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.

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 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.

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.

Analysis of variable (diversity) joining recombination in DNAdependent protein kinase (DNA-PK)-deficient mice reveals DNA-PK-independent pathways for both signal and coding joint formation

Previous studies have suggested that ionizing radiation causes irreparable DNA double-strand breaks in mice and cell lines harboring mutations in any of the three subunits of DNA-dependent protein kinase (DNA-PK) (the catalytic subunit, DNA-PKcs, or one of the DNA-binding subunits, Ku70 or Ku86). In actuality, these mutants vary in their ability to resolve double-strand breaks generated during variable (diversity) joining [V(D)J] recombination. Mutant cell lines and mice with targeted deletions in Ku70 or Ku86 are severely compromised in their ability to form coding and signal joints, the products of V(D)J recombination. It is noteworthy, however, that severe combined immunodeficient (SCID) mice, which bear a nonnull mutation in DNA-PKcs, are substantially less impaired in forming signal joints than coding joints. The current view holds that the defective protein encoded by the murine SCID allele retains enough residual function to support signal joint formation. An alternative hypothesis proposes that DNA-PKcs and Ku perform different roles in V(D)J recombination, with DNA-PKcs required only for coding joint formation. To resolve this issue, we examined V(D)J recombination in DNA-PKcs-deficient (SLIP) mice. We found that the effects of this mutation on coding and signal joint formation are identical to the effects of the SCID mutation. Signal joints are formed at levels 10-fold lower than in wild type, and one-half of these joints are aberrant. These data are incompatible with the notion that signal joint formation in SCID mice results from residual DNA-PKcs function, and suggest a third possibility: that DNA-PKcs normally plays an important but nonessential role in signal joint formation.

Genetic evidence for the involvement of DNA ligase IV in the DNA-PK-dependent pathway of non-homologous end joining in mammalian cells

Cells of vertebrates remove DNA double-strand breaks (DSBs) from their genome predominantly utilizing a fast, DNA-PKcs-dependent form of non-homologous end joining (D-NHEJ). Mutants with inactive DNA-PKcs remove the majority of DNA DSBs utilizing a slow, DNA-PKcs-independent pathway that does not utilize genes of the RAD52 epistasis group, is error-prone and can therefore be classified as a form of NHEJ (termed basic or B-NHEJ). We studied the role of DNA ligase IV in these pathways of NHEJ. Although biochemical studies show physical and functional interactions between the DNA-PKcs/Ku and the DNA ligase IV/Xrcc4 complexes suggesting operation within the same pathway, genetic evidence to support this notion is lacking in mammalian cells. Primary human fibroblasts (180BR) with an inactivating mutation in DNA ligase IV, rejoined DNA DSBs predominantly with slow kinetics similar to those observed in cells deficient in DNA-PKcs, or in wild-type cells treated with wortmannin to inactivate DNA-PK. Treatment of 180BR cells with wortmannin had only a small effect on DNA DSB rejoining and no effect on cell radiosensitivity to killing although it sensitized control cells to 180BR levels. This is consistent with DNA ligase IV functioning as a component of the D-NHEJ, and demonstrates the unperturbed operation of the DNA-PKcs-independent pathway (B-NHEJ) at significantly reduced levels of DNA ligase IV. In vitro, extracts of 180BR cells supported end joining of restriction endonuclease-digested plasmid to the same degree as extracts of control cells when tested at 10 mM Mg2+. At 0.5 mM Mg2+, where only DNA ligase IV is expected to retain activity, low levels of end joining (∼10% of 10 mM) were seen in the control but there was no detectable activity in 180BR cells. Antibodies raised against DNA ligase IV did not measurably inhibit end joining at 10 mM Mg2+ in either cell line. Thus, in contrast to the situation in vivo, end joining in vitro is dominated by pathways with properties similar to B-NHEJ that do not display a strong dependence on DNA ligase IV, with D-NHEJ retaining only a limited contribution. The implications of these observations to studies of NHEJ in vivo and in vitro are discussed.

Ultrafast diffraction and structural dynamics: The nature of complex molecules far from equilibrium

Studies of molecular structures at or near their equilibrium configurations have long provided information on their geometry in terms of bond distances and angles. Far-from-equilibrium structures are relatively unknown—especially for complex systems—and generally, neither their dynamics nor their average geometries can be extrapolated from equilibrium values. For such nonequilibrium structures, vibrational amplitudes and bond distances play a central role in phenomena such as energy redistribution and chemical reactivity. Ultrafast electron diffraction, which was developed to study transient molecular structures, provides a direct method for probing the nature of complex molecules far from equilibrium. Here we present our ultrafast electron diffraction observations of transient structures for two cyclic hydrocarbons. At high internal energies of ≈4 eV, these molecules display markedly different behavior. For 1,3,5-cycloheptatriene, excitation results in the formation of hot ground-state structures with bond distances similar to those of the initial structure, but with nearly three times the average vibrational amplitude. Energy is redistributed within 5 ps, but with a negative temperature characterizing the nonequilibrium population. In contrast, the ring-opening reaction of 1,3-cyclohexadiene is shown to result in hot structures with a C—C bond distance of over 1.7 Å, which is 0.2 Å away from any expected equilibrium value. Even up to 400 ps, energy remains trapped in large-amplitude motions comprised of torsion and asymmetric stretching. These studies promise a new direction for studying structural dynamics in nonequilibrium complex systems.

