Determination of rate distributions from kinetic experiments

Maximum entropy determination of mammalian proteome dynamics

Full understanding of proteostasis and energy utilization in cells will require knowledge of the fraction of cell proteins being degraded with different half-lives and their rates of synthesis. We therefore developed a method to determine such information that combines mathematical analysis of protein degradation kinetics obtained in pulse-chase experiments with Bayesian data fitting using the maximum entropy principle. This approach will enable rapid analyses of whole-cell protein dynamics in different cell types, physiological states, and neurodegenerative disease. Using it, we obtained surprising insights about protein stabilities in cultured cells normally and upon activation of proteolysis by mTOR inhibition and increasing cAMP or cGMP. It revealed that >90% of protein content in dividing mammalian cell lines is long-lived, with half-lives of 24 to 200 h, and therefore comprises much of the proteins in daughter cells. The well-studied short-lived proteins (half-lives < 10 h) together comprise <2% of cell protein mass, but surprisingly account for 10 to 20% of measurable newly synthesized protein mass. Evolution thus appears to have minimized intracellular proteolysis except to rapidly eliminate misfolded and regulatory proteins.

Transcending Markov: non-Markovian rate processes of thermosensitive TRP ion channels

The Markov state model (MSM) is a popular theoretical tool for describing the hierarchy of time scales involved in the function of many proteins especially ion channel gating. An MSM is a particular case of the general non-Markovian model, where the rate of transition from one state to another does not depend on the history of state occupancy within the system, i.e. it only includes reversible, non-dissipative processes. However, an MSM requires knowledge of the precise conformational state of the protein and is not predictive when those details are not known. In the case of ion channels, this simple description fails in real (non-equilibrium) situations, for example when local temperature changes, or when energy losses occur during channel gating. Here, we show it is possible to use non-Markovian equations (i.e. offer a general description that includes the MSM as a particular case) to develop a relatively simple analytical model that describes the non-equilibrium behaviour of the temperature-sensitive transient receptor potential (TRP) ion channels, TRPV1 and TRPM8. This model accurately predicts asymmetrical opening and closing rates, infinite processes and the creation of new states, as well as the effect of temperature changes throughout the process. This approach therefore overcomes the limitations of the MSM and allows us to go beyond a mere phenomenological description of the dynamics of ion channel gating towards a better understanding of the physics underlying these processes.

Image segmentation and separation of spectrally similar dyes in fluorescence microscopy by dynamic mode decomposition of photobleaching kinetics

Background

Image segmentation in fluorescence microscopy is often based on spectral separation of fluorescent probes (color-based segmentation) or on significant intensity differences in individual image regions (intensity-based segmentation). These approaches fail, if dye fluorescence shows large spectral overlap with other employed probes or with strong cellular autofluorescence.

Results

Here, a novel model-free approach is presented which determines bleaching characteristics based on dynamic mode decomposition (DMD) and uses the inferred photobleaching kinetics to distinguish different probes or dye molecules from autofluorescence. DMD is a data-driven computational method for detecting and quantifying dynamic events in complex spatiotemporal data. Here, DMD is first used on synthetic image data and thereafter used to determine photobleaching characteristics of a fluorescent sterol probe, dehydroergosterol (DHE), compared to that of cellular autofluorescence in the nematode Caenorhabditis elegans. It is shown that decomposition of those dynamic modes allows for separating probe from autofluorescence without invoking a particular model for the bleaching process. In a second application, DMD of dye-specific photobleaching is used to separate two green-fluorescent dyes, an NBD-tagged sphingolipid and Alexa488-transferrin, thereby assigning them to different cellular compartments.

Conclusions

Data-based decomposition of dynamic modes can be employed to analyze spatially varying photobleaching of fluorescent probes in cells and tissues for spatial and temporal image segmentation, discrimination of probe from autofluorescence and image denoising. The new method should find wide application in analysis of dynamic fluorescence imaging data.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-022-04881-x.

Towards understanding non-equivalence of α and β subunits within human hemoglobin in conformational relaxation and molecular oxygen rebinding

Picosecond to millisecond laser time-resolved transient absorption spectroscopy was used to study molecular oxygen (O₂) rebinding and conformational relaxation following O₂ photodissociation in the α and β subunits within human hemoglobin in the quaternary R-like structure. Oxy-cyanomet valency hybrids, α₂(Fe²⁺-O₂)β₂(Fe³⁺-CN) and α₂(Fe³⁺-CN)β₂(Fe²⁺-O₂), were used as models for oxygenated R-state hemoglobin. An extended kinetic model for geminate O₂ rebinding in the ferrous hemoglobin subunits, ligand migration between the primary and secondary docking site(s), and nonexponential tertiary relaxation within the R quaternary structure, was introduced and discussed. Significant functional non-equivalence of the α and β subunits in both the geminate O₂ rebinding and concomitant structural relaxation was revealed. For the β subunits, the rate constant for the geminate O₂ rebinding to the unrelaxed tertiary structure and the tertiary transition rate were found to be greater than the corresponding values for the α subunits. The conformational relaxation following the O₂ photodissociation in the α and β subunits was found to decrease the rate constant for the geminate O₂ rebinding, this effect being more than one order of magnitude greater for the β subunits than for the α subunits. Evidence was provided for the modulation of the O₂ rebinding to the individual α and β subunits within human hemoglobin in the R-state structure by the intrinsic heme reactivity through a change in proximal constraints upon the relaxation of the tertiary structure on a picosecond to microsecond time scale. Our results demonstrate that, for native R-state oxyhemoglobin, O₂ rebinding properties and spectral changes following the O₂ photodissociation can be adequately described as the sum of those for the α and β subunits within the valency hybrids. The isolated β chains (hemoglobin H) show similar behavior to the β subunits within the valency hybrids and can be used as a model for the β subunits within the R-state oxyhemoglobin. At the same time, the isolated α chains behave differently to the α subunits within the valency hybrids.

