Solute binding curves for (two polymer + solute)-complex systems. I. "One-solute-stack" systems

The matrix method is used for calculating the grand partition function for the reaction: 2 polymer + solute = complex. The homogeneous polymers are assumed to have two types of sites within each nucleotide unit: sites for the polymer–polymer association, i.e., (p-p) sites; sites for polymer–solute association i.e., (p-s) sites. The respective binding parameters, P and F , and nearest-neighbor interaction parameters, W and S , are assumed independent. Complications due to ring entropy are avoided by rest riding the model to one-solute-stack systems, which are physically realizable when the reciprocal of the solute cooperativity parameter is much larger than the number of nucleotides in the polymer. The 4 × 4 generating matrix is shown to be a tensor product of two 2 × 2 matrices, each the generating matrix of a particular type of site. The scalar product of the 4 × 4 matrix is shown to be equivalent to the scalar product of a 2 × 2 matrix in the weak interaction limit, W ≈ 0. Calculations are presented for the general case which restricts the (p-s) association to occur only with (p-p) associated nucleotide units. The nature of the binding curve in relation to partitioning the total interaction energy (F + P + S + W ) among the parameters is discussed. Also presented is a criterion for neglecting possible states in the calculation of the grand partition function.     

On the adsorption of ions at charged sites in an electric field: I

Considering only nearest neighbor interactions, an expression is obtained for the grand partition function for the adsorption of two kinds of monovalent positive ions at a long chain of one type of monovalent negative fixed sites in an electric field. Expressions are obtained for the fractions of sites which are occupied by each kind of ion as well as of those which are unoccupied as a function of the potential of the electric field.     

Defects in associating systems: example of actin.

A statistical mechanical model is given for linear associating systems that contain defects, using the double-stranded actin polymer as an example. We treat the system as a one-dimensional lattice that can desorb monomers (giving defects) using grand partition function techniques. The main difference from a standard adsorption problem is that the monomer units are also responsible for the structural integrity of the lattice (polymer) and if too many desorb the polymer will be broken. We use literature data to estimate the density of defects in the actin polymer.     

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.     

Protein sorption and recovery by hydrogels using principles of aqueous two-phase extraction

Use of the thermodynamic principles of aqueous two-phase extraction (ATPE) to drive protein into a crosslinked gel is developed as a protein isolation and separation technique, and as a protein loading technique for drug delivery applications. A PEG/dextran gel system was chosen as a model system because PEG/dextran systems are widely used in aqueous two-phase extraction and dextran gels (Sephadex(R)) are common chromatographic media. The effects of polymer concentrations and molecular weights, salts, and pH on the partitioning of ovalbumin matched ATPE heuristics and data trends. Gel partition coefficients (Cgel/Csolution) increased with increasing PEG molecular weight and concentration and decreasing dextran concentration (increased gel swelling). The addition of PEG to the buffer solution yielded partition coefficients more than an order of magnitude greater than those obtained in systems with buffer alone, or added salt. A combined salt/PEG system yielded an additional order of magnitude increase. For example, when ovalbumin solution (2.3 mg/mL) was equilibrated with Sephadex(R) G-50 at pH 6.75, the partition coefficients were 0.13 in buffer, 0.11 in buffer with 0.22M KI, 2.3 in 12 wt% PEG-10,000 and 32.0 in 12 wt% PEG-10, 000 with 0.22M KI. The effect of anions and cations as well as ionic strength and pH on the partitioning of ovalbumin also matched ATPE heuristics. Using the heuristics established above, partition coefficients as high as 80 for bovine serum albumin and protein recoveries over 90% were achieved. In addition, the wide range of partition coefficients that were obtained for different proteins suggests the potential of the technique for separating proteins. Also, ovalbumin sorption capacities in dextran were as high as 450 mg/g dry polymer, and the sorption isotherms were linear over a broad protein concentration range.     

On the adsorption of ions at charged sites in an electric field: II

A study is made of the adsorption of one kind of monovalent positive ion at a long chain of alternating monovalent negative fixed charged (“lattice”) and uncharged (“interstitial”) sites both of one type in an electric field. Considering only nearest neighbor interactions an expression is obtained for the grand partition function. The fractions of sites of both types which are occupied and unoccupied are determined. It is shown that an equilibrium constant can be defined for the adsorption of ions at oppositely charged sites.     

