Protein dynamics and 1/f noise

Phosphorylation of rat brain purified mitochondrial Voltage-Dependent Anion Channel by c-Jun N-terminal kinase-3 modifies open-channel noise

The drift kinetic energy of ionic flow through single ion channels cause vibrations of the pore walls which are observed as open-state current fluctuations (open-channel noise) during single-channel recordings. Vibration of the pore wall leads to transitions among different conformational sub-states of the channel protein in the open-state. Open-channel noise analysis can provide important information about the different conformational sub-state transitions and how biochemical modifications of ion channels would affect their transport properties. It has been shown that c-Jun N-terminal kinase-3 (JNK3) becomes activated by phosphorylation in various neurodegenerative diseases and phosphorylates outer mitochondrion associated proteins leading to neuronal apoptosis. In our earlier work, JNK3 has been reported to phosphorylate purified rat brain mitochondrial voltage-dependent anion channel (VDAC) in vitro and modify its conductance and opening probability. In this article we have compared the open-state noise profile of the native and the JNK3 phosphorylated VDAC using Power Spectral Density vs frequency plots. Power spectral density analysis of open-state noise indicated power law with average slope value α ≈1 for native VDAC at both positive and negative voltage whereas average α value < 0.5 for JNK3 phosphorylated VDAC at both positive and negative voltage. It is proposed that 1/f¹ power law in native VDAC open-state noise arises due to coupling of ionic transport and conformational sub-states transitions in open-state and this coupling is perturbed as a result of channel phosphorylation.

Decoherence and spin echo in biological systems

The spin-echo approach is extended to include biocomplexes for which the interaction with dynamical noise, produced by the protein environment, is strong. Significant restoration of the free induction decay signal due to homogeneous (decoherence) and inhomogeneous (dephasing) broadening is demonstrated analytically and numerically for both an individual dimer of interacting chlorophylls and for an ensemble of dimers. Our approach does not require the use of small interaction constants between the electron states and the protein fluctuations. It is based on an exact and closed system of ordinary differential equations that can be easily solved for a wide range of parameters that are relevant for bioapplications.

Role of protein fluctuation correlations in electron transfer in photosynthetic complexes

We consider the dependence of the electron transfer in photosynthetic complexes on correlation properties of random fluctuations of the protein environment. The electron subsystem is modeled by a finite network of connected electron (exciton) sites. The fluctuations of the protein environment are modeled by random telegraph processes, which act either collectively (correlated) or independently (uncorrelated) on the electron sites. We derived an exact closed system of first-order linear differential equations with constant coefficients, for the average density matrix elements and for their first moments. Under some conditions, we obtained analytic expressions for the electron transfer rates and found the range of parameters for their applicability by comparing with the exact numerical simulations. We also compared the correlated and uncorrelated regimes and demonstrated numerically that the uncorrelated fluctuations of the protein environment can, under some conditions, either increase or decrease the electron transfer rates.

Frequency response of a protein to local conformational perturbations

Signals created by local perturbations are known to propagate long distances through proteins via backbone connectivity and nonbonded interactions. In the current study, signal propagation from the flexible ligand binding loop to the rest of Protein Tyrosine Phosphatase 1B (PTP1B) was investigated using frequency response techniques. Using restrained Targeted Molecular Dynamics (TMD) potential on WPD and R loops, PTP1B was driven between its crystal structure conformations at different frequencies. Propagation of the local perturbation signal was manifested via peaks at the fundamental frequency and upper harmonics of 1/f distributed spectral density of atomic variables, such as Cα atoms, dihedral angles, or polar interaction distances. Frequency of perturbation was adjusted high enough (simulation length >∼10×period of a perturbation cycle) not to be clouded by random diffusional fluctuations, and low enough (<∼0.8 ns(-1)) not to attenuate the propagating signal and enhance the contribution of the side-chains to the dissipation of the signals. Employing Discrete Fourier Transform (DFT) to TMD simulation trajectories of 16 cycles of conformational transitions at periods of 1.2 to 5 ns yielded Cα displacements consistent with those obtained from crystal structures. Identification of the perturbed atomic variables by statistical t-tests on log-log scale spectral densities revealed the extent of signal propagation in PTP1B, while phase angles of the filtered trajectories at the fundamental frequency were used to cluster collectively fluctuating elements. Hydrophobic interactions were found to have a higher contribution to signal transduction between side-chains compared to the role of polar interactions. Most of in-phase fluctuating residues on the signaling pathway were found to have high identity among PTP domains, and located over a wide region of PTP1B including the allosteric site. Due to its simplicity and efficiency, the suggested technique may find wide applications in identification of signaling pathways of different proteins.

