“Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

Twist grain boundaries in three-dimensional lamellar Turing structures

Steady spatial self-organization of three-dimensional chemical reaction-diffusion systems is discussed with the emphasis put on the possible defects that may alter the Turing patterns. It is shown that one of the stable defects of a three-dimensional lamellar Turing structure is a twist grain boundary embedding a Scherk minimal surface.

Neuropeptide release by efficient recruitment of diffusing cytoplasmic secretory vesicles

Neuropeptides are slowly released from a limited pool of secretory vesicles. Despite decades of research, the composition of this pool has remained unknown. Endocrine cell studies support the hypothesis that a population of docked vesicles supports the first minutes of hormone release. However, it has been proposed that mobile cytoplasmic vesicles dominate the releasable neuropeptide pool. Here, to determine the cellular basis of the releasable pool, single green fluorescent protein-labeled secretory vesicles were visualized in neuronal growth cones with the use of an inducible construct or total internal reflection fluorescence microscopy. We report that vesicle movement follows the diffusion equation. Furthermore, rapidly moving secretory vesicles are used more efficiently than stationary vesicles near the plasma membrane to support stimulated release. Thus, randomly moving cytoplasmic vesicles participate in the first minutes of neuropeptide release. Importantly, the preferential recruitment of diffusing cytoplasmic secretory vesicles contributes to the characteristic slow kinetics and limited extent of sustained neuropeptide release.

Microtubule self-organization is gravity-dependent

Although weightlessness is known to affect living cells, the manner by which this occurs is unknown. Some reaction-diffusion processes have been theoretically predicted as being gravity-dependent. Microtubules, a major constituent of the cellular cytoskeleton, self-organize in vitro by way of reaction-diffusion processes. To investigate how self-organization depends on gravity, microtubules were assembled under low gravity conditions produced during space flight. Contrary to the samples formed on an in-flight 1 × g centrifuge, the samples prepared in microgravity showed almost no self-organization and were locally disordered.

Theoretical prediction of a crystal structure.

The diffusion equation method of global minimization is applied to compute the crystal structure of S6, with no a priori knowledge about the system. The experimental lattice parameters and positions and orientations of the molecules in the unit cell are predicted correctly.

Ecological chaos in the wake of invasion.

Irregularities in observed population densities have traditionally been attributed to discretization of the underlying dynamics. We propose an alternative explanation by demonstrating the evolution of spatiotemporal chaos in reaction-diffusion models for predator-prey interactions. The chaos is generated naturally in the wake of invasive waves of predators. We discuss in detail the mechanism by which the chaos is generated. By considering a mathematical caricature of the predator-prey models, we go on to explain the dynamical origin of the irregular behavior and to justify our assertion that the behavior we present is a genuine example of spatiotemporal chaos.

A coupled proton-transfer and twisting-motion fluorescence probe for lipid bilayers

A new and sensitive molecular probe, 2-(2′-hydroxyphenyl)imidazo[1,2-a]pyridine (HPIP), for monitoring structural changes in lipid bilayers is presented. Migration of HPIP from water into vesicles involves rupture of hydrogen (H) bonds with water and formation of an internal H bond once the probe is inside the vesicle. These structural changes of the dye allow the occurrence of a photoinduced intramolecular proton-transfer reaction and a subsequent twisting/rotational process upon electronic excitation of the probe. The resulting large Stokes-shifted fluorescence band depends on the twisting motion of the zwitterionic phototautomer and is characterized in vesicles of dimyristoyl-phosphatidylcholine and in dipalmitoyl-phosphatidylcholine at the temperature range of interest and in the presence of cholesterol. Because the fluorescence of aqueous HPIP does not interfere in the emission of the probe within the vesicles, HPIP proton-transfer/twisting motion fluorescence directly allows us to monitor and quantify structural changes within bilayers. The static and dynamic fluorescence parameters are sensitive enough to such changes to suggest this photostable dye as a potential molecular probe of the physical properties of lipid bilayers.

Low frequency vibrational modes in proteins: Changes induced by point-mutations in the protein-cofactor matrix of bacterial reaction centers

As a step toward understanding their functional role, the low frequency vibrational motions (<300 cm−1) that are coupled to optical excitation of the primary donor bacteriochlorophyll cofactors in the reaction center from Rhodobacter sphaeroides were investigated. The pattern of hydrogen-bonding interaction between these bacteriochlorophylls and the surrounding protein was altered in several ways by mutation of single amino acids. The spectrum of low frequency vibrational modes identified by femtosecond coherence spectroscopy varied strongly between the different reaction center complexes, including between different mutants where the pattern of hydrogen bonds was the same. It is argued that these variations are primarily due to changes in the nature of the individual modes, rather than to changes in the charge distribution in the electronic states involved in the optical excitation. Pronounced effects of point mutations on the low frequency vibrational modes active in a protein-cofactor system have not been reported previously. The changes in frequency observed indicate a strong involvement of the protein in these nuclear motions and demonstrate that the protein matrix can increase or decrease the fluctuations of the cofactor along specific directions.

