Structural changes and fluctuations of proteins: I. A statistical thermodynamic model

A general theory of the structural changes and fluctuations of proteins has been proposed based on statistical thermodynanic considerations at the chain level.The “structure” of protein was assumed to be characterized by the state of secondary bonds between unique pairs of specific sites on peptide chains. Every secondary bond changes between the bonded and unboned states by thermal agitation and the “structure” is continuously fluctuating. The free energy of the “structural state” that is defined by the fraction of secondary bonds in the bonded state has been expressed by the bond energy, the cooperative interaction between bonds, the mixing entropy of bonds, and the entropy of polypeptide chains. The most probable “structural state” can be simply determined by graphical analysis and the effect of temperature or solvent composition on it is discussed. The temperature dependence of the free energy, the probability distribution of structural states and the specific heat have been calculated for two examples of structural change.The theory predicts two different types of structural changes from the ordered to disordered state, a “structural transition” and a “gradual structural change” with rising temperature, In the “structural transition”, the probability distribution has two maxima in the temperature range of transition. In the “gradual structural change”, the probability distribution has only one maximum during the change.A considerable fraction of secondary bonds is in the unbonded state and is always fluctuating even in the ordered state at room temperature. Such structural fluctuations in a single protein molecule have been discussed quantitatively.The theory is extended to include small molecules which bind to the protein molecule and affect the structural state. The changes of structural state caused by specific and non-specific binding and allosteric effects are explained in a unified manner.     

Reverse-engineering of biochemical reaction networks from spatio-temporal correlations of fluorescence fluctuations

Recent developments of fluorescence labeling and highly advanced microscopy techniques have enabled observations of activities of biosignaling molecules in living cells. The high spatial and temporal resolutions of these video microscopy experiments allow detection of fluorescence fluctuations at the timescales approaching those of enzymatic reactions. Such fluorescence fluctuation patterns may contain information about the complex reaction-diffusion system driving the dynamics of the labeled molecule. Here, we have developed a method of identifying the reaction-diffusion system of fluorescently labeled signaling molecules in the cell, by combining spatio-temporal correlation function analysis of fluctuating fluorescent patterns, stochastic reaction-diffusion simulations, and an iterative system identification technique using a simulated annealing algorithm. In this report, we discuss the validity and usability of spatio-temporal correlation functions in characterizing the reaction-diffusion dynamics of biomolecules, and demonstrate application of our reaction-diffusion system identification method to a simple conceptual model for small GTPase activation.     

Effect of various physical factors on oscillations of light-dispersion in water and aqueous biopolymer solutions

Study of the dependence of light scattering fluctuation on temperature, mechanic perturbation and magnetic field in water and water hemoglobin and DNA solution has shown that an increase in temperature results in the decline of long-term fluctuation amplitude and in the increase of short-term fluctuation amplitude. Mechanical mixing removes long-term fluctuations and over 10 hours are spent for their recovery. Regular fluctuations appear when the constant magnetic field above 240 A/m is applied; the fluctuations are retained for many hours after the removal of the field (when the field is off). It was supposed that maintenance of long-range correlation of molecular rotation-translation fluctuation by the effect of long-range forces and external fields underlies the mechanism of long-term light scattering fluctuations.     

Population models in a periodically fluctuating environment

Summary We investigate the behavior of population models in the presence of a periodically fluctuating environment. We consider in particular single-species models and models of interspecific competition. It is shown that the fluctuations cause constant equilibrium states to be replaced by periodic equilibrium states, with a shift in the mean value relative to the constant-environment state. It is shown also that the locations of points of exchange of stability may be changed as a result of the fluctuations.     

The photon counting histogram in fluorescence fluctuation spectroscopy with non-ideal photodetectors

Fluorescence fluctuation spectroscopy utilizes the signal fluctuations of single molecules for studying biological processes. Information about the biological system is extracted from the raw data by statistical methods such as used in fluctuation correlation spectroscopy or photon counting histogram (PCH) analysis. Since detectors are never ideal, it is crucial to understand the influence of photodetectors on signal statistics to correctly interpret the experimental data. Here we focus on the effects of afterpulsing and detector dead-time on PCH statistics. We determine the dead-time and afterpulse probability for our detectors experimentally and show that afterpulsing can be neglected for most experiments. Dead-time effects on the PCH are concentration-dependent and become significant when more than one molecule is present in the excitation volume. We develop a new PCH theory that includes dead-time effects and verify it experimentally. Additionally, we derive a simple analytical expression that accurately predicts the effect of dead-time on the molecular brightness. Corrections for non-ideal detector effects extend the useful concentration range of PCH experiments and are crucial for the interpretation of titration and dilution experiments.     

