Low-frequency motions in protein molecules. Beta-sheet and beta-barrel.

Low-frequency internal motions in protein molecules play a key role in biological functions. Based on previous work with alpha-helical structure, the quasi-continuum model is extended to the beta-structure, another fundamental element in protein molecules. In terms of the equations derived here, one can easily calculate the low-frequency wave number of a beta-sheet in an accordionlike motion, and the low-frequency wave number of a beta-barrel in a breathing motion. The calculated results for immunoglobulin G and concanavalin A agree well with the observations. These findings further verify that the observed low-frequency motion (or the so-called dominant low-frequency mode) in a protein molecule is essentially governed by the collective fluctuations of its weak bonds, especially hydrogen bonds, and the internal displacement of the massive atoms therein, as described by the quasi-continuum model.     

Low-frequency vibrations of helical structures in protein molecules.

A physically intuitive and mathematically easily handled formula is presented for calculating the low-frequency vibrations of helical structures in protein molecules. alpha-Chymotrypsin is taken as an example, and the calculated result shows precise agreement with observations of the low-frequency Raman spectra. As reflected in the formula, this kind of low frequency is very sensitive to the conformation of a biomacromolecule, and can therefore serve as a vehicle to investigate the mechanism of action of a biomacromolecule from the viewpoint of dynamics. On this basis a feasible experiment is suggested by which one can examine the relationship between a presumed mode of low-frequency vibration in a biomacromolecule and its activity.     

Identification of low-frequency modes in protein molecules.

It is demonstrated that the observed low-frequency motions with wave numbers of 22 cm-1 and 25 cm-1 for insulin and lysozyme respectively originate from the accordion-like motions of the principal helices therein. The calculated results based on such a model are in good agreement with the observed values. During calculations the role of the internal microenvironment upon the low-frequency motion is naturally revealed, so as to elucidate as well why this kind of low-frequency motion is so sensitive to the conformations of proteins observed.     

Local dynamics of DNA probed with optical absorption spectroscopy of bound ethidium bromide.

We have studied the local dynamics of calf thymus double-helical DNA by means of an "optical labeling" technique. The study has been performed by measuring the visible absorption band of the cationic dye ethidium bromide, both free in solution and bound to DNA, in the temperature interval 360-30 K and in two different solvent conditions. The temperature dependence of the absorption line shape has been analyzed within the framework of the vibronic coupling theory, to extract information on the dynamic properties of the system; comparison of the thermal behavior of the absorption band of free and DNA-bound ethidium bromide gave information on the local dynamics of the double helix in the proximity of the chromophore. For the dye free in solution, large spectral heterogeneity and coupling to a "bath" of low-frequency (soft) modes is observed; moreover, anharmonic motions become evident at suitably high temperatures. The average frequency of the soft modes and the amplitude of anharmonic motions depend upon solvent composition. For the DNA-bound dye, at low temperatures, heterogeneity is decreased, the average frequency of the soft modes is increased, and anharmonic motions are hindered. However, a new dynamic regime characterized by a large increase in anharmonic motions is observed at temperatures higher than approximately 280 K. The DNA double helix therefore appears to provide, at low temperatures, a rather rigid environment for the bound chromophore, in which conformational heterogeneity is reduced and low-frequency motions (both harmonic vibrations and anharmonic contributions) are hindered. The system becomes anharmonic at approximately 180 K; however, above approximately 280 K, anharmonicity starts to increase much more rapidly than for the dye free in solution; this can be attributed to the onset of wobbling of the dye in its intercalation site, which is likely connected with the onset of (functionally relevant) DNA motions, involving local opening/unwinding of the double helix. As shown by parallel measurements of the melting curves, these motions precede the melting of the double helix and depend upon solvent composition much more than does the melting itself.     

RF absorption involving biological macromolecules

The fundamental intramolecular frequency of a globular protein can be obtained from the measurements of acoustic velocities of bulk protein matter. This lowest frequency for common size molecules is shown to be above several hundred GHz. All modes below this frequency would then be intermolecular modes or bulk modes of the molecule and surrounding matter or tissue. The lowest frequency modes of an extended DNA double helix are also shown to be bulk modes because of interaction with water. Only DNA modes, whose frequency is well above 4 GHz, can be intrahelical modes, that is, confined to the helix rather than in the helix plus surroundings. Near 4 GHz, they are heavily damped and, therefore, not able to resonantly absorb. Modes that absorb radio frequency (RF) below this frequency are bulk modes of the supporting matter. Bulk modes rapidly thermalize all absorbed energy. The implication of these findings for the possibility of athermal RF effects is considered. The applicability of these findings for other biological molecules is discussed.     

