Brownian dynamics simulations of protein folding: Access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion–collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion–collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Complete resolution of the solid-state NMR spectrum of a uniformly 15N-labeled membrane protein in phospholipid bilayers

Complete resolution of the amide resonances in a three-dimensional solid-state NMR correlation spectrum of a uniformly 15N-labeled membrane protein in oriented phospholipid bilayers is demonstrated. The three orientationally dependent frequencies, 1H chemical shift, 1H–15N dipolar coupling, and 15N chemical shift, associated with each amide resonance are responsible for resolution among resonances and provide sufficient angular restrictions for protein structure determination. Because the protein is completely immobilized by the phospholipids on the relevant NMR time scales (10 kHz), the linewidths will not degrade in the spectra of larger proteins. Therefore, these results demonstrate that solid-state NMR experiments can overcome the correlation time problem and extend the range of proteins that can have their structures determined by NMR spectroscopy to include uniformly 15N-labeled membrane proteins in phospholipid bilayers.

Control of time-dependent biological processes by temporally patterned input

Temporal patterning of biological variables, in the form of oscillations and rhythms on many time scales, is ubiquitous. Altering the temporal pattern of an input variable greatly affects the output of many biological processes. We develop here a conceptual framework for a quantitative understanding of such pattern dependence, focusing particularly on nonlinear, saturable, time-dependent processes that abound in biophysics, biochemistry, and physiology. We show theoretically that pattern dependence is governed by the nonlinearity of the input–output transformation as well as its time constant. As a result, only patterns on certain time scales permit the expression of pattern dependence, and processes with different time constants can respond preferentially to different patterns. This has implications for temporal coding and decoding, and allows differential control of processes through pattern. We show how pattern dependence can be quantitatively predicted using only information from steady, unpatterned input. To apply our ideas, we analyze, in an experimental example, how muscle contraction depends on the pattern of motorneuron firing.

Two-dimensional infrared spectroscopy of peptides by phase-controlled femtosecond vibrational photon echoes

Two-dimensional infrared spectra of peptides are introduced that are the direct analogues of two- and three-pulse multiple quantum NMR. Phase matching and heterodyning are used to isolate the phase and amplitudes of the electric fields of vibrational photon echoes as a function of multiple pulse delays. Structural information is made available on the time scale of a few picoseconds. Line narrowed spectra of acyl-proline-NH2 and cross peaks implying the coupling between its amide-I modes are obtained, as are the phases of the various contributions to the signals. Solvent-sensitive structural differences are seen for the dipeptide. The methods show great promise to measure structure changes in biology on a wide range of time scales.

A sampling problem in molecular dynamics simulations of macromolecules.

Correlations in low-frequency atomic displacements predicted by molecular dynamics simulations on the order of 1 ns are undersampled for the time scales currently accessible by the technique. This is shown with three different representations of the fluctuations in a macromolecule: the reciprocal space of crystallography using diffuse x-ray scattering data, real three-dimensional Cartesian space using covariance matrices of the atomic displacements, and the 3N-dimensional configuration space of the protein using dimensionally reduced projections to visualize the extent to which phase space is sampled.

Separation of the arterial wall from blood contact using hydrogel barriers reduces intimal thickening after balloon injury in the rat: The roles of medial and luminal factors in arterial healing

The objective of this study was to clarify the relative roles of medial versus luminal factors in the induction of thickening of the arterial intima after balloon angioplasty injury. Platelet-derived growth factor (PDGF) and thrombin, both associated with thrombosis, and basic fibroblast growth factor (bFGF), stored in the arterial wall, have been implicated in this process. To unequivocally isolate the media from luminally derived factors, we used a 20-μm thick hydrogel barrier that adhered firmly to the arterial wall to block thrombus deposition after balloon-induced injury of the carotid artery of the rat. Thrombosis, bFGF mobilization, medial repopulation, and intimal thickening were measured. Blockade of postinjury arterial contact with blood prevented thrombosis and dramatically inhibited both intimal thickening and endogenous bFGF mobilization. By blocking blood contact on the two time scales of thrombosis and of intimal thickening, and by using local protein release to probe, by reconstitution, the individual roles of PDGF-BB and thrombin, we were able to conclude that a luminally derived factor other than PDGF or thrombin is required for the initiation of cellular events leading to intimal thickening after balloon injury in the rat. We further conclude that a luminally derived factor is required for mobilization of medial bFGF.

Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts

We use residual-delay maps of observational field data for barometric pressure to demonstrate the structure of latitudinal gradients in nonlinearity in the atmosphere. Nonlinearity is weak and largely lacking in tropical and subtropical sites and increases rapidly into the temperate regions where the time series also appear to be much noisier. The degree of nonlinearity closely follows the meridional variation of midlatitude storm track frequency. We extract the specific functional form of this nonlinearity, a V shape in the lagged residuals that appears to be a basic feature of midlatitude synoptic weather systems associated with frontal passages. We present evidence that this form arises from the relative time scales of high-pressure versus low-pressure events. Finally, we show that this nonlinear feature is weaker in a well regarded numerical forecast model (European Centre for Medium-Range Forecasts) because small-scale temporal and spatial variation is smoothed out in the grided inputs. This is significant, in that it allows us to demonstrate how application of statistical corrections based on the residual-delay map may provide marked increases in local forecast accuracy, especially for severe weather systems.

