Transition modes in Ising networks: an approximate theory for macromolecular recognition.

For a statistical lattice, or Ising network, composed of N identical units existing in two possible states, 0 and 1, and interacting according to a given geometry, a set of values can be found for the mean free energy of the 0-->1 transition of a single unit. Each value defines a transition mode in an ensemble of nu N = 3N - 2N possible values and reflects the role played by intermediate states in shaping the energetics of the system as a whole. The distribution of transition modes has a number of intriguing properties. Some of them apply quite generally to any Ising network, regardless of its dimension, while others are specific for each interaction geometry and dimensional embedding and bear on fundamental aspects of analytical number theory. The landscape of transition modes encapsulates all of the important thermodynamic properties of the network. The free energy terms defining the partition function of the system can be derived from the modes by simple transformations. Classical mean-field expressions can be obtained from consideration of the properties of transition modes in a rather straightforward way. The results obtained in the analysis of the transition mode distributions have been used to develop an approximate treatment of the problem of macromolecular recognition. This phenomenon is modeled as a cooperative process that involves a number of recognition subsites across an interface generated by the binding of two macromolecular components. The distribution of allowed binding free energies for the system is shown to be a superposition of Gaussian terms with mean and variance determined a priori by the theory. Application to the analysis of the biologically interaction of thrombin with hirudin has provided some useful information on basic aspects of the interaction, such as the number of recognition subsites involved and the energy balance for binding and cooperative coupling among them. Our results agree quite well with information derived independently from analysis of the crystal structure of the thrombin-hirudin complex.     

Acid phosphatase reaction with peroxyl radicals: inactivation mechanism and behavior of the partially modified ensemble

Acid phosphatase (AP) is readily inactivated when exposed to the free radicals generated in the pyrolysis of 2, 2'-azobis(2-amidinopropane) hydrochloride (AAPH) under aerobic conditions. On average, a large number of tryptophan groups are modified by each protein molecule that loses its catalytic activity. In spite of this, the enzyme inactivation takes place without induction times, a result that indicates either that damage is progressive or that damage of a critical target is needed to inactivate the enzyme (all-or-nothing mechanism). A Lineweaver-Burk plot of the enzyme activity measured at pH 4.8 is not compatible with an all-or-nothing mechanism, showing that after exposure of the native protein ensemble to the free radical source there are partially damaged molecules whose affinity for the substrate is widely different from that of the native molecules. On the other hand, the partially damaged ensemble shows a normal Michaelis-Menten behavior when the activity is measured at pH 7.0, with only a reduced value of V(M), relative to that of the unmodified ensemble. These results show that the native protein and modified proteins that remain active constitute different populations, with different responses to pH changes. Comparative heat denaturation studies of the native and pretreated proteins support this proposal.     

Folding energetics of ligand binding proteins. I. Theoretical model

Heat capacity curves as obtained from differential scanning calorimetry are an outstanding source for molecular information on protein folding and ligand-binding energetics. However, deconvolution of C(p) data of proteins in the presence of ligands can be compromised by indeterminacies concerning the correct choice of the statistical thermodynamic ensemble. By convent, the assumption of constant free ligand concentration has been used to derive formulae for the enthalpy. Unless the ligand occurs at large excess, this assumption is incorrect. Still the relevant ensemble is the grand canonical ensemble. We derive formulae for both constraints, constancy of total or free ligand concentration and illustrate the equations by application to the typical equilibrium Nx <=> N + x <=> D + x. It is demonstrated that as long as the thermodynamic properties of the ligand can be completely corrected for by performing a reference measurement, the grand canonical approach provides the proper and mathematically significantly simpler choice. We demonstrate on the two cases of sequential or independent ligand-binding the fact, that similar binding mechanisms result in different and distinguishable heat capacity equations. Finally, we propose adequate strategies for DSC experiments as well as for obtaining first estimates of the characteristic thermodynamic parameters, which can be used as starting values in a global fit of DSC data.     