Trehalose matrix effects on electron transfer in Mn-depleted protein-pigment complexes of Photosystem II

The kinetics of flash-induced re-reduction of the Photosystem II (PS II) primary electron donor P₆₈₀ was studied in solution and in trehalose glassy matrices at different relative humidity. In solution, and in the re-dissolved glass, kinetics were dominated by two fast components with lifetimes in the range of 2-7 μs, which accounted for >85% of the decay. These components were ascribed to the direct electron transfer from the redox-active tyrosine Y_Z to P₆₈₀⁺. The minor slower components were due to charge recombination between the primary plastoquinone acceptor Q_A^- and P₆₈₀⁺. Incorporation of the PS II complex into the trehalose glassy matrix and its successive dehydration caused a progressive increase in the lifetime of all kinetic phases, accompanied by an increase of the amplitudes of the slower phases at the expense of the faster phases. At 63% relative humidity the fast components contribution dropped to ~50%. A further dehydration of the trehalose glass did not change the lifetimes and contribution of the kinetic components. This effect was ascribed to the decrease of conformational mobility of the protein domain between Y_Z and P₆₈₀, which resulted in the inhibition of Y_Z → P₆₈₀⁺ electron transfer in about half of the PS II population, wherein the recombination between Q_A^- and P₆₈₀⁺ occurred. The data indicate that PS II binds a larger number of water molecules as compared to PS I complexes. We conclude that our data disprove the "water replacement" hypothesis of trehalose matrix biopreservation.

Brownian motion-based nanoparticle sizing-A powerful approach for in situ analysis of nanoparticle-protein interactions

A key hurdle toward effective application of nanoparticles (NPs) in biomedicine is still the incomplete understanding of the biomolecular adsorption layer, the so-called protein corona, which inevitably forms around NPs when they are immersed in a biofluid. NP sizing techniques via the analysis of Brownian motions offer a powerful way to measure the thickness of the protein corona in situ. Here, the fundamentals of three techniques, dynamic light scattering, fluorescence correlation spectroscopy, and nanoparticle tracking analysis are briefly summarized. Then, experimental procedures for the determination of binding curves are presented in a tutorial fashion. Nanoparticle sizing experiments are illustrated with a selection of recent results on the interactions of transferrin with hydrophilic and hydrophobic polystyrene nanoparticles, and key insights gained from this work are discussed.

Control of electron transfer by protein dynamics in photosynthetic reaction centers

Trehalose and glycerol are low molecular mass sugars/polyols that have found widespread use in the protection of native protein states, in both short- and long-term storage of biological materials, and as a means of understanding protein dynamics. These myriad uses are often attributed to their ability to form an amorphous glassy matrix. In glycerol, the glass is formed only at cryogenic temperatures, while in trehalose, the glass is formed at room temperature, but only upon dehydration of the sample. While much work has been carried out to elucidate a mechanistic view of how each of these matrices interact with proteins to provide stability, rarely have the effects of these two independent systems been directly compared to each other. This review aims to compile decades of research on how different glassy matrices affect two types of photosynthetic proteins: (i) the Type II bacterial reaction center from Rhodobacter sphaeroides and (ii) the Type I Photosystem I reaction center from cyanobacteria. By comparing aggregate data on electron transfer, protein structure, and protein dynamics, it appears that the effects of these two distinct matrices are remarkably similar. Both seem to cause a "tightening" of the solvation shell when in a glassy state, resulting in severely restricted conformational mobility of the protein and associated water molecules. Thus, trehalose appears to be able to mimic, at room temperature, nearly all of the effects on protein dynamics observed in low temperature glycerol glasses.

Kinetic Study of Ligand Binding and Conformational Changes in Inducible Nitric Oxide Synthase

Nitric oxide synthases (NOSs) are heme enzymes that generate highly reactive nitric oxide from l-arginine (l-Arg) in a complex mechanism that is still only partially understood. We have studied carbon monoxide (CO) binding to the oxygenase domain of murine inducible NOS (iNOS) by using flash photolysis. The P420 and P450 forms of the enzyme, assigned to a protonated and unprotonated proximal cysteine, through which the heme is anchored to the protein, show markedly different CO rebinding properties. The data suggest that P420 has a widely open distal pocket that admits water. CO rebinding to the P450 form strongly depends on the presence of the substrate l-Arg, the intermediate N^ω-hydroxy-l-arginine, and the cofactor tetrahydrobiopterin. After adding these small molecules to the enzyme solution, the CO kinetics change slowly over the hours. This process can be described as a relaxation from a fast rebinding, metastable species to a slowly rebinding, thermodynamically stable species, which we associate with the enzymatically active form. Our results allow us to determine kinetic parameters of l-Arg binding to the ferrous deoxy iNOS protein for the first time and also provide clues regarding the nature of structural differences between the two interconverting species.

