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.     

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.     

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.     

Diffusion-controlled sensitization of photocleavage reactions on surfaces

The kinetic rate equation for the photosensitized cleavage reaction of surface-bound photolabile chromophores with free diffusion of sensitizer molecules from the bulk of a solution to the surface is derived by determining the stationary solution of a diffusion equation with suitable boundary conditions. The relation between the phenomenological rate constant for the photosensitized reaction at the surface and in the bulk is established. Applying the result to the analysis of an experimental example, the origin of the quasi zeroth-order kinetics of the sensitized reaction is revealed. A theoretical comparison of intramolecular sensitization in photocleavable protecting groups with a molecular antenna and sensitization with the freely diffusing sensitizer shows that in a typical case sensitization with free diffusion is more effective than intramolecular sensitization for sensitizer concentrations higher than 5 mM.     

The effect of diffusion on the binding of membrane-bound receptors to coated pits.

We have formulated a kinetic model for the primary steps that occur at the cell membrane during receptor-mediated endocytosis. This model includes the diffusion of receptor molecules, the binding of receptors to coated pits, the loss of coated pits by invagination, and random reinsertion of receptors and coated pits. Using the mechanistic statistical theory of nonequilibrium thermodynamics, we employ this mechanism to calculate the two-dimensional radial distribution of receptors around coated pits at steady state. From this we obtain an equation that describes the effect of receptor diffusion on the rate of binding to coated pits. Our equation does not assume that ligand binding is instantaneous and can be used to assess the effect of diffusion on the binding rate. Using experimental data for low density lipoprotein receptors on fibroblast cells, we conclude that the effect of diffusion on the binding of these receptors to coated pits is no more than 84% diffusion controlled. This corresponds to a dissociation rate constant for receptors on coated pits (k-) that is much less than the rate constant for invagination of the pits (lambda = 3.3 X 10(-3)/s) and a correlation length for the radial distribution function of six times the radius of a coated pit. Although the existing experimental data are compatible with any value of k-, we obtain a lower bound for the value of the binding constant (k+) of 2.3 X 10(-2)(micron)2/s. Comparison of the predicted radial distributions with experiment should provide a clear indication of the effect of diffusion on k+.     

Cellular level models as tools for cytokine design

Cytokines and growth factors are critical regulators that connect intracellular and extracellular environments through binding to specific cell‐surface receptors. They regulate a wide variety of immunological, growth, and inflammatory response processes. The overall signal initiated by a population of cytokine molecules over long time periods is controlled by the subtle interplay of binding, signaling, and trafficking kinetics. Building on the work of others, we abstract a simple kinetic model that captures relevant features from cytokine systems as well as related growth factor systems. We explore a large range of potential biochemical behaviors, through systematic examination of the model's parameter space. Different rates for the same reaction topology lead to a dramatic range of biochemical network properties and outcomes. Evolution might productively explore varied and different portions of parameter space to create beneficial behaviors, and effective human therapeutic intervention might be achieved through altering network kinetic properties. Quantitative analysis of the results reveals the basis for tensions among a number of different network characteristics. For example, strong binding of cytokine to receptor can increase short‐term receptor activation and signal initiation but decrease long‐term signaling due to internalization and degradation. Further analysis reveals the role of specific biochemical processes in modulating such tensions. For instance, the kinetics of cytokine binding and receptor activation modulate whether ligand–receptor dissociation can generally occur before signal initiation or receptor internalization. Beyond analysis, the same models and model behaviors provide an important basis for the design of more potent cytokine therapeutics by providing insight into how binding kinetics affect ligand potency. © 2010 American Institute of Chemical Engineers Biotechnol. Prog., 2010     

Fluorescence correlation spectroscopy. I. Conceptual basis and theory

We describe a method for determining chemical kinetic constants and diffusion coefficients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disappear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.     

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.     

