The Berg-Purcell Limit Revisited

Biological systems often have to measure extremely low concentrations of chemicals with high precision. When dealing with such small numbers of molecules, the inevitable randomness of physical transport processes and binding reactions will limit the precision with which measurements can be made. An important question is what the lower bound on the noise would be in such measurements. Using the theory of diffusion-influenced reactions, we derive an analytical expression for the precision of concentration estimates that are obtained by monitoring the state of a receptor to which a diffusing ligand can bind. The variance in the estimate consists of two terms, one resulting from the intrinsic binding kinetics and the other from the diffusive arrival of ligand at the receptor. The latter term is identical to the fundamental limit derived by Berg and Purcell (Biophys. J., 1977), but disagrees with a more recent expression by Bialek and Setayeshgar. Comparing the theoretical predictions against results from particle-based simulations confirms the accuracy of the resulting expression and reaffirms the fundamental limit established by Berg and Purcell.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor are internalized. Here we present a mathematical model for the combined binding and other reactions which gives useful information about the behaviour of such systems. The set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity on time of measurement of free and bound ligand, on degradation and internalization of ligand and on degradation and internalization of receptor is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations lead to possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions Where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor may be internalized. Here we present a mathematical model in which the binding and other reactions are combined. The resulting set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity - on time of measurement of free and bound ligand, on degradation and internalization of ligand, and on degradation and internalization of receptor - is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations suggest experimental possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.     

Cell-membrane hormone receptors: some perspectives on their structure and function relationship

There are three structural regions common to hormone receptors on the cell membrane: an external region containing the ligand-specific binding domain, a hydrophobic region transversing the membrane, and a cytoplasmic region involved in the expression of activity. The structural features of these regions are broadly reviewed in regard to their functional roles. It is suggested that the specificity of cellular response to hormonal stimulation lies not only in the binding of a specific ligand by the external domain but also in the cytoplasmic region of a receptor. The cytoplasmic region can either interact specifically with a regulatory system or express its own specific ligand-dependent catalytic activity. The fundamental question of how ligand binding can lead to the expression and regulation of receptor activity is considered.     

The nicotinic receptor ligand binding domain

The ligand binding domain (LBD) of the nicotinic acetylcholine receptor has served as a prototype for understanding molecular recognition in the family of neurotransmitter-gated ion channels. During the past fifty years, studies progressed from fundamental electrophysiological analyses of ACh-evoked ion flow, to biochemical purification of the receptor protein, pharmacological measurements of ligand binding, molecular cloning of receptor subunits, site-directed mutagenesis combined with functional analysis and recently, atomic structural determination. The emerging picture of the nicotinic receptor LBD is a specialized pocket of aromatic and hydrophobic residues formed at interfaces between protein subunits that changes conformation to convert agonist binding into gating of an intrinsic ion channel.     

Farseer-NMR: automatic treatment,analysis and plotting of large,multi-variable NMR data

Molecular dynamics play a significant role in how molecules perform their function. A critical method that provides information on dynamics, at the atomic level, is NMR-based relaxation dispersion (RD) experiments. RD experiments have been utilized for understanding multiple biological processes occurring at micro-to-millisecond time, such as enzyme catalysis, molecular recognition, ligand binding and protein folding. Here, we applied the recently developed high-power RD concept to the Carr–Purcell–Meiboom–Gill sequence (extreme CPMG; E-CPMG) for the simultaneous detection of fast and slow dynamics. Using a fast folding protein, gpW, we have shown that previously inaccessible kinetics can be accessed with the improved precision and efficiency of the measurement by using this experiment.     

smFRET studies of the ‘encounter’ complexes and subsequent intermediate states that regulate the selectivity of ligand binding

The selectivity with which a biomolecule can bind its cognate ligand when confronted by the vast array of structurally similar, competing ligands that are present in the cell underlies the fidelity of some of the most fundamental processes in biology. Because they collectively comprise one of only a few methods that can sensitively detect the ‘encounter’ complexes and subsequent intermediate states that regulate the selectivity of ligand binding, single-molecule fluorescence, and particularly single-molecule fluorescence resonance energy transfer (smFRET), approaches have revolutionized studies of ligand-binding reactions. Here, we describe a widely used smFRET strategy that enables investigations of a large variety of ligand-binding reactions, and discuss two such reactions, aminoacyl-tRNA selection during translation elongation and splice site selection during spliceosome assembly, that highlight both the successes and challenges of smFRET studies of ligand-binding reactions. We conclude by reviewing a number of emerging experimental and computational approaches that are expanding the capabilities of smFRET approaches for studies of ligand-binding reactions and that promise to reveal the mechanisms that control the selectivity of ligand binding with unprecedented resolution.     

