Ligand binding on ladder lattices

The ligand binding problems on two-dimensional ladders, which model many important binding phenomena in molecular biology, are studied in details. The model is represented by four parameters, the interactions between ligands when bound to adjacent sites on opposite legs of the ladder (tau), the interactions between bound ligands in the longitudinal direction of the ladder (sigma), the number of binding sites that are covered by a bound ligand (m), and the intrinsic binding constant (K). The partition functions of ring ladders are approached with the transfer matrix method. A general relation is derived which connects the partition function of a linear ladder with that of a ring ladder. The results obtained apply to the general situation of multivalent binding, in which m>1. Special attention is paid to the case where the ligand covers one site (m=1). In this case explicit formulas are given for the partition functions of ring and linear ladders. Closed-form expressions are obtained for various properties of the system, including the degree of binding (theta), the midpoint in the binding isotherm (1/square root(tau sigma)), the initial and end slopes of the Scatchard plots (2sigma + tau - 4 and -sigma2 tau, respectively). From these closed-form formulas, sigma and tau may be extracted from experimental data. The model reveals certain features which do not exist in one-dimensional models. Using the general method discussed in [1], the recurrence relation is found for the partition functions. The analytical solution found for this model provides test cases to verify the numerical results for more complex two-dimensional models.     

Map analysis of ligand binding to a linear lattice

A one-dimensional mapping of the binding properties of a linear lattice offers an exact analytical solution for the site-specific properties of the lattice once the length N and the parameter for nearest neighbor interactions are specified. The solution is derived independent of the definition of the partition function or the transfer matrix, nor does it involve combinatorial arguments. This result provides a simple and effective way of analyzing experimental data for protein-ligand interactions and broadens our understanding of site-specific properties in biological macromolecules.     

A note on the quantitative properties of McGhee-von Hippel model

Exact closed-form expressions are presented for the properties of the McGhee-von Hippel model of non-specific binding of large ligands to one-dimensional homogeneous lattices. These properties include the midpoint location and the slope at the middle point of the binding isotherms (v varies with ln L plots), the location and magnitude of the maximum, as well as the location of the inflection point, in the Scatchard plots (v/L varies with v plots).     

Application of the one- and two-dimensional Ising models to studies of cooperativity between ion channels.

The Ising model of statistical physics provides a framework for studying systems of protomers in which nearest neighbors interact with each other. In this article, the Ising model is applied to the study of cooperative phenomena between ligand-gated ion channels. Expressions for the mean open channel probability, rho o, and the variance, sigma 2, are derived from the grand partition function. In the one-dimensional Ising model, interactions between neighboring open channels give rise to a sigmoidal rho o versus concentration curve and a nonquadratic relationship between sigma 2 and rho o. Positive cooperativity increases the slope at the midpoint of the rho o versus concentration curve, shifts the apparent binding affinity to lower concentrations, and increases the variance for a given rho o. Negative cooperativity has the opposite effects. Strong negative cooperativity results in a bimodal sigma 2 versus rho o curve. The slope of the rho o versus concentration curve increases linearly with the number of binding sites on a protomer, but the sigma 2 versus rho o relationship is independent of the number of ligand binding sites. Thus, the sigma 2 versus rho o curve provides unambiguous information about channel interactions. In the two-dimensional Ising model, rho o and sigma 2 are calculated numerically from a series expansion of the grand partition function appropriate for weak interactions. Virtually all of the features exhibited by the one-dimensional model are qualitatively present in the two-dimensional model. These models are also applicable to voltage-gated ion channels.     

Transition modes in Ising networks: an approximate theory for macromolecular recognition.

For a statistical lattice, or Ising network, composed of N identical units existing in two possible states, 0 and 1, and interacting according to a given geometry, a set of values can be found for the mean free energy of the 0-->1 transition of a single unit. Each value defines a transition mode in an ensemble of nu N = 3N - 2N possible values and reflects the role played by intermediate states in shaping the energetics of the system as a whole. The distribution of transition modes has a number of intriguing properties. Some of them apply quite generally to any Ising network, regardless of its dimension, while others are specific for each interaction geometry and dimensional embedding and bear on fundamental aspects of analytical number theory. The landscape of transition modes encapsulates all of the important thermodynamic properties of the network. The free energy terms defining the partition function of the system can be derived from the modes by simple transformations. Classical mean-field expressions can be obtained from consideration of the properties of transition modes in a rather straightforward way. The results obtained in the analysis of the transition mode distributions have been used to develop an approximate treatment of the problem of macromolecular recognition. This phenomenon is modeled as a cooperative process that involves a number of recognition subsites across an interface generated by the binding of two macromolecular components. The distribution of allowed binding free energies for the system is shown to be a superposition of Gaussian terms with mean and variance determined a priori by the theory. Application to the analysis of the biologically interaction of thrombin with hirudin has provided some useful information on basic aspects of the interaction, such as the number of recognition subsites involved and the energy balance for binding and cooperative coupling among them. Our results agree quite well with information derived independently from analysis of the crystal structure of the thrombin-hirudin complex.     

