A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis

An exact theorem is proved and its implication is discussed. The theorem states that, if a large molecule, typically biological macromolecules such as proteins, undergoes small-amplitude conformational fluctuations around its native conformation in such a way that within the range of conformational fluctuations at thermal equilibrium the conformational energy surface can be approximated by a multidimensional parabola, then the mass-weighted mean-square displacement of constituent atoms is given by the sum of the contributions from each normal mode of conformational vibration, which in turn is proportional to the inverse of the square of its frequency. This theorem provides a firm theoretical basis for the fact hitherto empirically recognized in the conformational dynamics of, for instance, native proteins that very-low-frequency normal modes make dominant contributions to the conformational fluctuations at thermal equilibrium. Discussion is given on the implication of this theorem, especially on the importance of the concept of the low-frequency normal modes, even in the case where the basic assumption of the harmonicity of the energy surface does not hold.     

Fluctuation driven transport and models of molecular motors and pumps

Non-equilibrium fluctuations can drive vectorial transport along an anisotropic structure in an isothermal medium by biasing the effect of thermal noise (k _B T). Mechanisms based on this principle are often called Brownian ratchets and have been invoked as a possible explanation for the operation of biomolecular motors and pumps. We discuss the thermodynamics and kinetics for the operation of microscopic ratchet motors under conditions relevant to biology, showing how energy provided by external fluctuations or a non-equilibrium chemical reaction can cause unidirectional motion or uphill pumping of a substance. Our analysis suggests that molecular pumps such as Na,K-ATPase and molecular motors such as kinesin and myosin may share a common underlying mechanism. Received: 18 February 1998 / Revised version: 5 May 1998 / Accepted: 14 May 1998     

Allostery without conformational change

A general model is presented whereby lignand-induced changes in protein dynamics could produce allosteric communication between distinct binding sites, even in the absence of a macromolecular conformational change. Theoretical analysis, based on the statistical thermodynamics of ligand binding, shows that cooperative interaction free energies amounting to several kJ · mol^-1 may be generated by this means. The effect arises out of the possible changes in frequencies and amplitudes of macromolecular thermal fluctuations in response to ligand attachment, and can involve all forms of dynamic behaviour, ranging from highly correlated, low-frequency normal mode vibrations to random local anharmonic motions of individual atoms or groups. Dynamic allostery of this form is primarily an entropy effect, and we derive approximate expressions which might allow the magnitude of the interaction in real systems to be calculated directly from experimental observations such as changes in normal mode frequencies and mean-square atomic displacements. Long-range influence of kinetic processes at different sites might also be mediated by a similar mechanism. We suggest that proteins and other biological macromolecules may have evolved to take functional advantage not only of mean conformational states but also of the inevitable thermal fluctuations about the mean.     

Thermo-mechanical responses of a surface-coupled AFM cantilever

Atomic force microscopy (AFM) has been widely used for measuring mechanical properties of biological specimens such as cells, DNA, and proteins. This is usually done by monitoring deformations in response to controlled applied forces, which have to be at ultralow levels due to the extreme softness of the specimens. Consequently, such experiments may be susceptible to thermal excitations, manifested as force and displacement fluctuations that could reduce the measurement accuracy. To take advantage of, rather than to be limited by, such fluctuations, we have characterized the thermomechanical responses of an arbitrarily shaped AFM cantilever with the tip coupled to an elastic spring. Our analysis shows that the cantilever and the specimen behave as springs in parallel. This provides a method for determining the elasticity of the specimen by measuring the change in the tip fluctuations in the presence and absence of coupling. For rectangular and V-shaped cantilevers, we have derived a relationship between the mean-square deflection and the mean-square inclination and an approximate expression for the specimen spring constant in terms of contributions to the mean-square inclination from the first few vibration modes.     

Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.     

Cell motility driven by actin polymerization.

