Ligand binding distributions in nucleic acids

Statistical-mechanical lattice models for protein-DNA binding in chromatin

Statistical-mechanical lattice models for protein-DNA binding are well established as a method to describe complex ligand binding equilibria measured in vitro with purified DNA and protein components. Recently, a new field of applications has opened up for this approach since it has become possible to experimentally quantify genome-wide protein occupancies in relation to the DNA sequence. In particular, the organization of the eukaryotic genome by histone proteins into a nucleoprotein complex termed chromatin has been recognized as a key parameter that controls the access of transcription factors to the DNA sequence. New approaches have to be developed to derive statistical-mechanical lattice descriptions of chromatin-associated protein-DNA interactions. Here, we present the theoretical framework for lattice models of histone-DNA interactions in chromatin and investigate the (competitive) DNA binding of other chromosomal proteins and transcription factors. The results have a number of applications for quantitative models for the regulation of gene expression.

Intermediates in the melting transitions of aluminum nanoclusters

The author uses heat capacity data for aluminum cluster ions, Aln+, obtained in the laboratory of Breaux et al. [Phys. Rev. Lett. 94, 17340 (2005)] to determine whether or not intermediate species are present in the transition from the solidlike form of the clusters present at low temperatures to the liquidlike form present at high temperatures. He gives a general method on how to test for the presence of such intermediates and how to calculate their probabilities and thermodynamics as a function of temperature. In addition he uses energy distribution functions, using the maximum-entropy method that he developed previously, to substantiate the presence or absence of intermediates. As examples of the method he treats n=53 and n=79 clusters both of which exhibit marked maxima in the temperature dependence of their heat capacity curves, indicating strong order-disorder transitions. He find that in the melting transition n=53 clusters have no intermediates while the melting of n=79 clusters is dominated by intermediate species.

Enthalpy distribution functions for protein-DNA complexes: Example of the binding of AT-hooks to target DNA

In this article we use the published heat capacity data of Dragan et al. [A.I. Dragan, et al., The energetics of specific binding of AT-hooks from HMGA1 to target DNA, J. Mol. Biol. 327 (2003) 393-411] on the association of proteins with DNA duplexes to construct enthalpy probability distributions for the protein/DNA complexes formed in these systems. We first analyze the multistep equilibrium that determines the species concentrations in this system to determine whether or not the DNA-peptide complex goes cleanly to DNA single-strands and peptide. Using the heat capacity data for this case we employ the maximum-entropy method to construct enthalpy probability distribution functions for the species involved in this equilibrium. We find that the distribution functions for this system clearly show bimodal behavior indicating a two-state transition from complex to non-complex form.

Enthalpy distribution functions for the unwinding of a short DNA duplex

In this article we use the published heat capacity data of Dragan et al. (J Mol Biol 2003, 327, 293-411) for a short DNA duplex to calculate the enthalpy probability distribution for this species as a function of temperature. Our approach is based on a procedure that we developed (Poland, D. J Chem Phys 2000, 112, 6554) whereby one obtains moments of the enthalpy distribution from the temperature dependence of the heat capacity. One then uses the maximum-entropy method to construct the enthalpy probability distribution from the set of enthalpy moments. For the DNA duplex treated here the heat capacity goes through a maximum as a function of temperature reflecting the unwinding of the duplex structure. In the neighborhood of the heat capacity maximum, the enthalpy distribution functions show a clear bimodal structure, indicating the coexistence of two distinct states, the duplex and the single-strand state. The probabilities of theses two states can be estimated from the enthalpy distribution functions and can be used to calculate the temperature dependence of the equilibrium constant for the unwinding of the DNA duplex. This example illustrates that the temperature dependence of the heat capacity can be used to give a detailed picture of conformational transitions in biological macromolecules. In particular, the structure of the enthalpy distribution in this case allows one to see the temperature evolution of the two-state distribution in detail.