Scanning image correlation spectroscopy

Molecular interactions are at the origin of life. How molecules get at different locations in the cell and how they locate their partners is a major and partially unresolved question in biology that is paramount to signaling. Spatio-temporal correlations of fluctuating fluorescently tagged molecules reveal how they move, interact, and bind in the different cellular compartments. Methods based on fluctuations represent a remarkable technical advancement in biological imaging. Here we discuss image analysis methods based on spatial and temporal correlation of fluctuations, raster image correlation spectroscopy, number and brightness, and spatial cross-correlations that give us information about how individual molecules move in cells and interact with partners at the single molecule level. These methods can be implemented with a standard laser scanning microscope and produce a cellular level spatio-temporal map of molecular interactions.     

Total internal reflection with fluorescence correlation spectroscopy: nonfluorescent competitors

Total internal reflection with fluorescence correlation spectroscopy is a method for measuring the surface association/dissociation rate constants and absolute densities of fluorescent molecules at the interface of a planar substrate and solution. This method can also report the apparent diffusion coefficient and absolute concentration of fluorescent molecules very close to the surface. Theoretical expressions for the fluorescence fluctuation autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave, in solution, contribute to the fluorescence fluctuations have been published previously. In the work described here, the nature of the autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave contribute to the fluorescence fluctuations, and when fluorescent and nonfluorescent molecules compete for surface binding sites, is described. The autocorrelation function depends in general on the kinetic association and dissociation rate constants of the fluorescent and nonfluorescent molecules, the surface site density, the concentrations of fluorescent and nonfluorescent molecules in solution, the solution diffusion coefficients of the two chemical species, the depth of the evanescent field, and the size of the observed area on the surface. Both general and approximate expressions are presented.     

Fluorescence Correlation Spectroscopy Measures Molecular Transport in Cells

Fluorescence correlation spectroscopy (FCS) can measure dynamics of fluorescent molecules in cells. FCS measures the fluctuations in the number of fluorescent molecules in a small volume illuminated by a thin beam of excitation light. These fluctuations are processed statistically to yield an autocorrelation function from which rates of diffusion, convection, chemical reaction, and other processes can be extracted. The advantages of this approach include the ability to measure the mobility of a very small number of molecules, even down to the single molecule level, over a wide range of rates in very small regions of a cell. In addition to rates of diffusion and convection, FCS also provides unique information about the local concentration, states of aggregation and molecular interaction using fluctuation amplitude and cross-correlation methods. Recent advances in technology have rendered these once difficult measurements accessible to routine use in cell biology and biochemistry. This review provides a summary of the FCS method and describes current areas in which the FCS approach is being extended beyond its original scope.     

Nonequilibrium radiative properties in fluctuating plasmas1

A general kinetic model was developed to simulate the radiative properties of nonstationary fluctuating plasmas and characterize the relationship between the nonstationary fluctuation time and the atomic relaxation times. The developed theory is applied to the radiative line emission in the case of instabilities in tokamaks. It is shown by exact time dependent simulations that involve explicitly LSJ-split excited states that the radiation emission in fluctuating plasma can be larger than in the corresponding stationary limits. For regular fluctuations like the sawtooth activity, also the startup phase of sawtooth activity can lead to higher emission compared to the time dependent regular phase. It is demonstrated that the sawtooth crash can be almost exactly followed by resonance line emission like H-like Lyman-alpha and He-like Helium-alpha of, e.g., argon impurity ions, whereas the effective charge state distribution lags seriously behind.     

Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion

Total internal reflection with fluorescence correlation spectroscopy (TIR-FCS) is a method for measuring the surface association/dissociation rates and absolute densities of fluorescent molecules at the interface of solution and a planar substrate. This method can also report the apparent diffusion coefficient and absolute concentration of fluorescent molecules very close to the surface. An expression for the fluorescence fluctuation autocorrelation function in the absence of contributions from diffusion through the evanescent wave, in solution, has been published previously (N. L. Thompson, T. P. Burghardt, and D. Axelrod. 1981, Biophys. J. 33:435-454). This work describes the nature of the TIR-FCS autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave contribute to the fluorescence fluctuations. The fluorescence fluctuation autocorrelation function depends in general on the kinetic association and dissociation rate constants, the surface site density, the concentration of fluorescent molecules in solution, the solution diffusion coefficient, and the depth of the evanescent field. Both general and approximate expressions are presented.     

Structural fluctuations in the steady state of muscular contraction.