Certain kinds of cellular movements are apparently driven by actin polymerization. Examples include the lamellipodia of spreading and migrating embryonic cells, and the bacterium Listeria monocytogenes, that propels itself through its host's cytoplasm by constructing behind it a polymerized tail of cross-linked actin filaments. Peskin et al. (1993) formulated a model to explain how a polymerizing filament could rectify the Brownian motion of an object so as to produce unidirectional force (Peskin, C., G. Odell, and G. Oster. 1993. Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65:316-324). Their "Brownian ratchet" model assumed that the filament was stiff and that thermal fluctuations affected only the "load," i.e., the object being pushed. However, under many conditions of biological interest, the thermal fluctuations of the load are insufficient to produce the observed motions. Here we shall show that the thermal motions of the polymerizing filaments can produce a directed force. This "elastic Brownian ratchet" can explain quantitatively the propulsion of Listeria and the protrusive mechanics of lamellipodia. The model also explains how the polymerization process nucleates the orthogonal structure of the actin network in lamellipodia.     

Thermal fluctuations biased for directional motion in molecular motors

Recently developed single molecule measurements have demonstrated that the mechanisms for numerous protein functions involve thermal fluctuation, or Brownian motion. Protein interactions bias the random thermal noise in a manner such that the protein can perform its given functions. This phenomenon has been observed in molecular motor unidirectional movement where Brownian motion is used to preferentially bind the motor heads in one direction causing directional motility. This is analogous to that used by proteins in which spontaneous structural fluctuations are used to switch function. Seeing that two very different systems implement similar mechanisms suggests there exists a general scheme applied by diverse proteins that exploits thermal fluctuations in order to achieve their respective functions.     

Structural dynamics of liganded myoglobin.

X-ray crystallography can reveal the magnitudes and principal directions of the mean-square displacements of every atom in a protein. This structural information is complementary to the temporal information obtainable by spectroscopic techniques such as nuclear magnetic resonance. Determination of the temperature dependence of the mean-square displacements makes it possible to separate large conformational motions from simple thermal vibrations. The contribution of crystal lattice disorder to the overall apparent displacement can be estimated by Mössbauer spectroscopy. This technique has been applied to high resolution x-ray diffraction data from sperm whale myoglobin in its Met iron and oxy cobalt forms. Both crystal structures display regions of large conformational motions, particularly at the chain termini and in the region of the proximal histidine. Overall, the mean-square displacement increases with increasing distance from the center of gravity of the molecule. Some regions of the heme pocket in oxy cobalt myoglobin are more rigid than the corresponding regions in Met myoglobin.     

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm(-1) contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm(-1), but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis.     

Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes—namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another—are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.     

A Multiscale Red Blood Cell Model with Accurate Mechanics, Rheology, and Dynamics

Red blood cells (RBCs) have highly deformable viscoelastic membranes exhibiting complex rheological response and rich hydrodynamic behavior governed by special elastic and bending properties and by the external/internal fluid and membrane viscosities. We present a multiscale RBC model that is able to predict RBC mechanics, rheology, and dynamics in agreement with experiments. Based on an analytic theory, the modeled membrane properties can be uniquely related to the experimentally established RBC macroscopic properties without any adjustment of parameters. The RBC linear and nonlinear elastic deformations match those obtained in optical-tweezers experiments. The rheological properties of the membrane are compared with those obtained in optical magnetic twisting cytometry, membrane thermal fluctuations, and creep followed by cell recovery. The dynamics of RBCs in shear and Poiseuille flows is tested against experiments and theoretical predictions, and the applicability of the latter is discussed. Our findings clearly indicate that a purely elastic model for the membrane cannot accurately represent the RBC's rheological properties and its dynamics, and therefore accurate modeling of a viscoelastic membrane is necessary.     

Nonequilibrium Brownian Motion beyond the Effective Temperature

The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einstein’s relation between mobility and diffusivity for Brownian motion. Several recent theories relax the hypothesis of thermal equilibrium resulting in at least two main scenarios. With well separated timescales, as in aging glassy systems, equilibrium Fluctuation-Dissipation Theorem applies at each scale with its own “effective” temperature. With mixed timescales, as for example in active or granular fluids or in turbulence, temperature is no more well-defined, the dynamical nature of fluctuations fully emerges and a Generalized Fluctuation-Dissipation Theorem (GFDT) applies. Here, we study experimentally the mixed timescale regime by studying fluctuations and linear response in the Brownian motion of a rotating intruder immersed in a vibro-fluidized granular medium. Increasing the packing fraction, the system is moved from a dilute single-timescale regime toward a denser multiple-timescale stage. Einstein’s relation holds in the former and is violated in the latter. The violation cannot be explained in terms of effective temperatures, while the GFDT is able to impute it to the emergence of a strong coupling between the intruder and the surrounding fluid. Direct experimental measurements confirm the development of spatial correlations in the system when the density is increased.     

