Quantitative measurements of the cooperativity in an EF-hand protein with sequential calcium binding.

Positive cooperativity, defined as an enhancement of the ligand affinity at one site as a consequence of binding the same type of ligand at another site, is a free energy coupling between binding sites. It can be present both in systems with sites having identical ligand affinities and in systems where the binding sites have different affinities. When the sites have widely different affinities such that they are filled with ligand in a sequential manner, it is often difficult to quantify or even detect the positive cooperativity, if it occurs. This study presents verification and quantitative measurements of the free energy coupling between the two calcium binding sites in a mutant form of calbindin D9k. Wild-type calbindin D9k binds two calcium ions with similar affinities and positive cooperativity--the free energy coupling, delta delta G, is around -8 kJ.mol-1 (Linse S, et al., 1991, Biochemistry 30: 154-162). The mutant, with the substitution Asn 56-->Ala, binds calcium in a sequential manner. In the present work we have taken advantage of the variations among different metal ions in terms of their preferences for the two binding sites in calbindin D9k. Combined studies of the binding of Ca2+, Cd2+, and La3+ have allowed us to conclude that in this mutant delta delta G < -6.4 kJ.mol-1, and that Cd2+ and La3+ also bind to this protein with positive cooperativity. The results justify the use of the (Ca2+)1 state of the Asn 56-->Ala mutant, as well as the (Cd2+)1 state of the wild type, as models for the half-saturated states along the two pathways of cooperative Ca2+ binding in calbindin D9k.     

On morphometric measurement of oxygen diffusing capacity in middle ear gas exchange.

An accurate mathematical model of transmucosal gas exchange is prerequisite to understanding middle ear (ME) physiology. Current models require experimentally measured gas species time constants for all extant conditions as input parameters. However, studies on pulmonary gas exchange have shown that a morphometric model that incorporates more fundamental physiochemical and anatomic parameters accurately simulates transport from which the species time constants can be derived for all extant conditions. Here, we implemented a variant of that model for ME gas exchange that requires the measurement of diffusional length (tau) for the ME mucosa. That measure contributes to the mucosal diffusing capacity and reflects the resistance to gas flow between air space and capillary. Two methods for measuring tau have been proposed: linear distance between the air-mucosal boundary and capillary and the harmonic mean of all contributing pathway lengths. Oxygen diffusing capacity was calculated for different ME mucosal geometries by using the two tau measures, and the results were compared with those predicted by a detailed, two-dimensional finite element analysis. Predictive accuracy was improved by incorporating the harmonic tau measure, which captures important information regarding variations in capillary shape and distribution. However, compared with the oxygen diffusing capacity derived from the finite element analysis, both measures yielded nonlinear, positively biased estimates. The morphometric techniques underestimate diffusion length by failing to account for the curvilinear gas flow pathways predicted by the finite element model.     

PAR proteins diffuse freely across the anterior-posterior boundary in polarized C. elegans embryos

Polarization of cells by PAR proteins requires the segregation of antagonistic sets of proteins into two mutually exclusive membrane-associated domains. Understanding how nanometer scale interactions between individual PAR proteins allow spatial organization across cellular length scales requires determining the kinetic properties of PAR proteins and how they are modified in space. We find that PAR-2 and PAR-6, which localize to opposing PAR domains, undergo exchange between well mixed cytoplasmic populations and laterally diffusing membrane-associated states. Domain maintenance does not involve diffusion barriers, lateral sorting, or active transport. Rather, both PAR proteins are free to diffuse between domains, giving rise to a continuous boundary flux because of lateral diffusion of molecules down the concentration gradients that exist across the embryo. Our results suggest that the equalizing effects of lateral diffusion are countered by actin-independent differences in the effective membrane affinities of PAR proteins between the two domains, which likely depend on the ability of each PAR species to locally modulate the membrane affinity of opposing PAR species within its domain. We propose that the stably polarized embryo reflects a dynamic steady state in which molecules undergo continuous diffusion between regions of net association and dissociation.     

