Collective binding properties of receptor arrays

Sum Rate of MISO Neuro-Spike Communication Channel With Constant Spiking Threshold

Communication among neurons, known as neuro-spike communication, is the most promising technique for realization of a bio-inspired nanoscale communication paradigm to achieve biocompatible nanonetworks. In neuro-spike communication, the information, encoded into spike trains, is communicated to various brain regions through neuronal network. An output neuron needs to receive signal from multiple input neurons to generate a spike. Hence, in this paper, we aim to quantify the information transmitted through the multiple-input single-output (MISO) neuro-spike communication channel by considering models for axonal propagation, synaptic transmission, and spike generation. Moreover, the spike generation and propagation in each neuron requires opening and closing of numerous ionic channels on the cell membrane, which consumes considerable amount of ATP molecules called metabolic energy. Thus, we evaluate how applying a constraint on available metabolic energy affects the maximum achievable mutual information of this system. To this aim, we derive a closed form equation for the sum rate of the MISO neuro-spike communication channel and analyze it under the metabolic cost constraints. Finally, we discuss the impacts of changes in number of pre-synaptic neurons on the achievable rate and quantify the tradeoff between maximum achievable sum rate and the consumed metabolic energy.

Stochastic, structural and functional factors influencing AMPA and NMDA synaptic response variability: a review

Synaptic transmission is the basic mechanism of information transfer between neurons not only in the brain, but along all the nervous system. In this review we will briefly summarize some of the main parameters that produce stochastic variability in the synaptic response. This variability produces different effects on important brain phenomena, like learning and memory, and, alterations of its basic factors can cause brain malfunctioning.

Diffusion-Based Model for Synaptic Molecular Communication Channel

Computational methods have been extensively used to understand the underlying dynamics of molecular communication methods employed by nature. One very effective and popular approach is to utilize a Monte Carlo simulation. Although it is very reliable, this method can have a very high computational cost, which in some cases renders the simulation impractical. Therefore, in this paper, for the special case of an excitatory synaptic molecular communication channel, we present a novel mathematical model for the diffusion and binding of neurotransmitters that takes into account the effects of synaptic geometry in 3-D space and re-absorption of neurotransmitters by the transmitting neuron. Based on this model we develop a fast deterministic algorithm, which calculates expected value of the output of this channel, namely, the amplitude of excitatory postsynaptic potential (EPSP), for given synaptic parameters. We validate our algorithm by a Monte Carlo simulation, which shows total agreement between the results of the two methods. Finally, we utilize our model to quantify the effects of variation in synaptic parameters, such as position of release site, receptor density, size of postsynaptic density, diffusion coefficient, uptake probability, and number of neurotransmitters in a vesicle, on maximum number of bound receptors that directly affect the peak amplitude of EPSP.

Random dispersion in excitatory synapse response

The excitatory synaptic function is subject to a huge amount of researches and fairly all the structural elements of the synapse are investigated to determine their specific contribution to the response. A model of an excitatory (hippocampal) synapse, based on time discretized Langevin equations (time-step = 40 fs), was introduced to describe the Brownian motion of Glutamate molecules (GLUTs) within the synaptic cleft and their binding to postsynaptic receptors. The binding has been computed by the introduction of a binding probability related to the hits of GLUTs on receptor binding sites. This model has been utilized in computer simulations aimed to describe the random dispersion of the synaptic response, evaluated from the dispersion of the peak amplitude of the excitatory post-synaptic current. The results of the simulation, presented here, have been used to find a reliable numerical quantity for the unknown value of the binding probability. Moreover, the same results have shown that the coefficient of variation decreases when the number of postsynaptic receptors increases, all the other parameters of the process being unchanged. Due to its possible relationships with the learning and memory, this last finding seems to furnish an important clue for understanding the basic mechanisms of the brain activity.