Efficient exchange of the primary quinone acceptor QA in isolated reaction centers of Rhodopseudomonas viridis

A key step in the conversion of solar energy into chemical energy by photosynthetic reaction centers (RCs) occurs at the level of the two quinones, QA and QB, where electron transfer couples to proton transfer. A great deal of our understanding of the mechanisms of these coupled reactions relies on the seminal work of Okamura et al. [Okamura, M. Y., Isaacson, R. A., & Feher, G. (1975) Proc. Natl. Acad. Sci. USA 88, 3491–3495], who were able to extract with detergents the firmly bound ubiquinone QA from the RC of Rhodobacter sphaeroides and reconstitute the site with extraneous quinones. Up to now a comparable protocol was lacking for the RC of Rhodopseudomonas viridis despite the fact that its QA site, which contains 2-methyl-3-nonaprenyl-1,4-naphthoquinone (menaquinone-9), has provided the best x-ray structure available. Fourier transform infrared difference spectroscopy, together with the use of isotopically labeled quinones, can probe the interaction of QA with the RC protein. We establish that a simple incubation procedure of isolated RCs of Rp. viridis with an excess of extraneous quinone allows the menaquinone-9 in the QA site to be almost quantitatively replaced either by vitamin K1, a close analogue of menaquinone-9, or by ubiquinone. To our knowledge, this is the first report of quinone exchange in bacterial photosynthesis. The Fourier transform infrared data on the quinone and semiquinone vibrations show a close similarity in the bonding interactions of vitamin K1 with the protein at the QA site of Rp. viridis and Rb. sphaeroides, whereas for ubiquinone these interactions are significantly different. The results are interpreted in terms of slightly inequivalent quinone–protein interactions by comparison with the crystallographic data available for the QA site of the two RCs.

Molecular switch in signal transduction: Reaction paths of the conformational changes in ras p21

Conformational changes in ras p21 triggered by the hydrolysis of GTP play an essential role in the signal transduction pathway. The path for the conformational change is determined by molecular dynamics simulation with a holonomic constraint directing the system from the known GTP-bound structure (with the γ-phosphate removed) to the GDP-bound structure. The simulation is done with a shell of water molecules surrounding the protein. In the switch I region, the side chain of Tyr-32, which undergoes a large displacement, moves through the space between loop 2 and the rest of the protein, rather than on the outside of the protein. As a result, the charged residues Glu-31 and Asp-33, which interact with Raf in the homologous RafRBD–Raps complex, remain exposed during the transition. In the switch II region, the conformational changes of α2 and loop 4 are strongly coupled. A transient hydrogen bonding complex between Arg-68 and Tyr-71 in the switch II region and Glu-37 in switch I region stabilizes the intermediate conformation of α2 and facilitates the unwinding of a helical turn of α2 (residues 66–69), which in turn permits the larger scale motion of loop 4. Hydrogen bond exchange between the protein and solvent molecules is found to be important in the transition. Possible functional implications of the results are discussed.

Proton transfer from the bulk to the bound ubiquinone QB of the reaction center in chromatophores of Rhodobacter sphaeroides: Retarded conveyance by neutral water

The mechanism of proton transfer from the bulk into the membrane protein interior was studied. The light-induced reduction of a bound ubiquinone molecule QB by the photosynthetic reaction center is accompanied by proton trapping. We used kinetic spectroscopy to measure (i) the electron transfer to QB (at 450 nm), (ii) the electrogenic proton delivery from the surface to the QB site (by electrochromic carotenoid response at 524 nm), and (iii) the disappearance of protons from the bulk solution (by pH indicators). The electron transfer to QB− and the proton-related electrogenesis proceeded with the same time constant of ≈100 μs (at pH 6.2), whereas the alkalinization in the bulk was distinctly delayed (τ ≈ 400 μs). We investigated the latter reaction as a function of the pH indicator concentration, the added pH buffers, and the temperature. The results led us to the following conclusions: (i) proton transfer from the surface-located acidic groups into the QB site followed the reduction of QB without measurable delay; (ii) the reprotonation of these surface groups by pH indicators and hydronium ions was impeded, supposedly, because of their slow diffusion in the surface water layer; and (iii) as a result, the protons were slowly donated by neutral water to refill the proton vacancies at the surface. It is conceivable that the same mechanism accounts for the delayed relaxation of the surface pH changes into the bulk observed previously with bacteriorhodopsin membranes and thylakoids. Concerning the coupling between proton pumps in bioenergetic membranes, our results imply a tendency for the transient confinement of protons at the membrane surface.