Binding of Charged Ligands to Macromolecules. Anomalous Salt Dependence

Although interactions in biological systems occur in the presence of a large number of charged species, the binding of charged ligands to different biomolecules is often analyzed in a simplified model focusing only upon the receptor, ligand, and added salt. Here we demonstrate that the presence of charged macromolecules can affect binding to the receptor in an unexpected way. Experimental studies of the binding of barium ions to the chelator 5,5'-dibromo-1,2-bis(O-amino-phenoxy)-ethane-N,N,N',N'-tetraacetic acid in the presence of charged silica sols show that the binding affinity increases with added salt. The experimental findings are verified in Monte Carlo simulations using a dielectric continuum model with a uniform dielectric permittivity throughout the solution. The anomalous salt behavior is caused by a reduction of the chemical potential of the free ligand, which even in the absence of binding interacts strongly with the oppositely charged receptor. These results are also relevant for the interpretation of competition studies often used in the case of strong ligand binding.

Long-range correlations in the helix free energy distribution in DNA

In this paper we explore the free energy distribution in the helical form of DNA using the genome of the virus Rickettsia prowazekii Madrid E as an example. The genome of this organism has been determined by Andersson et al. (Nature 396 (1998) 133) and is available on the World Wide Web (www.tigr.org). Using the helix statistical weights based on nearest-neighbor base pairs of SantaLucia (Proc. Natl. Acad. Sci. USA 95 (1998) 1460), we calculate the free energy in consecutive blocks of m base pairs in the DNA sequence and then construct the free energy distribution for these values. Using the maximum-entropy method we can fit the distribution curves with a function based on the moments of the distribution. For blocks containing 10-20 base pairs the distribution is slightly skewed and we require four moments to accurately fit the function. For blocks containing 100 base pairs or more, the distribution is well approximated by a Gaussian function based on the first two moments of the distribution. We find that the free energy distribution for m=20 can be reproduced using random sequences that have the local (singlet, doublet or triplet) statistics of Rickettsia. However, for much larger blocks, for example m=500, the width of the free energy distribution based on the actual Rickettsia genome is broader by almost a factor of 3 than the distributions based on random local statistics. We find that the distribution functions for the C or G content in blocks of m base pairs have almost the same behavior as a function of block size as do the free energy distributions. In order to duplicate the width of the distribution functions based on the actual Rickettsia sequence, we need to introduce tables (matrices) that correlate the states of consecutive blocks hundreds of base pairs long. This indicates that correlations on the order of the number of base pairs contained in the average gene are required to give the actual widths for either the C or G content or the helix free energy distributions. Above a certain m value, the distributions for larger m can be accurately expressed in terms of the distribution functions for smaller m. Thus, for example, the distribution for m=5000 can be expressed in terms of the generating function for m=1000.

Combined affinity and rate constant distributions of ligand populations from experimental surface binding kinetics and equilibria

The present article considers the influence of heterogeneity in a mobile analyte or in an immobilized ligand population on the surface binding kinetics and equilibrium isotherms. We describe strategies for solving the inverse problem of calculating two-dimensional distributions of rate and affinity constants from experimental data on surface binding kinetics, such as obtained from optical biosensors. Although the characterization of a heterogeneous population of analytes binding to uniform surface sites may be possible under suitable experimental conditions, computational difficulties currently limit this approach. In contrast, the case of uniform analytes binding to heterogeneous populations of surface sites is computationally feasible, and can be combined with Tikhonov-Phillips and maximum entropy regularization techniques that provide the simplest distribution that is consistent with the data. The properties of this ligand distribution analysis are explored with several experimental and simulated data sets. The resulting two-dimensional rate and affinity constant distributions can describe well experimental kinetic traces measured with optical biosensors. The use of kinetic surface binding data can give significantly higher resolution than affinity distributions from the binding isotherms alone. The shape and the level of detail of the calculated distributions depend on the experimental conditions, such as contact times and the concentration range of the analyte. Despite the flexibility introduced by considering surface site distributions, the impostor application of this model to surface binding data from transport limited binding processes or from analyte distributions can be identified by large residuals, if a sufficient range of analyte concentrations and contact times are used. The distribution analysis can provide a rational interpretation of complex experimental surface binding kinetics, and provides an analytical tool for probing the homogeneity of the populations of immobilized protein.