153 Wonderful roles of the entropy in protein dynamics,binding and folding

The entropy, which is central to the second law of thermodynamics, determines that the thermal energy always flows spontaneously from regions of higher temperature to regions of lower temperature. In the protein–solvent thermodynamic system, the entropy is defined as a measure of how evenly the thermal energy would distribute over the entire system (Liu et al., 2012). Such tendency to distribute energy as evenly as possible will reduce the state of order of the initial system, and hence, the entropy can be regarded as an expression of the disorder, or randomness of the system (Yang et al., 2012). For a protein–solvent system under a constant solvent condition, the origin of entropy is the thermal energy stored in atoms, which makes atoms jostle around and bump onto one another, thus leading to vibrations of the covalent bonds connecting two atoms (occurring on the fs timescale) and the rotational and translational motions of amino acid side chain groups (occurring on ps timescale) and water molecules. These motions break the noncovalent bonds around structural regions that are weakly constrained thereby triggering the competitive interactions among residues or between residues and water molecules leading ultimately to the loop motions (occurring on ns timescale) around the protein surface. The loop motions can further transmit either through the water network around the protein surface or via specific structural components (such as the hinge-bending regions) over the entire protein molecule leading to large concerted motions (occurring on μs to s timescales) that are most relevant to protein functions (Amadei, Linssen & Berendsen, 1993; Tao, Rao & Liu, 2010). Thus, the multiple hierarchies of the protein dynamics on distinct timescales (Henzler-Wildman & Kern, 2007) are a consequence of the cascade amplification of the microscopic motions of atoms and groups for which the entropy originating from atomic thermal energy is most fundamental. In the case of protein–ligand binding, the importance of the entropy is embodied in the following aspects. (i) The release of the water molecule kinetic energy (which is a process of the solvent entropy maximization) will cause Brownian motions of individual water molecules which result in strong Brownian bombardments to solute molecules causing molecule wanders/diffusions and subsequent accident contacts/collisions between proteins and ligands. (ii) Such collisions will inevitably cause water molecule displacement and, if the contact interfaces are properly complementary, the requirement to increase the solvent entropy would further displace the water network around the binding interfaces thus leading to the formation the initial protein-ligand complex. (iii) In the initial complex, the loose association of the two partners provide the opportunity for protein to increase conformational entropy, thus triggering the conformational adjustments through competitive interaction between protein residues and ligand, leading ultimately to the formation of tightly associated complex (Liu et al., 2012). In the protein folding process, the first stage, i.e. the rapid hydrophobic collapse (Agashe, Shastry & Udgaonkar, 1995; Dill, 1985), is in fact driven by the effect of the solvent entropy maximization. Specifically, the requirement to maintain as many as possible the dynamic hydrogen bonds among the water molecules will squeeze/sequestrate the hydrophobic amino acid side chains into the interior of the folding intermediates and expose the polar/charged side chains onto the intermediate surface. This will minimize the solvent accessible surface area of the folding intermediates and as thus maximize the entropy of the solvent. The resulting molten globule states (Ohgushi & Wada, 1983) may contain a few secondary structural components and native tertiary contacts, while many native contacts, or close residue–residue interactions present in the native state have not yet formed. However, the nature to increase the protein conformational entropy can trigger a further conformational adjustment process, i.e. the conformational entropy increase breaks the transient secondary or tertiary contacts and triggers the competitive interactions among protein residues and between residues and water. This process may repeat many rounds until the negative enthalpy change resulting from the noncovalent formations can overcompensate for protein conformational entropy loss. In summary, we consider that the tendency to maximize the entropy of the protein–solvent system, which originates from the atomic thermal energy, is the most fundamental driving factor for protein folding, binding, and dynamics, whereas the enthalpy reduction, an opposing factor that tends to make the system become ordered, can compensate for the effect of entropy loss to ultimately allow the system to reach equilibrium at the free energy minima, either global or local.     