Structural thermodynamics of a random coil protein in guanidine hydrochloride

An important problem in protein folding is to understand the relationship between the structure of a denatured ensemble and its thermodynamics. Using 0 – 6M GdnHCl at fixed pH, we evaluated dimensional changes of an extensively denatured ensemble along with a thermodynamic parameter (Δυ) that monitors the proton inventory of the ensemble. Reduced and carboxyamidated ribonuclease A (RCAM) is a member of a class of disulfide‐free RNase A molecules believed to be random coils (extensively denatured) in aqueous solution. Because GdnHCl interacts more favorably with the protein than water does, this denaturant is observed to increase the Stokes radius of the random coil, with the greatest Stokes radius change occurring in the 0 – 1.5M GdnHCl range. Measurement of the degree of protonation (proton inventory) of the ensemble as a function of GdnHCl at the fixed pH shows that the thermodynamic character of the ensemble also changes markedly in the 0 – 1.5M GdnHCl range, but with little or no change beyond 1.5M GdnHCl. To obtain denaturant‐independent ΔG°_N–D values, the linear extrapolation method (LEM) requires the thermodynamic character of the native and denatured ensembles to be invariant in the transition zone. The results reported here indicate that proteins with a transition midpoint in the 0 – 1.5M GdnHCl range will not give denaturant‐concentration independent ΔG°_N–D values. Such LEM‐derived ΔG°_N–D quantities are a property of the protein and the denaturant, a condition that considerably limits their value in understanding structural energetics. Proteins 2000;41:44–49. © 2000 Wiley‐Liss, Inc.     

The Possible Role of Spike Patterns in Cortical Information Processing

When the same visual stimulus is presented across many trials, neurons in the visual cortex receive stimulus-related synaptic inputs that are reproducible across trials (S) and inputs that are not (N). The variability of spike trains recorded in the visual cortex and their apparent lack of spike-to-spike correlations beyond that implied by firing rate fluctuations, has been taken as evidence for a low S/N ratio. A recent re-analysis of in vivo cortical data revealed evidence for spike-to-spike correlations in the form of spike patterns. We examine neural dynamics at a higher S/N in order to determine what possible role spike patterns could play in cortical information processing. In vivo-like spike patterns were obtained in model simulations. Superpositions of multiple sinusoidal driving currents were especially effective in producing stable long-lasting patterns. By applying current pulses that were either short and strong or long and weak, neurons could be made to switch from one pattern to another. Cortical neurons with similar stimulus preferences are located near each other, have similar biophysical properties and receive a large number of common synaptic inputs. Hence, recordings of a single neuron across multiple trials are usually interpreted as the response of an ensemble of these neurons during one trial. In the presence of distinct spike patterns across trials there is ambiguity in what would be the corresponding ensemble, it could consist of the same spike pattern for each neuron or a set of patterns across neurons. We found that the spiking response of a neuron receiving these ensemble inputs was determined by the spike-pattern composition, which, in turn, could be modulated dynamically as a means for cortical information processing.     

A nonuniform sampling approach for fast ultrasonic flow imaging.

Conventional Pulsed Wave Doppler (PWD) systems acquire an ensemble of N echoes per beam line at a constant pulse repetition frequency fprf, so that the pulse repetition interval equals Tpri = 1/fpn. The total time span determines the velocity resolution, and Tpri the unambiguous velocity range. The ensemble size N is by approximation inversely proportional to the frame rate, assuming that the system performs interleaving. For a given frame rate, a tradeoff can only be made between velocity resolution and velocity range. We propose an approach that allows increasing velocity resolution or range while keeping the frame rate constant. The approach is based on nonuniform sampling, i.e. sampling with varying sampling intervals. Thus, for a given ensemble size N a larger total time span, which would increase velocity resolution, or a shorter minimal Tpn, which would increase the velocity range, may be chosen. The conventional Doppler signal processing techniques are not compatible with nonuniform sampling. We, therefore, developed a velocity estimation algorithm for arbitrary sampling that is based on cross correlation. Furthermore, an adaptive wall filter was implemented that differentiates between tissue motion and blood flow. The new approach was successfully tested with in vitro and in vivo data.     

Dynamics in the unfolded state of beta2-microglobulin studied by NMR

Many proteins form amyloid-like fibrils in vitro under conditions that favour the population of partially folded conformations or denatured state ensembles. Characterising the structural and dynamic properties of these states is crucial towards understanding the mechanisms of self-assembly in amyloidosis. The aggregation of beta2-microglobulin (beta2m) into amyloid fibrils in vivo occurs in the condition known as dialysis-related amyloidosis (DRA) and the protein has been shown to form amyloid-like fibrils under acidic conditions in vitro. We have used a number of 1H-15N nuclear magnetic resonance (NMR) experiments in conjunction with site-directed mutagenesis to study the acid-unfolded state of beta2m. 15N NMR transverse relaxation experiments reveal that the acid-denatured ensemble, although predominantly unfolded at the N and C termini, contains substantial non-native structure in the central region of the polypeptide chain, stabilised by long-range interactions between aromatic residues and by the single disulphide bond. Relaxation dispersion studies indicate that the acid-unfolded ensemble involves two or more distinct species in conformational equilibrium on the micro- to millisecond time-scale. One of these species appears to be hydrophobically collapsed, as mutations in an aromatic-rich region of the protein, including residues that are solvent-exposed in the native protein, disrupt this structure and cause a consequent decrease in the population of this conformer. Thus, acid-unfolded beta2m consists of a heterogeneous ensemble of rapidly fluctuating species, some of which contain stable, non-native hydrophobic clusters. Given that amyloid assembly of beta2m proceeds with lag kinetics under the conditions of this study, a rarely populated species such as a conformer with non-native aromatic clustering could be key to the initiation of amyloidosis.     