Thermodynamics of information transfer between subunits in oligomeric enzymes and kinetic cooperativity. 2. Thermodynamics of kinetic cooperativity

The principles of structural kinetics allow one to define the thermodynamic conditions that are sufficient to generate a certain type of kinetic behavior. If subunits are loosely coupled, that is if no quaternary constraint exists between them, the kinetic behavior of the polymeric enzyme is qualitatively defined by the behavior of an ideal dimer. The nature and the extent of the kinetic cooperativity are defined by the energy of interaction, delta G rho, between two subunits. This energy of interaction is that of an ideal dimer relative to that of the A2 and B2 states. This thermodynamic formulation of a given type of cooperativity holds whatever the degree of polymerization of the enzyme. Under these conditions of loose coupling between subunits, positive kinetic cooperativity cannot be associated with any sigmoidicity of the rate curve. The range of energy coupling where positive kinetic cooperativity must, of necessity, be observed becomes more and more narrow as the number of subunit interactions is increased. This range, however, is independent of the number of subunits. The same situation is not observed for negative cooperativity which appears to be independent of both the number of subunits and the number of subunit interactions. If the subunits are tightly coupled, that is if quaternary constraints exist between them, three thermodynamic parameters, delta G' rho, delta G lambda, delta G mu, are required to define the nature of kinetic cooperativity. delta G' rho is the free energy of an ideal strained dimer relative to that of strained A2 and B2 states. delta G lambda and delta G mu represent the difference of strain energies between conformations A and B and B and B relative to that existing between conformations A and A. One may determine in the parametric space (delta G' rho, delta G lambda, delta G mu) the boundaries between the sufficient conditions that generate a certain type of cooperativity and the lack of these conditions. The kinetic parameters of the rate equation are not all independent. A number of constraint conditions exist between them which depend upon the subunit design of the polymeric enzyme. The existence of these constraint conditions may be diagnostic of a certain type of subunit interactions.     

The stability and folding process of amyloidogenic mutant human lysozymes.

Amyloid deposits are frequently formed by mutant proteins that have a lower stability than the wild-type proteins. Some reports, however, have shown that mutant-induced thermodynamic destabilization is not always a general mechanism of amyloid formation. To obtain a better understanding of the mechanism of amyloid fibril formation, we show in this study that equilibrium and kinetic refolding-unfolding reaction experiments with two amyloidogenic mutant human lysozymes (I56T and D67H) yield folding pathways that can be drawn as Gibbs energy diagrams. The equilibrium stabilities between the native and denatured states of both mutant proteins were decreased, but the degrees of instability were different. The Gibbs energy diagrams of the folding process reveal that the Gibbs energy change between the native and folding intermediate states was similar for both proteins, and also that the activation Gibbs energy change from the native state to the transition state decreased. Our results confirm that the tendency to favor the intermediate of denaturation facilitates amyloid formation by the mutant human lysozymes more than equilibrium destabilization between the native and completely denatured states does.     

Site-specific electronic couplings in dyads with MLCT excited states. Intramolecular energy transfer in isomeric Ru(II)-Ru(II) cyclometalated complexes.

The rod-like binuclear complexes [(ttpy)Ru(tpy-ph(2)-phbpy)Ru(ttpy)](4+) and [(ttpy)Ru(tpy-ph(2)-tpy)Ru(phtbpy)](4+) (for abbreviations, see text) have been synthesized and characterized. In both complexes, the polypyridine Ru(II) centers have (N--N--N)Ru(N--N--N) and (N--N--N)Ru(C--N--N) coordination environment. The two isomeric species differ in whether the cyclometalating carbon resides on the bridging or on the terminal ligand. The two complexes have virtually identical energy levels, but MLCT excited states of different (bridging or terminal) ligand localization. They are thus ideally suited to investigate possible effects of excited-state localization on intramolecular energy transfer kinetics. In fact, ultrafast spectroscopic measurements yield different energy transfer time constants for the two isomers, with the bridge-cyclometalated complex (2.7 ps) being faster than the terminal-cyclometalated one (8.0 ps). This difference can be explained in terms of different electronic factors for Dexter energy transfer. The study highlights the peculiar intricacies of intramolecular energy transfer in inorganic dyads involving MLCT excited states.     