Kinetics of large-ligand binding to one-dimensional lattices: theory of irreversible binding

Exact solutions are obtained for the time dependence of the extent of irreversible binding of ligands that cover more than one lattice site to a homogeneous one-dimensional lattice. The binding may be cooperative or noncooperative and the lattice either finite or infinite. Although the form of the solution is most convenient when the ligand concentration is buffered, exact numerical or approximate analytical solutions, including upper and lower bounds, can be derived for the case of variable ligand concentration as well. The physical reason behind the relative simplicity of the kinetics of irreversible as opposed to reversible binding in such systems is discussed.     

Detecting Isolation by Distance Using Phylogenies of Genes

We introduce a method for analyzing phylogenies of genes sampled from a geographically structured population. A parsimony method can be used to compute s, the minimum number of migration events between pairs of populations sampled, and the value of s can be used to estimate the effective migration rate M, the value of Nm in an island model with local populations of size N and a migration rate m that would yield the same value of s. Extensive simulations show that there is a simple relationship between M and the geographic distance between pairs of samples in one- and two-dimensional models of isolation by distance. Both stepping-stone and lattice models were simulated. If two demes k steps apart are sampled, then, s, the average value of s, is a function only of k/(Nm) in a one-dimensional model and is a function only of k/(Nm)2 in a two-dimensional model. Furthermore, log(M) is approximately a linear function of log(k). In a one-dimensional model, the regression coefficient is approximately -1 and in a two-dimensional model the regression coefficient is approximately -0.5. Using data from several locations, the regression of log(M) on log(distance) may indicate whether there is isolation by distance in a population at equilibrium and may allow an estimate of the effective migration rate between adjacent sampling locations. Alternative methods for analyzing DNA sequence data from a geographically structured population are discussed. An application of our method to the data of R. L. Cann, M. Stoneking and A. C. Wilson on human mitochondrial DNA is presented.     

Packaging of single DNA molecules by the yeast mitochondrial protein Abf2p: reinterpretation of recent single molecule experiments

Brewer et al. (Biophys. J. 85 (2003) 2519-2524) have studied the compaction of dsDNA in a double flow cell by observing the extension of stained DNA tethered in buffer solutions with or without Abf2p. They use a Langmuir adsorption model in which one Abf2p molecule adsorbs on one site on the DNA, and the binding constant, K, is given as the ratio of the experimental rates of adsorption and desorption. This paper presents an improved interpretation. Instead of Langmuir adsorption we use the more appropriate McGhee-von Hippel (J. Mol. Biol. 86 (1974) 469-489) theory for the adsorption of large ligands to a one-dimensional lattice. We assume that each adsorbed molecule shortens the effective contour length of DNA by the foot print of Abf2p of 27 base pairs. When Abf2p adsorbs to DNA stretched in the flowing buffer solution, we account for a tension effect that decreases the adsorption rate and the binding constant by a factor of 2 to 4. The data suggest that the accessibility to Abf2p decreases significantly with increasing compaction of DNA, resulting in a lower adsorption rate and a lower binding constant. The kinetics reported by Brewer et al. (Biophys. J. 85 (2003) 2519-2524) lead to a binding constant K=3.6 x 10(6) M(-1) at the beginning, and to K=5 x 10(5) M(-1) near the end of a compaction run, more than an order of magnitude lower than the value K=2.57 x 10(7) M(-1) calculated by Brewer et al. (Biophys. J. 85 (2003) 2519-2524).     

Statistical fluctuations in processes of gene expression regulation: consideration of the problem from point of view of statistical mechanics

The regulation of gene expression is a basic problem of biology. In some cases, the gene activity is regulated by specific binding of regulatory proteins to DNA. In terms of statistical mechanics, this binding is described as the process of adsorption of ligands on the one-dimensional lattice and has a probability nature. As a random physical process, the adsorption of regulatory proteins on DNA introduces a noise to the regulation of gene activity. We derived equations, which make it possible to estimate this noise in the case of the binding of the lac repressor to the operator and showed that these estimates correspond to experimental data. Many ligands are able to bind nonspecifically to DNA. Nonspecific binding is characterized by a lesser equilibrium constant but a greater number of binding sites on the DNA, as compared with specific binding. Relations are presented, which enable one to estimate the probability of the binding of a ligand on a specific site and on nonspecific sites on DNA. The competition between specific and nonspecific binding of regulatory proteins plays a great role in the regulation of gene activity. Similar to the one-dimensional "lattice gas" of particles, ligands adsorbed on DNA produce "one-dimensional" pressure on proteins located at the termini of free regions of DNA. This pressure, an analog of osmotic pressure, may be of importance in processes leading to changes in chromatin structure and activation of gene expression.     