Recent studies of the intensity fluctuation spectra of coherent light scattered from striated muscle have demonstrated the existence of large scale fluctuations in position and polarizability at the level of the myofibrillar sarcomere and its major structural subunits during the steady state of contraction. The existence of these fluctuations implies a fluctuating driving force. Various possible fluctuating motions of the thick and thin filaments, A and I bands, and entire sarcomeres are described. The magnitude of the fluctuating forces associated with the making and breaking of cross bridges is estimated. A mechanical model is proposed for coupling structural elements of a single sarcomere to one another and for coupling myofibrillar sarcomeres to one another. It is shown that the fluctuating force generated by the spontaneous making and breaking of cross bridges in conjunction with the model accounts for some of the features of the observed intensity fluctuation spectra.     

Rotational relaxation of macromolecules determined by dynamic light scattering. I. Tobacco mosaic virus

The intensity autocorrelation function for the depolarized component of forward-scattered light from a solution of large polymeric molecules is derived in terms of the correlation function for the amplitudes of the Y₂,±₁(θ,?) fluctuations in the anglar distribution of segments in the solution without any assumptions regarding the statistical properties of the scatterad light field. Effects arising from the use of polychromatic incident light and from the mixing of the scattered and polychromatic incident light beams are examined in detail. Apparatus for observing the depolarized forward-scattered light, digitizing and storing the fluctuating phototube current at rates from 10 to 540,000 times per second, and computing the correlation functions directly in the time-domain is described herein. Correlation functions were obtained for 0.05 mg/ml solution of tobacco mosaic virus at pH 9.1 and also at pH 6. The degree of association of the virus appears to be independent of pH, and the monomer relaxation times (corrected to 25°C) extracted from the data by a least-squares procedure lie in the range 0.44–0.49 msec, also independent of pH. The absence of faster component in the correlation function between 6 μsec and 0.5 msec is used in conjunction with thermal fluctuation theory to infer a lower limit for the effective Young's modulaus of the rod, E ≤ 2.5 × 10⁷ dynes/cm².     

Conformational fluctuation of native-like and molten-globule-like cytochrome c observed by time-resolved hole burning

Nanosecond to millisecond conformational fluctuations of Zn-substituted cytochrome c (ZnCytc) have been studied by the time-resolved transient hole-burning method. The investigation of low-temperature dynamics has been made on the ZnCytc solution sample in a water-glycerol mixture. The conformational fluctuations in the native-like and the molten-globule (MG)-like states have been compared for the aqueous solution samples at room temperature. ZnCytc in the MG-like state has been prepared by adding 200 mM NaClO4 to the protein solution with a pH of 2.1, and the formation of the MG-like state has been confirmed by both the far-UV CD and the visible absorption spectra. The hole spectrum of ZnCytc has been found to consist of two nearly degenerate components, that is, the Qx and Qy bands. The temporal change in the Qx component hole spectrum has been extracted by fitting the observed hole spectrum to the three-Gaussian form. The experimental results for ZnCytc dissolved in a water-glycerol mixture have revealed that the conformational fluctuation of ZnCytc is suppressed around 200 K, which is nearly the same temperature as the glass-like transition point of Zn-substituted myoglobin (ZnMb) and also as the glass-transition point of the solvent. This supports the idea of the solvent-induced glass-like transition of a protein. It has been also found that at physiological temperatures the time scale of the conformational fluctuation of ZnCytc lies around a few tens of nanoseconds, which is 2-3 orders of magnitude faster than that of ZnMb. The experimental results for the aqueous solution samples have shown that the difference between the native-like and the MG-like states is not conspicuous. However, they are indicative of the appearance of the slower conformational fluctuation in the MG-like state.     

Imaging Barriers to Diffusion by Pair Correlation Functions

Molecular diffusion and transport are fundamental processes in physical, chemical, biochemical, and biological systems. However, current approaches to measure molecular transport in cells and tissues based on perturbation methods such as fluorescence recovery after photobleaching are invasive, fluctuation correlation methods are local, and single-particle tracking requires the observation of isolated particles for relatively long periods of time. We propose to detect molecular transport by measuring the time cross-correlation of fluctuations at a pair of locations in the sample. When the points are farther apart than two times the size of the point spread function, the maximum of the correlation is proportional to the average time a molecule takes to move from a specific location to another. We demonstrate the method by simulations, using beads in solution, and by measuring the diffusion of molecules in cellular membranes. The spatial pair cross-correlation method detects barriers to diffusion and heterogeneity of diffusion because the time of the correlation maximum is delayed in the presence of diffusion barriers. This noninvasive, sensitive technique follows the same molecule over a large area, thereby producing a map of molecular flow. It does not require isolated molecules, and thus many molecules can be labeled at the same time and within the point spread function.     