Gap junctions with varied permeability properties establish cell-type specific communication pathways in the rat seminiferous epithelium

Dye coupling experiments were performed to determine whether the gap junctions connecting Sertoli cells with other Sertoli cells and different germ cell stages in rats showed functional variations. Chop loading of adult rat seminiferous tubules was conducted using fluorescent dextran controls and a variety of low-molecular-weight tracers (lucifer yellow, biotin-X-cadaverine, biotin cadaverine, and neurobiotin) to evaluate dye coupling in situ, and scrape loading was used to study dye coupling in Sertoli-germ cell cocultures established using prepuberal rats. Sertoli-Sertoli coupling is relatively short range and nonselective in situ, whereas coupling between Sertoli cells and chains of spermatogonia is strongly selective for the positively charged biotin tracers relative to negatively charged lucifer yellow. Coupling between Sertoli cells and spermatogonia was also asymmetric; lucifer yellow in germ cells never diffused into Sertoli cells, and biotinylated tracers only weakly diffused from spermatogonia to Sertoli cells. Asymmetric coupling would facilitate the concentration in germ cells of molecules diffusing through junctions from Sertoli cells. Dye coupling between Sertoli cells and adluminal germ cells was too weak to detect by fluorescence microscopy, suggesting that the junctional communication between these cells may be functionally different from that between Sertoli and basal germ cells. The results show that there are multiple routes of gap junction communication in rat seminiferous tubules that differ in permeability properties and show alternative gating states. Functional diversity of gap junctions may permit regulated communication among the many interacting Sertoli cells and germ cell stages in the seminiferous epithelium.     

Tracer flow, permeability, and partial conductance.

It is often not possible to evaluate a permeability coefficient for net flow P from the small flows produced by physiological gradients of concentration or electrical potential. The common use of a tracer permeability coefficient P-x for this purpose, under the assumption that P-x = P, requires that the species be transported passively, and that there be no significant coupling between its flow and that of other chemical species, and between the flows of its tracer and abundant isotopes (isotope interaction). These conditions are often not satisfied. However, for passive transport in the absence of coupling of flows of different chemical species the measurement of tracer flow at two values of electrical potential difference evaluates (P-x/P) and thus P. In the presence of coupling of flows of different chemical species, although these measurements no longer evaluate P, they evaluate the partial conductance G. A graphical method of evaluating (p-x/P), P, and G is presented.     

A comparison of circatidal rhythmicity and entrainment by hydrostatic pressure cycles in the rock goby, Gobius paganellus L. and the shanny, Lipophrys pholis (L.)

The intertidal teleosts Gobius paganellus and Lipophrys pholis show endogenous circatidal activity rhythms when recorded in constant conditions. Under these conditions, the rhythm of L. pholis is the more precise which may indicate stronger coupling between underlying circalunadian oscillators in this species. In G. pagunellus the inter-oscillator coupling may be weaker and this could enable a more subtle interpretation of tidal fluctuations than in L. pholis . The oscillators may, however, be fundamentally different in the two species; circalunadian in G. paganellus and circatidal in L. pholis .
When exposed to hydrostatic pressure cycles of tidal frequency both species responded pre- dominantly to increasing pressure, which suggests that in the wild they are likely to be most active on the rising tide. Hydrostatic pressure cycles are confirmed as a zeitgeber for both species by the successful entrainment of some individuals. The lack of entrainment of others impIies that additional zeitgebers are required for complete entrainment.     

Structural bioenergetics and energy transduction mechanisms.

