Equilibrium constants and free energies in unfolding of proteins in urea solutions

A novel thermodynamic approach to the reversible unfolding of proteins in aqueous urea solutions has been developed based on the premise that urea ligands are bound cooperatively to the macromolecule. When successive stoichiometric binding constants have values larger than expected from statistical effects, an equation for moles of bound urea can be derived that contains imaginary terms. For a very steep unfolding curve, one can then show that the fraction of protein unfolded, B̄, depends on the square of the urea concentration, U, and is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\bar {B}=\frac{{\mathit{A}}^{{\mathit{2}}}_{{\mathit{1}}}{\mathit{e}}^{{\mathrm{{\lambda}}}n\bar {B}}{\mathit{U}}^{{\mathit{2}}}}{{\mathrm{1\hspace{.167em}+\hspace{.167em}}}{\mathit{A}}^{{\mathrm{2}}}_{{\mathrm{1}}}{\mathit{e}}^{{\mathrm{{\lambda}}}\bar {B}}{\mathit{U}}^{{\mathrm{2}}}}{\mathrm{.}}\end{equation*}\end{document} A12 is the binding constant as B̄→ 0, and λ is a parameter that reflects the augmentation in affinities of protein for urea as the moles bound increases to the saturation number, n. This equation provides an analytic expression that reproduces the unfolding curve with good precision, suggests a simple linear graphical procedure for evaluating A12 and λ, and leads to the appropriate standard free energy changes. The calculated ΔG° values reflect the coupling of urea binding with unfolding of the protein. Some possible implications of this analysis to protein folding in vivo are described.

Kinetic characterization of brush border myosin-I ATPase

Brush border myosin-I (BBM-I) is a single-headed unconventional myosin found in the microvilli of intestinal epithelial cells. We used stopped-flow kinetic analysis to measure the rate and equilibrium constants for several steps in the BBM-I ATPase cycle. We determined the rates for ATP binding to BBM-I and brush border actomyosin-I (actoBBM-I), the rate of actoBBM-I dissociation by ATP, and the rates for the steps in ADP dissociation from actoBBM-I. The rate and equilibrium constants for several of the steps in the actoBBM-I ATPase are significantly different from those of other members of the myosin superfamily. Most notably, dissociation of the actoBBM-I complex by ATP and release of ADP from actoBBM-I are both very slow. The slow rates of these steps may play a role in lengthening the time spent in force-generating states and in limiting the maximal rate of BBM-I motility. In addition, release of ADP from the actoBBM-I complex occurs in at least two steps. This study provides evidence for a member of the myosin superfamily with markedly divergent kinetic behavior.

Direct interaction of lignin and lignin peroxidase from Phanerochaete chrysosporium

Binding properties of lignin peroxidase (LiP) from the basidiomycete Phanerochaete chrysosporium against a synthetic lignin (dehydrogenated polymerizate, DHP) were studied with a resonant mirror biosensor. Among several ligninolytic enzymes, only LiP specifically binds to DHP. Kinetic analysis revealed that the binding was reversible, and that the dissociation equilibrium constant was 330 μM. The LiP–DHP interaction was controlled by the ionization group with a pKa of 5.3, strongly suggesting that a specific amino acid residue plays a role in lignin binding. A one-electron transfer from DHP to oxidized intermediates LiP compounds I and II (LiPI and LiPII) was characterized by using a stopped-flow technique, showing that binding interactions of DHP with LiPI and LiPII led to saturation kinetics. The dissociation equilibrium constants for LiPI–DHP and LiPII–DHP interactions were calculated to be 350 and 250 μM, and the first-order rate constants for electron transfer from DHP to LiPI and to LiPII were calculated to be 46 and 16 s−1, respectively. These kinetic and spectral studies strongly suggest that LiP is capable of oxidizing lignin directly at the protein surface by a long-range electron transfer process. A close look at the crystal structure suggested that LiP possesses His-239 as a possible lignin-binding site on the surface, which is linked to Asp-238. This Asp residue is hydrogen-bonded to the proximal His-176. This His–Asp⋅⋅⋅proximal-His motif would be a possible electron transfer route to oxidize polymeric lignin.