Crystal structures of apo- and holo-bovine alpha-lactalbumin at 2. 2-A resolution reveal an effect of calcium on inter-lobe interactions

High affinity binding of Ca(2+) to alpha-lactalbumin (LA) stabilizes the native structure and is required for the efficient generation of native protein with correct disulfide bonds from the reduced denatured state. A progressive increase in affinity of LA conformers for Ca(2+) as they develop increasingly native structures can account for the tendency of the apo form to assume a molten globule state and the large acceleration of folding by Ca(2+). To investigate the effect of calcium on structure of bovine LA, x-ray structures have been determined for crystals of the apo and holo forms at 2.2-A resolution. In both crystal forms, which were grown at high ionic strength, the protein is in a similar global native conformation consisting of alpha-helical and beta-subdomains separated by a cleft. Even though alternative cations and Ca(2+) liganding solvent molecules are absent, removal of Ca(2+) has only minor effects on the structure of the metal-binding site and a structural change was observed in the cleft on the opposite face of the molecule adjoining Tyr(103) of the helical lobe and Gln(54) of the beta-lobe. Changes include increased separation of the lobes, loss of a buried solvent molecule near the Ca(2+)-binding site, and the replacement of inter- and intra-lobe H-bonds of Tyr(103) by interactions with new immobilized water molecules. The more open cleft structure in the apo protein appears to be an effect of calcium binding transmitted via a change in orientation of helix H3 relative to the beta-lobe to the inter-lobe interface. Calcium is well known to promote the folding of LA. The results from the comparison of apo and holo structures of LA provide high resolution structural evidence that the acceleration of folding by Ca(2+) is mediated by an effect on interactions between the two subdomains.     

The biological functions of low-frequency vibrations (phonons). 4. Resonance effects and allosteric transition

Based on the internal structure of oligoprotein as well as the basic physical characteristics+ of vibrations, it is deduced that the low-frequency vibrations possess some exceptional functions in transmitting biological information at the molecular level. In particular, according to the viewpoint of energy exchange and intramolecular displacement, it is demonstrated that the low-frequency resonance plays a very significant role during the dynamic process of allosterism of an oligomeric protein molecule. Furthermore, the cooperative reaction between hemoglobins and ligands is taken as an example, through which it is seen that some observed phenomena, whose dynamic principle has thus far been unclear, can be explicitly interpreted in terms of the concept of low-frequency resonance.     

Origin of low-frequency motions in biological macromolecules. A view of recent progress in the quasi-continuity model

The recent progress in the quasi-continuity model and its applications in studying the low-frequency internal motions of biological macromolecules have been surveyed. Emphasis is placed on revealing the origin of this kind of internal collective motion, which involves many atoms and has significant biological functions. In light of such a line, the low-frequency motions in alpha-helix structure, beta-structure (including beta-sheet and beta-barrel), and DNA double-helix structure, the three most fundamental component elements in biological macromolecules, are discussed, and the corresponding physical pictures described. It turns out that the low-frequency motion in biological macromolecules originates from their two common intrinsic characteristics, i.e., they possess a series of weak bonds, such as hydrogen bonds and salt bridges, and a substantial mass distributed over the region containing those weak bonds.     

The biological functions of low-frequency vibrations (phonons). VI. A possible dynamic mechanism of allosteric transition in antibody molecules

By means of the quasi-continuity theory developed recently [Chou, K.-C. (1983) Biochem. J. 209 , 573–580; Chou, K.-C. (1984) Biochem. J. 221 , 27–31; Chou, K.-C. (1985) Biophys. J. 48 , 289–297], the low-frequency breathing motions of the β-barrels in the 12 domains of an IgG antibody molecule have been calculated. The results are in very good agreement with the observed low-frequency peaks in the Raman spectra. Based on this, the resonant couplings among the domains located at its different regions are discussed. It turns out that some very interesting functions of antibody molecules, such as the “chelate effect” and “trigger effect” whose dynamic principle has been so far unknown, can be clearly elucidated in terms of a fundamental natural law; i.e., it is through the channel of low-frequency resonance that the “signal” (or energy) in an antibody is transmitted from one location to the other, so as to be able to induce the relevant conformational change required for operating those important biological functions. The physical picture illustrated here once again provides us with a classic example that structure and function in nature are highly harmonized.     

