Relaxation folding and the principle of the minimum rate of energy dissipation for conformational motions in a viscous medium

A numerical simulation of the folding of a model polymer chain of 50 units with valence bonds of a fixed length and fixed valence angle values has been performed using the strong friction approximation. The rate of energy dissipation in the system has been analyzed for conformational motions along a trajectory determined by the equations of mechanics and the trajectories characterized by random and variable deviations from the mechanical path. The validity of the principle of the minimum average rate of the energy dissipation for the conformational relaxation of a macromolecule in a viscous medium has been demonstrated. A profile of the relaxation energy funnel for the folding of a macromolecular chain has been constructed. Slow and rapid stages of folding could be distinguished in the energy funnel profile; the final state was separated from the nearest conformations of the folded chain by an energy gap.     

Variational Principles in the Mechanics of Conformational Motions of Macromolecules in a Viscous Medium

The mechanics of the conformational motions of macromolecules due to rotations around the valence bonds in a viscous medium have been considered. The variational principles for the energy-dissipation rate during conformational motions in a viscous medium and the rate of the potential energy decrease of a macromolecule during conformational relaxation have been analyzed. The seeming contradiction between this principle and the principle of the minimum energy-dissipation rate is resolved. It is shown that the energy-dissipation rate must be optimal and minimal in order to simultaneously satisfy the conservation laws and fulfill the deterministic nature of classical trajectories. The generalization and analysis of the influence of thermal fluctuations and external forces on the variational principles for the conformational relaxation of macromolecules is carried out. A visual graphical geometric depiction has been developed using hyperspheres in the space of the velocities of chain nodes to describe conformational movements along many degrees of freedom in a viscous medium. The equipartition of the energy-dissipation rates (and the rates of potential energy decrease) among the conformational degrees of freedom is discussed.     

Formation of collective conformational degrees of freedom during macromolecular chain folding dynamics in a viscous medium

Computational methods were used to study the dynamics of the formation of the collective conformational degrees of freedom in the relaxation folding of a model biopolymer chain of 50 nodes in a viscous medium; the model has been described previously. Collective conformational motions of the nodes were shown to arise due to friction forces in a viscous medium. The collective motions have several typical forms, including a wave of differently directed motions of chain nodes that propagates from one end of the chain to another (like a soliton) in response to a pertubation in terminal group position. Individual nodes located at the middle of the chain make approximately equal contributions to the total energy dissipation rate. The end nodes contribute approximately 2–4 times more than internal nodes to the total energy dissipation. The results of numerical experiments are consistent with the theoretical concept developed earlier to describe the dynamics of linear macromolecular chains in a viscous medium in the limit of a very large number of nodes.     

Conformational motion correlations in the formation of polypeptide secondary structure in a viscous medium

The Langevin dynamics method and statistical correlation analysis were used to study the α-helical structure folding dynamics of the (Ala)₅₀, (AlaGly)₂₅, and (AlaGly)₇₅ polypeptides depending on the viscosity of the medium. Friction forces that arise when the effective viscosity of the medium is similar to the viscosity of water were found to result in strong correlations between the backbone torsion angles. The polypeptides under study folded mainly to produce α-helical structures. A structure of two contacting α-helices that were approximately equal in length and had a loop between them was observed for a longer chain of 150 residues. A method to visualize the correlation matrix of the dihedral angles of a polypeptide chain was developed for analyzing the effects of the dynamic correlation of conformational degrees of freedom. The analysis of the dynamics of the correlation matrix showed that rotations involving angles of the same type (φ–φ and ψ–ψ) occur predominantly in one direction. Rotations invoving different angles (φ–ψ) occur predominantly in opposite directions, so that the total macromolecule does not rotate. A significant reduction in the effective viscosity of the medium disrupts the correlation and makes the rotations stochastic, thus distorting the formation of the regular (helical) structure. The effects of correlated conformational motions are consequences of viscous friction forces. This conclusion agrees with our previous results that outlined the principle of the minimum rate of energy dissipation and the equipartition of energy dissipation rate between conformational degrees of freedom.     