Hybridization of single-stranded DNA targets to immobilized complementary DNA probes: comparison of hairpin versus linear capture probes

A microtiter-based assay system is described in which DNA hairpin probes with dangling ends and single-stranded, linear DNA probes were immobilized and compared based on their ability to capture single-strand target DNA. Hairpin probes consisted of a 16 bp duplex stem, linked by a T2-biotin·dT-T2 loop. The third base was a biotinylated uracil (UB) necessary for coupling to avidin coated microtiter wells. The capture region of the hairpin was a 3′ dangling end composed of either 16 or 32 bases. Fundamental parameters of the system, such as probe density and avidin adsorption capacity of the plates were characterized. The target DNA consisted of 65 bases whose 3′ end was complementary to the dangling end of the hairpin or to the linear probe sequence. The assay system was employed to measure the time dependence and thermodynamic stability of target hybridization with hairpin and linear probes. Target molecules were labeled with either a 5′-FITC, or radiolabeled with [γ-33P]ATP and captured by either linear or hairpin probes affixed to the solid support. Over the range of target concentrations from 10 to 640 pmol hybridization rates increased with increasing target concentration, but varied for the different probes examined. Hairpin probes displayed higher rates of hybridization and larger equilibrium amounts of captured targets than linear probes. At 25 and 45°C, rates of hybridization were better than twice as great for the hairpin compared with the linear capture probes. Hairpin–target complexes were also more thermodynamically stable. Binding free energies were evaluated from the observed equilibrium constants for complex formation. Results showed the order of stability of the probes to be: hairpins with 32 base dangling ends > hairpin probes with l6 base dangling ends > 16 base linear probes > 32 base linear probes. The physical characteristics of hairpins could offer substantial advantages as nucleic acid capture moieties in solid support based hybridization systems.

Significant discrepancies between van't Hoff and calorimetric enthalpies.

In this paper we show that the usual assumption in studies of the temperature variation of equilibrium constants for equilibria of the form A+B <-->AB that a plot of ln K vs. 1/T (K = equilibrium constant, T = temperature in degrees kelvin) is a straight line with slope equal to -delta HvH/R (delta HvH = van't Hoff or apparent enthalpy, R = gas constant) is not valid in many cases. In all the cases considered here, delta HvH is temperature dependent and is significantly different from the true or calorimetrically measured enthalpy, and the respective values for delta Cp are also significantly different.

Determination of the binding constants of the centromere protein Cbf1 to all 16 centromere DNAs of Saccharomyces cerevisiae

Cbf1p is a Saccharomyces cerevisiae chromatin protein belonging to the basic region helix–loop–helix leucine zipper (bHLHzip) family of DNA binding proteins. Cbf1p binds to a conserved element in the 5′-flanking region of methionine biosynthetic genes and to centromere DNA element I (CDEI) of S.cerevisiae centromeric DNA. We have determined the apparent equilibrium dissociation constants of Cbf1p binding to all 16 CDEI DNAs in gel retardation assays. Binding constants of full-length Cbf1p vary between 1.7 and 3.8 nM. However, the dissociation constants of a Cbf1p deletion variant that has been shown to be fully sufficient for Cbf1p function in vivo vary in a range between 3.2 and 12 nM. In addition, native polyacrylamide gel electrophoresis revealed distinct changes in the 3D structure of the Cbf1p/CEN complexes. We also show that the previously reported DNA binding stimulation activity of the centromere protein p64 functions on both the Cbf1 full-length protein and a deletion variant containing only the bHLHzip domain of Cbf1p. Our results suggest that centromeric DNA outside the consensus CDEI sequence and interaction of Cbf1p with adjacent centromere proteins contribute to the complex formation between Cbf1p and CEN DNA.

Simultaneous determination of helical unwinding angles and intrinsic association constants in ligand–DNA complexes: The interaction between DNA and calichearubicin B

We present a helical unwinding assay for reversibly binding DNA ligands that uses closed circular DNA, topoisomerase I (Topo I), and two-dimensional agarose gel electrophoresis. Serially diluted Topo I relaxation reactions at constant DNA/ligand ratio are performed, and the resulting apparent unwinding of the closed circular DNA is used to calculate both ligand unwinding angle (φ) and intrinsic association constant (Ka). Mathematical treatment of apparent unwinding is formally analogous to that of apparent extinction coefficient data for optical binding titrations. Extrapolation to infinite DNA concentration yields the true unwinding angle of a given ligand and its association constant under Topo I relaxation conditions. Thus this assay delivers simultaneous structural and thermodynamic information describing the ligand–DNA complex. The utility of this assay has been demonstrated by using calichearubicin B (CRB), a synthetic hybrid molecule containing the anthraquinone chromophore of (DA) and the carbohydrate domain of calicheamicin γ1I. The unwinding angle for CRB calculated by this method is −5.3 ± 0.5°. Its Ka value is 0.20 × 106 M−1. For comparison, the unwinding angles of ethidium bromide and DA have been independently calculated, and the results are in agreement with canonical values for these compounds. Although a stronger binder to selected sites, CRB is a less potent unwinder than its parent compound DA. The assay requires only small amounts of ligand and offers an attractive option for analysis of DNA binding by synthetic and natural compounds.