Breathing mode of conformational fluctuations in globular proteins.

Globular proteins in the native state are assumed to behave as continuous elastic spheres in the low frequency breathing motions. Reasonable values of Young's modulus E = 10(11) dyne/cm2 and the radius of the sphere ro = 20 A, yield a wave number of 26 cm-1 for the fundamental vibration of the sphere. The peak at around 30 cm-1 in the laser Raman spectra of native alpha-chymotrypsin and pepsin observed by Brown et al. might be assigned to the breathing motion which the native proteins undergo as continuous elastic bodies.     

Thermal motions of surface alpha-helices in the D-galactose chemosensory receptor. Detection by disulfide trapping.

The D-galactose chemosensory receptor of Escherichia coli is a .32 kDa globular protein possessing two distinct structural domains, each organized in an alpha/beta folding motif. Helices I and X lie at adjacent approximately parallel positions on the surface of the N-terminal domain, near the hinge region. In order to analyze the relative thermal motions of these two helices, the present study utilizes a generalizable disulfide trapping approach: first, site-directed mutagenesis is used to place a pair of cysteine residues at locations of interest on the protein surface, then disulfide bond formation is used to trap intramolecular cysteine-cysteine collisions resulting from thermal motions. Specifically, four engineered di-cysteine receptors have been constructed, each possessing one cysteine at position 26 on helix I, and a second cysteine at varying positions on helix X. A fifth control receptor possesses one cysteine at position 26, and a second on the opposite surface of the molecule. These surface cysteine substitutions have little or no effect on the measurable receptor parameters as judged by ligand binding equilibria and kinetics, protein stability, and 19F nuclear magnetic resonance, indicating that the engineered receptors are useful probes of native backbone dynamics. Spatial and kinetic features of backbone motions have been investigated by measuring intramolecular disulfide formation rates for cysteine pairs in the fully liganded receptor. The resulting rates decrease monotonically with increasing distance between cysteines in the crystal structure, while no disulfide formation is observed for the control pair unless the molecule is unfolded. The minimum translational amplitudes of the observed backbone motions range from 4.5 to 15.2 A, and the minimum rotational amplitudes are as large as 35 degrees. For each motion the rate of intramolecular sulfhydryl-sulfhydryl collision has been estimated from the measured rate of disulfide formation: the 4.5 and 15.2 A translations yield approximately 10(4) and approximately 10 collisions s-1 molecule-1, respectively. These collision rates, which are faster than ligand dissociation, likely underestimate the actual motional frequencies since only an undetermined fraction of the total motions yield collisions. The simplest plausible trajectory capable of producing such collisions is a rate-limiting translation of one or both helices along their long axes, coupled with minor helix rotations. When sugar is removed from the receptor, a substantial increase in backbone dynamics is observed, indicating the presence of new long-range backbone trajectories. Overall, the results suggest that internal motions in proteins may have larger amplitudes than previously observed.     

Calculation of the standard molal thermodynamic properties of aqueous biomolecules at elevated temperatures and pressures II. Unfolded proteins

Equations of state for completely unfolded proteins have been generated from group additivity algorithms and the revised Helgeson-Kirkham-Flowers (HKF) equations of state to compute the standard molal thermodynamic properties of these molecules at elevated temperatures and pressures. The requisite equations of state parameters were computed from those of groups retrieved by regression of experimental calorimetric and densimetric data reported in the literature. This approach permits calculation of the standard molal thermodynamic properties as a function of temperature and pressure for any completely unfolded protein for which the amino acid sequence is known. Calculations of this kind have been carried out for 11 thermophilic proteins. The thermodynamic properties reported below can be combined with those for protein unfolding to compute the corresponding properties of completely folded (i.e. native) proteins.     

