A statistical mechanical model for β-hairpin kinetics

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a β-hairpin, the minimal structural element of the antiparallel β-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, β-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 215 possible conformations and 15 × 215 possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the β-hairpin peptide.

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.

Exploring the origins of topological frustration: Design of a minimally frustrated model of fragment B of protein A

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This Gō-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht φ values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate φ, whereas highly frustrated systems display a bimodal distribution peaked at low and high φ values. The distribution of φ values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at φ = 0.3 with tails extending to low and high φ values, indicating the presence of a small amount of topological frustration. The contacts with high φ values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of φ values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

RNA folding kinetics regulates translation of phage MS2 maturation gene

The gene for the maturation protein of the single-stranded RNA coliphage MS2 is preceded by an untranslated leader of 130 nt, which folds into a cloverleaf, i.e., three stem–loop structures enclosed by a long distance interaction (LDI). This LDI prevents translation because its 3′ moiety contains the Shine–Dalgarno sequence of the maturation gene. Previously, several observations suggested that folding of the cloverleaf is kinetically delayed, providing a time window for ribosomes to access the RNA. Here we present direct evidence for this model. In vitro experiments show that ribosome binding to the maturation gene is faster than refolding of the denatured cloverleaf. This folding delay appears related to special properties of the leader sequence. We have replaced the three stem–loop structures by a single five nt loop. This change does not affect the equilibrium structure of the LDI. Nevertheless, in this construct, the folding delay has virtually disappeared, suggesting that now the RNA folds faster than ribosomes can bind. Perturbation of the cloverleaf by an insertion makes the maturation start permanently accessible. A pseudorevertant that evolved from an infectious clone carrying the insertion had overcome this defect. It showed a wild-type folding delay before closing down the maturation gene. This experiment reveals the biological significance of retarded cloverleaf formation.

From residue matching patterns to protein folding topographies: General model and bovine pancreatic trypsin inhibitor

A coarse-grained model for protein-folding dynamics is introduced based on a discretized representation of torsional modes. The model, based on the Ramachandran map of the local torsional potential surface and the class (hydrophobic/polar/neutral) of each residue, recognizes patterns of both torsional conformations and hydrophobic-polar contacts, with tolerance for imperfect patterns. It incorporates empirical rates for formation of secondary and tertiary structure. The method yields a topological representation of the evolving local torsional configuration of the folding protein, modulo the basins of the Ramachandran map. The folding process is modeled as a sequence of transitions from one contact pattern to another, as the torsional patterns evolve. We test the model by applying it to the folding process of bovine pancreatic trypsin inhibitor, obtaining a kinetic description of the transitions between the contact patterns visited by the protein along the dominant folding pathway. The kinetics and detailed balance make it possible to invert the result to obtain a coarse topographic description of the potential energy surface along the dominant folding pathway, in effect to go backward or forward between a topological representation of the chain conformation and a topographical description of the potential energy surface governing the folding process. As a result, the strong structure-seeking character of bovine pancreatic trypsin inhibitor and the principal features of its folding pathway are reproduced in a reasonably quantitative way.

Pressure-induced protein-folding/unfolding kinetics

We use an off-lattice minimalist model to describe the effects of pressure in slowing down the folding/unfolding kinetics of proteins when subjected to increasingly larger pressures. The potential energy function used to describe the interactions between beads in the model includes the effects of pressure on the pairwise interaction of hydrophobic groups in water. We show that pressure affects the participation of contacts in the transition state. More significantly, pressure exponentially decreases the chain reconfigurational diffusion coefficient. These results are consistent with experimental results on the kinetics of pressure-denaturation of staphylococcal nuclease.

Magnesium-dependent folding of self-splicing RNA: Exploring the link between cooperativity, thermodynamics, and kinetics

Folding of the Tetrahymena self-splicing RNA into its active conformation involves a set of discrete intermediate states. The Mg2+-dependent equilibrium transition from the intermediates to the native structure is more cooperative than the formation of the intermediates from the unfolded states. We show that the degree of cooperativity is linked to the free energy of each transition and that the rate of the slow transition from the intermediates to the native state decreases exponentially with increasing Mg2+ concentration. Monovalent salts, which stabilize the folded RNA nonspecifically, induce states that fold in less than 30 s after Mg2+ is added to the RNA. A simple model is proposed that predicts the folding kinetics from the Mg2+-dependent change in the relative stabilities of the intermediate and native states.