Influence of tissue fixation on the binding of (125)I-angiotensin receptor ligands in the rat,mouse and rabbit kidney

Aldehyde fixatives are often used to preserve tissue morphology and thereby aid in the identification of cellular structures expressing a target of interest. However, the effect of fixatives on target detection methods is unpredictable and it is currently unknown whether tissue fixation would allow the accurate detection of angiotensin AT(4) receptors in the kidney. In vitro receptor autoradiography on tissues fixed with 4% paraformaldehyde and 0.5% glutaraldehyde (+/-20% sucrose) had differing effects on the density of (125)I-AT(4) receptor ligand binding without affecting the tissue distribution of ligand binding in the rat and mouse kidney, whereas an increased expression of specific (125)I-AT(4) receptor ligand binding was found in the medulla region of the rabbit kidney. In contrast, such tissue fixation conditions dramatically decreased the renal binding of (125)I-angiotensin II receptor ligands, and altered the distribution of such ligand binding, in all three species. These results suggest that the method of tissue fixation and processing should be used cautiously in angiotensin receptor density measurements but can provide an accurate representation of kidney AT(4) receptor distribution only in the rat and mouse.     

Reconstructing potential energy functions from simulated force-induced unbinding processes.

One-dimensional stochastic models demonstrate that molecular dynamics simulations of a few nanoseconds can be used to reconstruct the essential features of the binding potential of macromolecules. This can be accomplished by inducing the unbinding with the help of external forces applied to the molecules, and discounting the irreversible work performed on the system by these forces. The fluctuation-dissipation theorem sets a fundamental limit on the precision with which the binding potential can be reconstructed by this method. The uncertainty in the resulting potential is linearly proportional to the irreversible component of work performed on the system during the simulation. These results provide an a priori estimate of the energy barriers observable in molecular dynamics simulations.     

Determination of binding parameters in the presence of coupled reactions

Components of a binding reaction may undergo nonbinding reactions: receptors may be degraded, internalized, or exchanged with cryptic sites; ligand may be degraded or compartmented. In such cases the parameters that characterize the system are not obtained from the usual equilibrium analyses. We have simulated the reactions of such systems and generated association curves, "Scatchard" plots, and "Scatchard-like" plots that permit the calculation of binding affinity and receptor number not normally calculable under nonequilibrium binding conditions. In particular, we show that certain coupled reactions produce local maxima and sigmoid shapes in association curves and that the maxima can be used to obtain affinities and receptor numbers.     

The effect of receptor density on the forward rate constant for binding of ligands to cell surface receptors.

For monovalent ligands interacting with cell surface receptors we have directly observed the functional dependence of the forward rate constant on the number of receptors per cell (N). The experimental system we studied consisted of monovalent ligand, 2,4-dinitrophenyl (DNP)-aminocaproyl-L-tyrosine (DCT), binding to bivalent, monoclonal anti-DNP immunoglobulin E (IgE) anchored to its high affinity receptor on rat basophilic leukemia (RBL) cells. To measure the fractional occupation of antibody combining sites by DNP we employed a recently developed fluorescence technique (Erickson, J., Kane, B. Goldstein, D. Holowka, and B. Baird, 1986, Mol. Immunol., 72:769-781). Our results are well fitted by the equation (Berg and Purcell, 1977, Biophys. J., 20:193-219) konc = 4 pi DaN kappa on/[4 pi Da + N kappa on] where konc is the forward rate constant for binding to the cell, D is the diffusion constant of the ligand, a is the radius of the cell, and kappa on is the intrinsic forward rate constant describing a single IgE combining site-DNP interaction. If D is fixed at 10(-5) cm2/s, the best fit of accumulated data predicts an average cell radius of approximately 4 microns and kappa on of approximately 1.8 x 10(-13) cm3/s [1.1 x 10(8)(M . s)-1]; both in excellent agreement with RBL cell size and the single-site forward rate constant for the binding of DCT to IgE in solution, respectively. We believe this is the first report of experimental evidence that directly illustrates the effect of surface density in determining the rates of binding for small molecules to membrane receptors.     

Role of extracellular disulfide-bonded cysteines in the ligand binding function of the beta 2-adrenergic receptor

Evidence is presented for a role of disulfide bridging in forming the ligand binding site of the beta 2-adrenergic receptor (beta AR). The presence of disulfide bonds at the ligand binding site is indicated by "competitive" inhibition by dithiothreitol (DTT) in radioligand binding assays, by specific protection by beta-adrenergic ligands of these effects, and by the requirement of disulfide reduction for limit proteolysis of affinity ligand labeled receptor. The kinetics of binding inhibition by DTT suggest at least two pairs of disulfide-bonded cysteines essential for normal binding. Through site-directed mutagenesis, we indeed were able to identify four cysteines which are critical for normal ligand binding affinities and for the proper expression of functional beta AR at the cell surface. Unexpectedly, the four cysteines required for normal ligand binding are not those located within the hydrophobic transmembrane domains of the receptor (where ligand binding is presumed to occur) but lie in the extracellular hydrophilic loops connecting these transmembrane segments. These findings indicate that, in addition to the well-documented involvement of the membrane-spanning domains of the receptor in ligand binding, there is an important and previously unsuspected role of the hydrophilic extracellular domains in forming the ligand binding site.     

