“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.

Configurational diffusion down a folding funnel describes the dynamics of DNA hairpins

Elucidating the mechanism of folding of polynucleotides depends on accurate estimates of free energy surfaces and a quantitative description of the kinetics of structure formation. Here, the kinetics of hairpin formation in single-stranded DNA are measured after a laser temperature jump. The kinetics are modeled as configurational diffusion on a free energy surface obtained from a statistical mechanical description of equilibrium melting profiles. The effective diffusion coefficient is found to be strongly temperature-dependent in the nucleation step as a result of formation of misfolded loops that do not lead to subsequent zipping. This simple system exhibits many of the features predicted from theoretical studies of protein folding, including a funnel-like energy surface with many folding pathways, trapping in misfolded conformations, and non-Arrhenius folding rates.

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.

A theory for controlling cell cycle dynamics using a reversibly binding inhibitor

We demonstrate, by using mathematical modeling of cell division cycle (CDC) dynamics, a potential mechanism for precisely controlling the frequency of cell division and regulating the size of a dividing cell. Control of the cell cycle is achieved by artificially expressing a protein that reversibly binds and inactivates any one of the CDC proteins. In the simplest case, such as the checkpoint-free situation encountered in early amphibian embryos, the frequency of CDC oscillations can be increased or decreased by regulating the rate of synthesis, the binding rate, or the equilibrium constant of the binding protein. In a more complex model of cell division, where size-control checkpoints are included, we show that the same reversible binding reaction can alter the mean cell mass in a continuously dividing cell. Because this control scheme is general and requires only the expression of a single protein, it provides a practical means for tuning the characteristics of the cell cycle in vivo.

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.

Dynamics and folding of single two-stranded coiled-coil peptides studied by fluorescent energy transfer confocal microscopy

We report single-molecule measurements on the folding and unfolding conformational equilibrium distributions and dynamics of a disulfide crosslinked version of the two-stranded coiled coil from GCN4. The peptide has a fluorescent donor and acceptor at the N termini of its two chains and a Cys disulfide near its C terminus. Thus, folding brings the two N termini of the two chains close together, resulting in an enhancement of fluorescent resonant energy transfer. End-to-end distance distributions have thus been characterized under conditions where the peptide is nearly fully folded (0 M urea), unfolded (7.4 M urea), and in dynamic exchange between folded and unfolded states (3.0 M urea). The distributions have been compared for the peptide freely diffusing in solution and deposited onto aminopropyl silanized glass. As the urea concentration is increased, the mean end-to-end distance shifts to longer distances both in free solution and on the modified surface. The widths of these distributions indicate that the molecules are undergoing millisecond conformational fluctuations. Under all three conditions, these fluctuations gave nonexponential correlations on 1- to 100-ms time scale. A component of the correlation decay that was sensitive to the concentration of urea corresponded to that measured by bulk relaxation kinetics. The trajectories provided effective intramolecular diffusion coefficients as a function of the end-to-end distances for the folded and unfolded states. Single-molecule folding studies provide information concerning the distributions of conformational states in the folded, unfolded, and dynamically interconverting states.

Constraints on nebular dynamics and chemistry based on observations of annealed magnesium silicate grains in comets and in disks surrounding Herbig Ae/Be stars

Understanding dynamic conditions in the Solar Nebula is the key to prediction of the material to be found in comets. We suggest that a dynamic, large-scale circulation pattern brings processed dust and gas from the inner nebula back out into the region of cometesimal formation—extending possibly hundreds of astronomical units (AU) from the sun—and that the composition of comets is determined by a chemical reaction network closely coupled to the dynamic transport of dust and gas in the system. This scenario is supported by laboratory studies of Mg silicates and the astronomical data for comets and for protoplanetary disks associated with young stars, which demonstrate that annealing of nebular silicates must occur in conjunction with a large-scale circulation. Mass recycling of dust should have a significant effect on the chemical kinetics of the outer nebula by introducing reduced, gas-phase species produced in the higher temperature and pressure environment of the inner nebula, along with freshly processed grains with “clean” catalytic surfaces to the region of cometesimal formation. Because comets probably form throughout the lifetime of the Solar Nebula and processed (crystalline) grains are not immediately available for incorporation into the first generation of comets, an increasing fraction of dust incorporated into a growing comet should be crystalline olivine and this fraction can serve as a crude chronometer of the relative ages of comets. The formation and evolution of key organic and biogenic molecules in comets are potentially of great consequence to astrobiology.

Ultrafast detection and control of molecular dynamics

Many elementary chemical and physical processes such as the breaking of a chemical bond or the vibrational motion of atoms within a molecule take place on a femtosecond (fs = 10−15 s) or picosecond (ps = 10−12 s) time scale. It is now possible to monitor these events as a function of time with temporal resolution well below 100 fs. This capability is based on the pump-probe technique where one optical pulse triggers a reaction and a second delayed optical pulse probes the changes that ensue. To illustrate this capability, the dynamics of ligand motion within a protein are presented. Moving beyond casual observation of a reaction to active control of its outcome requires additional experimental and theoretical effort. To illustrate the concept of control, the effect of optical pulse duration on the vibrational dynamics of a tri-atomic molecule are discussed. The experimental and theoretical resources currently available are poised to make the dream of reaction control a reality for certain molecular systems.