Fluorescence correlation spectroscopy: past, present, future

In recent years fluorescence correlation spectroscopy (FCS) has become a routine method for determining diffusion coefficients, chemical rate constants, molecular concentrations, fluorescence brightness, triplet state lifetimes, and other molecular parameters. FCS measures the spatial and temporal correlation of individual molecules with themselves and so provides a bridge between classical ensemble and contemporary single-molecule measurements. It also provides information on concentration and molecular number fluctuations for nonlinear reaction systems that complement single-molecule measurements. Typically implemented on a fluorescence microscope, FCS samples femtoliter volumes and so is especially useful for characterizing small dynamic systems such as biological cells. In addition to its practical utility, however, FCS provides a window on mesoscopic systems in which fluctuations from steady states not only provide the basis for the measurement but also can have important consequences for the behavior and evolution of the system. For example, a new and potentially interesting field for FCS studies could be the study of nonequilibrium steady states, especially in living cells.     

Brownian motion, fluctuation and life

The measurements of dynamic behaviors of biomolecules in relation to their functions have been allowed using single molecule measurements. Thermal Brownian motion causes random step motion of motor proteins and structural fluctuation of protein molecules between multiple states. In hierarchic structure of life, the fluctuation is modulated. Random fluctuation is biased to directional motion and reactions as a result of interaction of proteins. The fluctuation of kinetic state of signaling proteins results in polarization and localization of cells. A recognition process in brain is also explained by the equation analogous to biochemical reaction at the molecular level. Thus dynamic processes originated from thermal motion may play an important role in activation processes in life.     

Field fluctuation in ionic solutions and its biological significance

The thermal fluctuation of the electric potential, the electric field and the charge density of various modes are theoretically analysed in a homogeneous solution of simple electrolytes. The mean square of the fluctuating potential difference between two points at a finite distance averaged in a finite time and the mean square of the fluctuating field at a point averaged in a finite time are calculated. The response of a macromolecule in the solution to such a time-space averaged potential or field is discussed. Numerical estimation shows that the fluctuating potential or field may be actually important in various biological systems. Connection of the present result with the macroscopic case is also shown.     

Theory of excitable membranes. I. A simple model for a three-state artificial membrane.

The effect of the thermal fluctuation on the orientation distribution pattern of globular protein molecules in a two-dimensional lattice was investigated by the method of computer simulation. A set of interaction parameters was assigned to interaction sites on each molecule and the interaction energy between two molecules was given by the product of the parameters of facing sites. Orientation fluctuation was assumed to take place with the probability proportional to the Boltzmann factor. Patterns having different degrees of order appeared with the change of temperature. The entropy and other thermodynamic quantities of these patterns were calculated, and gradual and transitional changes of the pattern were discussed in comparison with the case of simple atoms or molecules.     

Ortho/para spin-isomers of H2O molecules as a factor responsible for formation of two structural motifs in water

According to the latest results obtained by small-angle X-ray scattering and X-ray spectroscopy, it was suggested that water on a nanometer scale represents a fluctuating mixture of clusters with tetrahedral structure and a subphase with partially broken hydrogen bonds, whereas the nuclear configuration of the H₂O molecule corresponds to single tetrahedral coordination. The basic reason of such structural partition is not clear until now. Here we show that it can be associated with existence of two nuclear H₂O spin isomers that have different probability to be in one or the other subphase. The para molecule can transfer an excess of its rotational energy to the environment, up to complete stopping of rotation because its rotational quantum number J = 0 in the basic state. This property is favorable for formation of clusters with closed H-bonds. Ortho molecules with odd-numbered J states lack this property and thus should be predominantly present in the locations with impaired bonds.     

Fluctuation Induces Evolutionary Branching in a Mathematical Model of Ecosystems

Ecological systems are always subjected to various environmental fluctuations. They evolve under these fluctuations and the resulting systems are robust against them. The diversity in ecological systems is also acquired through the evolution. How do the fluctuations affect the evolutionary processes? Do the fluctuations have direct impact on the species diversity in ecological systems? In the present paper, we investigate the relation between the environmental fluctuation and the evolution of species diversity with a mathematical model of evolutionary ecology. In the model, individual organisms compete for a single restricted resource and the temporal fluctuation in the resource supply is introduced as the environmental fluctuation. The evolutionary process is represented by the mutational change of genotypes which determines their resource utilization strategies. We found that when the environmental state is switched form static to fluctuating conditions, the initial closely related population distributed around the genotype adapted for the static environment is destabilized and divided into two groups in the genotype space; i.e., the evolutionary branching is induced by the environmental fluctuation. The consequent multiple species structures is evolutionary stable at the presence of the fluctuation. We perform the evolutionary invasion analysis for the phenomena and illustrate the mechanisms of the branchings. The results indicate a novel process of increasing the species diversity via evolutionary branching, and the analysis reveals the mechanisims of the branching preocess as the response to the environmental fluctuation. The robustness of the evolutionary process is also discussed.