Theory of multivalent binding in one and two-dimensional lattices

Ligand binding to a linear lattice composed of N sites, under general conditions of cooperativity and number of sites covered upon binding, m, is approached in terms of the theory of contracted partition functions. The partition function of the system obeys a recursion relation leading to a generating function that provides an exact analytical solution for any case of interest. Site-specific properties of the lattice are derived from simple transformations of the analytical expressions. The McGhee-von Hippel model is obtained as a special case in the limit N --> infinity. The derivation is straightforward and involves no combinatorial arguments. Partition functions and site-specific properties are also derived for the case of non-cooperative binding to a two-dimensional torus of length N, containing s sites in its section for a total of sN sites. The torus provides a relevant model for ligand binding to double-stranded DNA (s = 2) or protein helices (s = 3,4). It is proved that non-cooperative binding to the two-dimensional torus can mimic cooperative binding to a one-dimensional linear lattice when m = s. The dimensional embedding of the lattice and the geometry of interaction of its sites play a crucial role in defining the binding properties of the system accessible to experimental measurements. Hence, caution must be exercised in the interpretation of Scatchard plots in terms of the one-dimensional McGhee-von Hippel model, especially when m < or = 4 and the geometry of the system is clearly two-dimensional.     

The binding of cyclic AMP receptor protein to two lactose promoter sites is not cooperative in vitro.

The lactose promoter-operator region of Escherichia coli contains two binding sites for cyclic AMP receptor protein (CAP), two for the lactose repressor, and two for RNA polymerase. The high density of binding sites makes cooperative interactions between these proteins likely. In this study, we used the gel electrophoresis mobility shift assay and binding partition analysis techniques to determine whether the secondary CAP site influences the binding of CAP to the principal CAP site in the lactose promoter when both are present on a linear DNA molecule. Such an effect could occur through the formation of a bridged DNA-CAP-DNA structure, through the interaction of CAP molecules bound to each of the sites, or through allosteric effects caused by CAP-mediated DNA bending. We found, however, that the interaction of CAP with these sites was not cooperative, indicating that CAP sites 1 and 2 bind CAP in an independent manner.     

Thermodynamic study of forces involved in bovine serum albumin and ovalbumin partitioning in aqueous two-phase systems

The partitioning of bovine serum albumin and ovalbumin in different two-phase aqueous polymer systems is investigated using a thermodynamic approach. Systems used were polyethylene glycols (PEGs) of molecular weights 1000 to 10,000 Da and Dextran T500 (500,000 Da). Ovalbumin transfer to the top phase is exothermic, which suggests an electrostatic interaction between the hydroxyl groups of PEG and the hydrophilic side chain of the protein, whereas the bovine serum albumin partition is an endothermic process that is entropically driven, which coincides with its high surface hydrophobicity. The effect of PEG molecular weight on enthalpy and heat capacity changes, associated with the partition of both proteins, is examined on the basis of a preferential interaction of low-molecular-weight PEG with the protein surface.     

Interaction of porin from Yersinia pseudotuberculosis with lipopolysaccharides. Effect of ionic strength, pH, and divalent cations on the binding parameters

Interaction of the pore-forming protein (porin) from Yersinia pseudotuberculosis with S- and R-forms of the endogenous lipopolysaccharide (LPS) was studied at various ionic strengths (20-600 mM NaCl), concentrations of divalent cations (5-100 mM CaCl2, MgCl2), and pH values from 3.0 to 9.0. The interaction of the R-LPS with porin has been shown in all experimental conditions to be in consensus with the model suggesting binding at independent sites of two types. S-LPS binds to interacting sites of relatively high affinity and to independent sites of low affinity at all pH values examined and at low NaCl concentration. The cooperative interaction of the S-LPS and porin is not observed at high ionic strength and in divalent cation-free medium. The number of binding sites of porin and association constants (Ka) for both LPS forms decrease significantly on increasing the solution ionic strength. The Ka values for the R- and S-LPS change oppositely on changing the pH: the Ka value for the R-LPS is maximal (Ka = 6.7 x 10(5) M-1), but that for S-LPS is minimal (Ka = 0.4 x 10(5) M(-1) at pH 5.0-5.5. The number of high-affinity and low-affinity binding sites for both LPS forms is maximal at pH 5.0-5.5. In this case, the numbers of high- and low-affinity sites for R-LPS are 3 and 10, respectively, and those for the S-LPS are 7 and 20, respectively. These data suggest an important role of electrostatic interactions on binding of LPS to porin. The contribution of conformational changes of the ligand and protein and hydrophobic interactions are discussed.     

