Temperature dependence of protein dynamics: computer simulation analysis of neutron scattering properties

The temperature dependence of the internal dynamics of an isolated protein, bovine pancreatic trypsin inhibitor, is examined using normal mode analysis and molecular dynamics (MD) simulation. It is found that the protein exhibits marked anharmonic dynamics at temperatures of approximately 100-120 K, as evidenced by departure of the MD-derived average mean square displacement from that of the harmonic model. This activation of anharmonic dynamics is at lower temperatures than previously detected in proteins and is found in the absence of solvent molecules. The simulation data are also used to investigate neutron scattering properties. The effects are determined of instrumental energy resolution and of approximations commonly used to extract mean square displacement data from elastic scattering experiments. Both the presence of a distribution of mean square displacements in the protein and the use of the Gaussian approximation to the dynamic structure factor lead to quantified underestimation of the mean square displacement obtained.     

Dynamical transition of myoglobin in a crystal: comparative studies of X-ray crystallography and Mössbauer spectroscopy

The crystallographic normal mode refinements of myoglobin at a wide range of temperature from 40 K to 300 K were carried out to study the temperature dependence of the internal atomic fluctuations. The refinement method decomposes the mean square displacement from the average position, (deltar2), into the contributions from the internal degrees of freedom and those from the external degrees of freedom. The internal displacements show linear temperature dependence as (deltar2)=alphaT+beta, throughout the temperature range measured here, and exhibit no obvious change in the slope alpha at the dynamical transition temperature (Tc=ca. 180 K). The slope alpha is practically the same as the value predicted theoretically by normal mode analysis. Such linear dependence is considered to be due to the following reason. The crystallographic Debye-Waller factor represents the static distribution caused by convolution of temperature-dependent normal mode motions and a temperature-independent set of the conformational substates. In contrast, M?ssbauer absorption spectroscopy shows a clear increase in the gradient alpha at Tc. This difference from X-ray diffraction originates from the incoherent nature of the M?ssbauer effect together with its high-energy resolution, which yields the self-correlation, and the temporal behavior of individual Fe atoms in the myoglobin crystal.     

Protein dynamics of a β-sheet protein

Rhodnius prolixus Nitrophorin 4 (abbreviated NP4) is an almost pure β-sheet heme protein. Its dynamics is investigated by X-ray structure determination at eight different temperatures from 122 to 304 K and by means of Mössbauer spectroscopy. A comparison of this β-sheet protein with the pure α-helical protein myoglobin (abbreviated Mbmet) is performed. The mean square displacement derived from the Mössbauer spectra increases linearly with temperature below a characteristic temperature T _c. It is about 10 K larger than that of myoglobin. Above T _c the mean square displacements increase dramatically. The Mössbauer spectra are analyzed by a two state model. The increased mean square displacements are caused by very slow motions occurring on a time scale faster than 140 ns. With respect to these motions NP4 shows the same protein specific modes as Mbmet. There is, however, a difference in the fast vibration regime. The B values found in the X-ray structures vary linearly over the entire temperature range. The mean square displacements in NP4 increase with slopes which are 60% larger than those observed for Mbmet. This indicates that nitrophorin has a larger structural distribution which makes it more flexible than myoglobin.     

Homology-modeled structure of the yeast mitochondrial citrate transport protein

We have used homology modeling to construct a three-dimensional model of the yeast mitochondrial citrate transport protein (CTP), based on the recently published x-ray crystal structure of another mitochondrial transport protein, the ADP/ATP carrier. Superposition of the backbone traces of the homology-modeled CTP onto the crystallographically determined ADP carrier structure indicates that the CTP transmembrane domains are well modeled (i.e., root mean square deviation of 0.94 A), whereas the loops facing the intermembrane space and the mitochondrial matrix are less certain (i.e., root mean square deviation values of 0.72-2.06 A). The homology-modeled CTP is consistent with our earlier de novo models of the transporter's transmembrane domains, with respect to residues which face into the transport path. Importantly, the resulting model is consistent with our previous experimental data obtained from measuring reactivity of 34 single cysteine mutants in transmembrane domains 3 and 4 with methanethiosulfonate reagents. The model also points to a likely dimer interface region. In conclusion, our data help to define the substrate translocation pathway in both the modeled CTP structure and the crystallographic ADP carrier structure.     