Maximum entropy analysis of polarized fluorescence decay of (E)GFP in aqueous solution

The maximum entropy method (MEM) was used for the analysis of polarized fluorescence decays of enhanced green fluorescent protein (EGFP) in buffered water/glycerol mixtures, obtained with time-correlated single-photon counting (Visser et al 2016 Methods Appl. Fluoresc. 4 035002). To this end, we used a general-purpose software module of MEM that was earlier developed to analyze (complex) laser photolysis kinetics of ligand rebinding reactions in oxygen binding proteins. We demonstrate that the MEM software provides reliable results and is easy to use for the analysis of both total fluorescence decay and fluorescence anisotropy decay of aqueous solutions of EGFP. The rotational correlation times of EGFP in water/glycerol mixtures, obtained by MEM as maxima of the correlation-time distributions, are identical to the single correlation times determined by global analysis of parallel and perpendicular polarized decay components. The MEM software is also able to determine homo-FRET in another dimeric GFP, for which the transfer correlation time is an order of magnitude shorter than the rotational correlation time. One important advantage utilizing MEM analysis is that no initial guesses of parameters are required, since MEM is able to select the least correlated solution from the feasible set of solutions.

Protein Immobilization Capabilities of Sucrose and Trehalose Glasses: The Effect of Protein/Sugar Concentration Unraveled by High-Field EPR

Disaccharide glasses are increasingly used to immobilize proteins at room temperature for structural/functional studies and long-term preservation. To unravel the molecular basis of protein immobilization, we studied the effect of sugar/protein concentration ratios in trehalose or sucrose matrixes, in which the bacterial photosynthetic reaction center (RC) was embedded as a model protein. The structural, dynamical, and H-bonding characteristics of the sugar-protein systems were probed by high-field W-band EPR of a matrix-dissolved nitroxide radical. We discovered that RC immobilization and thermal stabilization, being independent of the protein concentration in trehalose, occur in sucrose only at sufficiently low sugar/protein ratios. EPR reveals that only under such conditions does sucrose form a microscopically homogeneous matrix that immobilizes, via H-bonds, the nitroxide probe. We conclude that the protein immobilization capability depends critically on the propensity of the glass-forming sugar to create intermolecular H-bond networks, thus establishing long-range, homogeneous connectivity within the matrix.

Implementation and evaluation of data analysis strategies for time-resolved optical spectroscopy

Time-resolved optical spectroscopy plays a key role in illuminating the mechanisms of many fundamental processes in physics, chemistry, and biology. However, to extract the essential information from the highly complex time-resolved data, advanced data analysis techniques are required. Here we present the implementation strategies and the evaluation of the familiar global lifetime and target analysis as well as the not so widely adopted lifetime distribution analysis (LDA). Furthermore, we demonstrate the implementation of analysis strategies dealing with a number of artifacts inherently present in data from ultrafast optical experiments. The focus of the work is placed on LDA as it allows invaluable exploration depth of the kinetic information contained in the experimental data. We establish a clear regularization procedure for the use of LDA in ultrafast optical spectroscopy and evaluate the performance of a number of factors that play a role in the reliable reconstruction of lifetime distributions. Our results show that the optimal regularization factor can be determined well with the L-curve and the generalized cross-validation techniques. Moreover, the performance evaluations indicate that the most efficient regularization norm is the identity matrix. The analytical procedures described in this work can be readily implemented and used for the analysis of any time-resolved data.

Single molecule conformational memory extraction: p5ab RNA hairpin

Extracting kinetic models from single molecule data is an important route to mechanistic insight in biophysics, chemistry, and biology. Data collected from force spectroscopy can probe discrete hops of a single molecule between different conformational states. Model extraction from such data is a challenging inverse problem because single molecule data are noisy and rich in structure. Standard modeling methods normally assume (i) a prespecified number of discrete states and (ii) that transitions between states are Markovian. The data set is then fit to this predetermined model to find a handful of rates describing the transitions between states. We show that it is unnecessary to assume either (i) or (ii) and focus our analysis on the zipping/unzipping transitions of an RNA hairpin. The key is in starting with a very broad class of non-Markov models in order to let the data guide us toward the best model from this very broad class. Our method suggests that there exists a folding intermediate for the P5ab RNA hairpin whose zipping/unzipping is monitored by force spectroscopy experiments. This intermediate would not have been resolved if a Markov model had been assumed from the onset. We compare the merits of our method with those of others.

High-field EPR on membrane proteins - crossing the gap to NMR

In this review on advanced EPR spectroscopy, which addresses both the EPR and NMR communities, considerable emphasis is put on delineating the complementarity of NMR and EPR concerning the measurement of molecular interactions in large biomolecules. From these interactions, detailed information can be revealed on structure and dynamics of macromolecules embedded in solution- or solid-state environments. New developments in pulsed microwave and sweepable cryomagnet technology as well as ultrafast electronics for signal data handling and processing have pushed to new horizons the limits of EPR spectroscopy and its multifrequency extensions concerning the sensitivity of detection, the selectivity with respect to interactions, and the resolution in frequency and time domains. One of the most important advances has been the extension of EPR to high magnetic fields and microwave frequencies, very much in analogy to what happens in NMR. This is exemplified by referring to ongoing efforts for signal enhancement in both NMR and EPR double-resonance techniques by exploiting dynamic nuclear or electron spin polarization via unpaired electron spins and their electron-nuclear or electron-electron interactions. Signal and resolution enhancements are particularly spectacular for double-resonance techniques such as ENDOR and PELDOR at high magnetic fields. They provide greatly improved orientational selection for disordered samples that approaches single-crystal resolution at canonical g-tensor orientations - even for molecules with small g-anisotropies. Exchange of experience between the EPR and NMR communities allows for handling polarization and resolution improvement strategies in an optimal manner. Consequently, a dramatic improvement of EPR detection sensitivity could be achieved, even for short-lived paramagnetic reaction intermediates. Unique structural and dynamic information is thus revealed that can hardly be obtained by any other analytical techniques. Micromolar quantities of sample molecules have become sufficient to characterize stable and transient reaction intermediates of complex molecular systems - offering highly interesting applications for chemists, biochemists and molecular biologists. In three case studies, representative examples of advanced EPR spectroscopy are reviewed: (I) High-field PELDOR and ENDOR structure determination of cation-anion radical pairs in reaction centers from photosynthetic purple bacteria and cyanobacteria (Photosystem I); (II) High-field ENDOR and ELDOR-detected NMR spectroscopy on the oxygen-evolving complex of Photosystem II; and (III) High-field electron dipolar spectroscopy on nitroxide spin-labelled bacteriorhodopsin for structure-function studies. An extended conclusion with an outlook to further developments and applications is also presented.

