Determination of the single chain density of states for membrane systems

We have derived the full single chain density of states for membrane systems ω(ε, a) where ε is the isomeric energy and a is the area of the configuration. Since there is no analytic relation between the area and a particular bond sequence it is necessary to assign an area for each chain configuration. Due to the large number of isomeric states (e.g. for a chain with 16 carbons there are ～5 × 106 distinct configurations), we found it necessary to perform the calculation on the computer. In order to avoid having to evaluate each distinct state, we used a Monte Carlo sampling technique which we found converged very rapidly. Our Monte Carlo calculation of the area distribution of the complete set of isomeric states is the first determination of the full single chain density of states. Due to the complexity of the internal coordinates of the chains and the importance of packing considerations, the knowledge of m (e, a) is very important to the understanding of the physical properties of the bilayer. For example, we found that chain configurations statistically tend to have an elongated shape quite independently of interactions between chains. As examples of the usefulness of ω (ε, a) we have calculated an upper bound on the entropy change of the phase transition given ω (ε, a), the X-ray long spacing of a system of non-interacting chains (i.e. no steric interactions), and the n.m.r. order parameter for non-interacting chains. The quantitative and qualitative agreement of these calculated values with experiment is discussed.     

The nature of the free energy barriers to two-state folding

We introduce a simple procedure to analyze the temperature dependence of the folding and unfolding rates of two-state proteins. We start from the simple transition-state-like rate expression: k = D(eff)exp(-DeltaG(TS)/RT), in which upper and lower bounds for the intra-chain effective diffusion coefficient (D(eff)) are obtained empirically using the timescales of elementary processes in protein folding. From the changes in DeltaG(TS) as a function of temperature, we calculate enthalpies and heat capacities of activation, together with the more elusive entropies of activation. We then estimate the conformational entropy of the transition state by extrapolation to the temperature at which the solvation entropy vanishes by cancellation between polar and apolar terms. This approach is based on the convergence temperatures for the entropy of solvating apolar (approximately 385 K) and polar groups (approximately 335 K), the assumption that the structural properties of the transition state are somewhere in between the unfolded and folded states, and the established relationship between observed heat capacity and solvent accessibility.1 To circumvent the lack of structural information about transition states, we use the empirically determined heat capacities of activation as constraints to identify the extreme values of the transition state conformational entropy that are consistent with experiment. The application of this simple approach to six two-state folding proteins for which there is temperature-dependent data available in the literature provides important clues about protein folding. For these six proteins, we obtain an average equilibrium cost in conformational entropy of -4.3 cal x mol(-1)K(-1)per residue, which is in close agreement to previous empirical and computational estimates of the same quantity. Furthermore, we find that all these proteins have a conformationally diverse transition state, with more than half of the conformational entropy of the unfolded state. In agreement with predictions from theory and computer simulations, the transition state signals the change from a regime dominated by loss in conformational entropy to one driven by the gain in stabilization free energy (i.e., including protein interactions and solvation effects). Moreover, the height of the barrier is determined by how much stabilization free energy is realized at that point, which is related to the relative contribution of local versus non-local interactions. A remarkable observation is that the fraction of conformational entropy per residue that is present in the transition state is very similar for the six proteins in this study. Based on this commonality, we propose that the observed change in thermodynamic regime is connected to a change in the pattern of structure formation: from one driven by formation of pairwise interactions to one dominated by coupling of the networks of interactions involved in forming the protein core. In this framework, the barrier to two-state folding is crossed when the folding protein reaches a "critical native density" that allows expulsion of remaining interstitial water and consolidation of the core. The principle of critical native density should be general for all two-state proteins, but can accommodate different folding mechanisms depending on the particularities of the structure and sequence.     

Relation of Raman order parameters to spin labeling parameters

For the quantitation of Raman and spin labeling data order parameters are commonly used. The spin label order parameter measured at any depth in the layer is a weighed sum of the segmental order since, due to fast conformational interconversions, each CH₂ segment is partly in trans and partly in non-trans, e.g. gauche, kink, jog, etc., conformation during the measurement. The weighing factor, the trans finding probability, varies along the chain (cf. flexibility profile) but its mean value should be equal to the Raman trans order parameter. This correlation is illustrated with the experimental data obtained for dipalmitoyl phosphatidylcholine and n-alcohol mixtures. The rate of rotational diffusion, a dynamical parameter from spin labeling studies, is correlated with the lateral packing density as measured by the Raman lateral order parameter. For the obtained linear correlation a qualitative explanation is given.The effect of a series of long chain alcohols on the phase transition characteristics of dipalmitoyl phosphatidylcholine was investigated. The possible role of hydrogen bonding in the interfacial region is emphasized.     