Kinetic analysis of chemotactic peptide receptor modulation

The dynamics of the chemotactic peptide receptor on rabbit peritoneal polymorphonuclear leucocytes were followed using the tritiated peptide N-formylnorleucylleucylphenylalanine (FNLLP). We have used a kinetic analysis to examine the possible interrelationships between receptor loss (down-regulation), receptor-mediated peptide uptake, and receptor recycling. We have previously demonstrated that cells incubated with FNLLP show a dose-dependent reduction in the number of receptors available on the surface. This receptor down-regulation is complete within 20 min and then the number of receptors available for binding remains at a plateau level. Peptide continues to be taken up in a receptor-mediated manner even after down-regulation is complete. If peptide is removed, receptor recovery occurs and does not require protein synthesis. In these studies we have investigated the kinetics of these processes. On the basis of this analysis, we propose that the plateau receptor level is a steady-state in which receptor internalization and return occur continuously. We demonstrate that the rate of receptor-mediated peptide uptake is approximately equal to the rate of receptor recovery measured after peptide removal. In addition, the rate of receptor recovery is proportional to the number of receptors missing from the surface, suggesting receptor recycling may be occurring.     

Kinetics and mechanics of cell adhesion

Cell adhesion is mediated by specific interaction between receptors and ligands. Such interaction provides not only physical linkage but also communication between the cell and its environment. The kinetics and mechanics of cell adhesion are coupled, because force can influence the formation and dissociation of receptor-ligand bonds. The kinetic rates and their force dependence determine how likely, how rapidly and how strongly cells bind as well as how long they remain bound. Since adhesion molecules are linked to apposing cellular membranes, their interaction is governed by two-dimensional (2D) kinetics. This is in contrast to the three-dimensional (3D) binding of soluble ligands to cell surface receptors. Unlike the 3D case in which many methods are available for measuring kinetic rates, not until recently have the 2D kinetic rates become experimentally measurable. In this review, I will discuss the recent progress in the experimental methods that enable quantification of the relevant kinetic and mechanical parameters, the fundamental concepts that underlie the physics of the biological phenomena, and the mathematical models that relate functions to the intrinsic properties of the adhesion molecules.     

Analysis of intracellular receptor/ligand sorting. Calculation of mean surface and bulk diffusion times within a sphere.

Cell surface receptors bind extracellular ligand molecules and transport those ligands into the cell by a process termed receptor-mediated endocytosis. Receptor and ligand molecules are sorted from one another after endocytosis, apparently within a structure consisting of intracellular vesicles and connected thin tubules. The experimental observation is that most free (unbound) ligand molecules are found in the lumen of the vesicles and receptors are located primarily within the tubules. Because equilibrium and geometric considerations do not explain this segregation, a kinetic scheme involving the passive diffusion of molecules from a vesicle into a tubule is investigated. Two possible sorting mechanisms are considered: first, that receptors are able to move into tubules more rapidly than ligand molecules due to an advantage in dimensionality and, second, that receptors diffusing into tubules are trapped there while ligands are not. Mean diffusion times for receptor and ligand movement into a tubule are calculated by solving Poisson's equation in two and three dimensions, respectively, on the surface of and within a sphere. Using estimated parameter values, we found that only the second scheme is able to account for the experimentally observed sorting. An estimate is obtained for the length of time a tubule and vesicle must be connected in order to remove a significant number of receptors into a tubule. The fraction of free ligand that is "mis-sorted" with the recycling receptor population and thus exocytosed is also determined.     

Modeling the effect of glutamate diffusion and uptake on NMDA and non-NMDA receptor saturation.

One- and two-dimensional models of glutamate diffusion, uptake, and binding in the synaptic cleft were developed to determine if the release of single vesicles of glutamate would saturate NMDA and non-NMDA receptors. Ranges of parameter values were used in the simulations to determine the conditions when saturation could occur. Single vesicles of glutamate did not saturate NMDA receptors unless diffusion was very slow and the number of glutamate molecules in a vesicle was large. However, the release of eight vesicles at 400 Hz caused NMDA receptor saturation for all parameter values tested. Glutamate uptake was found to reduce NMDA receptor saturation, but the effect was smaller than that of changes in the diffusion coefficient or in the number of glutamate molecules in a vesicle. Non-NMDA receptors were not saturated unless diffusion was very slow and the number of glutamate molecules in a vesicle was large. The release of eight vesicles at 400 Hz caused significant non-NMDA receptor desensitization. The results suggest that NMDA and non-NMDA receptors are not saturated by single vesicles of glutamate under usual conditions, and that tetanic input, of the type typically used to induce long-term potentiation, will increase calcium influx by increasing receptor binding as well as by reducing voltage-dependent block of NMDA receptors.     

