 〈y〉 = ?ln K₀ + (c/2); μ₂ = (π²/3) + c[(c/12) + 1] where c > ?4; and μ₃ = 0. General expressions for the moments in terms of ?(f) are derived. In general, μ₂ < (π²/3) for positive cooperativity and μ₂ > (π₂/3) for negative for negative cooperativity. Γ(y) will be symmetrical if and only if the cooperative free energy is introduced symmetrically about f = 0.5.     

Analysis of dynamic brain imaging data.

Modern imaging techniques for probing brain function, including functional magnetic resonance imaging, intrinsic and extrinsic contrast optical imaging, and magnetoencephalography, generate large data sets with complex content. In this paper we develop appropriate techniques for analysis and visualization of such imaging data to separate the signal from the noise and characterize the signal. The techniques developed fall into the general category of multivariate time series analysis, and in particular we extensively use the multitaper framework of spectral analysis. We develop specific protocols for the analysis of fMRI, optical imaging, and MEG data, and illustrate the techniques by applications to real data sets generated by these imaging modalities. In general, the analysis protocols involve two distinct stages: "noise" characterization and suppression, and "signal" characterization and visualization. An important general conclusion of our study is the utility of a frequency-based representation, with short, moving analysis windows to account for nonstationarity in the data. Of particular note are 1) the development of a decomposition technique (space-frequency singular value decomposition) that is shown to be a useful means of characterizing the image data, and 2) the development of an algorithm, based on multitaper methods, for the removal of approximately periodic physiological artifacts arising from cardiac and respiratory sources.     

Analysis of large-scale gene expression data.

DNA microarray technology has resulted in the generation of large complex data sets, such that the bottleneck in biological investigation has shifted from data generation, to data analysis. This review discusses some of the algorithms and tools for the analysis and organisation of microarray expression data, including clustering methods, partitioning methods, and methods for correlating expression data to other biological data.     

The ionization of clusters. II. Pairwise isotropic interactions and individual site binding data

Evaluation of the parameters describing the binding of protons to clusters of interacting sites requires some reasonable assumptions and procedures because it is impossible to observe an unperturbed site in its interacting environment. When the unperturbed sites are not identical, individual site binding data allow for the evaluation of the differences (or ratios) between the unperturbed (or intrinsic) binding constants but not their actual values (or the interaction energies). In this paper we extend our previous treatment of the ionization of clusters in order to generalize pairwise isotropic interactions and take into account the present availability of individual site binding data.     

Analysis of trp repressor-operator interaction by filter binding.

A filter binding assay was developed that allows measurement of specific binding of trp repressor to operator DNA. The most important feature of this procedure is the concentration and type of salt present in the binding buffer. Using this assay the dissociation constant of the repressor-operator complex was determined to be 2.6 X 10(-9) M, and 1.34 repressor dimers were found to be bound to each operator-containing DNA molecule. These values agree with those obtained by more complex methods. The dissociation constant of the repressor for the corepressor L-tryptophan in the presence of operator DNA was shown to be 2.5 X 10(-5) M. A synthetic 48 bp operator fragment was used to determine the repressor-operator dissociation constant in the presence of tryptophan or tryptophan analogs which have higher or lower affinities for aporepressor. The rate of dissociation of repressor from operator DNA also was determined. Our findings indicate that dissociation is influenced by the concentration of tryptophan or tryptophan analogs and suggest that release of the corepressor may be the first step in dissociation of the repressor-operator complex.     

Cooperative binding of oxygen to hemoglobin: analysis of accurate equilibrium binding data

Accurate equilibrium binding data for the oxygenation of hemoglobin are used (a) to show that various models for cooperativity are inconsistent with the best available experimental data, (b) to determine the equilibrium constants for binding of 2,3-diphosphoglycerate to hemoglobin molecules in intermediate stages of oxygenation, and (c) to deduce a mechanism for allosteric effects in hemoglobin which is consistent with the best available experimental data. The total free energy of cooperativity is defined and discussed.     

Analysis techniques for microarray time-series data.

We address possible limitations of publicly available data sets of yeast gene expression. We study the predictability of known regulators via time-series analysis, and show that less than 20% of known regulatory pairs exhibit strong correlations in the Cho/Spellman data sets. By analyzing known regulatory relationships, we designed an edge detection function which identified candidate regulations with greater fidelity than standard correlation methods. We develop general methods for integrated analysis of coarse time-series data sets. These include 1) methods for automated period detection in a predominately cycling data set and 2) phase detection between phase-shifted cyclic data sets. We show how to properly correct for the problem of comparing correlation coefficients between pairs of sequences of different lengths and small alphabets. Finally, we note that the correlation coefficient of sequences over alphabets of size two can exhibit very counterintuitive behavior when compared with the Hamming distance.     

Analysis of longitudinal data with unmeasured confounders.