Dynamical principles in biological processes: A model of charge migration in proteins and DNA

The generalized master equations (GMEs) that contain multiple time scales have been derived quantum mechanically. The GME method has then been applied to a model of charge migration in proteins that invokes the hole hopping between local amino acid sites driven by the torsional motions of the floppy backbones. This model is then applied to analyze the experimental results for sequence-dependent long-range hole transport in DNA reported by Meggers et al. [Meggers, E., Michel-Beyerle, M. E., & Giese, B. (1998) J. Am. Chem. Soc. 120, 12950–12955]. The model has also been applied to analyze the experimental results of femtosecond dynamics of DNA-mediated electron transfer reported by Zewail and co-workers [Wan, C., Fiebig, T., Kelley, S. O., Treadway, C. R., Barton, J. K. & Zewail, A. H. (1999) Proc. Natl. Acad. Sci. USA 96, 6014–6019]. The initial events in the dynamics of protein folding have begun to attract attention. The GME obtained in this paper will be applicable to this problem.

Solvent effects on the energy landscapes and folding kinetics of polyalanine

The effect of a solvation on the thermodynamics and kinetics of polyalanine (Ala12) is explored on the basis of its energy landscapes in vacuum and in an aqueous solution. Both energy landscapes are characterized by two basins, one associated with α-helical structures and the other with coil and β-structures of the peptide. In both environments, the basin that corresponds to the α-helical structure is considerably narrower than the basin corresponding to the β-state, reflecting their different contributions to the entropy of the peptide. In vacuum, the α-helical state of Ala12 constitutes the native state, in agreement with common helical propensity scales, whereas in the aqueous medium, the α-helical state is destabilized, and the β-state becomes the native state. Thus solvation has a dramatic effect on the energy landscape of this peptide, resulting in an inverted stability of the two states. Different folding and unfolding time scales for Ala12 in hydrophilic and hydrophobic chemical environments are caused by the higher entropy of the native state in water relative to vacuum. The concept of a helical propensity has to be extended to incorporate environmental solvent effects.

Insect herbivory, plant defense, and early Cenozoic climate change

Insect damage on fossil leaves from the Central Rocky Mountains, United States, documents the response of herbivores to changing regional climates and vegetation during the late Paleocene (humid, warm temperate to subtropical, predominantly deciduous), early Eocene (humid subtropical, mixed deciduous and evergreen), and middle Eocene (seasonally dry, subtropical, mixed deciduous and thick-leaved evergreen). During all three time periods, greater herbivory occurred on taxa considered to have short rather than long leaf life spans, consistent with studies in living forests that demonstrate the insect resistance of long-lived, thick leaves. Variance in herbivory frequency and diversity was highest during the middle Eocene, indicating the increased representation of two distinct herbivory syndromes: one for taxa with deciduous, palatable foliage, and the other for hosts with evergreen, thick-textured, small leaves characterized by elevated insect resistance. Leaf galling, which is negatively correlated with moisture today, apparently increased during the middle Eocene, whereas leaf mining decreased.

Equilibration of the terrestrial water, nitrogen, and carbon cycles

Recent advances in biologically based ecosystem models of the coupled terrestrial, hydrological, carbon, and nutrient cycles have provided new perspectives on the terrestrial biosphere’s behavior globally, over a range of time scales. We used the terrestrial ecosystem model Century to examine relationships between carbon, nitrogen, and water dynamics. The model, run to a quasi-steady-state, shows strong correlations between carbon, water, and nitrogen fluxes that lead to equilibration of water/energy and nitrogen limitation of net primary productivity. This occurs because as the water flux increases, the potentials for carbon uptake (photosynthesis), and inputs and losses of nitrogen, all increase. As the flux of carbon increases, the amount of nitrogen that can be captured into organic matter and then recycled also increases. Because most plant-available nitrogen is derived from internal recycling, this latter process is critical to sustaining high productivity in environments where water and energy are plentiful. At steady-state, water/energy and nitrogen limitation “equilibrate,” but because the water, carbon, and nitrogen cycles have different response times, inclusion of nitrogen cycling into ecosystem models adds behavior at longer time scales than in purely biophysical models. The tight correlations among nitrogen fluxes with evapotranspiration implies that either climate change or changes to nitrogen inputs (from fertilization or air pollution) will have large and long-lived effects on both productivity and nitrogen losses through hydrological and trace gas pathways. Comprehensive analyses of the role of ecosystems in the carbon cycle must consider mechanisms that arise from the interaction of the hydrological, carbon, and nutrient cycles in ecosystems.