Molecular dynamics of myoglobin at 298 degrees K. Results from a 300-ps computer simulation.

We have carried out a very long (300 ps) molecular dynamics simulation of the protein myoglobin. This trajectory is approximately three times longer than the longest previous molecular dynamics simulation of a protein, and ten times longer than protein simulations of comparable size (1,423 atoms in our model). Here we report results from this long simulation concerning the average structure, the mean square fluctuations of atoms about the average structure, and the nuclear magnetic resonance order parameters for various groups in myoglobin. The results demonstrate that the average coordinates change very slowly during the simulation. The relative atomic mobilities are well described by the simulation. For both the mean square atomic fluctuations and the order parameters, however, there are significant quantitative differences when values calculated using shorter portions of the trajectory are compared with results obtained for the entire 300-ps simulation. The implications of this result for obtaining converged properties from protein molecular dynamics simulations for comparison with experiment are discussed.     

Mathematical modelling of whole chromosome replication

All chromosomes must be completely replicated prior to cell division, a requirement that demands the activation of a sufficient number of appropriately distributed DNA replication origins. Here we investigate how the activity of multiple origins on each chromosome is coordinated to ensure successful replication. We present a stochastic model for whole chromosome replication where the dynamics are based upon the parameters of individual origins. Using this model we demonstrate that mean replication time at any given chromosome position is determined collectively by the parameters of all origins. Combining parameter estimation with extensive simulations we show that there is a range of model parameters consistent with mean replication data, emphasising the need for caution in interpreting such data. In contrast, the replicated-fraction at time points through S phase contains more information than mean replication time data and allowed us to use our model to uniquely estimate many origin parameters. These estimated parameters enable us to make a number of predictions that showed agreement with independent experimental data, confirming that our model has predictive power. In summary, we demonstrate that a stochastic model can recapitulate experimental observations, including those that might be interpreted as deterministic such as ordered origin activation times.     

Dynamic properties of photosystem II membranes at physiological temperatures characterized by elastic incoherent neutron scattering. Increased flexibility associated with the inactivation of the oxygen evolving complex

Elastic incoherent neutron scattering (EINS), a non-invasive technique which is capable of measuring the mean square displacement of atoms in the sample, has been widely used in biology for exploring the dynamics of proteins and lipid membranes but studies on photosynthetic systems are scarce. In this study we investigated the dynamic characteristics of Photosystem II (PSII) membrane fragments between 280 and 340?K, i.e., in the physiological temperature range and in the range of thermal denaturation of some of the protein complexes. The mean square displacement values revealed the presence of a hydration-sensitive transition in the sample between 310 and 320?K, suggesting that the oxygen evolving complex (OEC) plays an important role in the transition. Indeed, in samples in which the OEC had been removed by TRIS- or heat-treatments (323 and 333?K) no such transition was found. Further support on the main role of OEC in these reorganizations is provided by data obtained from differential scanning calorimetry experiments, showing marked differences between the untreated and TRIS-treated samples. In contrast, circular dichroism spectra exhibited only minor changes in the excitonic interactions below 323?K, showing that the molecular organization of the pigment-protein complexes remains essentially unaffected. Our data, along with earlier incoherent neutron scattering data on PSII membranes at cryogenic temperatures (Pieper et al., Biochemistry 46:11398-11409, 2007), demonstrate that this technique can be applied to characterize the dynamic features of PSII membranes, and can be used to investigate photosynthetic membranes under physiologically relevant experimental conditions.     

