Markus Dittrich

Fast Monte Carlo Simulation Methods for Biological Reaction-Diffusion Systems in Solution and on Surfaces

Many important physiological processes operate at time and space scales far beyond those accessible to atom-realistic simulations, and yet discrete stochastic rather than continuum methods may best represent finite numbers of molecules interacting in complex cellular spaces. We describe and validate new tools and algorithms developed for a new version of the MCell simulation program (MCell3), which supports generalized Monte Carlo modeling of diffusion and chemical reaction in solution, on surfaces representing membranes, and combinations thereof. A new syntax for describing the spatial directionality of surface reactions is introduced, along with optimizations and algorithms that can substantially reduce computational costs (e.g., event scheduling, variable time and space steps). Examples for simple reactions in simple spaces are validated by comparison to analytic solutions. Thus we show how spatially realistic Monte Carlo simulations of biological systems can be far more cost-effective than often is assumed, and provide a level of accuracy and insight beyond that of continuum methods.

Single-Pixel Optical Fluctuation Analysis of Calcium Channel Function in Active Zones of Motor Nerve Terminals

We used high-resolution fluorescence imaging and single-pixel optical fluctuation analysis to estimate the opening probability of individual voltage-gated calcium (Ca2+) channels during an action potential and the number of such Ca2+ channels within active zones of frog neuromuscular junctions. Analysis revealed ∼36 Ca2+ channels within each active zone, similar to the number of docked synaptic vesicles but far less than the total number of transmembrane particles reported based on freeze-fracture analysis (∼200–250). The probability that each channel opened during an action potential was only ∼0.2. These results suggest why each active zone averages only one quantal release event during every other action potential, despite a substantial number of docked vesicles. With sparse Ca2+ channels and low opening probability, triggering of fusion for each vesicle is primarily controlled by Ca2+ influx through individual Ca2+ channels. In contrast, the entire synapse is highly reliable because it contains hundreds of active zones.

Are unreliable release mechanisms conserved from NMJ to CNS

The frog neuromuscular junction (NMJ) is a strong and reliable synapse because, during activation, sufficient neurotransmitter is released to trigger a postsynaptic action potential (AP). Recent evidence supports the hypothesis that this reliability emerges from the assembly of thousands of unreliable single vesicle release sites. The mechanisms that govern this unreliability include a paucity of voltage-gated calcium channels, a low probability of calcium channel opening during an AP, and the rare triggering of synaptic vesicle fusion even when a calcium channel does open and allows calcium flux. Here, we discuss the evidence that these unreliable single vesicle release sites may be the fundamental building blocks of many types of synapses in both the peripheral and central nervous system (PNS and CNS, respectively).

(R)-roscovitine prolongs the mean open time of unitary N-type calcium channel currents

(R)-roscovitine (Ros) is a cyclin-dependent kinase inhibitor that also has been shown to have direct agonist and antagonist actions on Ca(v)2.1 (P/Q-type) and Ca(v) 2.2 (N-type) families of voltage-gated calcium channels. These kinase-independent effects represent a novel opportunity to advance our understanding of calcium channel function and calcium-triggered neurotransmitter release. Furthermore, such actions on calcium channels may direct the development of Ros derivatives as new therapeutic agents. We used patch clamp recordings to characterize mechanisms that underlie the agonist effects of Ros on unitary N-type calcium channel gating. We found that N-type channels normally gate with either a short or long mean open time, that Ros significantly prolonged the mean open time of the long gating component and increased the probability of observing channels that gated with a long open time, but had no effect on single channel conductance. Using Monte Carlo simulations of a single channel kinetic model and Ros interactions, we were able to reproduce our experimental results and investigate the models microscopic dynamics. In particular, our simulations predicted that the longer open times generated by Ros were due to the appearance of a long open state combined with an increased amount of time spent in transitions between open states. Our results suggest a mechanism for agonist effects of Ros at the level of single channels, and provide a mechanistic explanation for previously reported agonist effects on whole cell calcium currents.

An Excess-Calcium-Binding-Site Model Predicts Neurotransmitter Release at the Neuromuscular Junction

Despite decades of intense experimental studies, we still lack a detailed understanding of synaptic function. Fortunately, using computational approaches, we can obtain important new insights into the inner workings of these important neural systems. Here, we report the development of a spatially realistic computational model of an entire frog active zone in which we constrained model parameters with experimental data, and then used Monte Carlo simulation methods to predict the Ca(2+)-binding stoichiometry and dynamics that underlie neurotransmitter release. Our model reveals that 20-40 independent Ca(2+)-binding sites on synaptic vesicles, only a fraction of which need to bind Ca(2+) to trigger fusion, are sufficient to predict physiological release. Our excess-Ca(2+)-binding-site model has many functional advantages, agrees with recent data on synaptotagmin copy number, and is the first (to our knowledge) to link detailed physiological observations with the molecular machinery of Ca(2+)-triggered exocytosis. In addition, our model provides detailed microscopic insight into the underlying Ca(2+) dynamics during synapse activation.

