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

Overview of the role of purinergic signaling and insights into its role in cancer therapy

Innovation of cancer therapy has received a dramatic acceleration over the last fifteen years thanks to the introduction of the novel immune checkpoint inhibitors (ICI). On the other hand, the conspicuous scientific knowledge accumulated in purinergic signaling since the early seventies is finally being transferred to the clinic. Several Phase I/II clinical trials are currently underway to investigate the effect of drugs interfering with purinergic signaling as stand-alone or combination therapy in cancer. This is supporting the novel concept of "purinergic immune checkpoint" (PIC) in cancer therapy. In the present review we will address a) the basic pharmacology and cell biology of the purinergic system; b) principles of its pathophysiology in human diseases; c) implications for cell death, cell proliferation and cancer; d) novel molecular tools to investigate nucleotide homeostasis in the extracellular environment; e) recent developments in the pharmacology of P1, P2 receptors and related ecto-enzymes; f) P1 and P2 ligands as novel diagnostic tools; g) current issues in PIC-based anti-cancer therapy. This review will provide an appraisal of the current status of purinergic signaling in cancer and will help identify future avenues of development.

Acetylcholine waves and dopamine release in the striatum

Striatal dopamine encodes reward, with recent work showing that dopamine release occurs in spatiotemporal waves. However, the mechanism of dopamine waves is unknown. Here we report that acetylcholine release in mouse striatum also exhibits wave activity, and that the spatial scale of striatal dopamine release is extended by nicotinic acetylcholine receptors. Based on these findings, and on our demonstration that single cholinergic interneurons can induce dopamine release, we hypothesized that the local reciprocal interaction between cholinergic interneurons and dopamine axons suffices to drive endogenous traveling waves. We show that the morphological and physiological properties of cholinergic interneuron - dopamine axon interactions can be modeled as a reaction-diffusion system that gives rise to traveling waves. Analytically-tractable versions of the model show that the structure and the nature of propagation of acetylcholine and dopamine traveling waves depend on their coupling, and that traveling waves can give rise to empirically observed correlations between these signals. Thus, our study provides evidence for striatal acetylcholine waves in vivo, and proposes a testable theoretical framework that predicts that the observed dopamine and acetylcholine waves are strongly coupled phenomena.

Dequalinium chloride is an antagonists of α7 nicotinic acetylcholine receptors

Dequalinium chloride has been used primarily as antiseptic compounds, but recently has been investigated for its effects on specific targets, including muscarinic acetylcholine receptors. Here we investigated dequalinium chloride as an antagonist to α7 nicotinic acetylcholine receptors. The pharmacological properties of dequalinium were established using cell lines stably co-transfected with the calcium-permeable human α7 nicotinic acetylcholine receptors and its chaperone NACHO, calcium dye fluorescent measurements or a calcium-sensitive protein reporter, and patch clamp recording of ionic currents. Using calcium dye fluorescence plate reader measurements, we find dequalinium chloride is an antagonist of α7 nicotinic acetylcholine receptors with an IC₅₀ of 672 nM in response to activation with 500 μM acetylcholine chloride and positive allosteric modulator PNU-120596. However, using a membrane-tethered GCAMP7s calcium reporter allowed detection of α7-mediated calcium flux in the absence of PNU-120596. Using this approach revealed an IC₅₀ of 157 nM for dequalinium on 300 μM acetylcholine-evoked currents. Using patch clamp recordings with 300 μM acetylcholine chloride and 10 μM PNU-120596, we find lower concentrations are sufficient to block ionic currents, with IC₅₀ of 120 nM for dequalinium chloride and 54 nM for the related UCL 1684 compound. In summary, we find that dequalinium chloride and UCL1684, which are generally used to block SK-type potassium channels, are also highly effective antagonists of α7 nicotinic acetylcholine receptors. This finding, in combination with previous studies of muscarinic acetylcholine receptors, clearly establishes dequalinium compounds within the class of general anti-cholinergic antagonists.