Active Nucleosome Displacement: A Theoretical Approach

Three-quarters of eukaryotic DNA are wrapped around protein cylinders forming so-called nucleosomes that block the access to the genetic information. Nucleosomes need therefore to be repositioned, either passively (by thermal fluctuations) or actively (by molecular motors). Here we introduce a theoretical model that allows us to study the interplay between a motor protein that moves along DNA (e.g., an RNA polymerase) and a nucleosome that it encounters on its way. We aim at describing the displacement mechanisms of the nucleosome and the motor protein on a microscopic level to understand better the intricate interplay between the active step of the motor and the nucleosome-repositioning step. Different motor types (Brownian ratchet versus power-stroke mechanism) that perform very similarly under a constant load are shown to have very different nucleosome repositioning capacities.     

A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility

In recent single-particle tracking (SPT) measurements on Listeria monocytogenes motility in cells [Kuo and McGrath (2000)], the actin-based stochastic dynamics of the bacterium movement has been analyzed statistically in terms of the mean-square displacement (MSD) of the trajectory. We present a stochastic analysis of a simplified polymerization Brownian ratchet (BR) model in which motions are limited by the bacterium movement. Analytical results are obtained and statistical data analyses are investigated. It is shown that the MSD of the stochastic bacterium movement is a monotonic quadratic function while the MSD for detrended trajectories is linear. Both the short-time relaxation and the long-time kinetics in terms the mean velocity and effective diffusion constant of the propelled bacterium are obtained from the MSD analysis. The MSD of the gap between actin tip and the bacterium exhibits an oscillatory behavior when there is a large resistant force from the bacterium. For comparison, a continuous diffusion formalism of the BR model with great analytical simplicity is also studied.     

Chromatin Hydrodynamics

Following recent observations of large scale correlated motion of chromatin inside the nuclei of live differentiated cells, we present a hydrodynamic theory—the two-fluid model—in which the content of a nucleus is described as a chromatin solution with the nucleoplasm playing the role of the solvent and the chromatin fiber that of a solute. This system is subject to both passive thermal fluctuations and active scalar and vector events that are associated with free energy consumption, such as ATP hydrolysis. Scalar events drive the longitudinal viscoelastic modes (where the chromatin fiber moves relative to the solvent) while vector events generate the transverse modes (where the chromatin fiber moves together with the solvent). Using linear response methods, we derive explicit expressions for the response functions that connect the chromatin density and velocity correlation functions to the corresponding correlation functions of the active sources and the complex viscoelastic moduli of the chromatin solution. We then derive general expressions for the flow spectral density of the chromatin velocity field. We use the theory to analyze experimental results recently obtained by one of the present authors and her co-workers. We find that the time dependence of the experimental data for both native and ATP-depleted chromatin can be well-fitted using a simple model—the Maxwell fluid—for the complex modulus, although there is some discrepancy in terms of the wavevector dependence. Thermal fluctuations of ATP-depleted cells are predominantly longitudinal. ATP-active cells exhibit intense transverse long wavelength velocity fluctuations driven by force dipoles. Fluctuations with wavenumbers larger than a few inverse microns are dominated by concentration fluctuations with the same spectrum as thermal fluctuations but with increased intensity.     

Brownian dynamics simulations of supercoiled DNA with bent sequences.