How does a β-hairpin fold/unfold? Competition between topology and heterogeneity in a solvable model

We study the competition between topological effects and sequence inhomogeneities in determining the thermodynamics and the un/folding kinetics of a β-hairpin. Our work utilizes a new exactly solvable model that allows for arbitrary configurations of native contacts. In general, the competition between heterogeneity and topology results in a crossover of the dominant transition state. Interestingly, near this crossover, the single reaction coordinate picture can be seriously misleading. Our results also suggest that inferring the folding pathway from unfolding simulations is not always justified.

Genetically altered AMPA-type glutamate receptor kinetics in interneurons disrupt long-range synchrony of gamma oscillation

Gamma oscillations synchronized between distant neuronal populations may be critical for binding together brain regions devoted to common processing tasks. Network modeling predicts that such synchrony depends in part on the fast time course of excitatory postsynaptic potentials (EPSPs) in interneurons, and that even moderate slowing of this time course will disrupt synchrony. We generated mice with slowed interneuron EPSPs by gene targeting, in which the gene encoding the 67-kDa form of glutamic acid decarboxylase (GAD67) was altered to drive expression of the α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) glutamate receptor subunit GluR-B. GluR-B is a determinant of the relatively slow EPSPs in excitatory neurons and is normally expressed at low levels in γ-aminobutyric acid (GABA)ergic interneurons, but at high levels in the GAD-GluR-B mice. In both wild-type and GAD-GluR-B mice, tetanic stimuli evoked gamma oscillations that were indistinguishable in local field potential recordings. Remarkably, however, oscillation synchrony between spatially separated sites was severely disrupted in the mutant, in association with changes in interneuron firing patterns. The congruence between mouse and model suggests that the rapid time course of AMPA receptor-mediated EPSPs in interneurons might serve to allow gamma oscillations to synchronize over distance.

Non-Arrhenius kinetics for the loop closure of a DNA hairpin

Intramolecular chain diffusion is an elementary process in the conformational fluctuations of the DNA hairpin-loop. We have studied the temperature and viscosity dependence of a model DNA hairpin-loop by FRET (fluorescence resonance energy transfer) fluctuation spectroscopy (FRETfs). Apparent thermodynamic parameters were obtained by analyzing the correlation amplitude through a two-state model and are consistent with steady-state fluorescence measurements. The kinetics of closing the loop show non-Arrhenius behavior, in agreement with theoretical prediction and other experimental measurements on peptide folding. The fluctuation rates show a fractional power dependence (β = 0.83) on the solution viscosity. A much slower intrachain diffusion coefficient in comparison to that of polypeptides was derived based on the first passage time theory of SSS [Szabo, A., Schulten, K. & Schulten, Z. (1980) J. Chem. Phys. 72, 4350–4357], suggesting that intrachain interactions, especially stacking interaction in the loop, might increase the roughness of the free energy surface of the DNA hairpin-loop.

Quantitative ER ↔ Golgi Transport Kinetics and Protein Separation upon Golgi Exit Revealed by Vesicular Integral Membrane Protein 36 Dynamics in Live CellsV⃞

To quantitatively investigate the trafficking of the transmembrane lectin VIP36 and its relation to cargo-containing transport carriers (TCs), we analyzed a C-terminal fluorescent-protein (FP) fusion, VIP36-SP-FP. When expressed at moderate levels, VIP36-SP-FP localized to the endoplasmic reticulum, Golgi apparatus, and intermediate transport structures, and colocalized with epitope-tagged VIP36. Temperature shift and pharmacological experiments indicated VIP36-SP-FP recycled in the early secretory pathway, exhibiting trafficking representative of a class of transmembrane cargo receptors, including the closely related lectin ERGIC53. VIP36-SP-FP trafficking structures comprised tubules and globular elements, which translocated in a saltatory manner. Simultaneous visualization of anterograde secretory cargo and VIP36-SP-FP indicated that the globular structures were pre-Golgi carriers, and that VIP36-SP-FP segregated from cargo within the Golgi and was not included in post-Golgi TCs. Organelle-specific bleach experiments directly measured the exchange of VIP36-SP-FP between the Golgi and endoplasmic reticulum (ER). Fitting a two-compartment model to the recovery data predicted first order rate constants of 1.22 ± 0.44%/min for ER → Golgi, and 7.68 ± 1.94%/min for Golgi → ER transport, revealing a half-time of 113 ± 70 min for leaving the ER and 1.67 ± 0.45 min for leaving the Golgi, and accounting for the measured steady-state distribution of VIP36-SP-FP (13% Golgi/87% ER). Perturbing transport with AlF4− treatment altered VIP36-SP-GFP distribution and changed the rate constants. The parameters of the model suggest that relatively small differences in the first order rate constants, perhaps manifested in subtle differences in the tendency to enter distinct TCs, result in large differences in the steady-state localization of secretory components.

