Transport effects on surface reaction arrays: biosensor applications

The ubiquity of surface-volume reactions in biological and industrial processes makes knowledge of their kinetics critical. This has spurred technological advances in several biosensors designed to measure rate constants, such as the Flexchip and the dotLab. These biosensors have multiple reacting zones in a single flow channel, and hence they also serve as good model systems for biochemical systems with multiple reacting zones, such as cell membranes. A correct mathematical model for such systems must incorporate the effects of transport and zone position. A basic unidirectional flow model is developed in general and solved for typical experimental parameters using perturbation methods. The effect of zone placement along the channel can be quantified in terms of an effective Damköhler number based upon position. Moreover, it is established that zone placement across the channel does not affect the measurements.     

Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking

Both enzyme (e.g., G-protein) activation via a collision coupling model and the formation of cross-linked receptors by a multivalent ligand involve reactions between two molecules diffusing in the plasma membrane. The diffusion of these molecules is thought to play a critical role in these two early signal transduction events. In reduced dimensions, however, diffusion is not an effective mixing mechanism; consequently, zones in which the concentration of particular molecules (e.g., enzymes, receptors) becomes depleted or enriched may form. To examine the formation of these depletion/ accumulation zones and their effect on reaction rates and ultimately the cellular response, Monte Carlo techniques are used to simulate the reaction and diffusion of molecules in the plasma membrane. The effective reaction rate at steady state is determined in terms of the physical properties of the tissue and ligand for both enzyme activation via collision coupling and the generation of cross-linked receptors. The diffusion-limited reaction rate constant is shown to scale with the mean square displacement of a receptor-ligand complex. The rate constants determined in the simulation are compared with other theoretical predictions as well as experimental data.     

Effective rate models for receptors distributed in a layer above a surface: application to cells and Biacore

In the Biacore biosensor, a widely used tool for studying the kinetics of ligand/receptor binding, receptors are commonly localized to the sensor surface through attachment to polymers that extend from the surface to form a layer. The importance of the polymeric layer in analyzing data is controversial. The question of the effect of a binding layer also arises in the case of ligands interacting with binding sites distributed in the extracellular matrix of cells. To identify and quantify the effects of a binding layer on the estimation of association and dissociation rate constants, we derived effective rate coefficients. The expressions show that rate constants determined under the standard assumption that binding takes place on a two-dimensional surface underestimate the true reaction rate constants by a factor that depends on the ratio of the height of the layer to the mean free path of the ligand within the layer. We show that, for typical biological ligands, receptors, cells, and Biacore conditions, the binding layer will affect the interpretation of data only if transport of the ligand in the layer is slowed substantially--by one or two orders of magnitude--relative to transport outside the layer. From existing experiments and theory, it is not clear which Biacore experiments, if any, have transport within the dextran layer reduced to such an extent. We propose a method, based on the effective rate coefficients we have derived, for the experimental determination of ligand diffusion coefficients in a polymeric matrix.     

Ligand-receptor interaction rates in the presence of convective mass transport.

The rate of binding of a ligand to receptors on the cell surface can be diffusion limited. We analyze the kinetics of binding, diffusion-limited in a stationary liquid, in the presence of convective mass transport. We derive a formula that expresses the reaction kinetics in terms of the mass transfer coefficient. A moderately transport-limited kinetics is not readily recognizable from the shape of the binding curve and may lead to erroneous estimates of the rate coefficients. We apply our results to practically important cases: a cell suspension in a stirred volume of liquid and a confluent cell colony under a laminar stream. Using typical numbers characterizing the ligand-receptor interactions, we show that stirring and perfusion can be important factors determining the reaction rates. With the confluent colony, the early reaction kinetics requires a different treatment, and we provide it for the case of low receptor occupancy. We show that, even with a fast perfusion, a cell monolayer can transiently generate a zone of depletion of the ligand, and that would affect the early stages of the reaction. Our results are expressed in a simple analytical form and can be used for the design and interpretation of experimental data.     

The Reactions of Hemoglobin in Various Media

The rate constants for the association of oxygen and of carbon monoxide with reduced hemoglobin and for the dissociation of oxygen from hemoglobin were studied in a variety of media. All three of these rates were essentially independent of diffusion rate and dielectric constant throughout a wide range. These results are very different from those found for other heme proteins whose rates are diffusion-controlled in very viscous solvents.