Fractal symmetry of protein interior: what have we learned?

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide-dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to 'α/α toroid' (all-α class) and 'PLP-dependent transferase-like' domains (α/β class) are highly conducive. In contrast, the interiors of 'zinc finger design' ('designed proteins') and 'knottins' ('small proteins') were identified as folds with the least conducive electrostatic environments. The fold 'conotoxins' (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide-dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide-dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Relaxation dynamics of nucleosomal DNA

Recent experimental and theoretical evidence demonstrates that proteins and water in the hydration layer can follow complex stretched exponential or power law relaxation dynamics. Here, we report on a 50 ns all atom molecular dynamics (MD) simulation of the yeast nucleosome, where the interactions between DNA, histones, surrounding water and ions are explicitly included. DNA interacts with the histone core in 14 locations, approximately every 10.4 base pairs. We demonstrate that all sites of interaction exhibit anomalously slow power law relaxation, extending up to 10 ns, while fast exponential relaxation dynamics of hundreds of picoseconds applies to DNA regions outside these locations. The appearance of 1/f(alpha) noise or pink noise in DNA dynamics is ubiquitous. For histone-bound nucleotide dynamics alpha --> 1 and is a signature of complexity of the protein-DNA interactions. For control purposes two additional DNA simulations free of protein are conducted. Both utilize the same sequence of DNA, as found the in the nucleosome. In one simulation the initial conformation of the double helix is a straight B-form. In the other, the initial conformation is super helical. Neither of these simulations exhibits the variation of alpha as a function of position, the measure of power law for dynamical behavior, which we observe in the nucleosome simulation. The unique correspondence (high alpha to DNA-histone interaction sites, low alpha to free DNA sites), suggests that alpha may be an important and new quantification of protein-DNA interactions for future experiments.

Network analysis of protein dynamics

The network paradigm is increasingly used to describe the topology and dynamics of complex systems. Here, we review the results of the topological analysis of protein structures as molecular networks describing their small-world character, and the role of hubs and central network elements in governing enzyme activity, allosteric regulation, protein motor function, signal transduction and protein stability. We summarize available data how central network elements are enriched in active centers and ligand binding sites directing the dynamics of the entire protein. We assess the feasibility of conformational and energy networks to simplify the vast complexity of rugged energy landscapes and to predict protein folding and dynamics. Finally, we suggest that modular analysis, novel centrality measures, hierarchical representation of networks and the analysis of network dynamics will soon lead to an expansion of this field.

Analyte–receptor binding on surface plasmon resonance biosensors: a fractal analysis of Cre–loxP interactions and the influence of Cl, O, and S on drug–liposome interactions

A fractal analysis of the association and dissociation (whereever applicable) of Cre-loxP interactions and drug-liposome interactions on a sensor chip surface is presented. In both of these cases a dual-fractal analysis is required to adequately describe the association kinetics. The dissociation kinetics for Cre-loxP interactions is adequately described by a single-fractal analysis. The dual-fractal analysis used to describe the association kinetics of Cre-loxP interactions is consistent with the original two-step mechanism presented using a surface plasmon resonance biosensor. Our analysis includes both diffusion and surface effects by introducing the fractal dimension which makes quantitative the degree of heterogeneity on the sensor chip surface. Affinities are provided. Only the association kinetics were analysed for drug-liposome interactions since the initial sections of the dissociation curves were too steep to obtain reasonable drug-liposome complex concentration values on the sensor chip with time. Attempts made to relate the association rate coefficients with the molecular weight of the drug were unsuccessful. On using desipramine and imipramine as "arbitrarily selected standards" or "references" (only C, H, and N atoms present), it was noticed from the data analysed that the inclusion of the O and S atoms in the drug leads to a decrease in the association rate coefficients, ka1 (or k1) and ka2 (or k2) (compared with the arbitrarily selected standards or references). Similarly, the addition of the Cl atom in the drug leads to an increase in the association rate coefficient (compared with the arbitrarily selected standards or references). More data needs to be analysed to determine whether this is true for other drugs also.