Effects of directional environmental change on extinction dynamics in experimental microbial communities are predicted by a simple model

Global temperatures are expected to rise between 1.1 and 6.4°C over the next 100 years, although the exact rate will depend on future greenhouse emissions, and will vary spatially. Temperature can alter an individual's metabolic rate, and consequently birth and death rates. In declining populations, these alterations may manifest as changes in the rate of that population's decline, and subsequently the timing of extinction events. Predicting such events could therefore be of considerable use. We use a small‐scale experimental system to investigate how the rate of temperature change can alter a population's time to extinction, and whether it is possible to predict this event using a simple phenomenological model that incorporates information about population dynamics at a constant temperature, published scaling of metabolic rates, and temperature. In addition, we examine 1) the relative importance of the direct effects of temperature on metabolic rate, and the indirect effects (via temperature driven changes in body size), on predictive accuracy (defined as the proximity of the predicted date of extinction to the mean observed date of extinction), 2) the combinations of model parameters that maximise accuracy of predictions, and 3) whether substituting temperature change through time with mean temperature produces accurate predictions. We find that extinction occurs earlier in environments that warm faster, and this can be accurately predicted (R² > 0.84). Increasing the number of parameters that were temperature‐dependent increased the model's accuracy, as did scaling these temperature‐dependent parameters with either the direct effects of temperature alone, or with the direct and indirect effects. Using mean temperature through time instead of actual temperature produces less accurate predictions of extinction. These results suggest that simple phenomenological models, incorporating metabolic theory, may be useful in understanding how environmental change can alter a population's rate of extinction. Synthesis Understanding how populations will respond to future climatic change is a key goal in ecology, however the exact rate of future warming will vary both spatially and temporally. Consequently, mathematical models must be used to understand the potential range of future population dynamics under various warming scenarios. We use a combination of experimentation and modelling to show that the effects of varying rates of environmental change on population dynamics can be predicted by a simple model. However, the accuracy of these predictions depends upon, amongst other things, a detailed knowledge of how temperature will change over time, rather than approximating this change to mean temperature.     

Multi-locus Analysis of Genomic Time Series Data from Experimental Evolution

Genomic time series data generated by evolve-and-resequence (E&R) experiments offer a powerful window into the mechanisms that drive evolution. However, standard population genetic inference procedures do not account for sampling serially over time, and new methods are needed to make full use of modern experimental evolution data. To address this problem, we develop a Gaussian process approximation to the multi-locus Wright-Fisher process with selection over a time course of tens of generations. The mean and covariance structure of the Gaussian process are obtained by computing the corresponding moments in discrete-time Wright-Fisher models conditioned on the presence of a linked selected site. This enables our method to account for the effects of linkage and selection, both along the genome and across sampled time points, in an approximate but principled manner. We first use simulated data to demonstrate the power of our method to correctly detect, locate and estimate the fitness of a selected allele from among several linked sites. We study how this power changes for different values of selection strength, initial haplotypic diversity, population size, sampling frequency, experimental duration, number of replicates, and sequencing coverage depth. In addition to providing quantitative estimates of selection parameters from experimental evolution data, our model can be used by practitioners to design E&R experiments with requisite power. We also explore how our likelihood-based approach can be used to infer other model parameters, including effective population size and recombination rate. Then, we apply our method to analyze genome-wide data from a real E&R experiment designed to study the adaptation of D. melanogaster to a new laboratory environment with alternating cold and hot temperatures.     

Myocardial oxygenation in isolated hearts predicted by an anatomically realistic microvascular transport model