Cu2+ binding triggers alphaBoPrP assembly into insoluble laminar polymers

Cu(2+) binding is so far the best characterized property of the prion protein. This interaction has been mapped to the N-terminal domain of the prion protein where multiple His residues occur largely embedded within the repetitive PHGGGWGQ sequence known as octarepeats. When Cu(2+) interaction is studied using a solution of full-length bovine prion protein containing six octarepeats at protein concentrations above 25 microM, a drastic increase in solution turbidity is observed due to the formation of insoluble cation-protein complexes that appear as bidimensional polymer meshes. These bidimensional meshes consist of a single layer of protein molecules crosslinked by Cu(2+) cations. Polymer formation is a cooperative process that proceeds by nucleation of protein molecules with a Cu(2+) site occupancy of above 2. These results support the hypothesis that the N-terminal domain of prion protein is a ligand binding module that promotes crosslinked assembly, and suggest the existence of inter-repeat Cu(2+) sites.     

Cooperative reactions of poly-L-lysine-heme complex with molecular oxygen, carbon monoxide, or cyanide ion.

The interaction of the alpha-helical poly-L-lysine-heme complex with molecular oxygen, carbon monoxide, or cyanide ion was studied. Binding equilibrium curve and activation parameters for the reactions were determined. Sigmoid responses were observed for the absorption of molecular oxygen or carbon monoxide by the complex and the cooperative parameter was found to be 2.1. This indicated a cooperative interaction between hemes situated on a cylindrical alpha-helix of poly-L-lysine. But those of other polymer-ligand-heme complexes were 1.0. The cooperative reaction mechanism, in which an alpha-helical poly-L-lysine plays an important role, was suggested.     

A dimer as a building block in assembling RNA. A hexamer that gears bacterial virus phi29 DNA-translocating machinery

Six RNA (pRNA) molecules form a hexamer, via hand-in-hand interaction, to gear bacterial virus phi29 DNA translocation machinery. Here we report the pathway and the conditions for the hexamer formation. Stable pRNA dimers and trimers were assembled in solution, isolated from native gels, and separated by sedimentation, providing a model system for the study of RNA dimers and trimers in a protein-free environment. Cryo-atomic force microscopy revealed that monomers displayed a check mark outline, dimers exhibited an elongated shape, and trimers formed a triangle. Dimerization of pRNA was promoted by a variety of cations including spermidine, whereas procapsid binding and DNA packaging required specific divalent cations, including Mg(2+), Ca(2+), and Mn(2+). Both the tandem and fused pRNA dimers with complementary loops designed to form even-numbered rings were active in DNA packaging, whereas those without complementary loops were inactive. We conclude that dimers are the building blocks of the hexamer, and the pathway of building a hexamer is: dimer --> tetramer --> hexamer. The Hill coefficient of 2.5 suggests that there are three binding sites with cooperative binding on the surface of the procapsid. The two interacting loops played a key role in recruiting the incoming dimer, whereas the procapsid served as the foundation for hexamer assembly.     

Effects of divalent cations on the ultrasonic absorption coefficient of negatively charged liposomes (LUV) near their phase transition temperature

The ultrasonic absorption coefficient per wavelength (alpha lambda), as a function of temperature and frequency, was determined for large unilamellar vesicles (LUV) in the vicinity of their phospholipid phase transition temperature, using a double crystal acoustic interferometer. (The vesicles were composed of a 4:1 (w/w) mixture of dipalmitoylphosphatidylcholine (DPPC) and dipalmitoylphosphatidylglycerol (DPPG). It has been found that alpha lambda reaches a maximum (alpha lambda)max at the phase transition temperature (tm) of the phospholipids in the bilayer, at an ultrasonic relaxation frequency of 2.1 MHz. Divalent cations (Ca2+ and Mg2+), added to LUV suspensions, shifted (alpha lambda)max to higher temperatures, dependent upon the concentration of divalent cation. It was also found that the shape of the alpha lambda versus t curve was significantly changed, representing changes in the Van't Hoff enthalpy of the transition, and therefore, the cooperative unit of the transition. This suggests that divalent cations interact individually with the negatively charged phospholipid headgroups of DPPG and with DPPC headgroups, thus decreasing the cooperative unit of the transition. The observed upward shift in tm suggests an interaction that increases the activation energy and, therefore, the temperature of the phase transition. However, alpha lambda as a function of frequency did not change with the addition of divalent cations and, thus, the relaxation time of the event responsible for the absorption of ultrasound is not changed by the addition of divalent cations.     