Ion channel gating: a first-passage time analysis of the Kramers type

The opening rate of voltage-gated potassium ion channels exhibits a characteristic knee-like turnover where the common exponential voltage dependence changes suddenly into a linear one. An explanation of this puzzling crossover is put forward in terms of a stochastic first passage time analysis. The theory predicts that the exponential voltage dependence correlates with the exponential distribution of closed residence times. This feature occurs at large negative voltages when the channel is predominantly closed. In contrast, the linear part of voltage dependence emerges together with a nonexponential distribution of closed dwelling times with increasing voltage, yielding a large opening rate. Depending on the parameter set, the closed-time distribution displays a power law behavior that extends over several decades.

Statistical analysis of the fractal gating motions of the enzyme acetylcholinesterase

The enzyme acetylcholinesterase has an active site that is accessible only by a "gorge" or main channel from the surface, and perhaps by secondary channels such as the "back door." Molecular-dynamics simulations show that these channels are too narrow most of the time to admit substrate or other small molecules. Binding of substrates is therefore "gated" by structural fluctuations of the enzyme. Here, we analyze the fluctuations of these possible channels, as observed in the 10.8-ns trajectory of the simulation. The probability density function of the gorge proper radius (defined in the text) was calculated. A double-peak feature of the function was discovered and therefore two states with a threshold were identified. The relaxation (transition probability) functions of these two states were also calculated. The results revealed a power-law decay trend and an oscillation around it, which show properties of fractal dynamics with a "complex exponent." The cross correlation of potential energy versus proper radius was also investigated. We discuss possible physical models behind the fractal protein dynamics; the dynamic hierarchical model for glassy systems is evaluated in detail.

Glasslike dynamical behavior of the plastocyanin hydration water

The dynamical behavior of water around plastocyanin has been investigated in a wide temperature range by molecular dynamics simulation. The mean square displacements of water oxygen atoms show, at long times, a t(alpha) trend for all temperatures. Below 150 K, alpha is constant and equal to 1; at higher temperatures it drops to a value significantly smaller than 1, and thereafter decreases with increasing temperature. The occurrence of such an anomalous diffusion matches the onset of the dynamical transition observed in the protein. The intermediate scattering function of water is characterized, at high temperature, by a stretched exponential decay evolving, at low temperature, toward a two step relaxation behavior, which becomes more evident on increasing the exchanged wave vector q. Both the mean square displacements and the intermediate scattering functions show, beyond the ballistic regime, a plateau, which progressively extends for longer times as long as the temperature is lowered, such behavior reflecting trapping of water molecules within a cage formed by the nearest neighbors. At low temperature, a low frequency broad inelastic peak is observed in the dynamical structure factor of hydration water; such an excess of vibrational modes being reminiscent of the boson peak, characteristic of disordered, amorphous systems. All these features, which are typical of complex systems, can be traced back to the glassy character of the hydration water and suggest a dynamical coupling occurring at the macromolecule-solvent interface.

Characterization of inositol-1,4,5-trisphosphate-gated channels in the plasma membrane of rat olfactory neurons

It is generally accepted that inositol-1,4,5-trisphosphate (InsP3) plays a role in olfactory transduction. However, the precise mode of action of InsP3 remains controversial. We have characterized the conductances activated by the addition of 10 microM InsP3 to excised patches of soma plasma membrane from rat olfactory neurons. InsP3 induced current fluctuations in 25 of 121 inside-out patches. These conductances could be classified into two groups according to the polarity of the current at a holding potential of +40 to +60 mV (with Ringer's in the pipette and pseudointracellular solution in the bath). Conductances mediating outward currents could be further divided into large- (64 +/- 4 pS, n = 4) and small- (16 +/- 1.7 pS, n = 11) conductance channels. Both small- and large-conductance channels were nonspecific cation channels. The large-conductance channel displayed bursting behavior at +40 mV, with flickering increasing at negative holding potentials to the point where single-channel currents were no longer discernible. The small-conductance channel did not display flickering behavior. The conductance mediating inward currents at +40 to +60 mV reversed at +73 +/- 4 mV (n = 4). The current traces displayed considerable fluctuations, and single-channel currents could not be discerned. The current fluctuations returned to baseline after removal of InsP3. The power density spectrum for the excess noise generated by InsP3 followed a 1/f dependence consistent with conductance fluctuations in the channel mediating this current, although other mechanisms are not excluded. These experiments demonstrate the presence of plasma membrane InsP3-gated channels of different ionic specificity in olfactory receptor cells.