Chemically Fueled Self-Assembly in Biology and Chemistry

Life is a non-equilibrium state of matter maintained at the expense of energy. Nature uses predominantly chemical energy stored in thermodynamically activated, but kinetically stable, molecules. These high-energy molecules are exploited for the synthesis of other biomolecules, for the activation of biological machinery such as pumps and motors, and for the maintenance of structural order. Knowledge of how chemical energy is transferred to biochemical processes is essential for the development of artificial systems with life-like processes. Here, we discuss how chemical energy can be used to control the structural organization of organic molecules. Four different strategies have been identified according to a distinguishable physical-organic basis. For each class, one example from biology and one from chemistry are discussed in detail to illustrate the practical implementation of each concept and the distinct opportunities they offer. Specific attention is paid to the discussion of chemically fueled non-equilibrium self-assembly. We discuss the meaning of non-equilibrium self-assembly, its kinetic origin, and strategies to develop synthetic non-equilibrium systems.

Rapsyn facilitates recovery from desensitization in fetal and adult acetylcholine receptors expressed in a muscle cell line

Key points

The physiological significance of the developmental switch from fetal to adult acetylcholine receptors in muscle (AChRs) and the functional impact of AChR clustering by rapsyn are not well studied.

Using patch clamp experiments, we show that recovery from desensitization is faster in the adult AChR isoform.

Recovery from desensitization is determined by the AChR isoform‐specific cytoplasmic M3–M4 domain.

The co‐expression of rapsyn in muscle cells induced AChR clustering and facilitated recovery from desensitization in both fetal and adult AChRs. In fetal AChRs, facilitation of recovery kinetics by rapsyn was independent of AChR clustering.

These effects could be crucial adaptations to motor neuron firing rates, which, in rodents, have been shown to increase around the time of birth when AChRs cluster at the developing neuromuscular junctions.

Abstract

The neuromuscular junction (NMJ) is the site of a number of autoimmune and genetic disorders, many involving the muscle‐type nicotinic acetylcholine receptor (AChR), although there are aspects of normal NMJ development and function that need to be better understood. In particular, there are still questions regarding the implications of the developmental switch from fetal to adult AChRs, as well as how their functions might be modified by rapsyn that clusters the AChRs. Desensitization of human muscle AChRs was investigated using the patch clamp technique to measure whole‐cell currents in muscle‐type (TE671/CN21) and non‐muscle (HEK293) cell lines expressing either fetal or adult AChRs. Desensitization time constants were similar with both AChR isoforms but recovery time constants were shorter in cells expressing adult compared to fetal AChRs (P < 0.0001). Chimeric experiments showed that recovery from desensitization was determined by the M3–M4 cytoplasmic loops of the γ‐ and ε‐subunits. Expression of rapsyn in TE671/CN21 cells induced AChR aggregation and also, surprisingly, shortened recovery time constants in both fetal and adult AChRs. However, this was not dependent on clustering because rapsyn also facilitated recovery from desensitization in HEK293 cells expressing a δ‐R375H AChR mutant that did not form clusters in C2C12 myotubes. Thus, rapsyn interactions with AChRs lead not only to clustering, but also to a clustering independent faster recovery from desensitization. Both effects of rapsyn could be a necessary adjustment to the motor neuron firing rates that increase around the time of birth.

The physiological significance of the developmental switch from fetal to adult acetylcholine receptors in muscle (AChRs) and the functional impact of AChR clustering by rapsyn are not well studied.

Using patch clamp experiments, we show that recovery from desensitization is faster in the adult AChR isoform.

Recovery from desensitization is determined by the AChR isoform‐specific cytoplasmic M3–M4 domain.

The co‐expression of rapsyn in muscle cells induced AChR clustering and facilitated recovery from desensitization in both fetal and adult AChRs. In fetal AChRs, facilitation of recovery kinetics by rapsyn was independent of AChR clustering.

These effects could be crucial adaptations to motor neuron firing rates, which, in rodents, have been shown to increase around the time of birth when AChRs cluster at the developing neuromuscular junctions.

Engineering molecular communications integrated with carbon nanotubes in neural sensor nanonetworks

There have been recent advances in the engineering of molecular communication (MC)‐based networks for nanomedical applications. However, the integration of MC with biomaterials such as carbon nanotubes (CNTs) presents various critical research challenges. In this study, the authors envisaged integrating MC‐based nanonetwork with CNTs to optimise nanonetwork performance. In neural networks, a chronic reduction in the concentration of the neurotransmitter acetylcholine (ACh) eventually leads to the development of neurodegenerative diseases; therefore, they used CNTs as a molecular switch to optimise ACh conductivity supported by artificial MC. Furthermore, MC enables communication between transmitter neurons and receiver neurons for fine‐tuning the ACh release rate according to the feedback concentration of ACh. Subsequently, they proposed a min/max feedback scheme to fine‐tune the expected throughput and ACh transmission efficiency. For demonstration purposes, they deduced analytical forms for the proposed schemes in terms of throughput, incurred traffic rates, and average packet delay.