An anatomically realistic model for oxygen transport in cardiac tissue is introduced for analyzing data measured from isolated perfused guinea pig hearts. The model is constructed to match the microvascular anatomy of cardiac tissue based on available morphometric data. Transport in the three-dimensional system (divided into distinct microvascular, interstitial, and parenchymal spaces) is simulated. The model is used to interpret experimental data on mean cardiac tissue myoglobin saturation and to reveal differences in tissue oxygenation between buffer-perfused and red blood cell-perfused isolated hearts. Interpretation of measured mean myoglobin saturation is strongly dependent on the oxygen content of the perfusate (e.g., red blood cell-containing vs. cell-free perfusate). Model calculations match experimental values of mean tissue myoglobin saturation, measured mean myoglobin, and venous oxygen tension and can be used to predict distributions of intracellular oxygen tension. Calculations reveal that approximately 20% of the tissue is hypoxic with an oxygen tension of <0.5 mmHg when the buffer is equilibrated with 95% oxygen to give an arterial oxygen tension of over 600 mmHg. The addition of red blood cells to give a hematocrit of only 5% prevents tissue hypoxia. It is incorrect to assume that the usual buffer-perfused Langendorff heart preparation is adequately oxygenated for flows in the range of < or =10 ml. min-1. ml tissue-1.     

Nuclear forward scattering of synchrotron radiation by deoxymyoglobin

Nuclear forward scattering of synchrotron radiation is used to determine the quadrupole splitting and the mean square displacement of the iron atom in deoxymyoglobin in the temperature range between 50 K and 243 K. Above 200 K an abnormally fast decay of the forward scattered intensity at short times after the synchrotron flash is observed, which is caused by protein-specific motions. The results strongly support the picture that protein dynamics seen at the position of the iron can be understood by harmonic motions in the low temperature regime while in the physiological regime diffusive motions in limited space are present. The shape of the resonance broadening function is investigated. An inhomogeneous broadening with a Lorentzian distribution indicating dipole interactions results in a better agreement with the experimental data than the common Gaussian distribution. Received: 30 August 1999 / Revised version: 22 October 1999 / Accepted: 6 December 1999     

Space-Time Data fusion Under Error in Computer Model Output: An Application to Modeling Air Quality

Summary We provide methods that can be used to obtain more accurate environmental exposure assessment. In particular, we propose two modeling approaches to combine monitoring data at point level with numerical model output at grid cell level, yielding improved prediction of ambient exposure at point level. Extending our earlier downscaler model (Berrocal, V. J., Gelfand, A. E., and Holland, D. M. (2010b) . A spatio‐temporal downscaler for outputs from numerical models. Journal of Agricultural, Biological and Environmental Statistics 15, 176–197), these new models are intended to address two potential concerns with the model output. One recognizes that there may be useful information in the outputs for grid cells that are neighbors of the one in which the location lies. The second acknowledges potential spatial misalignment between a station and its putatively associated grid cell. The first model is a Gaussian Markov random field smoothed downscaler that relates monitoring station data and computer model output via the introduction of a latent Gaussian Markov random field linked to both sources of data. The second model is a smoothed downscaler with spatially varying random weights defined through a latent Gaussian process and an exponential kernel function, that yields, at each site, a new variable on which the monitoring station data is regressed with a spatial linear model. We applied both methods to daily ozone concentration data for the Eastern US during the summer months of June, July and August 2001, obtaining, respectively, a 5% and a 15% predictive gain in overall predictive mean square error over our earlier downscaler model ( Berrocal et al., 2010b ). Perhaps more importantly, the predictive gain is greater at hold‐out sites that are far from monitoring sites.     

Harmonic behavior of trehalose-coated carbon-monoxy-myoglobin at high temperature.

Embedding biostructures in saccharide glasses protects them against extreme dehydration and/or exposure to very high temperature. Among the saccharides, trehalose appears to be the most effective bioprotectant. In this paper we report on the low-frequency dynamics of carbon monoxy myoglobin in an extremely dry trehalose glass measured by neutron spectroscopy. Under these conditions, the mean square displacements and the density of state function are those of a harmonic solid, up to room temperature, in contrast to D2O-hydrated myoglobin, in which a dynamical transition to a nonharmonic regime has been observed at approximately 180 K (Doster et al., 1989. Nature. 337:754-756). The protective effect of trehalose is correlated, therefore, with a trapping of the protein in a harmonic potential, even at relatively high temperature.     