Glutamate-AMPAR interaction in a model of synaptic transmission

Over the last several years we have investigated the excitatory synaptic response by means of a mathematical model based on a detailed description of the synapse geometry, the Brownian motion of Glutamate molecules and their binding to postsynaptic receptors. Recently, the basic model has been modified for the numbers, the size and the 3D structure of receptors according to new data from the literature. Some results of simulations performed with the updated model are shown here. They were aimed to study the synaptic response in relation to the binding probability, to the probable height of the receptors in the synaptic cleft, and to the space-time distribution of Glutamate/Receptor collisions. A first series of simulations permitted to determine a possible range of values for the binding probability of Glutamate to receptors. Other simulations, investigating the changes induced on the synaptic response by the variations of the height of AMPA receptors in synaptic cleft, allowed to identify the height producing the higher amplitude peak of the mEPSCs. Finally, two new statistical descriptors for analyzing the synaptic response were presented. The first is based on the study of the space distribution of the number of Glutamate/Receptor collisions. Simulations investigating the effects of an increasing eccentricity of the releasing vesicle allowed assessing this method. The second one considers the inter-collision times between Glutamate molecules and binding sites. The results of some of the last simulations demonstrated its capacity to highlight the subtleties and the randomness underlying the activation of the receptors. This article is part of a Special Issue entitled Neural Coding 2012.

Effects of AMPARs trafficking and glutamate-receptors binding probability on stochastic variability of EPSC

Mathematical models of the excitatory synapse are providing valuable information about the synaptic response. The effects of several synaptic components on EPSC variability have been tested by computer simulation. Our model, based on Brownian diffusion of glutamate in the synaptic cleft, is basically the same we have used in previous papers but parameters have been upgraded according to the new experimental findings. The presence of filaments into the synaptic cleft and the number and the ratio of AMPA and NMDA receptors have been the main parameters upgraded. A different way of computing the binding probability of glutamate molecules to receptors by means of geometrical considerations has been also used. The obtained results were more precise and they suggested that the new elements can play a significant role in the stochastic variability of the synaptic response. Nevertheless, new problems arise concerning the value of the lower limit of the binding probability.

Effect of filaments within the synaptic cleft on the response of excitatory synapses simulated by computer experiments

Mathematical models of the excitatory synapse are furnishing valuable information about the synaptic response. Based on Brownian-diffusion of glutamate molecules, a synapse model was utilized to investigate the synaptic response on a femto-second time scale by the use of a parallel computer. In particular, the presence of fibrils crossing the synaptic cleft was simulated, which could have a role in shaping the brain activity. To this aim the model of synapse was modified by considering trans-synaptic filaments with diameters ranging from 7 nm to 3 nm, disposed on a grid with spacing of 14 nm or 8 nm. The simulation demonstrated that the presence of filaments induced an increase in the synaptic response, most likely linked to an increment in the probability of encounter between glutamate molecules and receptors. The increase was small--from 5 to 20%, but metabolic and functional considerations provide substantive hints about the importance of these small changes for brain activity. Moreover, it was shown that the presence of filaments made more stable the response of the synapse to random variations of pre-synaptic elements. Originated by these computational results, some inferences about the biological bases of mind diseases such as autism, mental retardation and schizophrenia, are reported in the Discussion.

Realistic modelling of receptor activation in hippocampal excitatory synapses: analysis of multivesicular release, release location, temperature and synaptic cross-talk

