Two-state Equilibrium Model Research Articles

Observation of broken detailed balance in polymorphic transformation of bacterial flagellar filament

Living systems operate far from thermodynamic equilibrium, which usually manifests as broken detailed balance at the molecular scale. At larger scales with collective function of many molecules, the presence of non-equilibrium thermodynamics may not be evident. In bacterial motility, the switching dynamics of the flagellar rotary motor was recently discovered to be operating in non-equilibrium. However, the resulting motility pattern at the mesoscale, the run-and-tumble behavior, was normally considered to be a Poisson process that can be described by a two-state equilibrium model. Here, we studied the details of the run-and-tumble behavior by following the polymorphic transformation of the flagellar filaments, observing broken detailed balance that reveals its non-equilibrium nature. Evaluation of entropy production provided a direct measure of the lack of detailed balance and a quantification of the rate of energy dissipation for bacterial run-and-tumble regulation.

Ionic Strength-Mediated Phase Transitions of Surface-Adsorbed DNA on Single-Walled Carbon Nanotubes

Single-stranded DNA oligonucleotides have unique, and in some cases sequence-specific molecular interactions with the surface of carbon nanotubes that remain the subject of fundamental study. In this work, we observe and analyze a generic, ionic strength-mediated phase transition exhibited by over 25 distinct oligonucleotides adsorbed to single-walled carbon nanotubes (SWCNTs) in colloidal suspension. The phase transition occurs as monovalent salts are used to modify the ionic strength from 500 mM to 1 mM, causing a reversible reduction in the fluorescence quantum yield by as much as 90%. The phase transition is only observable by fluorescence quenching within a window of pH and in the presence of dissolved O2, but occurs independently of this optical quenching. The negatively charged phosphate backbone increases (decreases) the DNA surface coverage on an areal basis at high (low) ionic strength, and is well described by a two-state equilibrium model. The resulting quantitative model is able to describe and link, for the first time, the observed changes in optical properties of DNA-wrapped SWCNTs with ionic strength, pH, adsorbed O2, and ascorbic acid. Cytosine nucleobases are shown to alter the adhesion of the DNA to SWCNTs through direct protonation from solution, decreasing the driving force for this phase transition. We show that the phase transition also changes the observed SWCNT corona phase, modulating the recognition of riboflavin. These results provide insight into the unique molecular interactions between DNA and the SWCNT surface, and have implications for molecular sensing, assembly, and nanoparticle separations.

Ionic Strength-Mediated Phase Transitions of Surface-Adsorbed DNA on Single-Walled Carbon Nanotubes.

Single-stranded DNA oligonucleotides have unique, and in some cases sequence-specific molecular interactions with the surface of carbon nanotubes that remain the subject of fundamental study. In this work, we observe and analyze a generic, ionic strength-mediated phase transition exhibited by over 25 distinct oligonucleotides adsorbed to single-walled carbon nanotubes (SWCNTs) in colloidal suspension. The phase transition occurs as monovalent salts are used to modify the ionic strength from 500 mM to 1 mM, causing a reversible reduction in the fluorescence quantum yield by as much as 90%. The phase transition is only observable by fluorescence quenching within a window of pH and in the presence of dissolved O2, but occurs independently of this optical quenching. The negatively charged phosphate backbone increases (decreases) the DNA surface coverage on an areal basis at high (low) ionic strength, and is well described by a two-state equilibrium model. The resulting quantitative model is able to describe and link, for the first time, the observed changes in optical properties of DNA-wrapped SWCNTs with ionic strength, pH, adsorbed O2, and ascorbic acid. Cytosine nucleobases are shown to alter the adhesion of the DNA to SWCNTs through direct protonation from solution, decreasing the driving force for this phase transition. We show that the phase transition also changes the observed SWCNT corona phase, modulating the recognition of riboflavin. These results provide insight into the unique molecular interactions between DNA and the SWCNT surface, and have implications for molecular sensing, assembly, and nanoparticle separations.