The effects of turbulent jet flows on plant cell suspension cultures

Cell suspensions of Morinda citrifolia were subjected to turbulent flow conditions in a submerged jet apparatus, to investigate their hydrodynamic shear susceptibility. The suspensions were exposed to repeated, pressure-driven passages through a submerged jet. Two nozzles, of 1 mm and 2 mm diameter, were employed. Average energy dissipation rates were in the range 10(3)-10(5) W/kg and cumulative energy dissipation in the range 10(5)-10(7) J/m3. System response to the imposed conditions was evaluated in terms of suspension viability (determined using a dye exclusion technique) and variations in both chain length distribution and maximum chain length. Viability loss was well-described by a first-order model, and a linear relationship was identified between the specific death rate constant and the average energy dissipation rate. This relationship was consistent with results obtained using the same suspension cultures in a turbulent capillary flow device. Morphological measurements indicated that exposure to the hydrodynamic environment generated in the jet resulted in a significant reduction in both the average and maximum chain lengths, and the reduction in the maximum chain length was identified as an appropriate measure of sustained damage. Analysis of both viability and chain length in terms of cumulative energy dissipated revealed good agreement with results reported by other authors for morphologically different plant cell systems. Copyright 1998 John Wiley & Sons, Inc.     

Global Conformational Dynamics of a Y-Family DNA Polymerase during Catalysis

Replicative DNA polymerases are stalled by damaged DNA while the newly discovered Y-family DNA polymerases are recruited to rescue these stalled replication forks, thereby enhancing cell survival. The Y-family DNA polymerases, characterized by low fidelity and processivity, are able to bypass different classes of DNA lesions. A variety of kinetic and structural studies have established a minimal reaction pathway common to all DNA polymerases, although the conformational intermediates are not well defined. Furthermore, the identification of the rate-limiting step of nucleotide incorporation catalyzed by any DNA polymerase has been a matter of long debate. By monitoring time-dependent fluorescence resonance energy transfer (FRET) signal changes at multiple sites in each domain and DNA during catalysis, we present here a real-time picture of the global conformational transitions of a model Y-family enzyme: DNA polymerase IV (Dpo4) from Sulfolobus solfataricus. Our results provide evidence for a hypothetical DNA translocation event followed by a rapid protein conformational change prior to catalysis and a subsequent slow, post-chemistry protein conformational change. Surprisingly, the DNA translocation step was induced by the binding of a correct nucleotide. Moreover, we have determined the directions, rates, and activation energy barriers of the protein conformational transitions, which indicated that the four domains of Dpo4 moved in a synchronized manner. These results showed conclusively that a pre-chemistry conformational change associated with domain movements was too fast to be the rate-limiting step. Rather, the rearrangement of active site residues limited the rate of correct nucleotide incorporation. Collectively, the conformational dynamics of Dpo4 offer insights into how the inter-domain movements are related to enzymatic function and their concerted interactions with other proteins at the replication fork.     

Rhythmic Conformational Change in a SinglePolymer Chain Induced by Laser Irradiation

We have investigated a rhythmic conformational change in a single polymer chain (T4DNA, 166 kbp, the contour length = 56 m) between a folded state and a elongated state under thermodynamically open conditions using a focused continuons wave (cw) Nd:YAG laser beam (wavelength = 1064 nm), where the focused laser beam plays dual roles both of trapping of a polymer chain at the focus and making temperature gradient there. The oscillatory phenomenon is discuseed in terms of a limit-cycle oscillation with dissipation of photon energy.     