Fluorescence correlation analysis of probe diffusion simplifies quantitative pathogen detection by PCR

A sensitive, labor-saving, and easily automatable nonradioactive procedure named APEX-FCS (amplified probe extension detected by fluorescence correlation spectroscopy) has been established to detect specific in vitro amplification of pathogen genomic sequences. As an example, Mycobacterium tuberculosis genomic DNA was subjected to PCR amplification with the Stoffel fragment of Thermus aquaticus DNA polymerase in the presence of nanomolar concentrations of a rhodamine-labeled probe (third primer), binding to the target in between the micromolar amplification primers. The probe becomes extended only when specific amplification occurs. Its low concentration avoids false-positives due to unspecific hybridization under PCR conditions. With increasing portion of extended probe molecules, the probe’s average translational diffusion properties gradually change over the course of the reaction, reflecting amplification kinetics. Following PCR, this change from a stage of high to a stage of low mobility can directly be monitored during a 30-s measurement using a fluorescence correlation spectroscopy device. Quantitation down to 10 target molecules in a background of 2.5 μg unspecific DNA without post-PCR probe manipulations could be achieved with different primer/probe combinations. The assay holds the promise to concurrently perform amplification, probe hybridization, and specific detection without opening the reaction chamber, if sealable foils are used.

Limited internal friction in the rate-limiting step of a two-state protein folding reaction

Small, single-domain proteins typically fold via a compact transition-state ensemble in a process well fitted by a simple, two-state model. To characterize the rate-limiting conformational changes that underlie two-state folding, we have investigated experimentally the effects of changing solvent viscosity on the refolding of the IgG binding domain of protein L. In conjunction with numerical simulations, our results indicate that the rate-limiting conformational changes of the folding of this domain are strongly coupled to solvent viscosity and lack any significant “internal friction” arising from intrachain collisions. When compared with the previously determined solvent viscosity dependencies of other, more restricted conformational changes, our results suggest that the rate-limiting folding transition involves conformational fluctuations that displace considerable amounts of solvent. Reconciling evidence that the folding transition state ensemble is comprised of highly collapsed species with these and similar, previously reported results should provide a significant constraint for theoretical models of the folding process.

Uracil-DNA glycosylase–DNA substrate and product structures: Conformational strain promotes catalytic efficiency by coupled stereoelectronic effects

Enzymatic transformations of macromolecular substrates such as DNA repair enzyme/DNA transformations are commonly interpreted primarily by active-site functional-group chemistry that ignores their extensive interfaces. Yet human uracil–DNA glycosylase (UDG), an archetypical enzyme that initiates DNA base-excision repair, efficiently excises the damaged base uracil resulting from cytosine deamination even when active-site functional groups are deleted by mutagenesis. The 1.8-Å resolution substrate analogue and 2.0-Å resolution cleaved product cocrystal structures of UDG bound to double-stranded DNA suggest enzyme–DNA substrate-binding energy from the macromolecular interface is funneled into catalytic power at the active site. The architecturally stabilized closing of UDG enforces distortions of the uracil and deoxyribose in the flipped-out nucleotide substrate that are relieved by glycosylic bond cleavage in the product complex. This experimentally defined substrate stereochemistry implies the enzyme alters the orientation of three orthogonal electron orbitals to favor electron transpositions for glycosylic bond cleavage. By revealing the coupling of this anomeric effect to a delocalization of the glycosylic bond electrons into the uracil aromatic system, this structurally implicated mechanism resolves apparent paradoxes concerning the transpositions of electrons among orthogonal orbitals and the retention of catalytic efficiency despite mutational removal of active-site functional groups. These UDG/DNA structures and their implied dissociative excision chemistry suggest biology favors a chemistry for base-excision repair initiation that optimizes pathway coordination by product binding to avoid the release of cytotoxic and mutagenic intermediates. Similar excision chemistry may apply to other biological reaction pathways requiring the coordination of complex multistep chemical transformations.

Single-molecule protein folding: Diffusion fluorescence resonance energy transfer studies of the denaturation of chymotrypsin inhibitor 2

We report single-molecule folding studies of a small, single-domain protein, chymotrypsin inhibitor 2 (CI2). CI2 is an excellent model system for protein folding studies and has been extensively studied, both experimentally (at the ensemble level) and theoretically. Conformationally assisted ligation methodology was used to synthesize the proteins and site-specifically label them with donor and acceptor dyes. Folded and denatured subpopulations were observed by fluorescence resonance energy transfer (FRET) measurements on freely diffusing single protein molecules. Properties of these subpopulations were directly monitored as a function of guanidinium chloride concentration. It is shown that new information about different aspects of the protein folding reaction can be extracted from such subpopulation properties. Shifts in the mean transfer efficiencies are discussed, FRET efficiency distributions are translated into potentials, and denaturation curves are directly plotted from the areas of the FRET peaks. Changes in stability caused by mutation also are measured by comparing pseudo wild-type CI2 with a destabilized mutant (K17G). Current limitations and future possibilities and prospects for single-pair FRET protein folding investigations are discussed.