Conformational transition and polyelectrolyte behaviour of a succinoglycan polysaccharide

We report the chemical characterization and the relationship between the physicochemical properties and conformational change of a succinoglycan polysaccharide produced by Pseudomonas sp, NCIB 11592. The expected chemical structure is confirmed, with a ratio of D-glucose: D-galactose: pyruvate: succinate of 7:1:1:1. The molecular weight of the native form is 4.2 x 10(6) but after a single heating cycle through the disordered state the molecular weight is reduced to 3.0 x 10(6). The polymer has a polymolecularity index of 1.3 in both cases. The conformational change was studied by different methods which enabled us to define the exact nature of the ordered and disordered states. The conformational transition depends on the temperature, the ionic strength and the nature of the counterion. The polyelectrolyte behaviour is in favour of a single chain conformation with an intramolecular helix-coil transition. The enthalpy change during this transition is greater than that expected solely on the basis of the polyelectrolyte contribution. It may be associated with changes in solvation or a rearrangement of water molecules in close association with the polymer.     

In situ microscopic cytometry enables noninvasive viability assessment of animal cells by measuring entropy states

Current state of the art to determine the viability of animal cell suspension cultures is based on sampling and subsequent counting using specific staining assays. We demonstrate for the first time a noninvasive in situ imaging cytometry capable of determining the statistics of a morphologic transition during cell death in suspension cultures. To this end, we measure morphometric inhomogeneity—defined as information entropy—in cell in situ micrographs. We found that the cells are partitioned into two discrete entropy states broadened by phenotypical variability. During the normal course of a culture or by inducing cell death, we observe the transition of cells between these states. As shown by comparison with ex situ diagnostics, the entropy transition happens before or while the cytoplasmatic membrane is loosing its ability to exclude charged dyes. Therefore, measurement of morphometric inhomogeneity constitutes a noninvasive assessment of viability in real time. Biotechnol. Bioeng. 2011;108: 2884–2893. © 2011 Wiley Periodicals, Inc.     

Effects of mutations on the thermodynamics of a protein folding reaction: implications for the mechanism of formation of the intermediate and transition states

We have measured changes in heat capacity, entropy, and enthalpy for each step in the folding reaction of CD2.d1 and evaluated the effects of core mutations on these properties. All wild-type and mutant forms fold through a rapidly formed intermediate state that precedes the rate-limiting transition state. Mutations have a pronounced effect on the enthalpy of both the intermediate and folded states, but in all cases a compensatory change in entropy results in a small net free-energy change. While the enthalpy change in the folded state can be attributed to a loss of van der Waals interactions, it has already been shown that changes in the stability of the intermediate are dominated by changes in secondary structure propensity [Lorch et al. (1999) Biochemistry 38, 1377-1385]. It follows that the thermodynamic basis of beta-propensity is enthalpic in origin. The effects of mutations on the enthalpy and entropy of the transition state are smaller than on the ground states. This relative insensitivity to mutation is discussed in the light of theories concerning the nature of the rate-limiting barrier in folding reactions.     

Recent results on the connection between thermodynamics and dynamics in supercooled water

We review recent results on the connection between thermodynamics and dynamics in a model for water. We verify the Adam-Gibbs relation between entropy and dynamic properties using computer simulations, which allow direct access to the relevant properties. We combine experimental measurements of entropy with the Adam-Gibbs hypothesis to predict dynamic properties in deeply supercooled states, which are difficult to access experimentally. We find evidence suggesting that the glass transition temperature of water may be significantly higher than previously reported, but is still consistent with recent measurements. Finally, we discuss the hypothesis that the dynamical behavior of deeply supercooled water undergoes a crossover from 'fragile' to 'strong' behavior.     

Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

Interpretation of protein folding psi values

The structural characterization of transition states is essential for understanding the mechanism of protein folding. Analyzing the effect of mutations on protein stability and folding kinetics in phi-value analysis is commonly used to gain information about the presence of side-chain interactions in transition states. Recently, specific binding of ligands to engineered binding sites was applied to monitor the formation of local structures in transition states (psi analysis). A surprising result from psi analysis was the presence of parallel folding pathways in all reported studies and a major discrepancy between phi and psi values measured in the same protein. Here, we show that psi values cannot be analyzed in the same way as other rate-equilibrium free energy relationships due to the involvement of bimolecular reactions that may have different dissociation constants for the native, unfolded and transition state. As a consequence, psi values reflect the relative binding energy (kappa) of the transition state only for the extreme values of kappa=0 or kappa=1. In all other cases, non-linear rate-equilibrium free-energy relationships (Leffler plots) are observed. This apparently indicates the presence of parallel folding pathways even if folding occurs over a single homogeneous transition state. Consequently, the results from Leffler plots do not yield information about the structural properties of the transition state. This explains the lack of agreement between results from psi analysis and other methods used to characterize protein folding transition states. We further show that the same considerations apply for the analysis of the effect of pH on protein folding.     