The interpretation of the average measured rates described herein in terms of the four rate constants discussed by Gibson and Roughton indicates that our measurements reflect most strongly the constants for the first reacting molecule in each case. For the purposes of these reactions, the reactants may be regarded as neutral molecules.

     

Incorporating Conservation Zone Effectiveness for Protecting Biodiversity in Marine Planning

Establishing different types of conservation zones is becoming commonplace. However, spatial prioritization methods that can accommodate multiple zones are poorly understood in theory and application. It is typically assumed that management regulations across zones have differential levels of effectiveness (“zone effectiveness”) for biodiversity protection, but the influence of zone effectiveness on achieving conservation targets has not yet been explored. Here, we consider the zone effectiveness of three zones: permanent closure, partial protection, and open, for planning for the protection of five different marine habitats in the Vatu-i-Ra Seascape, Fiji. We explore the impact of differential zone effectiveness on the location and costs of conservation priorities. We assume that permanent closure zones are fully effective at protecting all habitats, open zones do not contribute towards the conservation targets and partial protection zones lie between these two extremes. We use four different estimates for zone effectiveness and three different estimates for zone cost of the partial protection zone. To enhance the practical utility of the approach, we also explore how much of each traditional fishing ground can remain open for fishing while still achieving conservation targets. Our results show that all of the high priority areas for permanent closure zones would not be a high priority when the zone effectiveness of the partial protection zone is equal to that of permanent closure zones. When differential zone effectiveness and costs are considered, the resulting marine protected area network consequently increases in size, with more area allocated to permanent closure zones to meet conservation targets. By distributing the loss of fishing opportunity equitably among local communities, we find that 84–88% of each traditional fishing ground can be left open while still meeting conservation targets. Finally, we summarize the steps for developing marine zoning that accounts for zone effectiveness.     

Kinetics of carbon monoxide and oxygen binding to hemoglobin in human red blood cell suspensions studied by laser flash photolysis

The reaction kinetics of the binding of CO and O₂ to hemoglobin (Hb) in human red blood cell (RBC) suspensions have been examined using a 300 ns dye laser to photodissociate HbCO or HbO₂. Fast (halftime1?0 μs) and slow (5?ms) processes were seen after photolysis. The results indicate that neither the rate constants nor the activation energies for the binding of CO to the fast reacting form of Hb in the RBC are significantly different from that measured in solution in spite of the different environments. Rate constants determined for O₂ binding in RBC were intermediate between rates observed for reaction with fast and slow reacting forms of Hb and probably consist of contributions from each. The slow recombination of CO and O₂ probably has contributions both from reaction with slow reacting forms of Hb and from ligand that had diffused away from the RBC after photolysis.     

Sedimentation velocity analysis of heterogeneous protein-protein interactions: Lamm equation modeling and sedimentation coefficient distributions c(s)

We describe algorithms for solving the Lamm equations for the reaction-diffusion-sedimentation process in analytical ultracentrifugation, and examine the potential and limitations for fitting experimental data. The theoretical limiting case of a small, uniformly distributed ligand rapidly reacting with a larger protein in a "constant bath" of the ligand is recapitulated, which predicts the reaction boundary to sediment with a single sedimentation and diffusion coefficient. As a consequence, it is possible to express the sedimentation profiles of reacting systems as c(s) distribution of noninteracting Lamm equation solutions, deconvoluting the effects of diffusion. For rapid reactions, the results are quantitatively consistent with the "constant bath" approximation, showing c(s) peaks at concentration-dependent positions. For slower reactions, the deconvolution of diffusion is still partially successful, with c(s) resolving peaks that reflect the populations of sedimenting species. The transition between c(s) peaks describing reaction boundaries of moderately strong interactions (K(D) approximately 10(-6) M) or resolving sedimenting species was found to occur in a narrow range of dissociation rate constant between 10(-3) and 10(-4) s(-1). The integration of the c(s) peaks can lead to isotherms of species populations or s-value of the reaction boundary, respectively, which can be used for the determination of the equilibrium binding constant.     