A spectroscopic study of structural heterogeneity and carbon monoxide binding in neuroglobin

Neuroglobin (Ngb) is a small globular protein that binds diatomic ligands like oxygen, carbon monoxide (CO) and nitric oxide at a heme prosthetic group. We have performed FTIR spectroscopy in the infrared stretching bands of CO and flash photolysis with monitoring in the electronic heme absorption bands to investigate structural heterogeneity at the active site of Ngb and its effects on CO binding and migration at cryogenic temperatures. Four CO stretching bands were identified; they correspond to discrete conformations that differ in structural details and CO binding properties. Based on a comparison of bound-state and photoproduct IR spectra of the wild-type protein, Ngb distal pocket mutants and myoglobin, we have provided structural interpretations of the conformations associated with the different CO bands. We have also studied ligand migration to the primary docking site, B. Rebinding from this site is governed by very low enthalpy barriers (∼1 kJ/mol), indicating an extremely reactive heme iron. Moreover, we have observed ligand migration to a secondary docking site, C, from which CO rebinding involves higher enthalpy barriers.

Extracting conformational memory from single-molecule kinetic data

Single-molecule data often come in the form of stochastic time trajectories. A key question is how to extract an underlying kinetic model from the data. A traditional approach is to assume some discrete state model, that is, a model topology, and to assume that transitions between states are Markovian. The transition rates are then selected according to which ones best fit the data. However, in experiments, each apparent state can be a broad ensemble of states or can be hiding multiple interconverting states. Here, we describe a more general approach called the non-Markov memory kernel (NMMK) method. The idea is to begin with a very broad class of non-Markov models and to let the data directly select for the best possible model. To do so, we adapt an image reconstruction approach that is grounded in maximum entropy. The NMMK method is not limited to discrete state models for the data; it yields a unique model given the data, it gives error bars for the model, and it does not assume Markov dynamics. Furthermore, NMMK is less wasteful of data by letting the entire data set determine the model. When the data warrants, the NMMK gives a memory kernel that is Markovian. We highlight, by numerical example, how conformational memory extracted using this method can be translated into useful mechanistic insight.

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.

Markov processes follow from the principle of maximum caliber

Markov models are widely used to describe stochastic dynamics. Here, we show that Markov models follow directly from the dynamical principle of maximum caliber (Max Cal). Max Cal is a method of deriving dynamical models based on maximizing the path entropy subject to dynamical constraints. We give three different cases. First, we show that if constraints (or data) are given in the form of singlet statistics (average occupation probabilities), then maximizing the caliber predicts a time-independent process that is modeled by identical, independently distributed random variables. Second, we show that if constraints are given in the form of sequential pairwise statistics, then maximizing the caliber dictates that the kinetic process will be Markovian with a uniform initial distribution. Third, if the initial distribution is known and is not uniform we show that the only process that maximizes the path entropy is still the Markov process. We give an example of how Max Cal can be used to discriminate between different dynamical models given data.

Filtering artifacts from lifetime distributions when maximizing entropy using a bootstrapped model

The maximum entropy method (MEM) has been used in many studies to reliably recover effective lifetimes from kinetics, whether measured experimentally or simulated computationally. Here, recent claims made by Mulligan et al. regarding MEM analyses of kinetics (Anal. Biochem. 421 (2012) 181-190) are shown to be unfounded. Their assertion that their software allows "analysis of datasets too noisy to process by existing iterative search algorithms" is refuted with a MEM analysis of their triexponential test case with increased noise. In addition, it is shown that lifetime distributions recovered from noisy kinetics data with the MEM can be improved by using a simple filter when bootstrapping the prior model. When deriving the bootstrapped model from the lifetime distribution obtained using a uniform model, only the slower processes are represented as Gaussians in the bootstrapped model. Using this new approach, results are clearly superior to those of Mulligan et al. despite the presence of increased noise. In a second example, ambiguity in the interpretation of Poisson kinetics in the presence of scattered excitation light is resolved by filtering the prior model.