States and transitions during forced unfolding of a single spectrin repeat

Spectrin is a vital and abundant protein of the cytoskeleton. It has an elongated structure that is made by a chain of so-called spectrin repeats. Each repeat contains three antiparallel alpha-helices that form a coiled-coil structure. Spectrin forms an oligomeric structure that is able to cross-link actin filaments. In red cells, the spectrin/actin meshwork underlying cell membrane is thought to be responsible for special elastic properties of the cell. In order to determine mechanical unfolding properties of the spectrin repeat, we have used single molecule force spectroscopy to study the states of unfolding of an engineered polymeric protein consisting of identical spectrin domains. We demonstrate that the unfolding of spectrin domains can occur in a stepwise fashion during stretching. The force-extension patterns exhibit features that are compatible with the existence of at least one intermediate between the folded and the completely unfolded conformation. Only those polypeptides that still contain multiple intact repeats display intermediates, indicating a stabilisation effect. Precise force spectroscopy measurements on single molecules using engineered protein constructs reveal states and transitions during the mechanical unfolding of spectrin. Single molecule force spectroscopy appears to open a new window for the analysis of transition probabilities between different conformational states.     

Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins

The characterization of elastic properties of biopolymers is crucial to understand many molecular reactions determined by conformational bending fluctuations of the polymer. Direct measurement of such elastic properties using single‐molecule methods is usually hindered by the intrinsic tendency of such biopolymers to form high‐order molecular structures. For example, single‐stranded deoxyribonucleic acids (ssDNA) tend to form secondary structures such as local double helices that prevent the direct measurement of the ideal elastic response of the ssDNA. In this work, we show how to extract the ideal elastic response in the entropic regime of short ssDNA molecules by mechanically pulling two‐state DNA hairpins of different contour lengths. This is achieved by measuring the force dependence of the molecular extension and stiffness on mechanically folding and unfolding the DNA hairpin. Both quantities are fit to the worm‐like chain elastic model giving values for the persistence length and the interphosphate distance. This method can be used to unravel the elastic properties of short ssDNA and RNA sequences and, more generally, any biopolymer that can exhibit a cooperative two‐state transition between mechanically folded and unfolded states (such as proteins). © 2014 Wiley Periodicals, Inc. Biopolymers 101: 1193–1199, 2014.     

The effect of cholesterol on the structure of phosphatidylcholine bilayers

The effect of cholesterol on the structure of phosphatidylcholine bilayers was investigated by X-ray diffraction methods. Electron density profiles at 5 Å resolution along with chain tilt and chain packing parameters were obtained and compared for phosphatidylcholine/cholesterol bilayers and for pure phosphatidylcholine bilayers in both the gel and liquid crystalline states. The cholesterol in the bilayer was localized by noting the position of discrete elevations in the electron density profiles. Cholesterol can either increase or decrease the width of the bilayer depending on the physical state and chain length of the lipid before the introduction of cholesterol. For saturated phosphatidylcholines containing 12–16 carbons per chain, cholesterol increases the width of the bilayer as it removes the chain tilt from gel state lipids or increases the trans conformations of the chains for liquid crystalline lipids. However, cholesterol reduces the width of 18 carbon chain bilayers below the phase transition temperature as the long phospholipid chains must deform or kink to accomodate the significantly shorter cholesterol molecule. Although cholesterol has a marked effect on hydrocarbon chain organization, it was found that, within the resolution limits of the data, the phosphatidylcholine head group conformation is unchanged by the addition of cholesterol to the bilayer. The head group is oriented parallel to the plane of the bilayer for phosphatidylcholine in the gel and liquid crystalline states and this orientation is not changed by the addition of cholesterol.     

Characterisation of the hydrophobic collapse of polystyrene in water using free energy techniques

We characterise the hydrophobic collapse of single polystyrene chains in water using molecular dynamics simulations. Specifically, we calculate the potential of mean force for the collapse of a single polystyrene chain in water using metadynamics, comparing the results between all atomistic with coarse-grained (CG) molecular simulation. We next explore the scaling behaviour of the collapsed globular shape at the minimum energy configuration, characterised by the radius of gyration, as a function of chain length. The exponent is close to one third, consistent with that predicted for a polymer chain in bad solvent. We also explore the scaling behaviour of the solvent accessible surface area (SASA) as a function of chain length, finding a similar exponent for both all atomistic and CG simulations. Furthermore, calculation of the local water density as a function of chain length near the minimum energy configuration suggests that intermediate chain lengths are more likely to form dewetted states, as compared to shorter or longer chain lengths.     