Regulation and possible function of the violaxanthin cycle

This paper discusses biochemical and regulatory aspects of the violaxanthin cycle as well as its possible role in photoprotection. The violaxanthin cycle responds to environmental conditions in the short-term and long-term by adjusting rates of pigment conversions and pool sizes of cycle pigments, respectively. Experimental evidence indicating a relationship between zeaxanthin formation and non-photochemical energy dissipation is reviewed. Zeaxanthin-associated energy dissipation appears to be dependent on transthylakoid pH. The involvement of light-harvesting complex II in this quenching process is indicated by several studies. The current hypotheses on the underlying mechanism of zeaxanthin-dependent quenching are alterations of membrane properties, including conformational changes of the light-harvesting complex II, and singlet-singlet energy transfer from chlorophyll to zeaxanthin     

Study of hydrodynamics in microcarrier culture spinner vessels: A particle tracking velocimetry approach

Three-dimensional particle tracking velocimetry (3-D PTV), a modern, quantitative, visualization tool, has been applied to the characterization of the flow field in the impeller region of cell culture reactor vessels. The experimental system used here is a 250-mL microcarrier spinner vessel. The studies were conducted at three different agitation rates, 90, 150, and 210 rpm, corresponding to healthy, mildly damaging, and severely damaging shear intensities, respectively. The flow can be classified into three regions: a predominantly tangential (azimuthal) flow generated by the impeller; a trailing vortex region coming off the impeller tip; and a converging flow region close to the center of the vessel. The latter two are the regions of highest velocity gradients. Energy dissipation rates due to mean velocity gradients were also calculated to characterize the impeller stream. Local specific energy dissipation rates > 10,000 erg/(cm(3)sec) . have been measured. It is proposed that the critical regions for microcarrier culture damage due to impeller hydrodynamics are the trailing vortex region and the high energy converging flow region. Graphical representation of the mean velocity flow fields and the distribution of energy dissipation rates in the impeller region are also presented here. The merits of using the dissipation function (measure of specific energy dissipation rate) as a possible scale-up parameter are also discussed. (c) 1996 John Wiley & Sons, Inc.     

Exploring zipping and assembly as a protein folding principle

It has been proposed that proteins fold by a process called "Zipping and Assembly" (Z&A). Zipping refers to the growth of local substructures within the chain, and assembly refers to the coming together of already-formed pieces. Our interest here is in whether Z&A is a general method that can fold most of sequence space, to global minima, efficiently. Using the HP model, we can address this question by enumerating full conformation and sequence spaces. We find that Z&A reaches the global energy minimum native states, even though it searches only a very small fraction of conformational space, for most sequences in the full sequence space. We find that Z&A, a mechanism-based search, is more efficient in our tests than the replica exchange search method. Folding efficiency is increased for chains having: (a) small loop-closure steps, consistent with observations by Plaxco et al. 1998;277;985-994 that folding rates correlate with contact order, (b) neither too few nor too many nucleation sites per chain, and (c) assembly steps that do not occur too early in the folding process. We find that the efficiency increases with chain length, although our range of chain lengths is limited. We believe these insights may be useful for developing faster protein conformational search algorithms.     

Protein conformational landscapes: Energy minimization and clustering of a long molecular dynamics trajectory

Using energy minimization and cluster analysis, we have analyzed a 1020 ps molecular dynamics trajectory of solvated bovine pancreatic trypsin inhibitor. Elucidation of conformational sub states in this way both illustrates the degree of conformational convergence in the simulation and reduces the structural data to a tractable subset. The relative movement of structures upon energy minimization was used to estimate the sizes of features on the protein potential energy surface. The structures were analyzed using their pairwise root-mean-square C_α deviations, which gave a global measure of conformational changes that would not be apparent by monitoring single degrees of freedom. At time scales of 0.1 ps, energy minimization detected sharp transitions between energy minima separated by 0.1 Å rms deviation. Larger conformational clusters containing these smaller minima and separated by 0.25 Å were seen at 1 ps time scales. Both of these small features of the conformational landscape were characterized by movements in loop regions associated with small, correlated backbone dihedral angle shifts. On a nanosecond time scale, the main features of the protein energy landscape were clusters separated by over 0.7 Å rms deviation, with only seven of these sub states visited over the 1 ns trajectory. These substates, discernible both before and after energy minimization, differ mainly in a monotonic pivot of the loop residues 11–18 over the course of the simulation. This loop contains lysine 17, which specifically binds to trypsin in the active site. The trajectory did not return to previously visited clusters, indicating that this trajectory has not been shown to have completely sampled the conformational substates available to it. Because the apparent convergence to a single region of conformation space depends on both the time scale of observation and the size of the conformational features examined, convergence must be operationally defined within the context of the simulation. © 1995 Wiley-Liss, Inc.     