Picosecond timescale rigid-helix and side-chain motions in deoxymyoglobin

The contribution of rigidbody motions to the atomic trajectories in a 100 ps molecular dynamics simulation of deoxymyoglobin is examined. Two typesof rigid-body motions are considered: one in which the helices are rigid units and one in which the side-chains are rigid units. Using a quaternionbased algorithm, fits of the rigid reference structures are made to each time frame of the simulation to derive trajectories of the rigid-body motions. The fitted trajectories are analysed in terms of atomic position fluctuations, mean-square displacements as a function of time, velocity autocorrelation functions and densities of states. The results are compared with the corresponding quantities calculated from the full trajectory. The relative contribution of the rigid helix motions to the helix atom dynamics depends on which quantity is examined and on which subset of atoms is chosen: rigid-helix motions contribute 86% of the rms helix backbone atomic position fluctuations, but 30% of the helix,: atom (backbone and side-chain) mean square displacements and only 1.1% of total kinetic energy. Only very low-frequency motions contribute to the rigid-helix dynamics; the rigid-body analysis allows characteristic rigid-helix vibrations to be identified and described. Treating the side-chains as rigid bodies is foundto be an excellent approximation to both their diffusive and vibrationalmean-square displacements: 96% of side-chain atom mean-square displacements originate from rigid side-Chain motions. However, the errors in theside-chain atomic positional fits are not always small. An analysis is madeof factors contributing to the positional error for different types of side-chain. © Wiley-Liss, Inc.     

A molecular dynamics study of the C-terminal fragment of the L7/L12 ribosomal protein. Secondary structure motion in a 150 picosecond trajectory

A 150 picosecond molecular dynamics computer simulation of the C-terminal fragment of the L7/L12 ribosomal protein from Escherichia coli is reported. The molecular dynamics results are compared with the available high-resolution X-ray data in terms of atomic positions, distances and positional fluctuations. Good agreement is found between the molecular dynamics results and the X-ray data. The form and parameters of the interaction potential energy function and the procedures for deriving it are discussed. Some current misunderstandings concerning the ways of evaluating the efficiency of molecular dynamics algorithms and of application of bond-length constraints in protein simulations are cleared up. The 150 picosecond trajectory has been scanned in a search for correlated motions within and between secondary structure elements. The beta-strands have diffusional stretching modes, and uncorrelated transversal displacements. The dynamic analysis of alpha-helices shows a variety of features. The atomic fluctuations differ between the helix ends; this effect reflects long time-scale motions. Two alpha-helices, alpha A and alpha C, show diffusive longitudinal stretching modes. The third helix, alpha B, has a correlated asymmetric longitudinal stretching; the N-terminal part dominates this behaviour. Furthermore, alpha B presents a librational motion with respect to the other parts of the molecule with a frequency of approximately 5 cm-1. This motion is coupled to helix stretching. Interestingly, the regions of highly conserved residues contain the most mobile parts of the molecule.     

Low-frequency vibrations of a Mg pyropheophorbide-histidine complex]

The spectrum of vibrations and normal model for the Mg piropheophorbide-histidine complex was calculated using the MNDO-PM3 (MOPAC) semiempirical quantum chemical method. The delocalization index and the distribution function were introduced to describe the shape of normal vibrations. The greatest part (approximately 65%) of the low-frequency vibrations (1-400 cm-1) was shown to delocalize over both the His and Mg piropheophorbide molecules. Leu, Met, and Asp were also studied as the fifth ligand to the Mg piropheophorbide molecule. It is concluded that the fifth amino acid ligand to porphyrin molecules causes marked geometrical distortions in porphyrin, and induces a new, compared to four coordinated pigment, spectrum of normal modes.     

Contribution of secondary structure to the heat capacity and enthalpy distribution of the unfolded state in proteins.

We have recently shown that one can construct the enthalpy distribution for protein molecules from experimental knowledge of the temperature dependence of the heat capacity. For many proteins the enthalpy distribution evaluated at the midpoint of the denaturation transition (corresponding to the maximum in the heat capacity vs temperature curve) is broad and biphasic, indicating two different populations of molecules (native and unfolded) with distinctly different enthalpies. At temperatures above the denaturation point, the heat capacity for the unfolded state in many proteins is quite large and using the analysis just mentioned, we obtain a gaussian-like enthalpy distribution that is very broad. A large value of the heat capacity indicates that there are structural changes going on in the unfolded state above the transition temperature. In the present paper we investigate the origin of this large heat capacity by considering the presence of changing amounts of secondary structure (specifically, alpha-helix) in the unfolded state. For this purpose we use the empirical estimates of the Zimm-Bragg sigma and s factors for all of the native amino acids in water as determined by Scheraga and co-workers. Using myoglobin as an example, we calculate probability profiles and distribution functions for the total number of helix states in the specific-sequence molecule. Given the partition function for the specific-sequence molecule, we can then calculate a set of enthalpy moments for the molecule from which we obtain a good estimate of the enthalpy distribution in the unfolded state. This distribution turns out to be quite narrow when compared with the distribution obtained from the raw heat capacity data. We conclude that there must be other major structural changes (backbone and solvent) that are not accounted for by the inclusion of alpha-helix in the unfolded state.     