A theoretical model for the prediction of sequence-dependent nucleosome thermodynamic stability

A theoretical model for predicting nucleosome thermodynamic stability in terms of DNA sequence is advanced. The model is based on a statistical mechanical approach, which allows the calculation of the canonical ensemble free energy involved in the competitive nucleosome reconstitution. It is based on the hypothesis that nucleosome stability mainly depends on the bending and twisting elastic energy to transform the DNA intrinsic superstructure into the nucleosomal structure. The ensemble average free energy is calculated starting from the intrinsic curvature, obtained by integrating the dinucleotide step deviations from the canonical B-DNA and expressed in terms of a Fourier series, in the framework of first-order elasticity. The sequence-dependent DNA flexibility is evaluated from the differential double helix thermodynamic stability. A large number of free-energy experimental data, obtained in different laboratories by competitive nucleosome reconstitution assays, are successfully compared to the theoretical results. They support the hypothesis that the stacking energies are the major factor in DNA rigidity and could be a measure of DNA stiffness. A dual role of DNA intrinsic curvature and flexibility emerges in the determination of nucleosome stability. The difference between the experimental and theoretical (elastic) nucleosome-reconstitution free energy for the whole pool of investigated DNAs suggests a significant role for the curvature-dependent DNA hydration and counterion interactions, which appear to destabilize nucleosomes in highly curved DNAs. This model represents an attempt to clarify the main features of the nucleosome thermodynamic stability in terms of physical-chemical parameters and suggests that in molecular systems with a large degree of complexity, the average molecular properties dominate over the local features, as in a statistical ensemble.     

Ensemble projections elucidate effects of uncertainty in terrestrial nitrogen limitation on future carbon uptake

The magnitude of the nitrogen (N) limitation of terrestrial carbon (C) storage over the 21st century is highly uncertain because of the complex interactions between the terrestrial C and N cycles. We use an ensemble approach to quantify and attribute process‐level uncertainty in C‐cycle projections by analysing a 30‐member ensemble representing published alternative representations of key N cycle processes (stoichiometry, biological nitrogen fixation (BNF) and ecosystem N losses) within the framework of one terrestrial biosphere model. Despite large differences in the simulated present‐day N cycle, primarily affecting simulated productivity north of 40°N, ensemble members generally conform with global C‐cycle benchmarks for present‐day conditions. Ensemble projections for two representative concentration pathways (RCP 2.6 and RCP 8.5) show that the increase in land C storage due to CO₂ fertilization is reduced by 24 ± 15% due to N constraints, whereas terrestrial C losses associated with climate change are attenuated by 19 ± 20%. As a result, N cycling reduces projected land C uptake for the years 2006–2099 by 19% (37% decrease to 3% increase) for RCP 2.6, and by 21% (40% decrease to 9% increase) for RCP 8.5. Most of the ensemble spread results from uncertainty in temperate and boreal forests, and is dominated by uncertainty in BNF (10% decrease to 50% increase for RCP 2.6, 5% decrease to 100% increase for RCP 8.5). However, choices about the flexibility of ecosystem C:N ratios and processes controlling ecosystem N losses regionally also play important roles. The findings of this study demonstrate clearly the need for an ensemble approach to quantify likely future terrestrial C–N cycle trajectories. Present‐day C‐cycle observations only weakly constrain the future ensemble spread, highlighting the need for better observational constraints on large‐scale N cycling, and N cycle process responses to global change.     