Photolysis of Hi-CO Nitrogenase - Observation of a Plethora of Distinct CO Species using Infrared Spectroscopy

Fourier transform infrared spectroscopy (FT-IR) was used to study the photochemistry of CO-inhibited Azotobacter vinelandii nitrogenase using visible light at cryogenic temperatures. The FT-IR difference spectrum of photolyzed hi-CO at 4 K comprises negative bands at 1973 cm^-1 and 1679 cm^-1 together with positive bands at 1711 cm^-1, 2135 and 2123 cm^-1. The negative bands are assigned to a hi-CO state that comprises 2 metal-bound CO ligands, one terminally bound, and one bridged and/or protonated species. The positive band at 1711 cm^-1 is assigned to a lo-CO product with a single bridged and/or protonated metal-CO group. We term these species 'Hi-1' and 'Lo-1' respectively. The high-energy bands are assigned to a liberated CO trapped in the protein pocket. Warming results in CO recombination, and the temperature dependence of the recombination rate yields an activation energy of 4 kJ mol^-1. Two α-H195 variant enzymes yielded additional signals. Asparagine substitution, α-H195N, gives a spectrum containing 2 negative 'Hi-2' bands at 1936 and 1858 cm^-1 with a positive 'Lo-2' band at 1780 cm^-1, while glutamine substitution, α-H195Q, produces a complex spectrum that includes a third CO species, with negative 'Hi-3' bands at 1938 and 1911 cm^-1 and a positive feature 'Lo-3' band at 1921 cm^-1. These species can be assigned to a combination of terminal, bridged, and possibly protonated CO groups bound to the FeMo-cofactor active site. The proposed structures are discussed in terms of both CO inhibition and the mechanism nitrogenase catalysis. Given the intractability of observing nitrogenase intermediates by crystallographic methods, IR-monitored photolysis appears to be a promising and information-rich probe of nitrogenase structure and chemistry.

Kinetics and mechanism of single-stranded DNA adsorption onto citrate-stabilized gold nanoparticles in colloidal solution

A variety of rapid biomolecular assays under development rely on the selective adsorption of single-stranded DNA onto unfunctionalized, negatively charged, citrate-stabilized gold nanoparticles. We investigate the adsorption mechanism with a study of the binding kinetics and find strong evidence for the dominance of hydrophobic effects including linear compensation between the activation energy and the natural log of the Arrhenius prefactor and the correlation of the adsorption rate in the presence of various salts with the Hofmeister series. These results explain the selectivity for single-stranded over double-stranded DNA adsorption and contradict previous work citing an electrostatic DLVO-like mechanism. Our understanding should facilitate improvements to the selective-adsorption-based assays and, more generally, contribute to the understanding of interactions between like-charged species in aqueous solution.

Fuzzy domains: new way of describing flexibility and interdependence of the protein domains

We proposed the innovative method of domain identification based on the concept of the fuzzy domains. In this method each residue of the protein can belong to several domains simultaneously with certain weights, which reflect to what extent this residue shares the motion pattern of the given domain. Our method allows describing the fuzzy boundaries between the domains and the gradual changes of the motion pattern from one domain to the other. It provides the reasonable compromise between the continuous change of the protein dynamics from one residue to the other and the discrete description of the structure in terms of small number of domains. We suggested quantitative criterion, which shows the overall degree of domain flexibility in the protein. The concept of the fuzzy domains provides an innovative way of visualization of domain flexibility, which makes the gradual transitions between the domains clearly visible and comparable to available experimental and structural data. In the future, the concept of the fuzzy domains can be used in the coarse-grained simulations of the domain dynamics in order to account for internal protein flexibility.

Computer simulation of ion cluster speciation in concentrated aqueous solutions at ambient conditions

Dynamic simulations are used to investigate ion cluster formation in unsaturated aqueous NaCl at 25 degrees C. Statistical, structural, and dynamic properties are reported. An effort is made to identify general behaviors that are expected to hold beyond the limitations of the force field. Above approximately 1 M, clusters with more than ten ions begin to form after approximately 10-20 ns of simulation time, but no evidence of irreversible ion aggregation is observed. Cluster survival times are estimated, showing that the kinetics become increasingly complex as salt is added, leading to multiple decay rates. Cluster dipole moment distributions show characteristic peaks that reflect the preferred conformations of clusters in solution. These are modulated by electrostatic and liquid-structure forces and are described in detail for clusters of up to five ions. For a given size and charge, the cluster morphology is independent of salt concentration. Below approximately 2 M, clusters affect the structure of water in their first hydration shells, so dipole moments parallel to the cluster macrodipoles are induced. These effects show a weak dependence with concentration below approximately 2 M, but vanish in the 2-3 M range. A possible connection with the structural transition recently suggested by NMR data in concentrated electrolytes is discussed. The effects of electrostatics on cluster speciation and morphology are discussed based on results from a set of simulations carried out with the ionic charges removed.

Protein dynamics control of electron transfer in photosynthetic reaction centers from Rps. sulfoviridis

In the cycle of photosynthetic reaction centers, the initially oxidized special pair of bacteriochlorophyll molecules is subsequently reduced by an electron transferred over a chain of four hemes of the complex. Here, we examine the kinetics of electron transfer between the proximal heme c-559 of the chain and the oxidized special pair in the reaction center from Rps. sulfoviridis in the range of temperatures from 294 to 40 K. The experimental data were obtained for three redox states of the reaction center, in which one, two, or three nearest hemes of the chain are reduced prior to special pair oxidation. The experimental kinetic data are analyzed in terms of a Sumi-Marcus-type model developed in our previous paper,1 in which similar measurements were reported on the reaction centers from Rps. viridis. The model allows us to establish a connection between the observed nonexponential electron-transfer kinetics and the local structural relaxation dynamics of the reaction center protein on the microsecond time scale. The activation energy for relaxation dynamics of the protein medium has been found to be around 0.1 eV for all three redox states, which is in contrast to a value around 0.4-0.6 eV in Rps. viridis.1 The possible nature of the difference between the reaction centers from Rps. viridis and Rps. sulfoviridis, which are believed to be very similar, is discussed. The role of the protein glass transition at low temperatures and that of internal water molecules in the process are analyzed.