Partitioning of Differently Sized Poly(ethylene glycol)s into OmpF Porin

To understand the physics of polymer equilibrium and dynamics in the confines of ion channel pores, we study partitioning of poly(ethylene glycol)s (PEGs) of different molecular weights into the bacterial porin, OmpF. Thermodynamic and kinetic parameters of partitioning are deduced from the effects of polymer addition on ion currents through single OmpF channels reconstituted into planar lipid bilayer membranes. The equilibrium partition coefficient is inferred from the average reduction of channel conductance in the presence of PEG; rates of polymer exchange between the pore and the bulk are estimated from PEG-induced conductance noise. Partition coefficient as a function of polymer weight is best fitted by a “compressed exponential” with the compression factor of 1.65. This finding demonstrates that PEG partitioning into the OmpF channel pore has sharper dependence on polymer molecular weight than predictions of hard-sphere, random-flight, or scaling models. A 1360-Da polymer separates regimes of partitioning and exclusion. Comparison of its characteristic size with the size of a 2200-Da polymer previously found to separate these regimes for the α-toxin shows good agreement with the x-ray structural data for these channels. The PEG-induced conductance noise is compatible with the polymer mobility reduced inside the OmpF pore by an order of magnitude relatively to its value in bulk solution.     

An oxygenation-sensitive dye binding to Carcinus maenas hemocyanin

The interaction of 4',6-diamidino-2-phenylindole (DAPI) with Carcinus maenas hemocyanin has been investigated by steady state fluorescence, dynamic fluorescence and circular dichroism measurements. The dye binds to apohemocyanin (without copper) as well as to oxygenated hemocyanin and to deoxygenated hemocyanin with very similar affinities (kd approximately equal to 1 microM ) and number of binding sites (one per subunit). In contrast, the fluorescence quantum yield enhancement of DAPI bound to oxygenated hemocyanin is nearly 60% lower than that observed for deoxygenated and apo forms. The decrease of fluorescence of the dye bound to deoxygenated hemocyanin is a sigmoidal function of the oxygen partial pressure, specular to that observed by following the absorbance of the copper-oxygen charge transfer band at 340 nm. This result provides preliminary evidence that DAPI may be used as a functional probe to monitor the cooperative binding of oxygen to the protein. The higher fluorescence quantum yield of DAPI bound to either apohemocyanin or deoxygenated protein is characterized by a single fluorescence decay with lifetime of about 3 ns, while with the oxygenated protein two components of about 1 ns and 3.0 ns are observed. This result is interpreted assuming the existence of two rotamers of DAPI in solution (Szabo et al. Photochem. Photobiol. 44 (1986) 143-150) both able to interact with oxygenated hemocyanin but only one to deoxygenated and apo forms. We conclude that the different fluorescence behaviour of the dye induced by the presence of oxygen bound to the protein is probably due to a structural change of hemocyanin in cooperative interaction with oxygen. Furthermore, the interaction is confirmed by the induced negative ellipticity of DAPI bound to apohemocyanin and deoxy-hemocyanin and by the increase of fluorescence anisotropy of DAPI bound to all forms of protein investigated.     

Binding under a molecular ‘umbrella’ as a cooperative statistical mechanical system: tropomyosin-actin-myosin as an example

We sludy models in which initial ligand molecules can bind on a linear array of sites only by lifting or moving an adjacent polymer molecule, over a range of a few binding sites, into a state of higher free eneragy. This inhibits the initial binding but subsequent ligand molecules can then bind more casily under the ‘umbrella’ already lifted in the initial binding (a cooperative effect). Three subjects are considered: cooperative properties of a general model in which, besides the ‘umbrella’ effect, the polymer is divided into m-site long segments. with interactions between the ends of these segments (section 3): application of this model to tropomyosin-troponin-myosin (section 2): and cooperative properties of simpler models in which the polymer is not divided into segments though the ‘umbrella’ can have various ranges (section 4).     

Equilibrium binding of carcinogens and antitumor antibiotics to DNA: site selectivity, cooperativity, allosterism.

The equilibrium binding of the carcinogens N-hydroxy-N-acetyl-2-amino-fluorene (HAAF) and 4-nitroquinoline-1-oxide (NQO) to phi X174RF DNA have been studied by phase partition techniques. Both molecules bind in a cooperative manner with only a few carcinogen molecules binding to each phi X174RF DNA molecule. The binding data for both HAAF and NQO fit a model in which two carcinogens cluster into a small number of sites--four sites for HAAF and twelve sites for NQO. Phase partition techniques were also used to study the binding of actinomycin D to both calf thymus DNA and poly (dG-dC) . poly (dG-dC) at much lower r values than had been previously reported. These data exhibit humped Scatchard plots which are indicative of cooperative binding; the overall shape of the Scatchard plots are consistent with a model for drug induced allosteric transitions in the DNA structure. The cooperativity in the actinomycin D binding to calf thymus DNA increases with decreasing sodium chloride concentration, suggesting a role for DNA flexibility in allosteric binding.