Probing excited states and activation energy for the integral membrane protein phospholamban by NMR CPMG relaxation dispersion experiments

Phospholamban (PLN) is a dynamic single-pass membrane protein that inhibits the flow of Ca 2+ ions into the sarcoplasmic reticulum (SR) of heart muscle by directly binding to and inhibiting the SR Ca 2+ATPase (SERCA). The PLN monomer is the functionally active form that exists in equilibrium between ordered (T state) and disordered (R state) states. While the T state has been fully characterized using a hybrid solution/solid-state NMR approach, the R state structure has not been fully portrayed. It has, however, been detected by both NMR and EPR experiments in detergent micelles and lipid bilayers. In this work, we quantitatively probed the μs to ms dynamics of the PLN excited states by observing the T state in DPC micelles using CPMG relaxation dispersion NMR spectroscopy under functional conditions for SERCA. The 15N backbone and 13C δ1 Ile-methyl dispersion curves were fit using a two-state equilibrium model, and indicate that residues within domain Ia (residues 1–16), the loop (17–22), and domain Ib (23–30) of PLN undergo μs-ms dynamics ( k ex = 6100 ±800 s - 1 at 17 °C). We measured k ex at additional temperatures, which allowed for a calculation of activation energy equal to ∼ 5 kcal/mol. This energy barrier probably does not correspond to the detachment of the amphipathic domain Ia, but rather the energy needed to unwind domain Ib on the membrane surface, likely an important mechanism by which PLN converts between high and low affinity states for its binding partners.

RNA 1997–2007: A Remarkable Decade of Discovery

The past decade has witnessed spectacular progress in the field of RNA biology and biochemistry; old problems have been solved, and new ones have emerged. This perspective briefly reviews where we are today and where we might be 10 years hence.

Crystal Structure of a Complex between the Phosphorelay Protein YPD1 and the Response Regulator Domain of SLN1 Bound to a Phosphoryl Analog

The crystal structure of the yeast SLN1 response regulator (RR) domain bound to both a phosphoryl analog [beryllium fluoride (BeF 3 − )] and Mg 2 + , in complex with its downstream phosphorelay signaling partner YPD1, has been determined at a resolution of 1.70 Å. Comparisons between the BeF 3 − -activated complex and the unliganded (or apo) complex determined previously reveal modest but important differences. The SLN1-R1·Mg 2 + ·BeF 3 − structure from the complex provides evidence for the first time that the mechanism of phosphorylation-induced activation is highly conserved between bacterial RR domains and this example from a eukaryotic organism. Residues in and around the active site undergo slight rearrangements in order to form bonds with the essential divalent cation and fluorine atoms of BeF 3 − . Two conserved switch-like residues (Thr1173 and Phe1192) occupy distinctly different positions in the apo versus BeF 3 − -bound structures, consistent with the “Y–T” coupling mechanism proposed for the activation of CheY and other bacterial RRs. Several loop regions and the α4–β5–α5 surface of the SLN1-R1 domain undergo subtle conformational changes (∼ 1–3 Å displacements relative to the apo structure) that lead to significant changes in terms of contacts that are formed with YPD1. Detailed structural comparisons of protein–protein interactions in the apo and BeF 3 − -bound complexes suggest at least a two-state equilibrium model for the formation of a transient encounter complex, in which phosphorylation of the RR promotes the formation of a phosphotransfer-competent complex. In the BeF 3 − -activated complex, the position of His64 from YPD1 needs to be within ideal distance of and in near-linear geometry with Asp1144 from the SLN1-R1 domain for phosphotransfer to occur. The ground-state structure presented here suggests that phosphoryl transfer will likely proceed through an associative mechanism involving the formation of a pentacoordinate phosphorus intermediate.

Fluorescent behaviour in host–guest interactions. Part 2.For Part 1 see ref. 16(b). Thermal and pH-dependent sensing properties of two geometric isomers of fluorescent amino-β-cyclodextrin derivatives

The fluorescent behaviour in aqueous solution of two types of amino-β-cyclodextrin (amino-β-CDx) isomers 1 and 2 bearing the naphthoamide group at the 1- and 2-position as a fluorescent-sensing unit was investigated using mainly fluorescence spectroscopy. The isomers 1 and 2 show both strong temperature- and pH-responsive photophysical properties in aqueous solution. The protonation of the amino groups of 1 and 2 in acidic media affects efficiently their fluorescence intensities. The equilibrium and thermodynamic quantities of the two-state equilibrium model for 1 and 2 have been determined spectrophotometrically. Solution NMR behaviour in a DMSO–D2O mixed solvent confirms the stereospecific relationship between the appended naphthalene fluorophore and the hydrophobic cavity of CDx. Restricted and/or flexible motion of the naphthalene moiety of 1 and 2 about the bond between the amide group and the amino group at the C-6 carbon atom in the CDx ring is evident from the NMR studies. The potential application of 1 to anion sensing is also discussed.