Influence of external concentration fluctuations on leukocyte chemotactic orientation

The quantitative dependence of leukocyte chemotactic orientation on imprecision in the measurement of chemoattractant concentrations from thermal fluctuations is analyzed. First, a mathematical model relating orientation to differences in receptor occupancy across cell dimensions is developed. This is then coupled with an extension of Berg and Purcell's analysis (1) of the precision of attractant concentration measurements by means of receptor occupancy. Our results show that thermal fluctuations in external concentrations can limit the accuracy of orientation, unless the measurement noise is reduced by averaging the measurements over a period of time. Comparison of our model predictions to experimental orientation data suggests that leukocytes do overcome this limitation, and allows estimation of the time-averaging period necessary to do so. For the orientation observed in a visual bridge assay by Zigmond (2) using the attractant peptide FNLLP, we estimate that receptor occupancy measurements for spatial comparison across cell dimensions must be averaged for a few minutes. Otherwise, the fluctuations in the attractant concentration near the cell will be too great to allow the observed degree of orientation. Our analysis also suggests that the ratio of signal-to-signal noise does not adequately characterize orientation accuracy. Accurate orientation can, in some situations, occur when this ratio is substantially less than unity; in other situations, a ratio much greater than unity is required for accurate orientation.     

Purification of the D-2 dopamine receptor from bovine striatum

The D-2 dopamine receptor has been purified 21500 fold from bovine striatal membranes. Solubilized receptor preparation was partially purified by affinity chromatography on a haloperidol adsorbent followed by gel filtration on a Sephacryl S-300 column. The fractions eluted from this column which contained the ligand binding activity were further chromatographed on wheat germ agglutinin conjugated to Sepharose. The resulting receptor preparation displays a major polypeptide band of an apparent molecular weight of 92 kDa, and exhibits a specific binding activity of 2490 pmol spiperone per mg protein. This purified receptor preparation can reabsorb specifically to the haloperidol affinity column indicating that the 92 kDa polypeptide represents the ligand binding unit of the D-2 dopamine receptor.     

Modeling Multivalent Ligand-Receptor Interactions with Steric Constraints on Configurations of Cell-Surface Receptor Aggregates

We use flow cytometry to characterize equilibrium binding of a fluorophore-labeled trivalent model antigen to bivalent IgE-FcεRI complexes on RBL cells. We find that flow cytometric measurements are consistent with an equilibrium model for ligand-receptor binding in which binding sites are assumed to be equivalent and ligand-induced receptor aggregates are assumed to be acyclic. However, this model predicts extensive receptor aggregation at antigen concentrations that yield strong cellular secretory responses, which is inconsistent with the expectation that large receptor aggregates should inhibit such responses. To investigate possible explanations for this discrepancy, we evaluate four rule-based models for interaction of a trivalent ligand with a bivalent cell-surface receptor that relax simplifying assumptions of the equilibrium model. These models are simulated using a rule-based kinetic Monte Carlo approach to investigate the kinetics of ligand-induced receptor aggregation and to study how the kinetics and equilibria of ligand-receptor interaction are affected by steric constraints on receptor aggregate configurations and by the formation of cyclic receptor aggregates. The results suggest that formation of linear chains of cyclic receptor dimers may be important for generating secretory signals. Steric effects that limit receptor aggregation and transient formation of small receptor aggregates may also be important.     

Kinetics of ligand binding to receptor immobilized in a polymer matrix, as detected with an evanescent wave biosensor. I. A computer simulation of the influence of mass transport.

The influence of mass transport on ligand binding to receptor immobilized in a polymer matrix, as detected with an evanescent wave biosensor, was investigated. A one-dimensional computer model for the mass transport of ligand between the bulk solution and the polymer gel and within the gel was employed, and the influence of the diffusion coefficient, the partition coefficient, the thickness of the matrix, and the distribution of immobilized receptor were studied for a variety of conditions. Under conditions that may apply to many published experimental studies, diffusion within the matrix was found to decrease the overall ligand transport significantly. For relatively slow reactions, small spatial gradients of free and bound ligand in the gel are found, whereas for relatively rapid reactions strong inhomogeneities of ligand within the gel occur before establishment of equilibrium. Several types of deviations from ideal pseudo-first-order binding progress curves are described that resemble those of published experimental data. Extremely transport limited reactions can in some cases be fitted with apparently ideal binding progress curves, although with apparent reaction rates that are much lower than the true reaction rates. Nevertheless, the ratio of the apparent rate constants can be semiquantitatively consistent with the true equilibrium constant. Apparently "cooperative" binding can result from high chemical on rates at high receptor saturation. Dissociation in the presence of transport limitation was found to be well described empirically by a single or a double exponential, with both apparent rate constants considerably lower than the intrinsic chemical rate constant. Transport limitations in the gel can introduce many generally unknown factors into the binding progress curve. The simulations suggest that unexpected deviations from ideal binding progress curves may be due to highly transport influenced binding kinetics. The use of a thinner polymer matrix could significantly increase the range of detectable rate constants.