Equilibrium distributions of topological states in circular DNA: Interplay of supercoiling and knotting

Two variables define the topological state of closed double-stranded DNA: the knot type, K, and ΔLk, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(ΔLk, K), is related to two conditional distributions: P(ΔLk|K), the distribution of ΔLk for a particular K, and P(K|ΔLk) and also to two simple distributions: P(ΔLk), the distribution of ΔLk irrespective of K, and P(K). We explored the relationships between these distributions. P(ΔLk, K), P(ΔLk), and P(K|ΔLk) were calculated from the simulated distributions of P(ΔLk|K) and of P(K). The calculated distributions agreed with previous experimental and theoretical results and greatly advanced on them. Our major focus was on P(K|ΔLk), the distribution of knot types for a particular value of ΔLk, which had not been evaluated previously. We found that unknotted circular DNA is not the most probable state beyond small values of ΔLk. Highly chiral knotted DNA has a lower free energy because it has less torsional deformation. Surprisingly, even at |ΔLk| > 12, only one or two knot types dominate the P(K|ΔLk) distribution despite the huge number of knots of comparable complexity. A large fraction of the knots found belong to the small family of torus knots. The relationship between supercoiling and knotting in vivo is discussed.

Effective intercellular communication distances are determined by the relative time constants for cyto/chemokine secretion and diffusion

A cell’s ability to effectively communicate with a neighboring cell is essential for tissue function and ultimately for the organism to which it belongs. One important mode of intercellular communication is the release of soluble cyto- and chemokines. Once secreted, these signaling molecules diffuse through the surrounding medium and eventually bind to neighboring cell’s receptors whereby the signal is received. This mode of communication is governed both by physicochemical transport processes and cellular secretion rates, which in turn are determined by genetic and biochemical processes. The characteristics of transport processes have been known for some time, and information on the genetic and biochemical determinants of cellular function is rapidly growing. Simultaneous quantitative analysis of the two is required to systematically evaluate the nature and limitations of intercellular signaling. The present study uses a solitary cell model to estimate effective communication distances over which a single cell can meaningfully propagate a soluble signal. The analysis reveals that: (i) this process is governed by a single, key, dimensionless group that is a ratio of biological parameters and physicochemical determinants; (ii) this ratio has a maximal value; (iii) for realistic values of the parameters contained in this dimensionless group, it is estimated that the domain that a single cell can effectively communicate in is ≈250 μm in size; and (iv) the communication within this domain takes place in 10–30 minutes. These results have fundamental implications for interpretation of organ physiology and for engineering tissue function ex vivo.

Kinetic equilibrium of forces and molecular events in muscle contraction

In biomolecular systems, the mechanical transfer of free energy occurs with both high efficiency and high speed. It is shown here that such a transfer can be achieved only if the participating free-energy-storing elements exhibit opposing relationships between their content of free energy and the force they exert in the transfer direction. A kinetic equilibrium of forces (KEF) results, in which the transfer of free energy is mediated essentially by thermal molecular motion. On the basis of present evidence, KEF is used as a guiding principle in developing a mechanical model of the crossbridge cycle in muscle contraction. The model allows the basic features of molecular events to be visualized in terms of plausible structures. Real understanding of the process will require identification of the elements that perform the functions described here. Besides chemomechanical energy transduction, KEF may have a role in other biomolecular processes in which free energy is transferred mechanically over large distances.