Analysis of the entire sequence of a single photon experiment on a flavin protein

The large amount of statistical data collected by single biomolecule experiments often demonstrates complex and non-Markovian relaxation over many time scales. Analyzing and interpreting these data is a major challenge because of the inherently statistical noise and the lack of definite theoretical descriptions or computer simulations on biologically relevant time scales. This paper reports one of the first complete sequence analyses of a single photon experiment on the flavin protein to determine an underlying physical picture for protein motions on the millisecond to second regimes. The robustness of Bayesian information analysis combined with the nonparametric maximum entropy method (MEM) incorporates all available information of the single-molecule data sequence and maximizes our ability to test the legitimacy of possible models. Our analysis of the experimental data is consistent with the stochastic Gaussian diffusion model where the slow protein motions are modeled as a collection of over-damped diffusive normal modes and reveals non-universal and distinct dynamic features that are specific for protein functions.

Conformational dependence of hemoglobin reactivity under high viscosity conditions: the role of solvent slaved dynamics

The concept of protein dynamic states is introduced. This concept is based on (i) protein dynamics being organized hierarchically with respect to solvent slaving and (ii) which tier of dynamics is operative over the time window of a given measurement. The protein dynamic state concept is used to analyze the kinetic phases derived from the recombination of carbon monoxide to sol-gel-encapsulated human adult hemoglobin (HbA) and select recombinant mutants. The temperature-dependent measurements are made under very high viscosity conditions obtained by bathing the samples in an excess of glycerol. The results are consistent with a given tier of solvent slaved dynamics becoming operative at a time delay (with respect to the onset of the measurement) that is primarily solvent- and temperature-dependent. However, the functional consequences of the dynamics are protein- and conformation-specific. The kinetic traces from both equilibrium populations and trapped allosteric intermediates show a consistent progression that exposes the role of both conformation and hydration in the control of reactivity. Iron-zinc symmetric hybrid forms of HbA are used to show the dramatic difference between the kinetic patterns for T state alpha and beta subunits. The overall results support a model for allostery in HbA in which the ligand-binding-induced transition from the deoxy T state to the high -affinity R state proceeds through a progression of T state intermediates.

Practical aspects of the maximum entropy inversion of the laplace transform for the quantitative analysis of multi-exponential data

The number, position, area, and width of the bands in a lifetime distribution give the number of exponentials present in time-resolved data and their time constants, amplitudes, and heterogeneities. The maximum entropy inversion of the Laplace transform (MaxEnt-iLT) provides a lifetime distribution from time-resolved data, which is very helpful in the analysis of the relaxation of complex systems. In some applications both positive and negative values for the lifetime distribution amplitudes are physical, but most studies to date have focused on positive-constrained solutions. In this work, we first discuss optimal conditions to obtain a sign-unrestricted maximum entropy lifetime distribution, i.e., the selection of the entropy function and the regularization value. For the selection of the regularization value we compared four methods: the chi2 criterion and Bayesian inference (already used in sign-restricted MaxEnt-iLT), and the L-curve and the generalized cross-validation methods (not yet used in MaxEnt-iLT to our knowledge). Except for the frequently used chi2 criterion, these methods recommended similar regularization values, providing close to optimum solutions. However, even when an optimal entropy function and regularization value are used, a MaxEnt lifetime distribution will contain noise-induced errors, as well as systematic distortions induced by the entropy maximization (regularization-induced errors). We introduce the concept of the apparent resolution function in MaxEnt, which allows both the noise and regularization-induced errors to be estimated. We show the capability of this newly introduced concept in both synthetic and experimental time-resolved Fourier transform infrared (FT-IR) data from the bacteriorhodopsin photocycle.

Nitrite reductase activity of sol-gel-encapsulated deoxyhemoglobin. Influence of quaternary and tertiary structure

Nitrite reductase activity of deoxyhemoglobin (HbA) in the red blood cell has been proposed as a non-nitric-oxide synthase source of deliverable nitric oxide (NO) within the vasculature. An essential element in this scheme is the dependence of this reaction on the quaternary/tertiary structure of HbA. In the present work sol-gel encapsulation is used to trap and stabilize deoxy-HbA in either the T or R quaternary state, thus allowing for the clear-cut monitoring of nitrite reductase activity as a function of quaternary state with and without effectors. The results indicate that reaction is not only R-T-dependent but also heterotropic effector-dependent within a given quaternary state. The use of the maximum entropy method to analyze carbon monoxide (CO) recombination kinetics from fully and partially liganded sol-gel-encapsulated T-state species provides a framework for understanding effector modulation of T-state reactivity by influencing the distribution of high and low reactivity T-state conformations.

A two-dimensional view of the folding energy landscape of cytochrome c

Time-correlated single photon counting (TCSPC) was combined with fluorescence correlation spectroscopy (FCS) to study the transition between acid-denatured states and the native structure of cytochrome c (Cyt c) from Saccharomyces cerevisiae. The use of these techniques in concert proved to be more powerful than either alone, yielding a two-dimensional picture of the folding energy landscape of Cyt c. TCSPC measured the distribution of distances between the heme of the protein and a covalently attached dye molecule at residue C102 (one folding reaction coordinate), whereas FCS measured the hydrodynamic radius (a second folding reaction coordinate) of the protein over a range of pH values. These two independent measurements provide complimentary information regarding protein conformation. We see evidence for a well defined folding intermediate in the acid renaturation folding pathway of this protein reflected in the distribution of lifetimes needed to fit the TCSPC data. Moreover, FCS studies revealed this intermediate state to be in dynamic equilibrium with unfolded structures, with conformational fluctuations into and out of this intermediate state occurring on an approximately 30-micros time scale.