Chemically mediated synaptic transmission results from fusion of synaptic vesicles with the presynaptic plasma membrane, subsequent release of the vesicular content into the cleft and binding to postsynaptic receptors. Previous modelling studies of excitatory neurotransmitter glutamate were based on simplified geometries failing to account for the biologically realistic synaptic environment, in particular, the presence of astrocytes, the geometry of extracellular space, and the neurotransmitter uptake mechanism. Using 3-dimensional reconstructions of hippocampal glutamatergic synapses including the surrounding astrocytic processes we have developed a biologically realistic model to analyse receptor activation in different conditions. We used the finite element method to simulate glutamate release, analyse glutamate diffusion following single and multiple vesicle release and binding at the postsynaptic site to AMPA and NMDA receptors. We demonstrate that: (1) the transmitter diffusion is highly temperature-sensitive; (2) release conditions and geometry more specifically affect AMPARs than NMDARs; (3) the sensitivities of AMPARs and NMDARs to simultaneous vesicular release are different; (4) in the case of multivesicle neurotransmitter release with variable delays, the binding of glutamate to AMPARs is additive up to 1 ms after the release, then becomes independent, but to NMDARs the binding is additive up to 33 ms; (5) the number of AMPARs varies more than the number of NMDRs in response to the input firing patterns; (6) the presence of astrocytes effectively blocks synaptic cross-talk; and (7) synaptic cross-talk, mediated by NMDARs but not AMPARs, is only possible after quasi-simultaneous multivesicular release at physiological temperature (35 degrees C) without intervening astrocytes, but not at 25 degrees C. Our simulations demonstrate the importance of temperature and ultrastructural synaptic environment in synaptic transmission and synaptic cross-talk.

Finite element analysis of drug electrostatic diffusion: inhibition rate studies in N1 neuraminidase

This article describes a numerical solution of the steady-state Poisson-Boltzmann-Smoluchowski (PBS) and Poisson-Nernst-Planck (PNP) equations to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate electrostatic interactions and ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the wild-type and several mutated avian influenza neurominidase crystal structures. The calculated rates show very good agreement with recent experimental studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the electrostatic steering plays the important role in the drug binding process of the neurominidase.

Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm

We present a general-purpose model for biomolecular simulations at the molecular level that incorporates stochasticity, spatial dependence, and volume exclusion, using diffusing and reacting particles with physical dimensions. To validate the model, we first established the formal relationship between the microscopic model parameters (timestep, move length, and reaction probabilities) and the macroscopic coefficients for diffusion and reaction rate. We then compared simulation results with Smoluchowski theory for diffusion-limited irreversible reactions and the best available approximation for diffusion-influenced reversible reactions. To simulate the volumetric effects of a crowded intracellular environment, we created a virtual cytoplasm composed of a heterogeneous population of particles diffusing at rates appropriate to their size. The particle-size distribution was estimated from the relative abundance, mass, and stoichiometries of protein complexes using an experimentally derived proteome catalog from Escherichia coli K12. Simulated diffusion constants exhibited anomalous behavior as a function of time and crowding. Although significant, the volumetric impact of crowding on diffusion cannot fully account for retarded protein mobility in vivo, suggesting that other biophysical factors are at play. The simulated effect of crowding on barnase-barstar dimerization, an experimentally characterized example of a bimolecular association reaction, reveals a biphasic time course, indicating that crowding exerts different effects over different timescales. These observations illustrate that quantitative realism in biosimulation will depend to some extent on mesoscale phenomena that are not currently well understood.

Serial rebinding of ligands to clustered receptors as exemplified by bacterial chemotaxis

Serial ligation is the repeated reversible binding of a ligand to one receptor after another. It is a widespread phenomenon throughout biochemical systems, occurring anytime receptors are clustered together and ligand binding is reversible. Computer simulations are used in this work to investigate a representative example, which is the serial ligation of an extracellular aspartate molecule to the membrane-bound chemotaxis receptors of an Escherichia coli bacterium. It is found that the initial binding site of a ligand to a cluster of receptors is more likely to be near the edge of the cluster than near the middle, although there is no overall bias when all rebindings are considered. Serial ligation does not lead directly to signal amplification or attenuation but instead causes binding events to be correlated in both space and time: a ligand is likely to bind many times in rapid succession in a small region of the receptor cluster, but there can also be long intervals between bindings. This leads to an increased level of noise in the received signal but may allow a single ligand to be sensed above a uniform level of background noise. The focus of this paper is on the interpretation of simulation results so they can be generalized to a wide variety of other systems and to allow the identification of systems in which serial ligation is likely to be important. In the process, several characteristic times are identified, as are scaling laws for the spatial and temporal dynamics.

Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations

This article describes the numerical solution of the time-dependent Smoluchowski equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.

Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis

As described previously, continuum models, such as the Smoluchowski equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski equation based on a posteriori error estimates. We also describe new, molecular-surface-based models, for diffusional reaction boundary criteria and compare results obtained from these models with the traditional spherical criteria. The new methods are validated by comparison of the calculated reaction rates with experimental values for wild-type and mutant forms of mouse acetylcholinesterase. The results show good agreement with experiment and help to define optimal reactive boundary conditions.

Saturation in excitatory synapses of hippocampus investigated by computer simulations

The standard view of the synaptic function in excitatory synapses has been deeply questioned by recent experimental data on hippocampal glutamate synapses both for possible receptor nonsaturation and for larger and non-Gaussian peak amplitude fluctuations. Our previous investigations of the mechanisms involved in the variability of the response of hippocampal glutamatergic synapses, carried out by computer simulation of simple Brownian models of glutamate diffusion, furnished initial evidence about their presynaptic character. A new, refined model, reported here, assumes a collision volume for the glutamate molecule and a more realistic description of receptors and their binding dynamics. Based on this model, conditions for AMPA and NMDA receptor saturation have been investigated and new miniature (or quantal) EPSC parameters have been computed. The results corroborate the hypothesis that the lack of AMPA and NMDA receptor saturation and the EPSC stochastic variability are attributable to the small volume of glutamatergic synaptic vesicles and hence to the small number of glutamate molecules diffusing in the cleft after a vesicle release. The investigations better characterize some not well-known elements of the synaptic structure, such as the fusion pore, and provide useful information on AMPA receptor dynamics. Indeed, a nice fit between computed EPSCs and some miniature EPSCs in recent experimental literature allowed for the computation of new transition time values among the different AMPA receptor states through a trial-and-error optimization procedure. Moreover, the model has been used to evaluate two hypotheses on the genesis of the long-term potentiation phenomenon.

Spatial range of autocrine signaling: modeling and computational analysis

Autocrine loops formed by growth factors and their receptors have been identified in a large number of developmental, physiological, and pathological contexts. In general, the spatially distributed and recursive nature of autocrine signaling systems makes their experimental analysis, and often even their detection, very difficult. Here, we combine Brownian motion theory, Monte Carlo simulations, and reaction-diffusion models to analyze the spatial operation of autocrine loops. Within this modeling framework, the ability of autocrine cells to recapture the endogenous ligand and the distances traveled by autocrine ligands are explicitly related to ligand diffusion coefficients, density of surface receptors, ligand secretion rate, and rate constants of ligand binding and endocytic internalization. Applying our models to study autocrine loops in the epidermal growth factor receptor system, we find that autocrine loops can be highly localized--even at the level of a single cell. We demonstrate how the variations in molecular and cellular parameters may "tune" the spatial range of autocrine signals over several orders of magnitude: from microns to millimeters. We argue that this versatile regulation of the spatial range of autocrine signaling enables autocrine cells to perceive a broad spectrum of environmental information.

Experimental evidence for a kinetic transition in reversible reactions

We provide a first experimental verification of a theoretical prediction of a kinetic transition in a reversible binding reaction, AB right harpoon over left harpoon A+B, driven by the difference in effective lifetimes of the bound and the unbound states. We consider the kinetics of excited-state proton transfer to solvent from a photoacid whose conjugate anionic base possesses an extremely short unbound anion lifetime. Its solvent variation relative to the overall dissociation rate coefficient induces a transition in the kinetics, from power law to exponential.