Gases in ice cores

Air trapped in glacial ice offers a means of reconstructing variations in the concentrations of atmospheric gases over time scales ranging from anthropogenic (last 200 yr) to glacial/interglacial (hundreds of thousands of years). In this paper, we review the glaciological processes by which air is trapped in the ice and discuss processes that fractionate gases in ice cores relative to the contemporaneous atmosphere. We then summarize concentration–time records for CO2 and CH4 over the last 200 yr. Finally, we summarize concentration–time records for CO2 and CH4 during the last two glacial–interglacial cycles, and their relation to records of global climate change.

A long marine history of carbon cycle modulation by orbital-climatic changes

Pacing of the marine carbon cycle by orbital forcing during the Pliocene and Pleistocene Ice Ages [past 2.5 million years (Myr)] is well known. As older deep-sea sediment records are being studied at greater temporal resolution, it is becoming clear that similar fluctuations in the marine carbon system have occurred throughout the late Mesozoic and Tertiary, despite the absence of large continental ice sheets over much of this time. Variations in both the organic and the calcium carbonate components of the marine carbon system seem to have varied cyclically in response to climate forcing, and carbon and carbonate time series appear to accurately characterize the frequency spectrum of ancient climatic change. For the past 35 Myr, much of the variance in carbonate content carries the “polar” signal of obliquity [41,000 years (41 kyr)] forcing. Over the past 125 Myr, there is evidence from marine sediments of the continued role of precessional (≈21 kyr) climatic cycles. Repeat patterns of sedimentation at about 100, 400, and 2,400 kyr, the modulation periods of precession, persistently enter into marine carbon cycle records as well. These patterns suggest a nonlinear response of climate and/or the sedimentation of organic carbon and carbonates to precessional orbital perturbations. Nonlinear responses of the carbon system may help to amplify relatively weak orbital insolation anomalies into more significant climatic perturbations through positive feedback effects. Nonlinearities in the carbon cycle may have transformed orbital-climatic cycles into long-wavelength features on time scales comparable to the residence times of carbon and nutrient elements in the ocean.

Two mechanisms for transducer adaptation in vertebrate hair cells

Deflection of the hair bundle atop a sensory hair cell modulates the open probability of mechanosensitive ion channels. In response to sustained deflections, hair cells adapt. Two fundamentally distinct models have been proposed to explain transducer adaptation. Both models support the notion that channel open probability is modulated by calcium that enters via the transduction channels. Both also suggest that the primary effect of adaptation is to shift the deflection-response [I(X)] relationship in the direction of the applied stimulus, thus maintaining hair bundle sensitivity. The models differ in several respects. They operate on different time scales: the faster on the order of a few milliseconds or less and the slower on the order of 10 ms or more. The model proposed to explain fast adaptation suggests that calcium enters and binds at or near the transduction channels to stabilize a closed conformation. The model proposed to explain the slower adaptation suggests that adaptation is mediated by an active, force-generating process that regulates the effective stimulus applied to the transduction channels. Here we discuss the evidence in support of each model and consider the possibility that both may function to varying degrees in hair cells of different species and sensory organs.

Synaptic pattern formation during cellular recognition

Cell–cell recognition often requires the formation of a highly organized pattern of receptor proteins (a synapse) in the intercellular junction. Recent experiments [e.g., Monks, C. R. F., Freiberg, B. A., Kupfer, H., Sciaky, N. & Kupfer, A. (1998) Nature (London) 395, 82–86; Grakoui, A., Bromley, S. K., Sumen, C., Davis, M. M., Shaw, A. S., Allen, P. M. & Dustin, M. L. (1999) Science 285, 221–227; and Davis, D. M., Chiu, I., Fassett, M., Cohen, G. B., Mandelboim, O. & Strominger, J. L. (1999) Proc. Natl. Acad. Sci. USA 96, 15062–15067] vividly demonstrate a complex evolution of cell shape and spatial receptor–ligand patterns (several microns in size) in the intercellular junction during immunological synapse formation. The current view is that this dynamic rearrangement of proteins into organized supramolecular activation clusters is driven primarily by active cytoskeletal processes [e.g., Dustin, M. L. & Cooper, J. A. (2000) Nat. Immunol. 1, 23–29; and Wulfing, C. & Davis, M. M. (1998) Science 282, 2266–2269]. Here, aided by a quantitative analysis of the relevant physico-chemical processes, we demonstrate that the essential characteristics of synaptic patterns observed in living cells can result from spontaneous self-assembly processes. Active cellular interventions are superimposed on these self-organizing tendencies and may also serve to regulate the spontaneous processes. We find that the protein binding/dissociation characteristics, protein mobilities, and membrane constraints measured in the cellular environment are delicately balanced such that the length and time scales of spontaneously evolving patterns are in near-quantitative agreement with observations for synapse formation between T cells and supported membranes [Grakoui, A., Bromley, S. K., Sumen, C., Davis, M. M., Shaw, A. S., Allen, P. M. & Dustin, M. L. (1999) Science 285, 221–227]. The model we present provides a common way of analyzing immunological synapse formation in disparate systems (e.g., T cell/antigen-presenting cell junctions with different MHC-peptides, natural killer cells, etc.).