Interactions of bovine neurophysins with neurohypophyseal hormones. On the role of tyrosine-49.

Reaction of tetranitromethane with the lone tyrosine residue of bovine neurophysin I and II, tyrosine-49, gave nitro derivatives of these proteins which were obtained in a highly purified form by preparative electrophoresis. Equilibrium dialysis experiments indicated clearly that oxytocin binding remained essentially unaffected by the chemical modification of tyrosine-49. However, in the case of (8-lysine)vasopressin, the nitrated protein was found to bind only 1 hormone molecule in contrast to the 2 vasopressin molecules bound by the native protein. Ultraviolet absorption difference spectroscopy measurements between 250 nm and 300 nm indicated that upon binding of (2-phenylalanine, 8-lysine)vasopressin, tyrosine-49 of native neurophysin undergoes a change of microenvironment from less to more polar surroundings. Studies of the nitrotyrosyl-49 chromophore of neurophysin by ab sorption spectroscopy in the absence and presence of oxytocin or (8-lysine)vasopressin confirmed this finding. Since dimethylsulfoxide solvent perturbation studies suggested that in the Cys(Me)-Phe-Ile-NH2-neurophysin I complex, tyrosine-49 is more exposed to solvent than in neurophysin I alone, it is concluded that this residue is unmasked by conformational changes upon complex formation.     

The 'glass transition' in protein dynamics: what it is, why it occurs, and how to exploit it

All proteins undergo a dramatic change in their dynamical properties at approximately 200 K. Above this temperature, their dynamic behavior is dominated by large-scale collective motions of bonded and nonbonded groups of atoms. At lower temperatures, simple harmonic vibrations predominate. The transition has been described as a 'glass transition' to emphasize certain similarities between the change in dynamic behavior of individual protein molecules and the changes in viscosity and other properties of liquids when they form a glass. The glass transition may reflect the intrinsic temperature dependence of the motions of atoms in the protein itself, in the bound solvent on the surface of the protein, or it may reflect contributions from both. Protein function is significantly altered below this transition temperature; a fact that can be exploited to trap normally unstable intermediates in enzyme-catalyzed reactions and stabilize them for periods long enough to permit their characterization by high-resolution protein crystallography.     

Diagonalization in a mixed basis: A method to compute low-frequency normal modes for large macromolecules

Low-frequency collective motions in proteins are generally very important for their biological functions. To study such motions, harmonic dynamics proved most useful since it is a straightforward method; it consists of the diagonalization of the Hessian matrix of the potential energy, yielding the vibrational spectrum and the directions of internal motions. Unfortunately, the diagonalization of this matrix requires a large computer memory, which is a limiting factor when the protein contains several thousand atoms. To circumvent this limitation we have developed three methods that enable us to diagonalize large matrices using much less computer memory than the usual harmonic dynamics. The first method is approximate; it consists of diagonalizing small blocks of the Hessian matrix, followed by the coupling of the low-frequency modes obtained for each block. It yields the low-frequency vibrational spectrum with a maximum error of 20%. The second method consists, after diagonalizing small blocks, of coupling the high- and low-frequency modes using an iterative procedure. It yields the exact low-frequency normal modes, but requires a long computational time with convergence problems. The third method, DIMB (Diagonalization in a Mixed Basis), which has the best performance, consists of coupling the approximate low-frequency modes with the mass-weighted cartesian coordinates, also using an iterative procedure. It reduces significantly the required computer memory and converges rapidly. The eigenvalues and eigenvectors obtained by this method are without significant error in the chosen frequency range. Moreover, it is a general method applicable to any problem of diagonalization of a large matrix. We report the application of these methods to a deca-alanine helix, trypsin inhibitor, a neurotoxin, and lysozyme. © 1993 John Wiley & Sons, Inc.