Partial molar volume, surface area, and hydration changes for equilibrium unfolding and formation of aggregation transition state: High-pressure and cosolute studies on recombinant human IFN-γ

The equilibrium dissociation of recombinant human IFN-γ was monitored as a function of pressure and sucrose concentration. The partial molar volume change for dissociation was −209 ± 13 ml/mol of dimer. The specific molar surface area change for dissociation was 12.7 ± 1.6 nm2/molecule of dimer. The first-order aggregation rate of recombinant human IFN-γ in 0.45 M guanidine hydrochloride was studied as a function of sucrose concentration and pressure. Aggregation proceeded through a transition-state species, N*. Sucrose reduced aggregation rate by shifting the equilibrium between native state (N) and N* toward the more compact N. Pressure increased aggregation rate through increased solvation of the protein, which exposes more surface area, thus shifting the equilibrium away from N toward N*. The changes in partial molar volume and specific molar surface area between the N* and N were −41 ± 9 ml/mol of dimer and 3.5 ± 0.2 nm2/molecule, respectively. Thus, the structural change required for the formation of the transition state for aggregation is small relative to the difference between N and the dissociated state. Changes in waters of hydration were estimated from both specific molar surface area and partial molar volume data. From partial molar volume data, estimates were 25 and 128 mol H2O/mol dimer for formation of the aggregation transition state and for dissociation, respectively. From surface area data, estimates were 27 and 98 mol H2O/mol dimer. Osmotic stress theory yielded values ≈4-fold larger for both transitions.

Electric-field-induced Schiff-base deprotonation in D85N mutant bacteriorhodopsin.

The application of an external electric field to dry films of Asp-85-->Asn mutant bacteriorhodopsin causes deprotonation of the Schiff base, resulting in a shift of the optical absorption maximum from 600 nm to 400 nm. This is in marked contrast to the case of wild-type bacteriorhodopsin films, in which electric fields produce a red-shifted product whose optical properties are similar to those of the acid-blue form of the protein. This difference is due to the much weaker binding of the Schiff-base proton in the mutant protein, as indicated by its low pK of approximately 9, as compared with the value pK approximately 13 in the wild type. Other bacteriorhodopsins with lowered Schiff-base pK values should also exhibit a field-induced shift in the protonation equilibrium of the Schiff base. We propose mechanisms to account for these observations.

Phosphorylation by protein kinase C of serine-23 of the alpha-1 subunit of rat Na+,K(+)-ATPase affects its conformational equilibrium.

Phosphorylation of the alpha-1 subunit of rat Na+,K(+)-ATPase by protein kinase C has been shown previously to decrease the activity of the enzyme in vitro. We have now undertaken an investigation of the mechanism by which this inhibition occurs. Analysis of the phosphorylation of recombinant glutathione S-transferase fusion proteins containing putative cytoplasmic domains of the protein, site-directed mutagenesis, and two-dimensional peptide mapping indicated that protein kinase C phosphorylated the alpha-1 subunit of the rat Na+,K(+)-ATPase within the extreme NH2-terminal domain, on serine-23. The phosphorylation of this residue resulted in a shift in the equilibrium toward the E1 form, as measured by eosin fluorescence studies, and this was associated with a decrease in the apparent K+ affinity of the enzyme, as measured by ATPase activity assays. The rate of transition from E2 to E1 was apparently unaffected by phosphorylation by protein kinase C. These results, together with previous studies that examined the effects of tryptic digestion of Na+,K(+)-ATPase, suggest that the NH2-terminal domain of the alpha-1 subunit, including serine-23, is involved in regulating the activity of the enzyme.

Protonatable residues at the cytoplasmic end of transmembrane helix-2 in the signal transducer HtrI control photochemistry and function of sensory rhodopsin I.

Neutral residue replacements were made of 21 acidic and basic residues within the N-terminal half of the Halobacterium salinarium signal transducer HtrI [the halobacterial transducer for sensory rhodopsin I (SRI)] by site-specific mutagenesis. The replacements are all within the region of HtrI that we previously concluded from deletion analysis to contain sites of interaction with the phototaxis receptor SRI. Immunoblotting shows plasmid expression of the htrI-sopI operon containing the mutations produces SRI and mutant HtrI in cells at near wild-type levels. Six of the HtrI mutations perturb photochemical kinetics of SRI and one reverses the phototaxis response. Substitution with neutral amino acids of Asp-86, Glu-87, and Glu-108 accelerate, and of Arg-70, Arg-84, and Arg-99 retard, the SRI photocycle. Opposite effects on photocycle rate cancel in double mutants containing one replaced acidic and one replaced basic residue. Laser flash spectroscopy shows the kinetic perturbations are due to alteration of the rate of reprotonation of the retinylidene Schiff base. All of these mutations permit normal attractant and repellent signaling. On the other hand, the substitution of Glu-56 with the isosteric glutamine converts the normally attractant effect of orange light to a repellent signal in vivo at neutral pH (inverted signaling). Low pH corrects the inversion due to Glu-56 -> Gln and the apparent pK of the inversion is increased when arginine is substituted at position 56. The results indicate that the cytoplasmic end of transmembrane helix-2 and the initial part of the cytoplasmic domain contain interaction sites with SRI. To explain these and previous results, we propose a model in which (i) the HtrI region identified here forms part of an electrostatic bonding network that extends through the SRI protein and includes its photoactive site; (ii) alteration of this network by photoisomerization-induced Schiff base deprotonation and reprotonation shifts HtrI between attractant and repellent conformations; and (iii) HtrI mutations and extracellular pH alter the equilibrium ratios of these conformations.