A closed-form analytic expression for FRAP formula for the binding diffusion model

One of the most dominant methods cells use for a large class of cellular processes is reaction (or binding) diffusion kinetics, which are controlled by kinetic constants such as diffusion coefficients and on/off binding rate constants. Fluorescence recovery after photobleaching (FRAP) can be used to determine these kinetic constants in living cells. While an analytic expression for FRAP formulae for pure diffusion has been available for some time, an analytic FRAP formula for the binding diffusion model has not been reported yet. Here, we present an analytic FRAP formula for the binding diffusion model in an explicit form allowing for diffusion of the bound complex for either a uniform circle laser profile or a Gaussian laser profile.     

Active Site Formation, Not Bond Kinetics, Limits Adhesion Rate between Human Neutrophils and Immobilized Vascular Cell Adhesion Molecule 1

The formation of receptor ligand bonds at the interface between different cells and between cells and substrates is a widespread phenomenon in biological systems. Physical measurements of bond formation rates between cells and substrates have been exploited to increase our understanding of the biophysical mechanisms that regulate bond formation at interfaces. Heretofore, these measurements have been interpreted in terms of simple bimolecular reaction kinetics. Discrepancies between this simple framework and the behavior of neutrophils adhering to surfaces expressing vascular cell adhesion molecule 1 (VCAM-1) motivated the development of a new kinetic framework in which the explicit formation of active bond formation sites (reaction zones) are a prerequisite for bond formation to occur. Measurements of cells interacting with surfaces having a wide range of VCAM-1 concentrations, and for different durations of contact, enabled the determination of novel kinetic rate constants for the formation of reaction zones and for the intrinsic bond kinetics. Comparison of these rates with rates determined previously for other receptor-ligand pairs points to a predominant role of extrinsic factors such as surface topography and accessibility of active molecules to regions of close contact in determining forward rates of bond formation at cell interfaces.     

Diffusion-limited forward rate constants in two dimensions. Application to the trapping of cell surface receptors by coated pits.

A variety of receptors are known to aggregate in specialized cell surface structures called coated pits, prior to being internalized when the coated pits close off. At 37 degrees C on human fibroblasts, as well as on other cell types, a recycling process maintains a constant number of coated pits on the cell surface. In this paper, we explore implications for receptor aggregation and internalization of the two types of recycling models that have been proposed for the maintenance of the coated pit concentration. In one model, coated pits alternate between accessible and inaccessible states at fixed locations on the cell surface, while in the other model, coated pits recycle to random locations on the cell surface. We consider receptors that are randomly inserted in the membrane, move by pure diffusion with diffusion coefficient D, and are instantly and irreversibly trapped when they reach a coated pit boundary (the diffusion limit). For such receptors, we calculate for each of the two models: the mean time tau to reach a coated pit, the forward rate constant k+ for the interaction of a receptor with a coated pit, and the fraction phi of receptors aggregated in coated pits. We show that for the parameters that characterize coated pits on human fibroblasts, the way in which coated pits return to the surface has a negligible effect on the values of tau, k+, and phi for mobile receptors, D greater than or equal to 1.0 X 10(-11) cm2/s, but has a substantial effect for "immobile" receptors, D much less than 1 X 10(-11) cm2/s. We present numerical examples to show that it may be possible to distinguish between these models if one can monitor slowly diffusing receptors (D less than 1 X 10(-11) cm2/s) on cells whose coated pits have relatively short lifetimes (less than or equal to 1 min). Finally, we show that for the low-density lipoprotein (LDL) receptor on human fibroblasts (D = 4.5 X 10(-11) cm2/s), the predicted and observed values of K+ and phi are in close agreement. Therefore, even for slowly diffusing LDL receptor, unaided diffusion as the transport mechanism of receptors to coated pits is consistent with measured rates of LDL internalization.     