Life depends on transduction processes that couple cellular metabolism to environmental energy sources such as light or reduced compounds. These primary energy sources must be efficiently converted into forms that can be utilized by cells for biosynthesis, motility, transport, regulation, and other metabolic functions. In recent years, there has been an explosive increase in the determination of structures for proteins mediating energy transduction processes. These developments provide the opportunity to evaluate the structural basis for the efficient coupling of two energetic processes, which defines the area of structural bioenergetics. Here, we present some general features of energy transduction processes, including arguments that effective coupling of two processes by a transduction protein occurs by way of conformational states that are common to the catalysis of each process. This is illustrated by examples from the nucleotide switch family of proteins, with emphasis on the nitrogenase system where ATP hydrolysis is coupled to an electron transfer reaction.     

A channel model with fluctuating barrier structures.

Following the theory 'Fluctuations of barrier structure in ionic channels' (L?uger, P., Stephan, W. and Frehland, E. (1980) Biochim. Biophys. Acta 602, 167-180), we constructed a model of a channels with several conformational states. The origin of these conformational states and the source for the transitions from one to the other are given explicitly for the presented model. In this work the effect of multiple conformational states on the ion transport process is analyzed. We considered a channel protein with two main barriers and one binding site. The site is surrounded by dipolar groups. The dipole moment of these groups can be reoriented by thermal activity and also by electrical interaction with the transported ions. Differently polarized states generate different activation energy barriers for the ions. The set of conformational states of the channel is constituted by all the possible polarized states of the binding site. Using the rate-theory analysis of ion transport (Gl?sstone, S., Laider, K.J. and Eyring, H. (1941) The theory of rate processes, McGraw-Hill, New York), the possible coupling between ion flux and the channel conformational transitions has been incorporated into the model by considering the dependence of the rate constants on the heights of the energy barriers. The resulting multistate kinetic equations have been solved numerically. It was shown that the simple saturation characteristic of the flux-concentration curve was obtained. For certain values of the model parameters, the channel shows a strongly different conductance for anions compared to cations. In fact, the model contains an interesting mechanism that exhibits selectivity with respect to the charge of the ions.     

Dynamics of molecular motors and polymer translocation with sequence heterogeneity

The effect of sequence heterogeneity on polynucleotide translocation across a pore and on simple models of molecular motors such as helicases, DNA polymerase/exonuclease, and RNA polymerase is studied in detail. Pore translocation of RNA or DNA is biased due to the different chemical environments on the two sides of the membrane, whereas the molecular motor motion is biased through a coupling to chemical energy. An externally applied force can oppose these biases. For both systems we solve lattice models exactly both with and without disorder. The models incorporate explicitly the coupling to the different chemical environments for polymer translocation and the coupling to the chemical energy (as well as nucleotide pairing energies) for molecular motors. Using the exact solutions and general arguments, we show that the heterogeneity leads to anomalous dynamics. Most notably, over a range of forces around the stall force (or stall tension for DNA polymerase/exonuclease systems) the displacement grows sublinearly as t(micro), with micro < 1. The range over which this behavior can be observed experimentally is estimated for several systems and argued to be detectable for appropriate forces and buffers. Similar sequence heterogeneity effects may arise in the packing of viral DNA.     

Single-channel properties of ionic channels gated by cyclic nucleotides.

This paper presents an extensive analysis of single-channel properties of cyclic nucleotide gated (CNG) channels, obtained by injecting into Xenopus laevis oocytes the mRNA encoding for the alpha and beta subunits from bovine rods. When the alpha and beta subunits of the CNG channel are coexpressed, at least three types of channels with different properties are observed. One type of channel has well-resolved, multiple conductive levels at negative voltages, but not at positive voltages. The other two types of channel are characterized by flickering openings, but are distinguished because they have a low and a high conductance. The alpha subunit of CNG channels has a well-defined conductance of about 28 pS, but multiple conductive levels are observed in mutant channels E363D and T364M. The conductance of these open states is modulated by protons and the membrane voltage, and has an activation energy around 44 kJ/mol. The relative probability of occupying any of these open states is independent of the cGMP concentration, but depends on extracellular protons. The open probability in the presence of saturating cGMP was 0.78, 0.47, 0.5, and 0.007 in the w.t. and mutants E363D, T364M, and E363G, and its dependence on temperature indicates that the thermodynamics of the transition between the closed and open state is also affected by mutations in the pore region. These results suggest that CNG channels have different conductive levels, leading to the existence of multiple open states in homomeric channels and to the flickering behavior in heteromeric channels, and that the pore is an essential part of the gating of CNG channels.     