The recently presented Brownian dynamics model for superhelical DNA is extended to include local curvature of the DNA helix axis. Here we analyze the effect of a permanent bend on the structure and dynamics of an 1870-bp superhelix with delta Lk = -10. Furthermore, we define quantitative expressions for computing structural parameters such as loop positions, superhelix diameter, and plectonemic content for trajectories of superhelical DNA, and assess the convergence toward global equilibrium. The structural fluctuations in an interwound superhelix, as reflected in the change in end loop positions, seem to occur by destruction/creation of loops rather than by a sliding motion of the DNA around its contour. Their time scale is on the order of 30-100 microseconds. A permanent bend changes the structure and the internal motions of the DNA drastically. The position of the end loop is fixed at the permanent bend, and the local motions of the chain are enhanced near the loops. A displacement of the bend from the end loop to a position inside the plectonemic part of the superhelix results in the formation of a new loop and the disappearance of the old one; we estimate the time involved in this process to be about 0.5 ms.     

Chromatin Hydrodynamics

Following recent observations of large scale correlated motion of chromatin inside the nuclei of live differentiated cells, we present a hydrodynamic theory—the two-fluid model—in which the content of a nucleus is described as a chromatin solution with the nucleoplasm playing the role of the solvent and the chromatin fiber that of a solute. This system is subject to both passive thermal fluctuations and active scalar and vector events that are associated with free energy consumption, such as ATP hydrolysis. Scalar events drive the longitudinal viscoelastic modes (where the chromatin fiber moves relative to the solvent) while vector events generate the transverse modes (where the chromatin fiber moves together with the solvent). Using linear response methods, we derive explicit expressions for the response functions that connect the chromatin density and velocity correlation functions to the corresponding correlation functions of the active sources and the complex viscoelastic moduli of the chromatin solution. We then derive general expressions for the flow spectral density of the chromatin velocity field. We use the theory to analyze experimental results recently obtained by one of the present authors and her co-workers. We find that the time dependence of the experimental data for both native and ATP-depleted chromatin can be well-fitted using a simple model—the Maxwell fluid—for the complex modulus, although there is some discrepancy in terms of the wavevector dependence. Thermal fluctuations of ATP-depleted cells are predominantly longitudinal. ATP-active cells exhibit intense transverse long wavelength velocity fluctuations driven by force dipoles. Fluctuations with wavenumbers larger than a few inverse microns are dominated by concentration fluctuations with the same spectrum as thermal fluctuations but with increased intensity.     

Deriving protein dynamical properties from weighted protein contact number

It has recently been shown that in proteins the atomic mean-square displacement (or B-factor) can be related to the number of the neighboring atoms (or protein contact number), and that this relationship allows one to compute the B-factor profiles directly from protein contact number. This method, referred to as the protein contact model, is appealing, since it requires neither trajectory integration nor matrix diagonalization. As a result, the protein contact model can be applied to very large proteins and can be implemented as a high-throughput computational tool to compute atomic fluctuations in proteins. Here, we show that this relationship can be further refined to that between the atomic mean-square displacement and the weighted protein contact-number, the weight being the square of the reciprocal distance between the contacting pair. In addition, we show that this relationship can be utilized to compute the cross-correlation of atomic motion (the B-factor is essentially the auto-correlation of atomic motion). For a nonhomologous dataset comprising 972 high-resolution X-ray protein structures (resolution <2.0 A and sequence identity <25%), the mean correlation coefficient between the X-ray and computed B-factors based on the weighted protein contact-number model is 0.61, which is better than those of the original contact-number model (0.51) and other methods. We also show that the computed correlation maps based on the weighted contact-number model are globally similar to those computed through normal model analysis for some selected cases. Our results underscore the relationship between protein dynamics and protein packing. We believe that our method will be useful in the study of the protein structure-dynamics relationship.     

A chemically driven fluctuating ratchet model for actomyosin interaction

With reference to the experimental observations by Yanagida and his co-workers on actomyosin interaction, a Brownian motor of fluctuating ratchet kind is designed with the aim to describe the interaction between a Myosin II head and a neighboring actin filament. Our motor combines the dynamics of the myosin head with a chemical external system related to the ATP cycle, whose role is to provide the energy supply necessary to bias the motion. Analytical expressions for the duration of the ATP cycle, for the Gibbs free energy and for the net displacement of the myosin head are obtained. Finally, by exploiting a method due to Sekimoto [J. Phys. Soc. Jpn. 66 (1997) 1234], a formula is worked out for the amount of energy consumed during the ATP cycle.