Hybrid finite element and Brownian dynamics method for charged particles

Diffusion is often the rate-determining step in many biological processes. Currently, the two main computational methods for studying diffusion are stochastic methods, such as Brownian dynamics, and continuum methods, such as the finite element method. A previous study introduced a new hybrid diffusion method that couples the strengths of each of these two methods, but was limited by the lack of interactions among the particles; the force on each particle had to be from an external field. This study further develops the method to allow charged particles. The method is derived for a general multidimensional system and is presented using a basic test case for a one-dimensional linear system with one charged species and a radially symmetric system with three charged species.

Modeling and simulation of organophosphate-induced neurotoxicity: Prediction and validation by experimental studies

Exposure to organophosphorus (OP) compounds, either pesticides or chemical warfare agents, represents a major health problem. As potent irreversible inhibitors of cholinesterase, OP may induce seizures, as in status epilepticus, and occasionally brain lesions. Although these compounds are extremely toxic agents, the search for novel antidotes remains extremely limited. In silico modeling constitutes a useful tool to identify pharmacological targets and to develop efficient therapeutic strategies. In the present work, we developed a new in silico simulator in order to predict the neurotoxicity of irreversible inhibitors of acetyl- and/or butyrylcholinesterase (ChE) as well as the potential neuroprotection provided by antagonists of cholinergic muscarinic and glutamate N-methyl-d-aspartate (NMDA) receptors. The simulator reproduced firing of CA1 hippocampal neurons triggered by exposure to paraoxon (POX), as found in patch-clamp recordings in in vitro mouse hippocampal slices. In the case of POX intoxication, it predicted a preventing action of the muscarinic receptor antagonist atropine sulfate, as well as a synergistic action with the non-competitive NMDA receptor antagonist memantine. These in silico predictions relative to beneficial effects of atropine sulfate combined with memantine were recapitulated experimentally in an in vivo model of POX in adult male Swiss mice using electroencephalic (EEG) recordings. Thus, our simulator is a new powerful tool to identify protective therapeutic strategies against OP central effects, by screening various combinations of muscarinic and NMDA receptor antagonists.

Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics

Background

The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors.

Methods

We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) BC, is considered on the reactive boundaries. This new BC treatment allows for the analysis of enzymes with “imperfect” reaction rates.

Results

The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to a mouse acetylcholinesterase (mAChE) monomer. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates.

Conclusions

Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.

Synaptic bouton properties are tuned to best fit the prevailing firing pattern

The morphology of presynaptic specializations can vary greatly ranging from classical single-release-site boutons in the central nervous system to boutons of various sizes harboring multiple vesicle release sites. Multi-release-site boutons can be found in several neural contexts, for example at the neuromuscular junction (NMJ) of body wall muscles of Drosophila larvae. These NMJs are built by two motor neurons forming two types of glutamatergic multi-release-site boutons with two typical diameters. However, it is unknown why these distinct nerve terminal configurations are used on the same postsynaptic muscle fiber. To systematically dissect the biophysical properties of these boutons we developed a full three-dimensional model of such boutons, their release sites and transmitter-harboring vesicles and analyzed the local vesicle dynamics of various configurations during stimulation. Here we show that the rate of transmission of a bouton is primarily limited by diffusion-based vesicle movements and that the probability of vesicle release and the size of a bouton affect bouton-performance in distinct temporal domains allowing for an optimal transmission of the neural signals at different time scales. A comparison of our in silico simulations with in vivo recordings of the natural motor pattern of both neurons revealed that the bouton properties resemble a well-tuned cooperation of the parameters release probability and bouton size, enabling a reliable transmission of the prevailing firing-pattern at diffusion-limited boutons. Our findings indicate that the prevailing firing-pattern of a neuron may determine the physiological and morphological parameters required for its synaptic terminals.

