Fig 1.
One initial trajectory is iteratively resampled until its progeny trajectories are distributed equally around the state-space, according to user-defined bins. Excess trajectories are pruned if necessary. This process is statistically exact if the probabilistic weight of each trajectory is properly accounted for in the resampling (splitting/merging) process. In between splitting and merging events, each trajectory follows the natural dynamics of the system, and splitting events occur when a trajectory naturally wanders into an underpopulated region of space. Reprinted with permission from Efficient stochastic simulation of chemical kinetics networks using a weighted ensemble of trajectories, The Journal of Chemical Physics 139: 115105.
Fig 2.
Distribution of sampling power.
Brute-force sampling, by definition, concentrates sampling power on the most probable events. By contrast, weighted ensemble samples a distribution more evenly, and compared to brute-force it applies more resources to hard-to-sample regions of interest.
Fig 3.
Software pipeline for realistic cell geometry simulations.
Geometries are learned from images by CellOrganizer. Chemical reaction networks are generated from rule-based models in BioNetGen. Geometries and reaction networks are imported to MCell via the CellBlender visual editor. The spatial stochastic model is then simulated in MCell, with WESTPA managing a weighted ensemble of MCell trajectories.
Fig 4.
This toy system has receptors at the top and bottom of a cubical “cell”. The receptors at the top are initially bound by ligands, that are free to unbind and diffuse around the cell, and bind to receptors at the bottom.
Fig 5.
Realistic cell geometry generated from microscopy images by CellOrganizer. The geometry explicitly models the compartmentalization of the cell, by forcing molecules to diffuse through membranes to transition from, for example, the cytoplasm (grey) to the nucleus (blue). Also modeled are endosomes (green), and the extracellular environment (transparent).
Fig 6.
This rule-based model is translated into a system of chemical kinetics reactions by BioNetGen, and then simulated in a spatially realistic geometry by MCell. Figure adapted from [61]. Rate constants and further details are given in the SI.
Fig 7.
Schematic of the model of an active zone of a frog neuromuscular junction.
On the left is an example snapshot from a simulation, and on the right is a zoomed-in view of the model. Calcium is released into the presynaptic space and is free to diffuse around the geometry and bind to the synaptic vesicles at the bottom of the active zone.
Fig 8.
Sampling rare states of the toy binding model.
Shown is the probability distribution of the number of bound receptors on the target side of the cell after 10 milliseconds of simulation. The weighted ensemble data (blue circles) is plotted with brute-force data (red squares) generated using equal computational effort, i.e. 611 brute-force runs. Brute-force sampling is confined to the peak of the distribution, whereas weighted ensemble sampling captures more of the full probability distribution. To compare both approaches to an authoritative value, an exhaustive set of 64 large weighted ensemble simulations was performed (grey circles), from which a bootstrapped 95% confidence interval for the mean probability distribution was calculated (dark grey bars).
Fig 9.
Accelerated sampling of high P2 levels.
After 400 simulated seconds (∼1 week of wall time for one trajectory), we plot the histogram of the number of P2 molecules in the cell. The blue and green circles are the result of two independent weighted ensemble simulations. Note that some data points are missing, because not all weighted ensemble bins are necessarily populated by trajectory segments at all times.
Fig 10.
Steady-state estimate of time to produce five P2.
The last event in the complex signaling network of Fig 6 is the production of P2. Using a steady-state approach, weighted ensemble is able to estimate the mean time to producing 5 molecules of P2 as approximately 5,000 seconds. The graph shows the probability flux arriving at the target state (left axis) in each iteration (points), and a running average of that flux computed from the most recent 50% of the data (lines). The mean time to reach the target state (right axis) is obtained via the Hill relation (Eq 1), as the inverse of the steady-state flux. Note that during most iterations, zero weight reaches the target state, as evidenced by the nearly continuous band of points at the bottom of the figure. The running average of the flux is dominated by these uneventful iterations, and hence is less than the nonzero instantaneous values.
Fig 11.
Enhanced sampling of the first fusion time distribution in the NMJ Model in low calcium conditions.
Shown are measurements of the first fusion (i.e., first passage) time distribution for calcium concentrations of 0.5 mM (left), and 0.3 mM (right). In both plots, weighted ensemble estimate of the distribution of first fusion times for the NMJ model (blue), are compared to brute-force estimates (red) made using the same computational power as the weighted ensemble estimate. Points at the bottom of the plot indicate no fusion events in that time period. Single points with no error bars indicate only one sample, yielding no ability to estimate uncertainties. Brute-force sampling sees very few events, and gives a poor estimate of the shape of the distribution and yields poor confidence intervals (it is unable to exclude zero from any time point at one standard error). Weighted ensemble is able to capture the shape of the fusion time distribution, as well as providing good estimates in the uncertainty of the measured values.
Fig 12.
Verification of empirical fusion rate law extended to low calcium regime.
The probability for the model to release a synaptic vesicle in the simulation window is plotted vs. the calcium concentration in the simulation. Weighted ensemble is able to efficiently estimate these probabilities in the low calcium regime (0.5 mM and below). WE data points and the power-law fit are shown with 1-σ confidence intervals.