In search of a thermodynamic description of biomass yields for the chemotrophic growth of microorganisms

Correlations for the prediction of biomass yields are valuable, and many proposals based on a number of parameters (Y(ATP), Y(Ave), eta(o), Y(c), Gibbs energy efficiencies, and enthalpy efficiencies) have been published. This article critically examines the properties of the proposed parameters with respect to the general applicability to chemotrophic growth systems, a clear relation to the Second Law of Thermodynamics, the absence of intrinsic problems, and a requirement of only black box information. It appears that none of the proposed parameters satisfies all these requirements. Particularly, the various energetic efficiency parameters suffer from major intrinsic problems. However, this article will show that the Gibbs energy dissipation per amount of produced biomass (kJ/C-mod) is a parameter which satisfies the requirements without having intrinsic problems. A simple correlation is found which provides the Gibbs energy dissipation/C-mol biomass as a function of the nature of the C-source (expressed as the carbon chain length and the degree of reduction). This dissipation appears to be nearly independent of the nature of the electron acceptor (e.g., O(2), No(3) (-), fermentation). Hence, a single correlation can describe a very wide range of microbial growth systems. In this respect, Gibbs energy dissipation is much more useful than heat production/C-mol biomass, which is strongly dependent on the electron acceptor used. Evidence is presented that even a net heat-uptake can occur in certain growth systems.The correlation of Gibbs energy dissipation thus obtained shows that dissipation/C-mol biomass increases for C-sources with smaller chain length (C(6) --> C(1)), and increases for both higher and lower degrees of reduction than 4. It appears that the dissipation/C-mol biomass can be regarded as a simple thermodynamic measure of the amount of biochemical "work" required to convert the carbon source into biomass by the proper irreversible carbon-carbon coupling and oxidation/reduction reactions. This is supported by the good correlation between the theoretical ATP requirement for biomass formation on different C-sources and the dissipation values (kJ/C-mol biomass) found. The established correlation for the Gibbs energy dissipation allows the prediction of the chemotrophic biomass yield on substrate with an error of 13% in the yield range 0.01 to 0.80 C-mol biomass/(C)-mol substrate for aerobic/anaerobic/denitrifying growth systems.     

Entropy and barrier-controlled fluctuations determine conformational viscoelasticity of single biomolecules

Biological macromolecules have complex and nontrivial energy landscapes, endowing them with a unique conformational adaptability and diversity in function. Hence, understanding the processes of elasticity and dissipation at the nanoscale is important to molecular biology and emerging fields such as nanotechnology. Here we analyze single molecule fluctuations in an atomic force microscope, using a generic model of biopolymer viscoelasticity that includes local "internal" conformational dissipation. Comparing two biopolymers, dextran and cellulose (polysaccharides with and without local bistable transitions), demonstrates that signatures of simple conformational change are minima in both the elastic and internal friction constants around a characteristic force. A novel analysis of dynamics on a bistable energy landscape provides a simple explanation: an elasticity driven by the entropy, and friction by a barrier-controlled hopping time of populations between states, which is surprisingly distinct to the well-known relaxation time. This nonequilibrium microscopic analysis thus provides a means of quantifying new dynamical features of the energy landscape of the glucopyranose ring, revealing an unexpected underlying roughness and information on the shape of the barrier of the chair-boat transition in dextran. The results presented herein provide a basis toward probing the viscoelasticity of macromolecular conformational transitions on more complex energy landscapes, such as during protein folding.     

Exploring the conformational space of protein side chains using dead-end elimination and the A* algorithm

We describe an algorithm which enables us to search the conformational space of the side chains of a protein to identify the global minimum energy combination of side chain conformations as well as all other conformations within a specified energy cutoff of the global energy minimum. The program is used to explore the side chain conformational energy surface of a number of proteins, to investigate how this surface varies with the energy model used to describe the interactions within the system and the rotamer library. Enumeration of the rotamer combinations enables us to directly evaluate the partition function, and thus calculate the side chain contribution to the conformational entropy of the folded protein. An investigation of these conformations and the relationships between them shows that most of the conformations near to the global energy minimum arise from changes in side chain conformations that are essentially independent; very few result from a concerted change in conformation of two or more residues. Some of the limitations of the approach are discussed. Proteins 33:227–239, 1998. © 1998 Wiley-Liss, Inc.     