Microsecond dynamics of protein-DNA interactions: direct observation of the wrapping/unwrapping kinetics of single-stranded DNA around the E. coli SSB tetramer

The Escherichia coli single-stranded DNA binding protein (SSB) binds selectively to single-stranded (ss) DNA intermediates during DNA replication, recombination and repair. Each subunit of the homo-tetrameric protein contains a potential ssDNA binding site, thus the protein can bind to ssDNA in multiple binding modes, one of which is the (SSB)(65) mode, in which a 65 nucleotide stretch of ssDNA interacts with and wraps around all four subunits of the tetramer. Previous stopped-flow kinetic studies of (SSB)(65) complex formation using the oligodeoxynucleotide, (dT)70, were unable to resolve the initial binding step from the rapid wrapping of ssDNA around the tetramer. Here we report a laser temperature-jump study with resolution in the approximately 500 ns to 4 ms time range, which directly detects these ssDNA wrapping/unwrapping steps. Biphasic time courses are observed with a fast phase that is concentration-independent and which occurs on a time-scale of tens of microseconds, reflecting the wrapping/unwrapping of ssDNA around the SSB tetramer. Analysis of the slower binding phase, in combination with equilibrium binding and stopped-flow kinetic studies, also provides evidence for a previously undetected intermediate along the pathway to forming the (SSB)(65) complex.

Modulation of reactivity and conformation within the T-quaternary state of human hemoglobin: the combined use of mutagenesis and sol-gel encapsulation

A range of conformationally distinct functional states within the T quaternary state of hemoglobin are accessed and probed using a combination of mutagenesis and sol-gel encapsulation that greatly slow or eliminate the T --> R transition. Visible and UV resonance Raman spectroscopy are used to probe the proximal strain at the heme and the status of the alpha(1)beta(2) interface, respectively, whereas CO geminate and bimolecular recombination traces in conjunction with MEM (maximum entropy method) analysis of kinetic populations are used to identify functionally distinct T-state populations. The mutants used in this study are Hb(Nbeta102A) and the alpha99-alpha99 cross-linked derivative of Hb(Wbeta37E). The former mutant, which binds oxygen noncooperatively with very low affinity, is used to access low-affinity ligated T-state conformations, whereas the latter mutant is used to access the high-affinity end of the distribution of T-state conformations. A pattern emerges within the T state in which ligand reactivity increases as both the proximal strain and the alpha(1)beta(2) interface interactions are progressively lessened after ligand binding to the deoxy T-state species. The ligation and effector-dependent interplay between the heme environment and the stability of the Trp beta37 cluster in the hinge region of the alpha(1)beta(2) interface appears to determine the distribution of the ligated T-state species generated upon ligand binding. A qualitative model is presented, suggesting that different T quaternary structures modulate the stability of different alphabeta dimer conformations within the tetramer.

Role of Heme Iron Coordination and Protein Structure in the Dynamics and Geminate Rebinding of Nitric Oxide to the H93G Myoglobin Mutant

The influence of the heme iron coordination on nitric oxide binding dynamics was investigated for the myoglobin mutant H93G (H93G-Mb) by picosecond absorption and resonance Raman time-resolved spectroscopies. In the H93G-Mb, the glycine replacing the proximal histidine does not interact with the heme iron so that exogenous substituents like imidazole may coordinate to the iron at the proximal position. Nitrosylation of H93G-Mb leads to either 6- or 5-coordinate species depending on the imidazole concentration. At high concentrations, (imidazole)-(NO)-6-coordinate heme is formed, and the photoinduced rebinding kinetics reveal two exponential picosecond phases ( approximately 10 and approximately 100 ps) similar to those of wild type myoglobin. At low concentrations, imidazole is displaced by the trans effect leading to a (NO)-5-coordinate heme, becoming 4-coordinate immediately after photolysis as revealed from the transient Raman spectrum. In this case, NO rebinding kinetics remain bi-exponential with no change in time constant of the fast component whose amplitude increases with respect to the 6-coordinate species. Bi-exponential NO geminate rebinding in 5-coordinate H93G-Mb is in contrast with the single-exponential process reported for nitrosylated soluble guanylate cyclase (Negrerie, M., Bouzhir, L., Martin, J. L., and Liebl, U. (2001) J. Biol. Chem. 276, 46815-46821). Thus, our data show that the iron coordination state or the heme iron out-of-plane motion are not at the origin of the bi-exponential kinetics, which depends upon the protein structure, and that the 4-coordinate state favors the fast phase of NO geminate rebinding. Consequently, the heme coordination state together with the energy barriers provided by the protein structure control the dynamics and affinity for NO-binding enzymes.

Transformation of time-resolved spectra to lifetime-resolved spectra by maximum entropy inversion of the laplace transform

We present a method for the analysis of time-resolved spectroscopic data following first-order kinetics. The time traces at all the available spectroscopic channels (e.g., wavelength or wavenumber) are inverse Laplace transformed. The transformation is stabilized by the maximum entropy method generalized for solutions without sign-restriction. In this way, time-resolved spectra can be converted to lifetime-resolved spectra, where bands appear at coordinates corresponding to their spectroscopic maxima and time constant of appearance (negative amplitude) or disappearance (positive amplitude). From the lifetime-resolved spectra, the number of exponentially decaying components, their time constants, and their decay-associated spectra are readily available. Moreover, since bands are spread in two dimensions extra band-resolution is possible. We named this method of transforming time-resolved spectra into lifetime-resolved spectra multi-spectroscopic channel maximum entropy inversion of the Laplace transform (M-MaxEnt-iLT). The basis of M-MaxEnt-iLT is presented in detail and its properties and limitations are thoroughly discussed. We also show how the combination of M-MaxEnt-iLT with spectral smoothing or deconvolution can improve the appearance and/or band resolution of the obtained lifetime-resolved spectra.