Variability of excitatory currents due to single-channel noise, receptor number and morphological heterogeneity

Patch clamp recordings of excitatory postsynaptic currents (EPSCs) in central neurons reveal large fluctuations in amplitudes and decay times of AMPA-receptor-mediated EPSCs. By using Monte Carlo simulations of synaptic transmission in brainstem interneurons, we tested several hypothesis that could account for the observed variability. The coefficient of variation (CV) of 0.5 for miniature amplitudes cannot be explained by fluctuations in vesicle content or receptor distribution, but is traced to variations in receptor number, which is estimated as 77+/-39 receptors per bouton. As the variability of rise times reflects fluctuations in size of the post-synaptic density and heterogeneity of the receptor distribution, the relatively small CV=0.37 of experimentally determined values points to a homogeneous arrangement of receptors. Within our model the large variability of decay times (CV=0.49) can only be explained by fluctuations in the transmitter time course (mean residence times of 0.4+/-0.13 ms), presumably resulting from heterogeneities in synaptic morphology. Hence, our simulations indicate that different noise sources control the variability of amplitudes, rise and decay times. In particular, the distribution of decay times yields information about the synaptic transmission process, which cannot be obtained from other observables.

A Brownian model of glutamate diffusion in excitatory synapses of hippocampus

We simulated the diffusion of glutamate, following the release of a single vesicle from a pre-synaptic terminal, in the synaptic cleft by using a Brownian diffusion model based on Langevin equations. The synaptic concentration time course and the time course of quantal excitatory post-synaptic current have been analyzed. The results showed that they depend on the number of receptors located at post-synaptic membrane. Their time course are dependent both on the total number of the post-synaptic receptors and on the eccentricity of the pre-synaptic glutamate vesicle.

Rebinding of IgE Fabs at haptenated planar membranes: measurement by total internal reflection with fluorescence photobleaching recovery

In previous work, a general analytical theory for ligand rebinding at cell surfaces was developed for a reversible bimolecular reaction between ligands in solution and receptors on a membrane surface [Lagerholm, B. C., and Thompson, N. L. (1998) Biophys. J. 74, 1215-1228]. This theory can be used to predict theoretical forms for data obtained by using total internal reflection with fluorescence photobleaching recovery (TIR-FPR) [Thompson, N. L., Burghardt, T. P., and Axelrod, D. (1981) Biophys. J. 33, 435-454]. Thus, one method by which the rebinding theory can be tested is to use TIR-FPR. In the work described herein, the reversible kinetics of mouse monoclonal anti-dinitrophenyl (DNP) IgE Fabs at substrate-supported planar membranes composed of 25 mol % DNP-conjugated phosphatidylethanolamine and 75 mol % dipalmitoylphosphatidylcholine have been examined by using TIR-FPR. Data were obtained as a function of the Fab solution concentration. Higher Fab concentrations reduce rebinding (and increase the fluorescence recovery rate) because different Fab molecules compete for the same surface-binding sites. Data were also obtained for solutions containing different volume fractions of glycerol. In these measurements, higher glycerol concentrations increase rebinding (and decrease the fluorescence recovery rate) because the solution viscosity is increased and the Fab diffusion coefficient in solution is decreased. The TIR-FPR data were quantitatively compared with theoretical predictions which follow from the general theory for rebinding at the membrane surface. The data were consistent with the theoretical predictions and, therefore, provide experimental verification of the previously developed theory.

Analysis of synaptic transmission in the neuromuscular junction using a continuum finite element model

There is a steadily growing body of experimental data describing the diffusion of acetylcholine in the neuromuscular junction and the subsequent miniature endplate currents produced at the postsynaptic membrane. To gain further insights into the structural features governing synaptic transmission, we have performed calculations using a simplified finite element model of the neuromuscular junction. The diffusing acetylcholine molecules are modeled as a continuum, whose spatial and temporal distribution is governed by the force-free diffusion equation. The finite element method was adopted because of its flexibility in modeling irregular geometries and complex boundary conditions. The resulting simulations are shown to be in accord with experiment and other simulations.