Effects of receptor clustering on ligand dissociation kinetics: theory and simulations

Receptor-ligand binding is a critical first step in signal transduction and the duration of the interaction can impact signal generation. In mammalian cells, clustering of receptors may be facilitated by heterogeneous zones of lipids, known as lipid rafts. In vitro experiments show that disruption of rafts significantly alters the dissociation of fibroblast growth factor-2 (FGF-2) from heparan sulfate proteoglycans (HSPGs), co-receptors for FGF-2. In this article, we develop a continuum stochastic formalism to address how receptor clustering might influence ligand rebinding. We find that clusters reduce the effective dissociation rate dramatically when the clusters are dense and the overall surface density of receptors is low. The effect is much less pronounced in the case of high receptor density and shows nonmonotonic behavior with time. These predictions are verified via lattice Monte Carlo simulations. Comparison with FGF-2-HSPG experimental results is made and suggests that the theory could be used to analyze similar biological systems. We further present an analysis of an additional cooperative internal-diffusion model that might be used by other systems to increase ligand retention when simple rebinding is insufficient.     

Receptor-mediated cell attachment and detachment kinetics. I. Probabilistic model and analysis.

The kinetics of receptor-mediated cell adhesion to a ligand-coated surface play a key role in many physiological and biotechnology-related processes. We present a probabilistic model of receptor-ligand bond formation between a cell and surface to describe the probability of adhesion in a fluid shear field. Our model extends the deterministic model of Hammer and Lauffenburger (Hammer, D.A., and D.A. Lauffenburger. 1987. Biophys. J. 52:475-487) to a probabilistic framework, in which we calculate the probability that a certain number of bonds between a cell and surface exists at any given time. The probabilistic framework is used to account for deviations from ideal, deterministic behavior, inherent in chemical reactions involving relatively small numbers of reacting molecules. Two situations are investigated: first, cell attachment in the absence of fluid stress; and, second, cell detachment in the presence of fluid stress. In the attachment case, we examine the expected variance in bond formation as a function of attachment time; this also provides an initial condition for the detachment case. Focusing then on detachment, we predict transient behavior as a function of key system parameters, such as the distractive fluid force, the receptor-ligand bond affinity and rate constants, and the receptor and ligand densities.(ABSTRACT TRUNCATED AT 250 WORDS)     

Interaction of acetazolamide and 4-nitrothiophenolate ion with bivalent metal ion derivatives of bovine carbonic anhydrase.

The stability and rate constants for the interaction of acetazolamide (diamox) and 4-nitrothiophenolate ion (NTP) with the bivalent Mn, Co, Ni, Cu and Cd forms of bovine carbonic anhydrase have been measured by utilizing the distinct visible spectra of each metalloenzyme-NTP adduct. Differing stabilities of the various NTP and (particularly) diamox complexes reside mainly in varying values for the dissociation rate constants (kd). Intrinsic formation rate constants (for the acid form of the enzyme reacting with the basic form of the ligand) are uniformly high (greater than or equal 2 X 10(7) M-1 s-1 at 25 degrees C). Invariance of kd with pH and a bell-shaped log K-pH profile with the Cu-enzyme adducts are features observed previously with the native enzyme. Binding of NTP with the Cu and Cd metalloenzymes is stronger than to the native form.     

Theory of equilibrium binding of a bivalent ligand to cell surface antibody: The effect of antibody heterogeneity on cross-linking

We investigate the equilibrium binding of symmetric bivalent ligands to a heterogeneous population of symmetric bivalent cell surface receptors. The receptors are heterogeneous in their binding affinities (equilibrium binding constants) for the ligand. For any distribution of receptor binding affinities we show how to calculate the total concentration of receptors that are cross-linked by the ligand, i.e., the concentration of cell surface aggregates composed of two or more receptors, as well as the concentration of any given aggregate. We show that certain qualitative properties of cross-linking which hold for homogeneous antibody populations fail to hold in the heterogeneous case. We use our results to interpret certain in vitro experiments in which synthetic bivalent haptens are used to trigger histamine release from basophils which have on their surface antibody specific for the hapten.This work was performed under the auspices of the Department of Energy and supported by Grant AI 16465 from the National Institute of Allergy and Infectious Diseases.     