Mechanical components of motor enzyme function.

Motor enzymes use energy from ATP dephosphorylation to generate movement by a mechanical cycle, moving and pushing in one direction while attached to their cytoskeletal substrate, and recovering by moving relative to their substrate to a new attachment site. Mainstream models assert that movement while attached to the substrate results from preexisting strain in the attached motor. The additional underlying ideas can be described in terms of three components for strain amplification: a rotating lever arm, multiple attached states, and elastic compliance. These components determine how energy is recovered during the mechanical cycle and stored in a strained motor. They may coexist in a real motor; the challenge is to determine the contributions of each component. Because these components can generate similar relationships between strain energy and strain, standard measurements of motor function do not discriminate easily between these components. However, important information could be is provided by observations that suggest weak coupling between chemical and mechanical cycles, observations of negative force and movement events in single motor experiments, and the discovery that two motors that move in opposite directions have very similar structures. In models incorporating changes in conformation between attached states, these observations are only explained easily if the conformational changes are tightly coupled to changes in the strength of motor-substrate binding.     

The cooperativity of burst phase reactions explored.

The denaturant-dependence of the major, observable relaxation rates for folding (kobs) of ribonuclease HI from Escherichia coli (RNase H) and phage T4 lysozyme (T4L) reveal that, for both proteins, folding begins with the rapid and transient accumulation of intermediate species in a "burst phase" which precedes the rate-limiting formation of the native state; this is evidenced by a "rollover" in the folding limb of the rate profiles (kobs versus denaturant, or chevron plot). These rate profiles are most simply described by a three-state mechanism (unfolded-to-intermediate-to-native), which implies that the burst phase represents a transition between two distinct thermodynamic states. It is shown here that the equilibrium properties of these burst phase reactions can be equally well modeled by a mechanism involving a continuum of states where the free energy of each state is linearly related to its m-value (the parameter describing the linear relationship between free energy and denaturant). A numerical model is also developed to describe the time evolution of such a system, which exhibits nearly perfect exponential behavior. Both models emphasize how a continuum of states operating under a linear free energy relationship may behave like a two state system. Such a scheme finds experimental justification from an interpretation of recent native state hydrogen exchange data. The analytical model described for a continuum can account for the observed kinetic profiles of several other model proteins. The results, however, appear context specific, suggesting that burst phase reactions are not entirely random and non-specific. The results reported in this study have important implications for the concept of cooperativity in protein folding reactions.     

Current-Voltage Curves of Porous Membranes in the Presence of Pore-Blocking Ions: I. Narrow Pores Containing No More Than One Moving Ion

We propose a physical model for voltage-dependent conductance changes of excitable cell membranes. It is based on competition of uni- and bivalent ions for chains of stable sites extending through the membrane. These one-dimensional pathways (pores) have different profiles of chemical potential for the two ionic species so that bivalent ions can block the passage of univalent ions at large membrane potentials. We treat the special case that each pore is either empty or, because of electrostatic repulsion, contains no more than one uni- or bivalent ion at a time. A system of linear differential equations describes the time-dependent probabilities of the various possible pore states. The states are limited by transition rate constants involving the profile of the chemical potential, the membrane voltage, the ionic concentrations in the adjacent baths, and electrostatic interactions between the ions. The steady-state solutions (Kirchhoff-Hill theorem) yield expressions for the relationship between the small signal conductance of univalent ions and the concentration of these ions in the external bathing medium (a saturation curve) and for the ionic currents and the steady-state current-voltage curve (N-shaped). From the latter curve we compute the shift of theshold potential caused by concentration changes of the external bathing medium. The model yields a number of predictions which can be tested experimentally.     