Organization and function of transmitter release sites at the neuromuscular junction

Abstract  The neuromuscular junction is known as a strong and reliable synapse. It is strong because it releases an excess of chemical transmitter, beyond what is required to bring the postsynaptic muscle cell to threshold. Because the synapse can sustain suprathreshold muscle activation during short trains of action potentials, it is also reliable. The presynaptic mechanisms that lead to reliability during short trains of activity have only recently been elucidated. It appears that there are relatively few calcium channels in individual active zones, that channels open with a low probability during action potential stimulation and that even if channels open the resulting calcium flux only rarely triggers vesicle fusion. Thus, each synaptic vesicle may only associate with a small number of calcium channels, forming an unreliable single vesicle release site. Strength and reliability of the neuromuscular junction emerge as a result of its assembly from thousands of these unreliable single vesicle release sites. Hence, these synapses are strong while at the same time only releasing a small subset of available docked vesicles during each action potential, thus conserving transmitter release resources. This prevents significant depression during short trains of action potential activity and confers reliability.

Transmitter release is evoked with low probability predominately by calcium flux through single channel openings at the frog neuromuscular junction

The quantitative relationship between presynaptic calcium influx and transmitter release critically depends on the spatial coupling of presynaptic calcium channels to synaptic vesicles. When there is a close association between calcium channels and synaptic vesicles, the flux through a single open calcium channel may be sufficient to trigger transmitter release. With increasing spatial distance, however, a larger number of open calcium channels might be required to contribute sufficient calcium ions to trigger vesicle fusion. Here we used a combination of pharmacological calcium channel block, high-resolution calcium imaging, postsynaptic recording, and 3D Monte Carlo reaction-diffusion simulations in the adult frog neuromuscular junction, to show that release of individual synaptic vesicles is predominately triggered by calcium ions entering the nerve terminal through the nearest open calcium channel. Furthermore, calcium ion flux through this channel has a low probability of triggering synaptic vesicle fusion (∼6%), even when multiple channels open in a single active zone. These mechanisms work to control the rare triggering of vesicle fusion in the frog neuromuscular junction from each of the tens of thousands of individual release sites at this large model synapse.

Rapid Creation, Monte Carlo Simulation, and Visualization of Realistic 3D Cell Models

Spatially realistic diffusion-reaction simulations supplement traditional experiments and provide testable hypotheses for complex physiological systems. To date, however, the creation of realistic 3D cell models has been difficult and time-consuming, typically involving hand reconstruction from electron microscopic images. Here, we present a complementary approach that is much simpler and faster, because the cell architecture (geometry) is created directly in silico using 3D modeling software like that used for commercial film animations. We show how a freely available open source program (Blender) can be used to create the model geometry, which then can be read by our Monte Carlo simulation and visualization softwares (MCell and DReAMM, respectively). This new workflow allows rapid prototyping and development of realistic computational models, and thus should dramatically accelerate their use by a wide variety of computational and experimental investigators. Using two self-contained examples based on synaptic transmission, we illustrate the creation of 3D cellular geometry with Blender, addition of molecules, reactions, and other run-time conditions using MCells Model Description Language (MDL), and subsequent MCell simulations and DReAMM visualizations. In the first example, we simulate calcium influx through voltage-gated channels localized on a presynaptic bouton, with subsequent intracellular calcium diffusion and binding to sites on synaptic vesicles. In the second example, we simulate neurotransmitter release from synaptic vesicles as they fuse with the presynaptic membrane, subsequent transmitter diffusion into the synaptic cleft, and binding to postsynaptic receptors on a dendritic spine.

Unbiased Rare Event Sampling in Spatial Stochastic Systems Biology Models Using a Weighted Ensemble of Trajectories.

The long-term goal of connecting scales in biological simulation can be facilitated by scale-agnostic methods. We demonstrate that the weighted ensemble (WE) strategy, initially developed for molecular simulations, applies effectively to spatially resolved cell-scale simulations. The WE approach runs an ensemble of parallel trajectories with assigned weights and uses a statistical resampling strategy of replicating and pruning trajectories to focus computational effort on difficult-to-sample regions. The method can also generate unbiased estimates of non-equilibrium and equilibrium observables, sometimes with significantly less aggregate computing time than would be possible using standard parallelization. Here, we use WE to orchestrate particle-based kinetic Monte Carlo simulations, which include spatial geometry (e.g., of organelles, plasma membrane) and biochemical interactions among mobile molecular species. We study a series of models exhibiting spatial, temporal and biochemical complexity and show that although WE has important limitations, it can achieve performance significantly exceeding standard parallel simulation—by orders of magnitude for some observables.

Design Automation for Biological Models: A Pipeline that Incorporates Spatial and Molecular Complexity

Understanding the dynamics of biochemical networks is a major goal of systems biology. Due to the heterogeneity of cells and the low copy numbers of key molecules, spatially resolved approaches are required to fully understand and model these systems. Until recently, most spatial modeling was performed using geometries obtained either through manual segmentation or manual fabrication both of which are time-consuming and tedious. Similarly, the system of reactions associated with the model had to be manually defined, a process that is both tedious and error-prone for large networks. As a result, spatially resolved simulations have typically only been performed in a limited number of geometries, which are often highly simplified, and with small reaction networks.