Reconstruction of the src-SH3 protein domain transition state ensemble using multiscale molecular dynamics simulations

We use an integrated computational approach to reconstruct accurately the transition state ensemble (TSE) for folding of the src-SH3 protein domain. We first identify putative TSE conformations from free energy surfaces generated by importance sampling molecular dynamics for a fully atomic, solvated model of the src-SH3 protein domain. These putative TSE conformations are then subjected to a folding analysis using a coarse-grained representation of the protein and rapid discrete molecular dynamics simulations. Those conformations that fold to the native conformation with a probability (P(fold)) of approximately 0.5, constitute the true transition state. Approximately 20% of the putative TSE structures were found to have a P(fold) near 0.5, indicating that, although correct TSE conformations are populated at the free energy barrier, there is a critical need to refine this ensemble. Our simulations indicate that the true TSE conformations are compact, with a well-defined central beta sheet, in good agreement with previous experimental and theoretical studies. A structured central beta sheet was found to be present in a number of pre-TSE conformations, however, indicating that this element, although required in the transition state, does not define it uniquely. An additional tight cluster of contacts between highly conserved residues belonging to the diverging turn and second beta-sheet of the protein emerged as being critical elements of the folding nucleus. A number of commonly used order parameters to identify the transition state for folding were investigated, with the number of native Cbeta contacts displaying the most satisfactory correlation with P(fold) values.     

Large-scale phylogenies and measuring the performance of phylogenetic estimators

Performance measures of phylogenetic estimation methods such as accuracy, consistency, and power are an attempt at summarizing an ensemble of a given estimator's behavior. These summaries characterize an ensemble behavior with a single number, leading to a variety of definitions. In particular, the relationships between different performance measures such as accuracy and consistency or accuracy and error depend on the exact definition of these measures. In addition, it is relatively common to use large-sample behavior to infer similar behavior for small samples. In fact, large-sample results such as the claimed asymptotic efficiency of the maximum-likelihood estimator are often uninformative for small samples. Conversely, small-sample behavior using simulations is sometimes used to imply large-sample behavior such as consistency. However, such extrapolation is often difficult. How the performance of a phylogenetic estimator scales with the addition of taxa must be qualified with respect to whether the whole tree is being estimated or a fixed subset of taxa is being estimated. It must also be qualified with respect to how tree models are sampled. Over the ensemble of all possible trees of a given size, the performance of the estimators for the whole tree estimate suffers when the tree size becomes larger. However, under certain models of cladogenesis, the estimate can improve with the addition of taxa. In fact, at all numbers of taxa there are subsets of tree models that are easier to estimate than others. This suggests that with judicious addition or subtraction of taxa we can move from tree models that are more difficult to estimate at one number of taxa to those that are easier to estimate at another number of taxa.     

Thermodynamics and kinetics of a molecular motor ensemble

If, contrary to conventional models of muscle, it is assumed that molecular forces equilibrate among rather than within molecular motors, an equation of state and an expression for energy output can be obtained for a near-equilibrium, coworking ensemble of molecular motors. These equations predict clear, testable relationships between motor structure, motor biochemistry, and ensemble motor function, and we discuss these relationships in the context of various experimental studies. In this model, net work by molecular motors is performed with the relaxation of a near-equilibrium intermediate step in a motor-catalyzed reaction. The free energy available for work is localized to this step, and the rate at which this free energy is transferred to work is accelerated by the free energy of a motor-catalyzed reaction. This thermodynamic model implicitly deals with a motile cell system as a dynamic network (not a rigid lattice) of molecular motors within which the mechanochemistry of one motor influences and is influenced by the mechanochemistry of other motors in the ensemble.     

Background sampling for multi-scale ensemble habitat selection modeling: Does the number of points matter?

Ensemble habitat selection modeling is becoming a popular approach among ecologists to answer different questions. Since we are still in the early stages of development and application of ensemble modeling, there remain many questions regarding performance and parameterization. One important gap, which this paper addresses, is how the number of background points used to train models influences the performance of the ensemble model. We used an empirical presence-only dataset and three different selections of background points to train scale-optimized habitat selection models using six modeling algorithms (GLM, GAM, MARS, ANN, Random Forest, and MaxEnt). We tested four ensemble models using different combinations of the component models: (a) equal numbers of background points and presences, (b) background points equaled ten times the number of presences, (c) 10,000 background points, and (d) optimized background points for each component model. Among regression-based approaches, MARS performed best when built with 10,000 background points. Among machine learning models, RF performed the best when built with equal presences and background points. Among the four ensemble models, AUC indicated that the best performing model was the ensemble with each component model including the optimized number of background points, while TSS increased as the number of background points models increased. We found that an ensemble of models, each trained with an optimal number of background points, outperformed ensembles of models trained with the same number of background points, although differences in performance were slight. When using a single modeling method, RF with equal number of presences and background points can perform better than an ensemble model, but the performance fluctuates when the number of background points is not properly selected. On the other hand, ensemble modeling provides consistently high accuracy regardless of background point sampling approach. Further, optimizing the number of background points for each component model within an ensemble model can provide the best model improvement. We suggest evaluating more models across multiple species to investigate how background point selection might affect ensemble models in different scenarios.     