Free energy of proton binding in proteins

In this article we use literature data on the titration of denatured ribonuclease to test the accuracy of proton-binding distributions obtained using our recent approach employing moments. We find that using only the local slope of the titration curve at a small number of points (five, for example) we can reproduce the detailed proton-binding distribution at all pH values. Our method gives the complete proton-binding polynomial for a given protein and each coefficient in this polynomial in turn yields the free energy for binding a given number of protons in all ways to the protein. Using these net free energies, we can then compute the average proton-binding free energy per proton as a function of the fraction of protons bound. We find that this function is remarkably similar for different proteins, even for proteins that exhibit quite different titration behavior. For the special case of binding to independent sites, we obtain simple relations for the first and last terms in the free energy per-proton function. For this special case we also can calculate the distribution functions giving the probability that a molecule has a given number of positive or negative charges and the joint distribution that a molecule simultaneously has a given number of positive and negative charge.

Maximum-entropy calculation of free energy distributions for two forms of myoglobin

The temperature dependence of the heat capacity of myoglobin depends dramatically on pH. At low pH (near 4.5), there are two weak maxima in the heat capacity at low and intermediate temperatures, respectively, whereas at high pH (near 10.7), there is one strong maximum at high temperature. Using literature data for the low-pH form (Hallerbach and Hinz, 1999) and for the high-pH form (Makhatadze and Privalov, 1995), we applied a recently developed technique (Poland, 2001d) to calculate the free energy distributions for the two forms of the protein. In this method, the temperature dependence of the heat capacity is used to calculate moments of the protein enthalpy distribution function, which in turn, using the maximum-entropy method, are used to construct the actual distribution function. The enthalpy distribution function for a protein gives the fraction of protein molecules in solution having a given value of the enthalpy, which can be interpreted as the probability that a molecule picked at random has a given enthalpy value. Given the enthalpy distribution functions at several temperatures, one can then construct a master free energy function from which the probability distributions at all temperatures can be calculated. For the high-pH form of myoglobin, the enthalpy distribution function that is obtained exhibits bimodal behavior at the temperature corresponding to the maximum in the heat capacity (Poland, 2001a), reflecting the presence of two populations of molecules (native and unfolded). For this form of myoglobin, the temperature evolution of the relative probabilities of the two populations can be obtained in detail from the master free energy function. In contrast, the enthalpy distribution function for the low-pH form of myoglobin does not show any special structure at any temperature. In this form of myoglobin the enthalpy distribution function simply exhibits a single maximum at all temperatures, with the position of the maximum increasing to higher enthalpy values as the temperature is increased, indicating that in this case there is a continuous evolution of species rather than a shift between two distinct population of molecules.

Maximum-entropy determination of self-association distribution functions; daunorubicin and ATP

In the present paper we show how one can use the perturbation of some molecular optical property (for example circular dichroism or chemical shift) as a function of concentration to construct cluster distribution functions describing the self-association of molecules in solution. The optical data are first converted into data giving the variation of the average extent of clustering as a function of the total concentration and then, using straightforward thermodynamics, a set of moments of the cluster distribution function can be obtained. Utilizing the maximum-entropy method, the moments are then used to calculate approximate distribution functions, where the more moments that are used the better the approximation obtained. Given the probability distribution for clusters of different sizes one can then calculate the equilibrium constant for each stage of association. Thus one converts average degree of association into equilibrium constants without having to use any specific model. By this method one can clearly tell whether the equilibrium constants remain constant, increase, or decrease with the number of molecules in a cluster. We apply the method to literature data for two systems, namely daunorubicin, which has a strong tendency to cluster in solution, and Mg(ATP)(2-) which forms weaker clusters. We find that the successive equilibrium constants for adding a monomer to a cluster are approximately constant for daunorubicin but clearly decrease as a function of increasing cluster size for Mg(ATP)(2-).