SPOT synthesis: reliability of array-based measurement of peptide binding affinity

Peptide arrays prepared by the SPOT synthesis technology have emerged as a proteomic tool to study molecular recognition and identify biologically active peptides. However, it was previously not clear how accurately signal intensities obtained by probing peptide arrays for protein binding really reflect the dissociation constants of the protein-peptide complexes. Using the monoclonal antibody CB4-1 as a model system, we systematically compared dissociation constants of antibody-peptide complexes with signal intensities obtained using the SPOT technology. By analyzing a set of peptides possessing different affinities to the antibody, we determined the strengths of the SPOT screening method. The accuracy of the measured results was improved by taking regional trends in the membrane surface into account. A model based on the mass action law compares well with the experimental results. Interestingly, the applied concentrations of the binding partners do not directly correspond to the effective concentrations in the assay. We show that the SPOT technology is an accurate method for assigning the spots' measured signal intensities to three different binding affinity classes. The dissociation constants of the intermediate region were found to be between pK(dis)=5 and pK(dis)=7. Altering the experimental parameters causes a directed change of this region.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

A comparative study on the thermal decomposition of four cereal straws in an oxidizing atmosphere

The thermal decomposition characteristics of four types (wheat, barley, oats and rye) of cereal straws were studied. Two varieties from each type of straw were used. The thermal degradation behaviours and kinetic parameters (order of reaction, activation energy and preexponential factor) of the straws were compared. Two distinct reaction zones were observed for all types and varieties of straws. Thermal degradation rates in the first reaction zone were significantly higher than those in the second reaction zone. The activation energy was in the ranges of 80–102 kJ/mol and 34–75 kJ/mol, whereas the order of reaction was in the ranges of 1·3–2·3 and 0·1–0·7 for the first and second reaction zones, respectively. The Shaw variety of oats straw had the highest activation energies (102 and 75 kJ/mol) and reaction orders (2·3 and 0·7) in both the first and second reaction zones, respectively. The lowest activation energy (80 kJ/mol) and order of reaction (1·3), in the first reaction zone, corresponded to Absolvant and Monopol wheat straws. The activation energies and reaction orders of barley and rye straws were in the ranges of 85–94 kJ/mol and 1·9–2·3, respectively. There was not any significant difference between the rate constants of the straw varieties, in the first reaction zone. However, oats straws had significantly higher rate constants in the second reaction zone as compared to the rate constants of wheat, barley and rye straws.     

The effects of E7 and E11 mutations on the kinetics of ligand binding to R state human hemoglobin

Association and dissociation rate constants for O2, CO, and methyl isocyanide binding to native and distal pocket mutants of R state human hemoglobin were measured using ligand displacement and partial photolysis techniques. Individual rate constants for the alpha and beta subunits were resolved by comparisons between the kinetic behavior of the native and mutant proteins. His-E7 was replaced with Gly and Gln in both alpha and beta subunits and with Phe in beta subunits alone. In separate experiments Val-E11 was replaced with Ala, Leu, and Ile in each globin chain. The parameters describing ligand binding to R state alpha subunits are sensitive to the size and polarity of the amino acids at positions E7 and E11. The distal histidine in this subunit inhibits the bimolecular rate of binding of both O2 and CO, sterically hinders bound CO and methyl isocyanide, and stabilizes bound O2 by hydrogen bonding. The Val-E11 side chain in alpha chains also appears to be part of the kinetic barrier to O2 and CO binding since substitution with Ala causes approximately 10-fold increases in the association rate constants for the binding of these diatomic ligands. However, substitution of Val-E11 by Ile produces only small decreases in the rates of ligand binding to alpha subunits. For R state beta subunits, the bimolecular rates of O2 and CO binding are intrinsically large, approximately 2-5-fold greater than those for alpha subunits, and with the exception of Val-E11----Ile mutation, little affected by substitutions at either the E7 or E11 positions. For the beta Val-E11----Ile mutant the association rate and equilibrium constants for all three ligands decreased 10-50-fold. All of these results agree with Shaanan's conclusions that the distal pocket in liganded beta subunits is more open whereas in alpha subunits bound ligands are more sterically hindered by adjacent distal residues (Shaanan, B. (1983) J. Mol. Biol. 171, 31-59). In the case of O2 binding to alpha subunits, the unfavorable steric effects are compensated by the formation of a hydrogen bond between the nitrogen atom of His-E7 and bound dioxygen.