Analysis of Reaction-Diffusion Systems with Anomalous Subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.     

Lipid flow through fusion pores connecting membranes of different tensions.

When two membranes fuse, their components mix; this is usually described as a purely diffusional process. However, if the membranes are under different tensions, the material will spread predominantly by convection. We use standard fluid mechanics to rigorously calculate the steady-state convective flux of lipids. A fusion pore is modeled as a toroid shape, connecting two planar membranes. Each of the membrane monolayers is considered separately as incompressible viscous media with the same shear viscosity, etas. The two monolayers interact by sliding past each other, described by an intermonolayer viscosity, etar. Combining a continuity equation with an equation that balances the work provided by the tension difference, Deltasigma, against the energy dissipated by flow in the viscous membrane, yields expressions for lipid velocity, upsilon, and area of lipid flux, Phi. These expressions for upsilon and Phi depend on Deltasigma, etas, etar, and geometrical aspects of a toroidal pore, but the general features of the theory hold for any fusion pore that has a roughly hourglass shape. These expressions are readily applicable to data from any experiments that monitor movement of lipid dye between fused membranes under different tensions. Lipid velocity increases nonlinearly from a small value for small pore radii, rp, to a saturating value at large rp. As a result of velocity saturation, the flux increases linearly with pore radius for large pores. The calculated lipid flux is in agreement with available experimental data for both large and transient fusion pores.     

Inter-Subunit Interactions across the Upper Voltage Sensing-Pore Domain Interface Contribute to the Concerted Pore Opening Transition of Kv Channels

The tight electro-mechanical coupling between the voltage-sensing and pore domains of Kv channels lies at the heart of their fundamental roles in electrical signaling. Structural data have identified two voltage sensor pore inter-domain interaction surfaces, thus providing a framework to explain the molecular basis for the tight coupling of these domains. While the contribution of the intra-subunit lower domain interface to the electro-mechanical coupling that underlies channel opening is relatively well understood, the contribution of the inter-subunit upper interface to channel gating is not yet clear. Relying on energy perturbation and thermodynamic coupling analyses of tandem-dimeric Shaker Kv channels, we show that mutation of upper interface residues from both sides of the voltage sensor-pore domain interface stabilizes the closed channel state. These mutations, however, do not affect slow inactivation gating. We, moreover, find that upper interface residues form a network of state-dependent interactions that stabilize the open channel state. Finally, we note that the observed residue interaction network does not change during slow inactivation gating. The upper voltage sensing-pore interaction surface thus only undergoes conformational rearrangements during channel activation gating. We suggest that inter-subunit interactions across the upper domain interface mediate allosteric communication between channel subunits that contributes to the concerted nature of the late pore opening transition of Kv channels.     

Free energy distributions in proteins.

Protein molecules in solution have a broad distribution of enthalpy states. A good approximation to the distribution function for enthalpy states can be calculated, using the maximum-entropy method, from the moments of the distribution that, in turn, are obtained from the experimental temperature dependence of the heat capacity. In the present paper, we show that the enthalpy probability distribution can then be formulated in terms of a free energy function that gives the free energy of the protein corresponding to a particular value of the enthalpy. By the location of the minima in this function, the free energy distribution graphically indicates the most probable values of the enthalpy for the protein. We find that the behavior of the free energy functions for proteins falls somewhere between two different cases: a two-state like function with two minima, the relative levels of the two states changing with temperature; and, a single-minimum function where the position of the minimum shifts to higher enthalpy values as the temperature is increased. We show that the temperature dependence of the free energy function can be expressed in terms of a central free energy distribution for a given, fixed temperature (which is most conveniently chosen as the temperature of the maximum in the heat capacity). The nature of this central free energy function for a given protein thus yields all of the thermodynamic behavior of that protein over the temperature range of the denaturation process.