Kinetics and mechanism of DNA uptake into the cell nucleus

Gene transfer to eukaryotic cells requires the uptake of exogenous DNA into the cell nucleus. Except during mitosis, molecular access to the nuclear interior is limited to passage through the nuclear pores. Here we demonstrate the nuclear uptake of extended linear DNA molecules by a combination of fluorescence microscopy and single-molecule manipulation techniques, using the latter to follow uptake kinetics of individual molecules in real time. The assays were carried out on nuclei reconstituted in vitro from extracts of Xenopus eggs, which provide both a complete complement of biochemical factors involved in nuclear protein import, and unobstructed access to the nuclear pores. We find that uptake of DNA is independent of ATP or GTP hydrolysis, but is blocked by wheat germ agglutinin. The kinetics are much slower than would be expected from hydrodynamic considerations. A fit of the data to a simple model suggests femto-Newton forces and a large friction relevant to the uptake process.

Magnetic resonance microscopy and spectroscopy reveal kinetics of cryoprotectant permeation in a multicompartmental biological system.

Successful cryopreservation of most multicompartmental biological systems has not been achieved. One prerequisite for success is quantitative information on cryoprotectant permeation into and amongst the compartments. This report describes direct measurements of cryoprotectant permeation into a multicompartmental system using chemical shift selective magnetic resonance (MR) microscopy and MR spectroscopy. We used the developing zebrafish embryo as a model for studying these complex systems because these embryos are composed of two membrane-limited compartments: (i) a large yolk (surrounded by the yolk syncytial layer) and (ii) differentiating blastoderm cells (each surrounded by a plasma membrane). MR images of the spatial distribution of three cryoprotectants (dimethyl sulfoxide, propylene glycol, and methanol) demonstrated that methanol permeated the entire embryo within 15 min. In contrast, the other cryoprotectants exhibited little or no permeation over 2.5 h. MR spectroscopy and microinjections of cryoprotectants into the yolk inferred that the yolk syncytial layer plays a critical role in limiting the permeation of some cryoprotectants throughout the embryo. This study demonstrates the power of MR technology combined with micromanipulation for elucidating key physiological factors in cryobiology.

Kinetics of self-assembling microtubules: an "inverse problem" in biochemistry.

Experimental time series for a nonequilibrium reaction may in some cases contain sufficient data to determine a unique kinetic model for the reaction by a systematic mathematical analysis. As an example, a kinetic model for the self-assembly of microtubules is derived here from turbidity time series for solutions in which microtubules assemble. The model may be seen as a generalization of Oosawa's classical nucleation-polymerization model. It reproduces the experimental data with a four-stage nucleation process and a critical nucleus of 15 monomers.

A general two-process model describes the hydrogen exchange behavior of RNase A in unfolding conditions.

When NMR hydrogen exchange was used previously to monitor the kinetics of RNase A unfolding, some peptide NH protons were found to show EX2 exchange (detected by base catalysis) in addition to the expected EX1 exchange, whose rate is limited by the kinetic unfolding process. In earlier work, two groups showed independently that a restricted two-process model successfully fits published hydrogen exchange rates of native RNase A in the range 0-0.7 M guanidinium chloride. We find that this model predicts properties that are very different from the observed properties of the EX2 exchange reactions of RNase A in conditions where guanidine-induced unfolding takes place. The model predicts that EX2 exchange should be too fast to measure by the technique used, whereas it is readily measurable. Possible explanations for the contradiction are considered here, and we show that removing the restriction from the earlier two-process model is sufficient to resolve the contradiction; instead of specifying that exchange caused by global unfolding occurs by the EX2 mechanism, we allow it to occur by the general mechanism, which includes both the EX1 and EX2 cases. It is logical to remove this restriction because global unfolding of RNase A is known to give rise to EX1 exchange in these unfolding conditions. Resolving the contradiction makes it possible to determine whether populated unfolding intermediates contribute to the EX2 exchange, and this question is considered elsewhere. The results and simulations indicate that moderate or high denaturant concentrations readily give rise to EX1 exchange in native proteins. Earlier studies showed that hydrogen exchange in native proteins typically occurs by the EX2 mechanism but that high temperatures or pH values above 7 may give rise to EX1 exchange. High denaturant concentrations should be added to the list of variables likely to cause EX1 exchange.