Dominant features of protein reaction dynamics: conformational relaxation and ligand migration

Here, we review the dominant aspects of protein dynamics as revealed by studying hemoproteins using the combination of laser flash photolysis, kinetic spectroscopy and low temperature. The first breakthrough was the finding that geminate ligand rebinding with myoglobin is highly non-exponential at temperature T<200 K, providing evidence for the trapping of a large number of protein statistical substates. Another major advance was the introduction of a "model free" approach to analyze polychromatic kinetics in terms of their rate spectrum rather than to fit the data to some arbitrarily predefined kinetic scheme. Kinetic processes are identified and quantified directly from the rate spectrum without a priori assumptions. In recent years, further progresses were achieved by using xenon gas as a soft external perturbing agent that competes with ligand rebinding pathways by occupying hydrophobic protein cavities. The first part of this paper introduces several basic principles that are spread throughout a vast literature. The second part describes the main conclusions regarding conformational relaxation and ligand migration in hemoproteins obtained by combining these approaches.

Internal dynamics and protein–matrix coupling in trehalose-coated proteins

We review recent studies on the role played by non-liquid, water-containing matrices on the dynamics and structure of embedded proteins. Two proteins were studied, in water-trehalose matrices: a water-soluble protein (carboxy derivative of horse heart myoglobin) and a membrane protein (reaction centre from Rhodobacter sphaeroides). Several experimental techniques were used: Mossbauer spectroscopy, elastic neutron scattering, FTIR spectroscopy, CO recombination after flash photolysis in carboxy-myoglobin, kinetic optical absorption spectroscopy following pulsed and continuous photoexcitation in Q(B) containing or Q(B) deprived reaction centre from R. sphaeroides. Experimental results, together with the outcome of molecular dynamics simulations, concurred to give a picture of how water-containing matrices control the internal dynamics of the embedded proteins. This occurs, in particular, via the formation of hydrogen bond networks that anchor the protein surface to the surrounding matrix, whose stiffness increases by lowering the sample water content. In the conclusion section, we also briefly speculate on how the protein-matrix interactions observed in our samples may shed light on the protein-solvent coupling also in liquid aqueous solutions.

Structural dynamics of myoglobin: an infrared kinetic study of ligand migration in mutants YQR and YQRF

Recombination of carbon monoxide to myoglobin mutants YQR and YQRF was studied using transient infrared absorption spectroscopy and Fourier transform infrared-temperature derivative spectroscopy (FTIR-TDS). Photoproduct states B, C', C" and D associated with ligands residing in different protein cavities have been identified. After photolysis, ligands migrate to primary docking site B and subsequently rebind or escape to a secondary site (C) within the Xe4 cavity. For YQR, a global analysis of the isothermal rebinding kinetics below 160 K and the TDS data reveal a correlation between the enthalpy barriers governing the two processes. Above 120 K, a protein conformational change in both YQR and YQRF converts photoproduct C' into C" with markedly slowed kinetics. Above approximately 180 K, ligands migrate to the proximal Xe1 site (D) and also exit into the solvent, from where they rebind in a bimolecular reaction.

Bacteriorhodopsin photocycle kinetics analyzed by the maximum entropy method

A maximum entropy method (MEM) was developed for the study of the bacteriorhodopsin photocycle kinetics. The method can be applied directly to experimental kinetic absorption data without any assumption for the number of the intermediate states taking part in the photocycle. Though this method does not give a specific kinetics, its result is very useful for selection between possible photocycle kinetics. Using simulated data, it is shown that MEM gives correct results for the number of the intermediate states and the amplitude distributions around the characteristic lifetimes. Analyzing experimental absorption data at five different wavelengths, MEM gives seven or eight characteristic lifetimes, which means that at least so many distinct intermediate states exist during the photocycle. Many possible photocycle kinetic models were studied and compared with the MEM result. The best agreement was found with a branching photocycle model of eight intermediate states (K, L, M(1), M(2), M(3), M(4), N, O). The branching occurs at the L intermediate state (M(1) and M(2) being in one branch and M(3) and M(4) in the other branch), but at high pH it occurs already at the K state.

Competition with xenon elicits ligand migration and escape pathways in myoglobin

Evidence for ligand migration toward the xenon-binding cavities in myoglobin comes from a number of laser photolysis studies of MbO2 including mutants and from cryo- and time-resolved crystallography of MbCO. To explore ligand migration in greater detail, we investigated the rebinding kinetics of both MbO2 and MbCO under a xenon partial pressure ranging from 1 to 16 atm over the temperature range (293-77 K). Below 180 K xenon affects to a significant, but minor, extent the thermodynamic parameters for rebinding from the primary docking site in each Mb taxonomic substate. Above 200 K the ligand migrates to the proximal Xe1 site but when the latter is occupied by xenon a new kinetic process appears. It is attributed to rebinding from transient docking sites located on the path between the primary and the secondary docking site of both ligands. Ligand escape exhibits a more complicated pattern than expected. At room temperature O2 and CO escape appears to take place exclusively from the primary site. In contrast, at T approximately 250 K, roughly 50% of the CO molecules that have escaped from the protein originate from the Xe1 secondary site.