Selection of a human butyrylcholinesterase-like antibody single-chain variable fragment resistant to AChE inhibitors from a phage library expressed in E. coli

Organophosphates are potent poisoning agents that cause severe cholinergic toxicity. Current treatment has been reported to be unsatisfactory and novel antidotes are needed. In this study, we used a single-chain variable fragment (scFv) library to select a recombinant antibody fragment (WZ1-14.2.1) with butyrylcholinesterase-like catalytic activity by using an innovative method integrating genetic selection and the bait-and-switch strategy. Ellman assay demonstrated that WZ1-14.2.1 has Michaelis-Menten kinetics in the hydrolysis of all the three substrates used, acetylthiocholine, propionylthiocholine and butyrylthiocholine. Notably, the catalytic activity was resistant to the following acetylcholinesterase inhibitors: neostigmine, iso-OMPA, chlorpyrifos oxon, dichlorvos, and paraoxon ethyl. Otherwise, the enzymatic activity of WZ1-14.2.1 was inhibited by the selective butyrylcholinesterase inhibitor, ethopropazine, and by the Ser-blocking agent phenylmethanesuphonyl fluoride. A hypothetical 3D structure of the WZ1-14.2.1 catalytic site, compatible with functional results, is proposed on the basis of a molecular modeling analysis.

Hybrid finite element and Brownian dynamics method for diffusion-controlled reactions

Diffusion is often the rate determining step in many biological processes. Currently, the two main computational methods for studying diffusion are stochastic methods, such as Brownian dynamics, and continuum methods, such as the finite element method. This paper proposes a new hybrid diffusion method that couples the strengths of each of these two methods. The method is derived for a general multidimensional system, and is presented using a basic test case for 1D linear and radially symmetric diffusion systems.

Multi-Scale Continuum Modeling of Biological Processes: From Molecular Electro-Diffusion to Sub-Cellular Signaling Transduction

This article provides a brief review of multi-scale modeling at the molecular to cellular scale, with new results for heart muscle cells. A finite element-based simulation package (SMOL) was used to investigate the signaling transduction at molecular and sub-cellular scales (http://mccammon.ucsd.edu/smol/, http://FETK.org) by numerical solution of time-dependent Smoluchowski equations and a reaction-diffusion system. At the molecular scale, SMOL has yielded experimentally-validated estimates of the diffusion-limited association rates for the binding of acetylcholine to mouse acetylcholinesterase using crystallographic structural data. The predicted rate constants exhibit increasingly delayed steady-state times with increasing ionic strength and demonstrate the role of an enzyme's electrostatic potential in influencing ligand binding. At the sub-cellular scale, an extension of SMOL solves a non-linear, reaction-diffusion system describing Ca²⁺ ligand buffering and diffusion in experimentally-derived rodent ventricular myocyte geometries. Results reveal the important role for mobile and stationary Ca²⁺ buffers, including Ca²⁺ indicator dye. We found that the alterations in Ca²⁺-binding and dissociation rates of troponin C (TnC) and total TnC concentration modulate subcellular Ca²⁺ signals. Model predicts that reduced off-rate in whole troponin complex (TnC, TnI, TnT) versus reconstructed thin filaments (Tn, Tm, actin) alters cytosolic Ca²⁺ dynamics under control conditions or in disease-linked TnC mutations. The ultimate goal of these studies is to develop scalable methods and theories for integration of molecular-scale information into simulations of cellular-scale systems.

Hydrodynamic flow in a synaptic cleft during exocytosis

It is shown that exocytosis in a chemical synapse may be accompanied by "microjet" formation due to the overpressure that exists in the vesicles. This mechanism may take place either at complete fusion of a vesicle with the presynaptic membrane or in the so-called kiss-and-run mode of neurotransmitter release. A simple hydrodynamic model of the viscous incompressible flow arising in the synaptic cleft is suggested. The occurrence of hydrodynamic flow (microjet) leads to more efficient transport of neurotransmitter than in the case of classical diffusive transport.