Cooperative and non-cooperative binding of large ligands to a finite one-dimensional lattice: A model for ligand-ougonucleotide interactions

A combinatorial approach is employed to calculate exact expressions for the extent of binding to a finite one dimensional lattice of ligands which cover more than one lattice site. The binding may be either cooperative or non-cooperative. It is found that the assumption of an effectively infinite lattice is generally a good one, except with relatively low concentrations of strongly cooperative ligands. An approach to analyzing experimental data is suggested which makes explicit use of the lattice length dependence of binding to extract more information about the binding parameters than can be obtained using the infinite lattice approximation. It is shown that irreversible binding cannot be viewed as a limiting case of reversible binding- The reasons for this difference are discussed, and expressions for the extent of irreversible binding are derived.     

Characteristic patterns of N Oct-3 binding to a set of neuronal promoters

N Oct-3, a neurospecific POU protein, homodimerizes in a non-cooperative fashion on the neuronal aromatic l-amino acid decarboxylase gene promoter and generates heterodimers with HNF-3beta. Several other neuronal gene promoters, the corticotropin releasing hormone and the aldolase C gene promoters also contain overlapping binding sites for N Oct-3 and HNF-3beta. We have demonstrated that N Oct-3 presents a non-cooperative homodimerization on these two additional targets and can also give rise to heterodimers with HNF-3beta. Surprisingly, despite the high degree of conservation of the respective POU subunits, the ubiquitous POU protein Oct-1 can only form monomers even in the presence of either N Oct-3 or HNF-3beta on these DNA targets. Our data indicate that this difference is correlated with the specific ability of a portion of the N Oct-3 linker to fold as an alpha-helix, a property shared by class III POU proteins. These results suggest that this novel binding pattern permits the heterodimerization of N Oct-3 and HNF-3beta on the neuronal promoters, which could be a key issue in the development of the nervous system and possibly tumors of neural origin.     

Nucleotide binding to the chaperonin GroEL: non-cooperative binding of ATP analogs and ADP, and cooperative effect of ATP

Chaperonin-assisted protein folding proceeds through cycles of ATP binding and hydrolysis by GroEL, which undergoes a large structural change by the ATP binding or hydrolysis. One of the main concerns of GroEL is the mechanism of the productive and cooperative structural change of GroEL induced by the nucleotide. We studied the cooperative nature of GroEL by nucleotide titration using isothermal titration calorimetry and fluorescence spectroscopy. Our results indicated that the binding of ADP and ATP analogs to a single ring mutant (SR1), as well as that to GroEL, was non-cooperative. Only ATP induces an apparently cooperative conformational change in both proteins. Furthermore, the fluorescence changes of pyrene-labeled GroEL indicated that GroEL has two kinds of nucleotide binding sites. The fluorescence titration result fits well with a model in which two kinds of binding sites are both non-cooperative and independent of each other. These results suggest that the binding and hydrolysis of ATP may be necessary for the cooperative transition of GroEL.     

Steady-state kinetics of bifunctional enzymes. Taking into account kinetic hierarchy of fast and slow catalytic cycles in a generalized model

A steady-state approximation of the generalized two-dimensional model of a bifunctional enzyme catalyzing independent proceeding of two one-pathway reactions is considered in a case of mutual influence of the active sites. Coexistence of fast and slow catalytic cycles in the reaction mechanism is analyzed. Conditions when the hierarchy of fast and slow catalytic cycles allows simplification of a two-dimensional model and its reduction to the one-dimensional cyclic schemes were determined. Kinetic equations describing these simplified schemes are presented.     

A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

We have combined two distinct but related stochastic approaches to model the Escherichia coli chemotaxis pathway. Reactions involving cytosolic components of the pathway were assumed to obey the laws of conventional stochastic chemical kinetics, while the clustered membrane receptors were represented in two-dimensional arrays similar to the Ising model. Receptors were assumed to flip between an active and an inactive state with probabilities dependent upon three energy inputs: ligand binding, methylation level due to adaptation, and the activity of neighbouring receptors. Examination of models with different lattice size and geometry showed that the sensitivity to stimuli increases with lattice size and the nearest-neighbour coupling strength up to a critical point, but this amplification was also accompanied by a proportional increase in steady-state noise. Multiple methylation of receptors resulted in diminished signal-to-noise ratio, but showed improved stability to variation in the coupling strength and increased gain. Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria. The model also has some of the properties of a cellular automaton and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.     