Universal Algorithm for Identification of Fractional Brownian Motion. A?Case of Telomere Subdiffusion

We present a systematic statistical analysis of the recently measured individual trajectories of fluorescently labeled telomeres in the nucleus of living human cells. The experiments were performed in the U2OS cancer cell line. We propose an algorithm for identification of the telomere motion. By expanding the previously published data set, we are able to explore the dynamics in six time orders, a task not possible earlier. As a result, we establish a rigorous mathematical characterization of the stochastic process and identify the basic mathematical mechanisms behind the telomere motion. We find that the increments of the motion are stationary, Gaussian, ergodic, and even more chaotic—mixing. Moreover, the obtained memory parameter estimates, as well as the ensemble average mean square displacement reveal subdiffusive behavior at all time spans. All these findings statistically prove a fractional Brownian motion for the telomere trajectories, which is confirmed by a generalized p-variation test. Taking into account the biophysical nature of telomeres as monomers in the chromatin chain, we suggest polymer dynamics as a sufficient framework for their motion with no influence of other models. In addition, these results shed light on other studies of telomere motion and the alternative telomere lengthening mechanism. We hope that identification of these mechanisms will allow the development of a proper physical and biological model for telomere subdynamics. This array of tests can be easily implemented to other data sets to enable quick and accurate analysis of their statistical characteristics.     

Scattering of ultrasound in cancellous bone: predictions from a theoretical model

An understanding of the interaction between acoustic waves and cancellous bone is needed in order to realize the full clinical potential of ultrasonic bone measurements. Scattering is likely to be of central importance but has received little attention to date. In this study, we adopted a theoretical model from the literature in which scattering was assumed to be proportional to the mean fluctuation in sound speed, and bone was considered to be a random continuum containing identical scatterers. The model required knowledge only of sound speeds in bone and marrow, porosity, and scatter size. Predicted attenuation, broadband ultrasonic attenuation (BUA) and backscatter coefficient were obtained for a range of porosities and scatterer sizes, and were found to be comparable to published values for cancellous bone. Trends in predicted BUA with porosity agreed with previous experimental observations. All three predicted acoustic parameters showed a non-linear dependence on scatterer size which was independent of porosity. These data confirm the value of the scattering approach and provide the first quantitative predictions of the independent influence of structure and porosity on bone acoustic properties.     

Estimating quality of template-based protein models by alignment stability

The error in protein tertiary structure prediction is unavoidable, but it is not explicitly shown in most of the current prediction algorithms. Estimated error of a predicted structure is crucial information for experimental biologists to use the prediction model for design and interpretation of experiments. Here, we propose a method to estimate errors in predicted structures based on the stability of the optimal target-template alignment when compared with a set of suboptimal alignments. The stability of the optimal alignment is quantified by an index named the SuboPtimal Alignment Diversity (SPAD). We implemented SPAD in a profile-based threading algorithm and investigated how well SPAD can indicate errors in threading models using a large benchmark dataset of 5232 alignments. SPAD shows a very good correlation not only to alignment shift errors but also structure-level errors, the root mean square deviation (RMSD) of predicted structure models to the native structures (i.e. global errors), and local errors at each residue position. We have further compared SPAD with seven other quality measures, six from sequence alignment-based measures and one atomic statistical potential, discrete optimized protein energy (DOPE), in terms of the correlation coefficient to the global and local structure-level errors. In terms of the correlation to the RMSD of structure models, when a target and a template are in the same SCOP family, the sequence identity showed a best correlation to the RMSD; in the superfamily level, SPAD was the best; and in the fold level, DOPE was best. However, in a head-to-head comparison, SPAD wins over the other measures. Next, SPAD is compared with three other measures of local errors. In this comparison, SPAD was best in all of the family, the superfamily and the fold levels. Using the discovered correlation, we have also predicted the global and local error of our predicted structures of CASP7 targets by the SPAD. Finally, we proposed a sausage representation of predicted tertiary structures which intuitively indicate the predicted structure and the estimated error range of the structure simultaneously.     