An equilibrium statistical model for the spreading phase of open-ocean convection

A “most probable state” equilibrium statistical theory for random distributions of hetons in a closed basin is developed here in the context of two-layer quasigeostrophic models for the spreading phase of open-ocean convection. The theory depends only on bulk conserved quantities such as energy, circulation, and the range of values of potential vorticity in each layer. The simplest theory is formulated for a uniform cooling event over the entire basin that triggers a homogeneous random distribution of convective towers. For a small Rossby deformation radius typical for open-ocean convection sites, the most probable states that arise from this theory strongly resemble the saturated baroclinic states of the spreading phase of convection, with a stabilizing barotropic rim current and localized temperature anomaly.

A quantitative method for evaluating the stabilities of nucleic acids

We report a general method for screening, in solution, the impact of deviations from canonical Watson-Crick composition on the thermodynamic stability of nucleic acid duplexes. We demonstrate how fluorescence resonance energy transfer (FRET) can be used to detect directly free energy differences between an initially formed “reference” duplex (usually a Watson-Crick duplex) and a related “test” duplex containing a lesion/alteration of interest (e.g., a mismatch, a modified, a deleted, or a bulged base, etc.). In one application, one titrates into a solution containing a fluorescently labeled, FRET-active, reference duplex, an unlabeled, single-stranded nucleic acid (test strand), which may or may not compete successfully to form a new duplex. When a new duplex forms by strand displacement, it will not exhibit FRET. The resultant titration curve (normalized fluorescence intensity vs. logarithm of test strand concentration) yields a value for the difference in stability (free energy) between the newly formed, test strand-containing duplex and the initial reference duplex. The use of competitive equilibria in this assay allows the measurement of equilibrium association constants that far exceed the magnitudes accessible by conventional titrimetric techniques. Additionally, because of the sensitivity of fluorescence, the method requires several orders of magnitude less material than most other solution methods. We discuss the advantages of this method for detecting and characterizing any modification that alters duplex stability, including, but not limited to, mutagenic lesions. We underscore the wide range of accessible free energy values that can be defined by this method, the applicability of the method in probing for a myriad of nucleic acid variations, such as single nucleotide polymorphisms, and the potential of the method for high throughput screening.

NMR investigation of ferricytochrome c unfolding: Detection of an equilibrium unfolding intermediate and residual structure in the denatured state

Horse ferricytochrome c (cyt c) undergoes exchange of one of its axial heme ligands (Met-80) for one or more non-native ligands under denaturing conditions. We have used 1H NMR spectroscopy to detect two conformations of paramagnetic cyt c with non-native heme ligation through a range of urea concentrations. One non-native form is an equilibrium unfolding intermediate observed under partially denaturing conditions and is attributed to replacement of Met-80 with one or more Lys side chains. The second non-native form, in which the native Met ligand is replaced by a His, is observed under strongly denaturing conditions. Thermodynamic analysis of these data indicates a relatively small ΔG (17 kJ/mol) for the transition from native to the Lys-ligated intermediate and a significantly larger ΔG (47 kJ/mol) for the transition from native to the His-ligated species. Although CD and fluorescence data indicate that the equilibrium unfolding of cyt c is a two-state process, these NMR results implicate an intermediate with His-Lys ligation.

Temperature, ordering, and equilibrium with time-dependent confining forces

The concepts of temperature and equilibrium are not well defined in systems of particles with time-varying external forces. An example is a radio frequency ion trap, with the ions laser cooled into an ordered solid, characteristic of sub-mK temperatures, whereas the kinetic energies associated with the fast coherent motion in the trap are up to 7 orders of magnitude higher. Simulations with 1,000 ions reach equilibrium between the degrees of freedom when only aperiodic displacements (secular motion) are considered. The coupling of the periodic driven motion associated with the confinement to the nonperiodic random motion of the ions is very small at low temperatures and increases quadratically with temperature.

Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations

We describe and test a Markov chain model of microsatellite evolution that can explain the different distributions of microsatellite lengths across different organisms and repeat motifs. Two key features of this model are the dependence of mutation rates on microsatellite length and a mutation process that includes both strand slippage and point mutation events. We compute the stationary distribution of allele lengths under this model and use it to fit DNA data for di-, tri-, and tetranucleotide repeats in humans, mice, fruit flies, and yeast. The best fit results lead to slippage rate estimates that are highest in mice, followed by humans, then yeast, and then fruit flies. Within each organism, the estimates are highest in di-, then tri-, and then tetranucleotide repeats. Our estimates are consistent with experimentally determined mutation rates from other studies. The results suggest that the different length distributions among organisms and repeat motifs can be explained by a simple difference in slippage rates and that selective constraints on length need not be imposed.