Analyte-Receptor Binding Kinetics for Biosensor Applications: A Single-Fractal and a Dual-Fractal Analysis of the Influence of the Fractal Dimension on the Binding Rate Coefficient

The diffusion-limited binding kinetics of antigen (analyte) in solution to antibody (receptor) immobilized on a biosensor surface is analyzed within a fractal framework. Most of the data presented are adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot ("Scientific Graphing Procedure, User's Manual," Jandel Scientific, San Rafael, CA, 1993). A couple of examples of a dual-fractal analysis are also presented. It is of interest to note that the binding rate coefficient and the fractal dimension both exhibit changes in the same direction for the analyte-receptor systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the analyte-receptor binding systems indicate the high sensitivity of the binding rate coefficient on the fractal dimension when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis and for the binding of cell surface proteins from Helicobacter pylori strain in solution to sialyl-(alpha-2,3)-lactose-conjugated (20 mol%) polyacrylamide immobilized on a resonant mirror biosensor (S. Hirmo et al., Anal. Biochem. 257, 63, 1998), the order of dependence of the binding rate coefficient, k, on the fractal dimension, Df, was 14.15. The fractional order of dependence of the binding rate coefficient(s) on the fractal dimension(s) further reinforces the fractal nature of the system. The binding rate coefficient(s) expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface and further emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance. Copyright 1998 Academic Press.

An analysis of analyte-receptor binding kinetics for biosensor applications: influence of the fractal dimension on the binding rate coefficient

The diffusion-limited binding kinetics of analyte in solution to receptor immobilized on a biosensor surface is analysed within a fractal framework. Both a single- as well as a dual-fractal analysis are utilized. Antigen-antibody and analyte-receptor systems are analysed. For the antigen-antibody and analyte-receptor systems where a single- or a dual-fractal analysis was used, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction. The binding rate coefficient expressions obtained as a function of the fractal dimension indicate the high sensitivity of the binding rate coefficient with respect to the fractal dimension. For example, for a single-fractal analysis and for the binding of (a) 1 microM BSA in solution to the anti-BSA-protein fused to a biosensor surface, and for (b) the binding of m-xylene-saturated STE buffer solution to the microorganism immobilized to the fiber-optic end and covered with a polycarbonate membrane, the orders of dependence of the binding rate coefficient on the fractal dimension were 5.535 and 3.314, respectively. This emphasizes the importance of the degree of heterogeneity on the biosensor surface and its impact on the binding rate coefficient, k. This high sensitivity is also indicated for a dual-fractal analysis, at least for the binding rate coefficient, k2. For example, during regeneration runs and for the binding of polymerase chain-reaction amplified DNA in solution to DNA capture protein immobilized on a fiber-optic biosensor, the order of dependence of k2 on Df2 was 3.399. The fractional order of dependence of the binding rate coefficient(s) on the fractal dimension(s) further reinforces the fractal nature of the system. The binding rate coefficient expressions developed as a function of the fractal dimension for both single-fractal analysis and dual-fractal analysis systems are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface. Also, the importance of the nature of the surface on biosensor performance is emphasized in a quantitative sense.

A synthetic picture of intramolecular dynamics of proteins. Towards a contemporary statistical theory of biochemical processes

An increasing body of experimental evidence indicates the slow character of internal dynamics of native proteins. The important consequence of this is that theories of chemical reactions, used hitherto, appear inadequate for description of most biochemical reactions. Construction of a contemporary, truly advanced statistical theory of biochemical processes will need simple but realistic models of microscopic dynamics of biomolecules. In this review, intended to be a contribution towards this direction, three topics are considered. First, an intentionally simplified picture of dynamics of native proteins which emerges from recent investigations is presented. Fast vibrational modes of motion, of periods varying from 10(-14) to 10(-11) s, are contrasted with purely stochastic conformational transitions. Significant evidence is adduced that the relaxation time spectrum of the latter spreads in the whole range from 10(-11) to 10(5) s or longer, and up to 10(-7) s it is practically quasi-continuous. Next, the essential ideas of the theory of reaction rates based on stochastic models of intramolecular dynamics are outlined. Special attention is paid to reactions involving molecules in the initial conformational substrates confirmed to the transition state, which is realized in actual experimental situations. And finally, the two best experimentally justified classes of models of conformational transition dynamics, symbolically referred to as "protein glass" and "protein machine", are described and applied to the interpretation of a few simple biochemical processes, perhaps the most important result reported is the demonstration of the possibility of predominance of the short initial condition-dependent stage of protein involved reactions over the main stage described by the standard kinetics. This initial stage, and not the latter, is expected to be responsible for the coupling of component reactions in the complete enzymatic cycles as well as more complex processes of biological free energy transduction.