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.

Excited-state dynamics in photosystem II: Insights from the x-ray crystal structure

The heart of oxygenic photosynthesis is photosystem II (PSII), a multisubunit protein complex that uses solar energy to drive the splitting of water and production of molecular oxygen. The effectiveness of the photochemical reaction center of PSII depends on the efficient transfer of excitation energy from the surrounding antenna chlorophylls. A kinetic model for PSII, based on the x-ray crystal structure coordinates of 37 antenna and reaction center pigment molecules, allows us to map the major energy transfer routes from the antenna chlorophylls to the reaction center chromophores. The model shows that energy transfer to the reaction center is slow compared with the rate of primary electron transport and depends on a few bridging chlorophyll molecules. This unexpected energetic isolation of the reaction center in PSII is similar to that found in the bacterial photosystem, conflicts with the established view of the photophysics of PSII, and may be a functional requirement for primary photochemistry in photosynthesis. In addition, the model predicts a value for the intrinsic photochemical rate constant that is 4 times that found in bacterial reaction centers.

Protonation dynamics of the extracellular and cytoplasmic surface of bacteriorhodopsin in the purple membrane.

The dynamics of proton binding to the extracellular and the cytoplasmic surfaces of the purple membrane were measured by laser-induced proton pulses. Purple membranes, selectively labeled by fluorescein at Lys-129 of bacteriorhodopsin, were pulsed by protons released in the aqueous bulk from excited pyranine (8-hydroxy-1,3,6-pyrenetrisulfonate) and the reaction of protons with the indicators was measured. Kinetic analysis of the data imply that the two faces of the membrane differ in their buffer capacities and in their rates of interaction with bulk protons. The extracellular surface of the purple membrane contains one anionic proton binding site per protein molecule with pK = 5.1. This site is within a Coulomb cage radius (approximately 15 A) from Lys-129. The cytoplasmic surface of the purple membrane bears 4-5 protonable moieties (pK = 5.1) that, due to close proximity, function as a common proton binding site. The reaction of the proton with this cluster is at a very fast rate (3.10(10) M-1.s-1). The proximity between the elements is sufficiently high that even in 100 mM NaCl they still function as a cluster. Extraction of the chromophore retinal from the protein has a marked effect on the carboxylates of the cytoplasmic surface, and two to three of them assume positions that almost bar their reaction with bulk protons. The protonation dynamics determined at the surface of the purple membrane is of relevance both for the vectorial proton transport mechanism of bacteriorhodopsin and for energy coupling, not only in halobacteria, but also in complex chemiosmotic systems such as mitochondrial and thylakoid membranes.

Ultra-fast excited state dynamics in green fluorescent protein: multiple states and proton transfer.

The green fluorescent protein (GFP) of the jellyfish Aequorea Victoria has attracted widespread interest since the discovery that its chromophore is generated by the autocatalytic, posttranslational cyclization and oxidation of a hexapeptide unit. This permits fusion of the DNA sequence of GFP with that of any protein whose expression or transport can then be readily monitored by sensitive fluorescence methods without the need to add exogenous fluorescent dyes. The excited state dynamics of GFP were studied following photo-excitation of each of its two strong absorption bands in the visible using fluorescence upconversion spectroscopy (about 100 fs time resolution). It is shown that excitation of the higher energy feature leads very rapidly to a form of the lower energy species, and that the excited state interconversion rate can be markedly slowed by replacing exchangeable protons with deuterons. This observation and others lead to a model in which the two visible absorption bands correspond to GFP in two ground-state conformations. These conformations can be slowly interconverted in the ground state, but the process is much faster in the excited state. The observed isotope effect suggests that the initial excited state process involves a proton transfer reaction that is followed by additional structural changes. These observations may help to rationalize and motivate mutations that alter the absorption properties and improve the photo stability of GFP.

Diffusion and formation of microtubule asters: physical processes versus biochemical regulation.

Microtubule asters forming the mitotic spindle are assembled around two centrosomes through the process of dynamic instability in which microtubules alternate between growing and shrinking states. By modifying the dynamics of this assembly process, cell cycle enzymes, such as cdc2 cyclin kinases, regulate length distributions in the asters. It is believed that the same enzymes control the number of assembled microtubules by changing the "nucleating activity" of the centrosomes. Here we show that assembly of microtubule asters may be strongly altered by effects connected with diffusion of tubulin monomers. Theoretical analysis of a simple model describing assembly of microtubule asters clearly shows the existence of a region surrounding the centrosome depleted in GTP tubulin. The number of assembled microtubules may in some cases be limited by this depletion effect rather than by the number of available nucleation sites on the centrosome.