Mode of insertion of chlorophyll a in a lipid layer. A conformational approach

A new procedure for conformational analysis is described to define the orientation of chlorophyll a in model membranes. The method allows to predict the configuration of assembled amphiphilic molecules. The minimal conformational energy is calculated as the sum of the contributions, resulting from the Van der Waals interactions, the torsional potentials, the electrostatic interactions and the transfer energy. In the calculated conformation, the porphyrin ring is orientated at an angle of 45° ± 5° to the membrane interface, with the phytol chain inserted into the lipid layer. This orientation of the porphyrin ring is in excellent agreement with the experimental value obtained with model membranes. The method could allow to define the orientation of other components in the thylakoid membrane.     

Energetics of subdomain movements and fluorescence probe solvation environment change in ATP-bound myosin

Energetics of conformational changes experienced by an ATP-bound myosin head detached from actin was studied by all-atom explicit water umbrella sampling simulations. The statistics of coupling between large scale domain movements and smaller scale structural features were examined, including the closing of the ATP binding pocket, and a number of key hydrogen bond formations shown to play roles in structural and biochemical studies. The statistics for the ATP binding pocket open/close transition show an evolution of the relative stability from the open state in the early stages of the recovery stroke to the stable closed state after the stroke. The change in solvation environment of the fluorescence probe Trp507 (scallop numbering; 501 in Dictyostelium discoideum) indicates that the probe faithfully reflects the closing of the binding pocket as previously shown in experimental studies, while being directly coupled to roughly the early half of the overall large scale conformational change of the converter domain rotation. The free energy change of this solvation environment change, in particular, is −1.3 kcal/mol, in close agreement with experimental estimates. In addition, our results provide direct molecular level data allowing for interpretations of the fluorescence experiments of myosin conformational change in terms of the de-solvation of Trp side chain.     

Structural basis of hierarchical multiple substates of a protein. III: Side chain and main chain local conformations

An analysis is carried out of differences in the minimum energy conformations obtained in the previous paper by energy minimization starting from conformations sampled by a Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. Main conformational differences in each pair of energy minima are found usually localized in several side chains and in a few local main chain segments. Such side chains and local main chain segments are found to take a few distinct local conformations in the minimum energy conformations. Energy minimum conformations can thus be described in terms of combinations of these multiple local conformations.     

The magnitude of the backbone conformational entropy change in protein folding

The magnitude of the conformational entropy change experienced by the peptide backbone upon protein folding was investigated experimentally and by computational analysis. Experimentally, two different pairs of mutants of a 33 amino acid peptide corresponding to the leucine zipper region of GCN4 were used for high-sensitivity microcalorimetric analysis. Each pair of mutants differed only by having alanine or glycine at a specific solvent-exposed position under conditions in which the differences in stability could be attributed to differences in the conformational entropy of the unfolded state. The mutants studied were characterized by different stabilities but had identical heat capacity changes of unfolding (ΔC_p), identical solvent-related entropies of unfolding (ΔS_solv), and identical enthalpies of unfolding (ΔH) at equivalent temperatures. Accordingly, the differences in stability between the different mutants could be attributed to differences in conformational entropy. The computational studies were aimed at generating the energy profile of backbone conformations as a function of the main chain dihedral angles ϕ and ϱ. The energy profiles permit a direct calculation of the probability distribution of different conformers and therefore of the conformational entropy of the backbone. The experimental results presented in this paper indicate that the presence of the methyl group in alanine reduces the conformational entropy of the peptide backbone by 2.46 ± 0.2 cal/K · mol with respect to that of glycine, consistent with a 3.4-fold reduction in the number of allowed conformations in the alanine-containing peptides. Similar results were obtained from the energy profiles. The computational analysis also indicates that the addition of further carbon atoms to the side chain had only a small effect as long as the side chains were unbranched at position β. A further reduction with respect to Ala of only 0.61 and 0.81 cal/K · mol in the backbone entropy was obtained for leucine and lysine, respectively. β-branching (Val) produces the largest decrease in conformational entropy (1.92 cal/K · mol less than Ala). Finally, the backbone entropy change associated with the unfolding of an α-helix is 6.51 cal/K · mol for glycine. These and previous results have allowed a complete estimation of the conformational entropy changes associated with protein folding. © 1996 Wiley-Liss, Inc.     