Analyte-receptor binding kinetics for biosensor applications. An analysis of the influence of the fractal dimension on the binding rate coefficient

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (Df) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the Df developed for the antigen-antibody binding systems, indicate the high sensitivity of the k on the Df when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5 beta in solution to gp120 peptide immobilized on a BIAcore biosensor, the order of dependence on the Df was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k1 and k2, on the fractal dimensions, Df1 and Df2, were 7.6335 and -11.55, respectively. The fractional order of dependence of the k(s) on the Df(s) further reinforces the fractal nature of the system. The k(s) expressions developed as a function of the Df(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance.

Antigen-antibody binding kinetics for biosensors. Changes in the fractal dimension (surface roughness) and in the binding rate coefficient

The diffusion-limited binding kinetics of antigen in solution to antibody immobilized on a biosensor surface is analyzed within a fractal framework. Changes in the fractal dimension, Df observed are in the same and in the reverse directions as the forward binding rate coefficient k. For example, an increase in the concentration of the isoenzyme human creatine kinase isoenzyme MB form (CK-MB) (antigen) solution from 0.1 to 50 ng/mL and bound to anti-CK-MB antibody immobilized on fused silica fiber rods leads to increases in the fractal dimension Df from 0.294 to 0.5080, and in the forward binding rate coefficient k from 0.1194 to 9.716, respectively. The error in the fractal dimension Df decreases with an increase in the CK-MB isoenzyme concentration in solution. An increase in the concentration of human chorionic gonadotrophin (hCG) in solution from 4000 to 6000 mIU/mL hCG and bound to anti-hCG antibody immobilized on a fluorescence capillary fill device leads to a decrease in the fractal dimension Df from 2.6806 to 2.6164, and to an increase in the forward binding rate coefficient k from 3.571 to 4.033, respectively. The different examples analyzed and presented together indicate one means by which the forward binding rate coefficient k may be controlled, that is by changing the fractal dimension or the "disorder' on the surface. The analysis should assist in helping to improve the stability, the sensitivity, and the response time of biosensors.

A fractal analysis of the influence of non-specific binding on antigen-antibody binding kinetics for biosensor applications

A fractal analysis of the influence of non-specific binding on the specific binding of antigen in solution to antibody immobilized on a biosensor surface is presented for first-, one and a half-, second, and other-order reactions occurring under external diffusion-limited conditions. Both single-step and dual-step binding of antigen in solution to antibody immobilized on the surface is considered. For a first-order reaction, an increase in the fractal parameters, b, leads to a decrease in the amount of antigen in solution bound specifically to the antibody on the surface when non-specific binding is either absent or present. The presence of non-specific binding leads to a decrease in the amount of antigen bound to the antibody on the surface. For a one and a half- and for second-order reactions and when non-specific binding is either absent or present to a small degree (alpha = 0.01), an increase in the fractal parameter, b, leads to a decrease in the amount of antigen bound specifically to the antibody immobilized on the biosensor surface. However, for an alpha value of 0.1, the maximum rate and the amount of antigen bound specifically to the antibody immobilized on the biosensor surface is obtained for fractal parameter values of 0.2 and 0.4, and 0.4 for the one and a half- and for second-order reactions, respectively. Apparently, some amount of heterogeneity is helpful in obtaining the optimum amount and rate of antigen in solution bound specifically to the antibody on the surface for reaction orders higher than one. The applicability of the approach to real antibody surfaces is demonstrated.

Nonlinear kinetics of ferritin adsorption

The adsorption of ferritin at a methylized quartz surface was measured with off-null ellipsometry and transmission electron microscopy. An initial lag-phase was seen, followed by an accelerating adsorption leading to mass transport limitation of the reaction. The rate of adsorption then decreased at a surface concentration far below monolayer coverage, and a continuously decreasing rate of binding was seen. The slope of the binding rate was linear with the logarithm of time (fractal kinetics). The adsorbed ferritin molecules were distributed in clusters as seen by transmission electron microscopy. Clusters grown during the mass transport limited adsorption had crystalline structure at short range and low fractal dimensions (df = 0.89) over long range. Clusters grown during adsorption with fractal kinetics showed random structure at short range and a high fractal dimension df = 1.86 over all ranges. These findings indicate some new important mechanisms responsible for the complex kinetics of macromolecular reactions at solid-liquid interfaces. The results are discussed in relation to recently developed theories of self-organized criticality.