Biological assessment of robust noise models in microarray data analysis

MOTIVATION: Although several recently proposed analysis packages for microarray data can cope with heavy-tailed noise, many applications rely on Gaussian assumptions. Gaussian noise models foster computational efficiency. This comes, however, at the expense of increased sensitivity to outlying observations. Assessing potential insufficiencies of Gaussian noise in microarray data analysis is thus important and of general interest. RESULTS: We propose to this end assessing different noise models on a large number of microarray experiments. The goodness of fit of noise models is quantified by a hierarchical Bayesian analysis of variance model, which predicts normalized expression values as a mixture of a Gaussian density and t-distributions with adjustable degrees of freedom. Inference of differentially expressed genes is taken into consideration at a second mixing level. For attaining far reaching validity, our investigations cover a wide range of analysis platforms and experimental settings. As the most striking result, we find irrespective of the chosen preprocessing and normalization method in all experiments that a heavy-tailed noise model is a better fit than a simple Gaussian. Further investigations revealed that an appropriate choice of noise model has a considerable influence on biological interpretations drawn at the level of inferred genes and gene ontology terms. We conclude from our investigation that neglecting the over dispersed noise in microarray data can mislead scientific discovery and suggest that the convenience of Gaussian-based modelling should be replaced by non-parametric approaches or other methods that account for heavy-tailed noise.     

Development and validation of an atlas-based finite element brain model

Traumatic brain injury is a leading cause of disability and injury-related death. To enhance our ability to prevent such injuries, brain response can be studied using validated finite element (FE) models. In the current study, a high-resolution, anatomically accurate FE model was developed from the International Consortium for Brain Mapping brain atlas. Due to wide variation in published brain material parameters, optimal brain properties were identified using a technique called Latin hypercube sampling, which optimized material properties against three experimental cadaver tests to achieve ideal biomechanics. Additionally, falx pretension and thickness were varied in a lateral impact variation. The atlas-based brain model (ABM) was subjected to the boundary conditions from three high-rate experimental cadaver tests with different material parameter combinations. Local displacements, determined experimentally using neutral density targets, were compared to displacements predicted by the ABM at the same locations. Error between the observed and predicted displacements was quantified using CORrelation and Analysis (CORA), an objective signal rating method that evaluates the correlation of two curves. An average CORA score was computed for each variation and maximized to identify the optimal combination of parameters. The strongest relationships between CORA and material parameters were observed for the shear parameters. Using properties obtained through the described multiobjective optimization, the ABM was validated in three impact configurations and shows good agreement with experimental data. The final model developed in this study consists of optimized brain material properties and was validated in three cadaver impacts against local brain displacement data.     

Extending the Conditions of Application of an Inversion of the Hodgkin–Huxley Gating Model

We present an inversion of the Hodgkin–Huxley formalism to estimate initial conditions and model parameters, including functions of voltage, from the solutions of the underlying ordinary differential equation (ODE) subjected to multiple voltage step stimulations. As such, the procedure constitutes a means to estimate the parameters including functions of voltage of an Hodgkin–Huxley formalism from experimental data. The basic idea was developed in a previous communication (SIAM J. Appl. Math. 64:1264–1274, 2009). The inversion in question applies to currents exhibiting activation and inactivation, but the version, as published previously, cannot estimate the unknowns for channels that rapidly inactivate just after a brief opening. In such cases, the amplitude of the current, in a given voltage range, is too small to be detectable by the instrumentation using previously applied experimental protocols. This is, for example, the case for the sodium channels in a number of excitable tissue for potential in the vicinity of the cell resting potential. The current communication extends the inversion procedure in a manner to overcome this limitation. Furthermore, within the inversion framework, we can determine whether the data at the basis of the estimation sufficiently constrains the estimation problem, i.e., whether it is complete. We exploit this element of our method to document a set of stimulation protocols that constitute a complete data set for the purpose of inverting the Hodgkin–Huxley formalism.