Short term synaptic depression model--analytical solution and analysis

In this article we present analytical solutions of the single and pair pulse time evolution of a plastic neocortical synapse described by the TM-model. We show that this model is equivalent to the receptor-desensitization model with three kinetic states. For the TM-model we derive the analytical form of a measure of paired pulse depression. We analyze the sensitivity of the synaptic depression phenomenon on model parameters and derive the relative importance of each of the parameters. The closed form of the measure of synaptic depression allows fitting the model to experimental data. The fitted parameters are used to make predictions about the asymptotic properties of the postsynaptic currents. We show that for synapses with the ratio of inactivation and recovery rates of the same order, the synaptic depression does not preclude the rate-coding of information: e.g. in the pyramid-pyramid connections of adult rat neocortex, rate-coding is possible for higher frequencies.     

Pressure-jump small-angle x-ray scattering detected kinetics of staphylococcal nuclease folding

The kinetics of chain disruption and collapse of staphylococcal nuclease after positive or negative pressure jumps was monitored by real-time small-angle x-ray scattering under pressure. We used this method to probe the overall conformation of the protein by measuring its radius of gyration and pair-distance-distribution function p(r) which are sensitive to the spatial extent and shape of the particle. At all pressures and temperatures tested, the relaxation profiles were well described by a single exponential function. No fast collapse was observed, indicating that the rate limiting step for chain collapse is the same as that for secondary and tertiary structure formation. Whereas refolding at low pressures occurred in a few seconds, at high pressures the relaxation was quite slow, approximately 1 h, due to a large positive activation volume for the rate-limiting step for chain collapse. A large increase in the system volume upon folding implies significant dehydration of the transition state and a high degree of similarity in terms of the packing density between the native and transition states in this system. This study of the time-dependence of the tertiary structure in pressure-induced folding/unfolding reactions demonstrates that novel information about the nature of protein folding transitions and transition states can be obtained from a combination of small-angle x-ray scattering using high intensity synchrotron radiation with the high pressure perturbation technique.     

From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence–absence data to recover information on the population dynamics of predator–prey systems. We use a highly complex and spatially explicit simulation model of a predator–prey mite system to generate simple presence–absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the "footprints" of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator's functional response and the dispersal rates of both species) are reflected by different footprints
The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.     

Shape of the free energy barriers for protein folding probed by multiple perturbation analysis

The characterization of the free energy barriers has been a major goal in studies on the mechanism of protein folding. Testing the effect of mutations or denaturants on protein folding reactions revealed that transition state movement is rare, suggesting that folding barriers are robust and narrow maxima on the free energy landscape. Here we demonstrate that the application of multiple perturbations allows the observation of small transition state movements that escape detection in single perturbation experiments. We used tendamistat as a model protein to test the broadness of the free energy barriers. Tendamistat folds over two consecutive transition states and through a high-energy intermediate. Measuring the combined effect of temperature and denaturant on the position of the transition state in the wild-type protein and in several mutants revealed that the early transition state shows significant transition state movement. Its accessible surface area state becomes more native-like with destabilization of the native state by temperature. To the same extent, the entropy of the early transition state becomes more native-like with increasing denaturant concentration, in accordance with Hammond behavior. The position of the late transition state, in contrast, is much less sensitive to the applied perturbations. These results suggest that the barriers in protein folding become increasingly narrow as the folding polypeptide chain approaches the native state.     

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.     

Point mutations in membrane proteins reshape energy landscape and populate different unfolding pathways

Using single-molecule force spectroscopy, we investigated the effect of single point mutations on the energy landscape and unfolding pathways of the transmembrane protein bacteriorhodopsin. We show that the unfolding energy barriers in the energy landscape of the membrane protein followed a simple two-state behavior and represent a manifestation of many converging unfolding pathways. Although the unfolding pathways of wild-type and mutant bacteriorhodopsin did not change, indicating the presence of same ensemble of structural unfolding intermediates, the free energies of the rate-limiting transition states of the bacteriorhodopsin mutants decreased as the distance of those transition states to the folded intermediate states decreased. Thus, all mutants exhibited Hammond behavior and a change in the free energies of the intermediates along the unfolding reaction coordinate and, consequently, their relative occupancies. This is the first experimental proof showing that point mutations can reshape the free energy landscape of a membrane protein and force single proteins to populate certain unfolding pathways over others.