Insulin receptor recycling in vascular endothelial cells. Regulation by insulin and phorbol ester

Endothelial cell insulin receptors mediate the transcytosis of insulin from luminal to abluminal cell surface. We have investigated the kinetics of insulin receptor translocation by immunoprecipitation of radiolabeled receptors at various times before and after trypsin treatment of intact endothelial cells. Insulin receptors were constitutively internalized with t1/2 = 18 +/- 2 min and were recycled to the cell surface. Insulin stimulated receptor internalization and externalization rates 2.6- and 2.4-fold, respectively. Changes in cell-surface binding of 125I-insulin were consistent with the receptor translocation rates observed in surface-labeling experiments. Phorbol myristate acetate (PMA) treatment increased the rate of insulin-stimulated receptor externalization 1.7-fold. PMA treatment increased the constitutive externalization rate 3.5-fold without affecting the constitutive internalization rate, suggesting that recycling might occur via a mobilization of receptors from intracellular sites in a manner independent of internalization rate. Analysis of the intracellular distribution of receptors by 125I-insulin binding and immunogold electron microscopy revealed that less than one-third of the total insulin receptor pool resided on the cell surface. In summary, endothelial cell insulin receptors are constitutively recycled, and internalization and externalization rates are increased by receptor occupancy and PMA treatment.     

Extracting Transition Rates in Particle Tracking Using Analytical Diffusion Distribution Analysis

Single-particle tracking is an important technique in the life sciences to understand the kinetics of biomolecules. The analysis of apparent diffusion coefficients in vivo, for example, enables researchers to determine whether biomolecules are moving alone, as part of a larger complex, or are bound to large cellular components such as the membrane or chromosomal DNA. A remaining challenge has been to retrieve quantitative kinetic models, especially for molecules that rapidly switch between different diffusional states. Here, we present analytical diffusion distribution analysis (anaDDA), a framework that allows for extracting transition rates from distributions of apparent diffusion coefficients calculated from short trajectories that feature less than 10 localizations per track. Under the assumption that the system is Markovian and diffusion is purely Brownian, we show that theoretically predicted distributions accurately match simulated distributions and that anaDDA outperforms existing methods to retrieve kinetics, especially in the fast regime of 0.1–10 transitions per imaging frame. AnaDDA does account for the effects of confinement and tracking window boundaries. Furthermore, we added the option to perform global fitting of data acquired at different frame times to allow complex models with multiple states to be fitted confidently. Previously, we have started to develop anaDDA to investigate the target search of CRISPR-Cas complexes. In this work, we have optimized the algorithms and reanalyzed experimental data of DNA polymerase I diffusing in live Escherichia coli. We found that long-lived DNA interaction by DNA polymerase are more abundant upon DNA damage, suggesting roles in DNA repair. We further revealed and quantified fast DNA probing interactions that last shorter than 10 ms. AnaDDA pushes the boundaries of the timescale of interactions that can be probed with single-particle tracking and is a mathematically rigorous framework that can be further expanded to extract detailed information about the behavior of biomolecules in living cells.     

Characteristics and metabolism of alpha 1 adrenergic receptors in a nonfusing muscle cell line

The BC3H1 nonfusing muscle cell line possesses binding sites for [3H]prazosin. These binding sites are typically alpha 1 adrenergic receptors as shown by their greater affinity (3700-fold) for prazosin than for yohimbine. Both kinetic and equilibrium analyses indicated that [3H]prazosin interacted with only one category of independent binding sites with the following characteristics. KD = 0.13 +/- 0.01 nM. Bmax = 97 +/- 5 fmol/mg of protein corresponding to 25,000 sites/cell (n = 17). Biosynthesis of the alpha 1 adrenergic receptor was investigated at cell confluency (when the number of cells and their total protein content were constant). Phenoxybenzamine (10(-9) M) irreversibly blocked 50% of the alpha 1 receptors in intact cells. More than 95% blockade of receptors was obtained with 10(-7) M phenoxybenzamine. After this blockade, new alpha 1 adrenergic receptors reappeared in the cells with monoexponential kinetics. These new receptors corresponded to synthesized receptors since their appearance was blocked by cycloheximide (1 micrograms/ml). The cycloheximide action was reversible. If one makes the simple and probable hypotheses that the receptor production is constant and that degradation is a monoexponential process, the analysis of the kinetics of reappearance allows the determination of the rate constant for receptor degradation (k = 0.03 h-1) and the rate of receptor production (r = 3.2 fmol/mg/h) corresponding to the synthesis of about 760 receptors/cell/h. The half-life of the receptor was 23 h.