Interaction between wind-induced seiches and convective cooling governs algal distribution in a canyon-shaped reservoir

1. Wind is considered the dominant factor controlling phytoplankton distribution in lentic environments. In canyon‐shaped reservoirs, wind tends to blow along the main axis generating internal seiches and advective water movements that jointly with biological features of algae can produce a heterogeneous phytoplankton distribution. Turbulence generated by wind stress and convection will also affect the vertical distribution of algae, depending on their sinking properties. 2. We investigated the vertical and horizontal distribution of phytoplankton during the stratification period in Sau Reservoir (NE Spain). Sites along the main reservoir axis were sampled every 4 h for 3 days, and profiles of chlorophyll‐a and temperature were made using a fluorescent FluoroProbe, which can discriminate among the main algal groups. Convective and wind shear velocity scales, and energy dissipation were calculated from meteorological data, and simulation experiments were performed to describe non‐measured processes, like vertical advection and sinking velocity of phytoplankton. 3. Wind direction changed from day to night, producing a diel thermocline oscillation and an internal seiche. Energy dissipation was moderate during the night, and mainly attributed to convective cooling. During the day the energy dissipation was entirely attributed to wind shear, but values indicated low turbulence intensity. 4. The epilimnetic algal community was mainly composed of diatoms and chlorophytes. Chlorophytes showed a homogeneous distribution on the horizontal and vertical planes. Diatom horizontal pattern was also homogeneous, because the horizontal advective velocities generated by wind forcing were not high enough to develop phytoplankton gradients along the reservoir. 5. Diatom vertical distribution was heterogeneous in space and time. Different processes dominated in different regions of the reservoir, due to the interaction between seiching and the daily cycle of convective‐mediated turbulence. As the meteorological forcing followed a clear daily pattern, we found very different diatom sedimentation dynamics between day and night. Remarkably, these dynamics were asynchronous in the extremes of the seiche, implying that under the same meteorological forcing a diatom population can show contrasting sedimentation dynamics at small spatial scales (approximately 10³ m). This finding should be taken into account when interpreting paleolimnological records from different locations in a lake. 6. Vertical distribution of non‐motile algae is a complex process including turbulence, vertical and horizontal advection, variations in the depth of the mixing layer and the intrinsic sinking properties of the organisms. Thus, simplistic interpretations considering only one of these factors should be regarded with caution. The results of this work also suggest that diatoms can persist in stratified water because of a synergistic effect between seiching and convective turbulence.     

A physical derivation of nutrient-uptake rates in coral reefs: effects of roughness and waves

Mass-transfer rates between water and benthos are derived based on the dissipation of energy by the benthic communities of coral reefs. Roughness of the benthic communities causes currents and waves to dissipate energy on reef flats at rates which far exceed ocean values of energy dissipation. The derivation here shows that first-order rate constants for nutrient uptake are (1) proportional to energy dissipation to the 0.25 root, (2) proportional to the bottom shear stress to the 0.4 root, and (3) proportional to current speed to the 0.75 root (decreasing to the 0.4 root under extreme wave activity). The shear stress, thus nutrient uptake, is positively correlated to the large-scale roughness, and to excess wave height (above the breaking height) of incoming waves. These causal relationships between nutrient-uptake rates and dissipation of energy support the general observations of reef zonation and reef metabolic rates, and are the paramount reason that coral reefs can maintain high productivity in low-nutrient tropical waters.