The binding capacity is a probability density function.

The binding capacity of a system, or equivalently, the fluctuations of the number of ligands bound around the average value defined by the binding isotherm, can be regarded as a probability density function for the chemical potential of the ligand. The first moment of this density function is the mean ligand activity as defined by Wyman and gives the average free energy (in kT units) of binding per site. The second moment is directly related to the cooperativity of the system. These and higher moments can be obtained from numerical integration of experimental data in a direct way. An analytical expression for the moment generating function shows that the N independent coefficients of the partition function of a system containing N sites are uniquely defined by the first N moments of the binding capacity.     

Cooperative ligand-lattice binding. Approximate Gaussian binding distribution

Nearest-neighbor cooperative binding of a ligand covering n sites and binding with equilibrium constant K and cooperativity factor omega to a large molecule with m binding sites (m much greater than n omega, n/omega) can be approximately described by a Gaussian distribution P(q-qmax), where q is the number of ligands bound and qmax the most probable value of q. The variance of the Gaussian is equal to the derivative dqmax/d ln(L), where L is the free ligand concentration. This variance, sigma 2, is a complicated function of qmax. However, in the limits of very large cooperativity, omega much greater than 1, very large anticooperativity, omega much less than 1, or noncooperativity, omega = 1, simpler expressions for sigma 2 can be given. For qmax = m/(n + 1), where the most probable number of bound ligands equals the number of free binding sites, sigma 2 has a particularly simple form: sigma 2 = 2m omega 1/2/(n + 1)3. The Gaussian and the infinite lattice approximations for the average number of ligands bound are good approximations only if sigma is much smaller than the number of binding sites. The variance may therefore provide an easy check on the validity of the infinite lattice approximation, which is commonly used to analyze experimental binding data.     

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.     

Structure of ferricytochrome c' from Rhodospirillum molischianum at 1.67 A resolution

The structure of ferricytochrome c' from Rhodospirillum molischianum has been crystallographically refined to 1.67 A resolution using a combination of reciprocal space and restrained least-squares refinement methods. The final crystallographic R-factor for 30,533 reflections measured with I greater than sigma (I) between infinity and 1.67 A is 0.188. The final model incorporates 1944 unique protein atoms (of a total of 1972) together with 194 bound solvent molecules. The structure has been analysed with respect to its detailed conformational properties, secondary structural features, temperature factor behavior, bound solvent sites, and heme geometry. The asymmetric unit of the cytochrome c' crystal contains a dimer composed of chemically identical 128-residue polypeptide chains. Although the refined structure shows the monomers to be very similar, examination of the differences that do occur allows an evaluation of how different lattice contacts affect protein conformation and solvent binding. In particular, comparison of solvent binding sites in the two subunits allows identification of a common set that are not altered by lattice interactions. The preservation of these solvent interactions in different lattice environments suggests that they play a structural role in protein stabilization in solution. The refined structure additionally reveals some new features that relate to the ligand binding properties and unusual mixed-spin state character of cytochrome c'. Finally, comparison of the heme binding geometry in cytochrome c' and other structurally unrelated c-type cytochromes shows that two alternative, but sterically favorable, conformational variants occur among the seven examples examined.     

Saxitoxin binding to sodium channels in head extracts from wild-type and tetrodotoxin-sensitive strains of Drosophila melanogaster

Extracts prepared from heads of Drosophila melanogaster show high-affinity binding (KD = 1.9 nM) of [3H]saxitonin, a compound known to bind to and block voltage-sensitive sodium channels in other organisms. The interaction between saxitoxin and the Drosophila saxitoxin receptor is non-cooperative and reversible with a half-life of 18.3 s for binding at 4 degrees C. The saturable binding is specifically inhibited by tetrodotoxin with a K1 = 0.30 nM. The number of saturable binding sites in the extract is 97 fmol/mg protein. Since approx. 50% of the binding activity is recovered in the extract, the number of binding sites in the head is estimated to be 6.4 fmol/mg head. Nerve conduction in Drosophila larvae is completely blocked after 20 min in a bathing solution containing 200 nM tetrodotoxin. A comparison between the binding and the electrophysiological studies in Drosophila and other organisms suggests that the Drosophila saxitoxin receptor is part of the voltage-sensitive sodium channel involved in the propagation of action potentials. A mutant (ttxs), which is abnormally sensitive to dietary tetrodotoxin, is shown to be indistinguishable from wild type with respect to [3H]saxitonin-binding properties and physiological sensitivity to tetrodotoxin. These studies provide techniques which can be used to identify mutants with defects in the saxitoxin-binding component of the sodium channel.