Confounding in longitudinal or clustered data creates special problems and opportunities because the relationship between the confounder and covariate of interest may differ across and within individuals or clusters. A well-known example of such confounding in longitudinal data is the presence of cohort and period effects in models of aging in epidemiologic research. We first formulate a data-generating model with confounding and derive the distribution of the response variable unconditional on the confounder. We then examine the properties of the regression coefficient for some analytic approaches when the confounder is omitted from the fitted model. The expected value of the regression coefficient differs in across- and within-individual regression. In the multivariate case, within- and between-individual information is combined and weighted according to the assumed covariance structure. We assume compound symmetry in the fitted covariance matrix and derive the variance, bias, and mean squared error of the slope estimate as a function of the fitted within-individual correlation. We find that even in this simplest multivariate case, the trade-off between bias and variance depends on a large number of parameters. It is generally preferable to fit correlations somewhat above the true correlation to minimize the effect of between-individual confounders or cohort effects. Period effects can lead to situations where it is advantageous to fit correlations that are below the true correlation. The results highlight the trade-offs inherent in the choice of method for analysis of longitudinal data, and show that an appropriate choice can be made only after determining whether within- or between-individual confounding is the major concern.     

Analysis of cohort effects from cross-sectional data.

A statistical model and computer program are described to analyze data fro the existence of cohort effects. The analysis leads to the separate assessments of cohort, time and age influences on a series of age-specific rates that are available in cross-sectional form. The assessment of these three factors can possibly yield insights into specific disease etiologies.     

Improving MHC binding peptide prediction by incorporating binding data of auxiliary MHC molecules

MOTIVATION: Various computational methods have been proposed to tackle the problem of predicting the peptide binding ability for a specific MHC molecule. These methods are based on known binding peptide sequences. However, current available peptide databases do not have very abundant amounts of examples and are highly redundant. Existing studies show that MHC molecules can be classified into supertypes in terms of peptide-binding specificities. Therefore, we first give a method for reducing the redundancy in a given dataset based on information entropy, then present a novel approach for prediction by learning a predictive model from a dataset of binders for not only the molecule of interest but also for other MHC molecules. RESULTS: We experimented on the HLA-A family with the binding nonamers of A1 supertype (HLA-A*0101, A*2601, A*2902, A*3002), A2 supertype (A*0201, A*0202, A*0203, A*0206, A*6802), A3 supertype (A*0301, A*1101, A*3101, A*3301, A*6801) and A24 supertype (A*2301 and A*2402), whose data were collected from six publicly available peptide databases and two private sources. The results show that our approach significantly improves the prediction accuracy of peptides that bind a specific HLA molecule when we combine binding data of HLA molecules in the same supertype. Our approach can thus be used to help find new binders for MHC molecules.     

Determination of netropsin-DNA binding constants from footprinting data

A theory for deriving drug-DNA site binding constants from footprinting data is presented. Plots of oligonucleotide concentration, as a function of drug concentration, for various cutting positions on DNA are required. It is assumed that the rate of cleavage at each nucleotide position is proportional to the concentration of enzyme at that nucleotide and to the probability that the nucleotide is not blocked by drug. The probability of a nucleotide position not being blocked is calculated by assuming a conventional binding equilibrium for each binding site with exclusions for overlapping sites. The theory has been used to evaluate individual site binding constants for the antiviral agent netropsin toward a 139 base pair restriction fragment of pBR-322 DNA. Drug binding constants, evaluated from footprinting data in the presence of calf thymus DNA and poly(dGdC) as carrier and in the absence of carrier DNA, were determined by obtaining the best fit between calculated and experimental footprinting data. Although the strong sites on the fragment were all of the type (T.A)4, the value of the binding constant was strongly sequence dependent. Sites containing the dinucleotide sequence 5'-TA-3' were found to have significantly lower binding constants than those without this sequence, suggesting that an adenine-adenine clash produces a DNA structural alteration in the minor groove which discourages netropsin binding to DNA. The errors, scope, and limitations associated with the method are presented and discussed.     

Cobalamin binding proteins in stomach and serum from various animal species data for B12 binding capacities and molecular sizes of the binding proteins.

1. The unsaturated cobalamin-binding capacity of stomach mucosa and serum from 37 animals and the size of the binders have been measured. 2. The binding capacity in stomach mucosa was from 1 to 600 nmole kg-1 wet wt, the highest values occurring in pig, guinea pig, porpoise and earthworm. In serum it varied from 0.4 to 270 nmole 1(-1), the highest values occurring in duck, grass snake, toad and spiny dogfish. 3. The size in terms of Stokes radius of the cobalamin binders in stomach mucosa varied from 1 nm in hagfish (Mr approximately 14,000) to 7 nm in lamprey (Mr approximately 210,000). In serum it was from 1.4 nm (Mr approximately 17,000) to 7.4 nm (Mr approximately 240,000) both in frog.