Formulating a new basis for the treatment against botulinum neurotoxin intoxication: 3,4-Diaminopyridine prodrug design and characterization

Botulism is a disease characterized by neuromuscular paralysis and is produced from botulinum neurotoxins (BoNTs) found within the Gram positive bacterium Clostridium botulinum. This bacteria produces the most deadliest toxin known, with lethal doses as low as 1 ng/kg. Due to the relative ease of production and transport, the use of these agents as potential bioterrorist weapons has become of utmost concern. No small molecule therapies against BoNT intoxication have been approved to date. However, 3,4-diaminopyridine (3,4-DAP), a potent reversible inhibitor of voltage-gated potassium channels, is an effective cholinergic agonist used in the treatment of neuromuscular degenerative disorders that require cholinergic enhancement. 3,4-DAP has also been shown to facilitate recovery of neuromuscular action potential post botulinum intoxication by blocking K(+) channels. Unfortunately, 3,4-DAP displays toxicity largely due to blood-brain-barrier (BBB) penetration. As a dual-action prodrug approach to cholinergic enhancement we have designed carbamate and amide conjugates of 3,4-DAP. The carbamate prodrug is intended to be a slowly reversible inhibitor of acetylcholinesterase (AChE) along the lines of the stigmines thereby allowing increased persistence of released acetylcholine within the synaptic cleft. As a secondary activity, cleavage of the carbamate prodrug by AChE will afford the localized release of 3,4-DAP, which in turn, will enhance the pre-synaptic release of additional acetylcholine. Being a competitive inhibitor with respect to acetylcholine, the activity of the prodrug will be greatest at the synaptic junctions most depleted of acetylcholine. Here we report upon the synthesis and biochemical characterization of three new classes of prodrugs intended to limit previously reported stability and toxicity issues. Of the prodrugs examined, compound 32, demonstrated the most clinically relevant half-life of 2.76 h, while selectively inhibiting AChE over butyrylcholinesterase--a plasma-based high activity esterase. Future in vivo studies could provide validation of prodrug 32 as a potential treatment against BoNT intoxication as well as other neuromuscular disorders.

Synergistic inhibition of butyrylcholinesterase by galantamine and citalopram

BACKGROUND

Many persons with Alzheimer's disease (AD) treated with galantamine appear to receive additional cognitive benefit from citalopram. Both drugs inhibit acetylcholinesterase (AChE) and butyrylcholinesterase (BuChE). These enzymes co-regulate acetylcholine catabolism. In AD brain, AChE is diminished while BuChE is not, suggesting BuChE inhibition may be important in raising acetylcholine levels. BuChE is subject to activation at high acetylcholine levels reached at the synaptic cleft. The present study explores one way combining galantamine and citalopram could be beneficial in AD.

METHODS

Spectrophotometric studies of BuChE catalysis in the absence or presence of galantamine or citalopram or both, were performed using the Ellman method. Data analysis involved expansion of our previous equation describing BuChE catalysis.

RESULTS

Galantamine almost completely inhibited BuChE at low substrate concentrations (V(S)=43.6 μM/min; V(S(gal))=0.34 μM/min) without influencing the substrate-activated form of the enzyme (V(SS)=64.0 μM/min;V(SS(gal))=62.3 μM/min). Conversely, citalopram inhibited both un-activated (V(S)=43.6 μM/min; V(S(cit))=10.2 μM/min) and substrate-activated (V(SS)=64.0 μM/min; V(SS(cit))=47.3 μM/min) forms of BuChE. Combined galantamine and citalopram increased inhibition of un-activated BuChE (V(S)=43.6 μM/min; V(S(gal)(cit))=2.73 μM/min) and substrate-activated form (V(SS)=64.0 μM/min; V(SS(gal)(cit))=42.2 μM/min).

CONCLUSION

Citalopram and galantamine produce a combined inhibition of BuChE that is considered to be synergistic.

GENERAL SIGNIFICANCE

Clinical benefit from combined galantamine and citalopram may be related to a synergistic inhibition of BuChE, facilitating cholinergic neurotransmission. This emphasizes the importance of further study into use of drug combinations in AD treatment.

A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

BACKGROUND

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.

RESULT

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

CONCLUSION

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

Diffusion-convection effects on drug distribution at the cell membrane level in a patch-clamp setup

We present a model-based method for estimating the effective concentration of the active drug applied by a pressure pulse to an individual cell in a patch-clamp setup, which could be of practical use in the analysis of ligand-induced whole-cell currents recorded in patch-clamp experiments. Our modelling results outline several important factors which may be involved in the high variability of the electric response of the cells, and indicate that with a pressure pulse duration of 1s and diameter of the perfusion tip of 600 μm, elevated amounts of drug can accumulate locally between the pipette tip and the cell. Hence, the effective agonist concentration at the cell membrane level can be consistently higher than the initial concentration inside the perfusion tubes. We performed finite-difference and finite-element simulations to investigate the diffusion/convection effects on the agonist distribution on the cell membrane. Our model can explain the delay between the commencement of acetylcholine application and the onset of the whole-cell current that we recorded on human rhabdomyosarcoma TE671 cells, and reproduce quantitatively the decrease of signal latency with the concentration of agonist in the pipette. Results also show that not only the geometry of the bath chamber and pipette tip, but also the transport parameters of the diffusive and convective phenomena in the bath solution are determinant for the amplitude and kinetics of the recorded currents and have to be accounted for when analyzing patch-clamp data.