The Weighting Game: Formula Apportionment as an Instrument of Public Policy

We propose an explanation for why states choose differ- ent apportionment formulas for corporate income tax purposes. Based on a two-state equilibrium model of location choice by firms, we show that aggregate social welfare is maximized when both states use the same formula, regardless of which formula is chosen. How- ever, at least one of the states can increase its welfare by deviating from this coordinated solution; thus, the Nash equilibrium features the states choosing different formulas. Importing states have in- centives to increase the sales factor, whereas exporting states will tend to increase the input factors. An empirical test of which states have deviated from the traditionally equally-weighted three factor formula supports the predictions of the model.

Stability and oligosaccharide binding of the N1 cellulose-binding domain of Cellulomonas fimi endoglucanase CenC.

Differential scanning calorimetry has been used to study the thermal stability and oligosaccharide-binding thermodynamics of the N-terminal cellulose-binding domain of Cellulomonas fimi beta-1,4-glucanase CenC (CBDN1). CBDN1 has a relatively low maximum stability (delta Gmax = 33 kJ/mol = 216 J/residue at 1 degree C and pH 6.1) compared to other small single-domain globular proteins. The unfolding is fully reversible between pH 5.5 and 9 and in accordance with the two-state equilibrium model between pH 5.5 and 11. When the single disulfide bond in CBDN1 is reduced, the protein remains unfolded at all conditions, as judged by NMR spectroscopy. This indicates that the intramolecular cross-link makes a major contribution to the stability of CBDN1. The measured heat capacity change of unfolding (delta Cp = 7.5 kJ mol-1 K-1) agrees well with that calculated from the predicted changes in the solvent accessible nonpolar and polar surface areas upon unfolding. Extrapolation of the specific enthalpy and entropy of unfolding to their respective convergence temperature indicates that per residue unfolding energies for CBDN1, an isolated domain, are in accordance with those found by Privalov (1) for many single-domain globular proteins. DSC thermograms of the unfolding of CBDN1 in the presence of various concentrations of cellopentaose were fit to a thermodynamic model describing the linkage between protein-ligand binding and protein unfolding. A global two-dimensional minimization routine is used to regress the binding enthalpy, binding constant, and unfolding thermodynamics for the CBDN1-cellopentaose system. Extrapolated binding constants are in quantitative agreement with those determined by isothermal titration calorimetry at 35 degrees C.

Thermodynamic nonideality as a probe of reversible protein unfolding effected by variations in pH and temperature: Studies of ribonuclease

Thermodynamic nonideality arising from the space-filling effect of added sucrose is employed to confirm that the reversible unfolding of ribonuclease A effected by acid may be described as an equilibrium between native and unfolded states of the enzyme. However, the extent of the volume change is far too small for the larger isomer to be the fully expanded state, a result signifying that the acid-mediated unfolding of ribonuclease does not conform with the two-state equilibrium model of protein denaturation. Although the thermal denaturation of ribonuclease A is characterized by a larger increase in volume, quantitative reappraisal of published results on the effects of glycerol on this transition at pH 2.8 (Gekko, K., and Timasheff, S. N., 1981 Biochemistry 20, 4677–4686) leads to an estimated volume increase that is much smaller than that inferred from hydrodynamic studies—a disparity attributed to the dual actions of glycerol as a space-filling solute and as a ligand that binds preferentially to the thermally unfolded form of the enzyme. Even in this unfavorable circumstance the fact that glycerol exerts a net excluded volume effect at least confirms that the thermal unfolding of ribonuclease A is an equilibrium transition between two discrete states. The strengths and limitations of using thermodynamic nonideality as a probe of the two-state equilibrium model of protein denaturation are discussed in the light of these findings.