Effects of the difference in the unfolded-state ensemble on the folding of Escherichia coli dihydrofolate reductase

The unfolded state of a protein is an ensemble of a large number of conformations ranging from fully extended to compact structures. To investigate the effects of the difference in the unfolded-state ensemble on protein folding, we have studied the structure, stability, and folding of "circular" dihydrofolate reductase (DHFR) from Escherichia coli in which the N and C-terminal regions are cross-linked by a disulfide bond, and compared the results with those of disulfide-reduced "linear" DHFR. Equilibrium studies by circular dichroism, difference absorption spectra, solution X-ray scattering, and size-exclusion chromatography show that whereas the native structures of both proteins are essentially the same, the unfolded state of circular DHFR adopts more compact conformations than the unfolded state of the linear form, even with the absence of secondary structure. Circular DHFR is more stable than linear DHFR, which may be due to the decrease in the conformational entropy of the unfolded state as a result of circularization. Kinetic refolding measurements by stopped-flow circular dichroism and fluorescence show that under the native conditions both proteins accumulate a burst-phase intermediate having the same structures and both fold by the same complex folding mechanism with the same folding rates. Thus, the effects of the difference in the unfolded state of circular and linear DHFRs on the refolding reaction are not observed after the formation of the intermediate. This suggests that for the proteins with close termini in the native structure, early compaction of a protein molecule to form a specific folding intermediate with the N and C-terminal regions in close proximity is a crucial event in folding. If there is an enhancement in the folding reflecting the reduction in the breadth of the unfolded-state ensemble for circular DHFR, this acceleration must occur in the sub-millisecond time-range.     

Determination of the interfacial water content in protein-protein complexes from free energy simulations

The question as to how many tightly or weakly bound water molecules are located in interfaces between protein-protein complex constituents is addressed from a phase equilibrium point of view by developing a theory in the canonical ensemble. A fast method based on free energy simulations is described for computing the number of water molecules in the interface regions. Results are given for 211 interfacial cavities of 26 antigen-antibody complexes for which experimentally determined structures are found in the Protein Data Bank. The accuracy of the method is assessed and the computational water content is compared with experimental data, revealing the amount of water molecules not resolved by experimental approaches.     

Simulation of DNA bases in water: Comparison of the Monte Carlo algorithm with molecular mechanics force fields

The interaction between the nucleic acid bases and solvent molecules has an important effect in various biochemical processes. We have calculated total energy and free energy of the solvation of DNA bases in water by Monte Carlo simulation. Adenine, guanine, cytosine, and thymine were first optimized in the gas phase and then placed in a cubic box of water. We have used the TIP3 model for water and OPLS for the nucleic acid bases. The canonical (T, V, N) ensemble at 25 degrees C and Metropolis sampling technique have been used. Good agreement with other available computational data was obtained. Radial distribution functions of water around each site of adenine, guanine, cytosine, and thymine have been computed and the results have shown the ability of the sites for hydrogen bonding and other interactions. The computations have shown that guanine has the highest value of solvation free energy and N7 and N6 in adenine and guanine, N3 in cytosine, and N3 and O4 in thymine have the largest radial distribution function. Monte Carlo simulation has also been performed using the CHARMM program under the same conditions, and the results of two procedures are compared.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Stationary-state distributions in membrane channels. I. Conservative flux systems.

The population distribution of cations at binding sites in membrane channels is determined for stationary states characterized by a net ion flux. For conservative systems, the net flux conserves the total intra-channel cation population for an ensemble of channels. The requisite matrix formalism is developed and illustrated for some homogeneous channels with N single-ion occupancy states with chemical and electrochemical transmembrane gradients. The lowest eigen-value approximation, which is used effectively for systems with no net cation flux, is applied to these stationary-state systems for comparison with the exact solutions. The stationary states are characterized by special thermodynamic functions for entropy, internal energy, and free energy defined by using information theory. The changes in these parameters for the transition from the equilibrium to the stationary state depend on the ratio of the next flux to the transition velocity for nearest-neighbor transitions within the channel. The free energy changes with the square of this ratio while the internal energy changes linearly.