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.

Tortuosity and anomalous diffusion in the neuromuscular junction

The signal transfer from nerve to muscle occurs by diffusion across the neuromuscular junction. The continuum level analysis of diffusion processes is based on the diffusion equation, which in one dimension is partial differential c/partial differential t=D(partial differential(2)c/partial differential x(2)) , where c is the molecular concentration and D is the diffusivity. However, in confined systems such as the neuromuscular junction, the diffusion equation may not be valid, and even if valid the value of D may be altered by the confinement. In this paper, Monte Carlo simulations are used to probe diffusion at the molecular level in a realistic model of a neuromuscular junction. The results show that diffusion is anomalous (i.e., not described by the diffusion equation) for time scales less than approximately 0.01 s, which is the time scale relevant for signaling processes in the synapse. At longer time scales, the diffusion is normal (i.e., described by the diffusion equation), but with a value of D that is reduced by a factor of approximately 5 times compared to the value for diffusion in open space. As the width of the synaptic cleft decreases, these effects become even more pronounced. The physical basis of these results is described in terms of the structure of the neuromuscular junction.

Continuum simulations of acetylcholine consumption by acetylcholinesterase: a Poisson-Nernst-Planck approach

The Poisson-Nernst-Planck (PNP) equation provides a continuum description of electrostatic-driven diffusion and is used here to model the diffusion and reaction of acetylcholine (ACh) with acetylcholinesterase (AChE) enzymes. This study focuses on the effects of ion and substrate concentrations on the reaction rate and rate coefficient. To this end, the PNP equations are numerically solved with a hybrid finite element and boundary element method at a wide range of ion and substrate concentrations, and the results are compared with the partially coupled Smoluchowski-Poisson-Boltzmann model. The reaction rate is found to depend strongly on the concentrations of both the substrate and ions; this is explained by the competition between the intersubstrate repulsion and the ionic screening effects. The reaction rate coefficient is independent of the substrate concentration only at very high ion concentrations, whereas at low ion concentrations the behavior of the rate depends strongly on the substrate concentration. Moreover, at physiological ion concentrations, variations in substrate concentration significantly affect the transient behavior of the reaction. Our results offer a reliable estimate of reaction rates at various conditions and imply that the concentrations of charged substrates must be coupled with the electrostatic computation to provide a more realistic description of neurotransmission and other electrodiffusion and reaction processes.

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

Array feature size influences nucleic acid surface capture in DNA microarrays

Analyte affinity capture by surface-immobilized diagnostic agents is a routinely used assay format for profiling numerous medically and technologically important target analytes. These assays suffer from numerous performance limitations, including sensitivity and rapidity. Assay miniaturization is advocated to improve surface-capture performance, specifically exploiting the inverse relationship between analyte flux and capture feature size under mass transfer-limiting capture conditions that characterize many such assay formats. Reduced capture feature sizes, e.g., microarrays, are proposed to overcome mass transfer limitations, yet this is difficult to achieve across several size scales. This study validates certain advantages advocated for capture spot miniaturization using a rationale to understand surface capture miniaturization strategies. Experimentally derived immobilized ligand and target capture densities as a function of microspot size for DNA oligomers immobilized on model gold substrates are compared directly with theoretical analysis, validating the hypothesis that miniaturization yields many practical assay advantages. Specifically, results show that transitions from assay mass transfer limiting to kinetically limiting conditions as feature size decreases identify an optimal microspot size range for a specific bioassay system. Analytical advantages realized from such assay miniaturization are more uniform target-spot coverage and substantially increased rate of capture (hybridization), increasing assay signal and rapidity.

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.

Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models

The reaction-diffusion system of the neuromuscular junction has been modeled in 3D using the finite element package FEtk. The numerical solution of the dynamics of acetylcholine with the detailed reaction processes of acetylcholinesterases and nicotinic acetylcholine receptors has been discussed with the reaction-determined boundary conditions. The simulation results describe the detailed acetylcholine hydrolysis process, and reveal the time-dependent interconversion of the closed and open states of the acetylcholine receptors as well as the percentages of unliganded/monoliganded/diliganded states during the neuro-transmission. The finite element method has demonstrated its flexibility and robustness in modeling large biological systems.

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.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.

A diffusion–reaction model for DNA microarray assays

DNA microarrays are extensively used for the quantification of the degree of differential mRNA expression. The assay involves hybridization of mobile DNA strands with immobilized complementary DNA strands to form duplexes. The overall duplex formation rate depends on the rate of transport of strands in solution to the corresponding spot on the surface, and the rate of the hybridization reaction. We present a theoretical model that incorporates both kinetics of the reversible hybridization reaction and diffusional transport of the labeled strands, and analyze DNA microarray hybridization using this model. Simulations are carried out in a geometrically realistic domain for labeled DNA concentrations corresponding to rare and abundant transcripts for typical assay conditions. The rate of strand diffusion in solution is shown to strongly affect the overall hybridization rate. We compute the minimum inter-spot spacing for replicate spots to enhance sensitivity. We also determine the hybridization time for which reliable estimates of the relative mRNA abundance of two species can be obtained using total fluorescence intensities. An analytical solution for the concentration distribution of mobile strands at intermediate hybridization times provides a convenient tool to calculate the mobile strand concentration profiles. This model provides a framework for the process analysis of all microarray assays currently used for genomic transcriptional analysis.

Glutamate escape from a tortuous synaptic cleft of the hippocampal mossy fibre synapse

The time course of neurotransmitter in the synaptic cleft contributes substantially to the fast kinetics of synaptic signalling. Hippocampal mossy fibres (MFs), a well-characterised excitatory pathway from dentate granule cells to the hippocampus proper, form large glutamatergic synapses at branched spiny structures in CA3 pyramidal cell dendrites. To what extent transmission at these synapses is affected by retarded glutamate clearance from the large tortuous synaptic cleft is not known. Here, we propose a simple geometrical approximation representing the 'typical' geometry of thorny excrescences that form the tortuous cleft interface at a MF synapse. We then employ Monte Carlo simulations to monitor movements of 3000 individual glutamate molecules released within the cleft. The results predict that, in the absence of neuronal glutamate transporters, it should take approximately 10 ms for 50% and 60-70 ms for 90% of glutamate molecules to escape the MF synapse.

Finite element simulations of acetylcholine diffusion in neuromuscular junctions

A robust infrastructure for solving time-dependent diffusion using the finite element package FEtk has been developed to simulate synaptic transmission in a neuromuscular junction with realistic postsynaptic folds. Simplified rectilinear synapse models serve as benchmarks in initial numerical studies of how variations in geometry and kinetics relate to endplate currents associated with fast-twitch, slow-twitch, and dystrophic muscles. The flexibility and scalability of FEtk affords increasingly realistic and complex models that can be formed in concert with expanding experimental understanding from electron microscopy. Ultimately, such models may provide useful insight on the functional implications of controlled changes in processes, suggesting therapies for neuromuscular diseases.

Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules

Rapid activation of synaptic receptor-channels evokes an ion current that flows through the narrow synaptic cleft; this exerts a significant voltage drop and therefore strong electric field (10(4) V/m range) directed towards the current sinks in the cleft. To what extent this field affects fast diffusion of charged neurotransmitter molecules is not known. We draw a theoretical framework for this complex electrodiffusion phenomenon and establish the basic relationships between the synaptic current and the time course of neurotransmitter in the cleft. The analyses predict that excitatory currents could significantly accelerate the dispersion of negatively charged molecules from the cleft while attracting the positively charged molecules towards the current sinks. This previously unrecognized mechanism should affect the kinetics of synaptic receptor currents, thus contributing to fast synaptic signaling in the brain.

Electrostatic steering of substrate to acetylcholinesterase: analysis of field fluctuations

Based on previous molecular dynamics simulation results for acetylcholinesterase dimer, we calculate and analyse the electrostatic field fluctuations around the enzyme. The results show that dynamic features of the electrostatic field favor attraction of the positively-charged substrate. An Internet link to an animation of the results is also provided.