B to Z transitions of the short DNA hairpins formed from the oligomer sequences: d[(CG)3×4(CG)3] (X = A, T, G, C)

Circular Dichroism (CD) spectra were collected as a function of sodium perchlorate concentration [NaClO4] for the set of DNA hairpins formed from the oligomer sequences d[(CG)3X4(CG)3] where X = A, T, G or C. Over the range in salt concentration from 0 to 4.0 M NaClO4, the CD spectra invert in a manner characteristic of the B to Z transition. A factor analysis routine is described and employed to determine the least number of basis spectra required to fit the measured spectra of each hairpin over the entire salt range examined. In every case, linear combinations of only two sub-spectra fit the experimental spectra of the hairpins with greater than 98% accuracy, indicating the spectrally monitored structural transitions are two-state. From the relative weights of the individual sub-spectra, B-Z transition curves are constructed. The transitions are analyzed in terms of a simple two-state equilibrium model which yields an evaluation of the transition free-energy, delta GB-Z, as a function of NaClO4 concentration. At 1.0 M NaClO4 and 21 degrees C, delta GB-Z = 5.4, 4.9, 3.6 and 2.3 kcal/mole for the G4, T4, A4 and C4 loop hairpins, respectively.

Hemoglobin - membrane interaction at physiological ionic strength and temperature

The spin-label electron paramagnetic resonance (EPR) technique has been used to study the interaction between human hemoglobin and erythrocyte membranes as a function of temperature and ionic strength. We show, for the first time, experimental evidence for the existence of the interaction at physiological pH, ionic strength and temperature. In addition to the pH dependence that we have previously reported, the interactions are also temperature and ionic strength dependent. Using a simple two-state equilibrium model to analyze the EPR data, we obtain an equilibrium dissociation constant of about 8.1 ± 5.6 × 10 −5 M for hemoglobin-membrane systems in 5 mM phosphate with 150 mM NaCl at pH 7.4 and 37 °C.

Anomalous hairpin formation in an oligodeoxyribonucleotide.

An accurate method for deriving molar absorptivity-temperature profiles applied to a set of single-stranded oligodeoxyribonucleotides shows that the undecamer CGAGTTTGACGp exists in a hairpin conformation involving Watson-Crick base pairing between the two terminal CG dinucleotides. The hairpin, which has a transition midpoint of 40 degrees C in 0.115 M Na+, is unusually stable in comparison with previously reported hairpins. A non-linear least squares analysis of the undecamer's profile in terms of a two-state equilibrium model indicates that the hairpin-to-coil transition occurs with an enthalpy change about twice that expected if only combinations of Watson-Crick base-paired stacking interactions are considered. The analogous hairpin structure (containing an identical CG/CG stem) assignable to the complementary strand CGTCAAACTCGp does not form above 0 degrees C. Measurements on the two undecamers indicate that variation in non Watson-Crick interactions within the loops of two similar hairpins can produce a difference in stability of at least 2.2 kcal/mol (25 degrees C, 0.115 M Na+), roughly equal to the amount contributed to a double helix by a 5'-CG-3'/5'-CG-3' base-paired stacking interaction.

Relaxation kinetics of dimer formation by self complementary oligonucleotides

We report relaxation measurements on the kinetics of dimer formation by self complementary oligoribonucleotides of the form A(pA) n−1 (pU) n , 4 ≤ n ≤ 7. The two strands combine with a bimolecular rate constant of about 2 × 10 6 liters per mole of strands per second. One relaxation accounts for most of the optical change, with a time constant varying from 100 μsec to 1 second, depending on conditions. Additional effects are much faster and arise mainly from unstacking of bases in the separated strands. We interpret the results with a two-state equilibrium model, an all-or-none transition between the dissociated strands and the fully bonded helix. The rate constant for dimerization decreases with increasing temperature. Consequently the kinetic model requires a transient steady-state intermediate with several bases paired before the rate-limiting step of helix growth to the fully bonded dimer. When the temperature is raised, this species melts out and the rate of dimer formation decreases. According to our analysis the critical transient intermediate has 2 or 3 base pairs; longer helices grow more rapidly than they dissociate (at the temperature of our experiments). Careful analysis of the transition curves allows estimation of the concentration of the intermediate, from which we further calculate a rate constant for the helix growth step of between 1 × 10 6 and 2 × 10 7 sec −1. This is the rate constant for adding a base pair to the end of an existing helical segment. The rate constant for dimer dissociation is strongly dependent on oligomer size and on temperature, with an activation energy roughly proportional to chain length.