Stochastic Reaction-diffusion Simulations Research Articles

A Kinetic Model for Predicting Trace Gas Uptake and Reaction.

A model is developed to describe trace gas uptake and reaction with applications to aerosols and microdroplets. Gas uptake by the liquid is formulated as a coupled equilibria that links gas, surface, and bulk regions of the droplet or solution. Previously, this framework was used in explicit stochastic reaction-diffusion simulations to predict the reactive uptake kinetics of ozone with droplets containing aqueous aconitic acid, maleic acid, and sodium nitrite. With the use of prior data and simulation results, a new equation for the uptake coefficient is derived, which accounts for both surface and bulk reactions. Lambert W functions are used to obtain closed form solutions to the integrated rate laws for the multiphase kinetics; similar to previous expressions that describe Michaelis-Menten enzyme kinetics. Together these equations couple interface and bulk processes over a wide range of conditions and do not require many of the limiting assumptions needed to apply resistor model formulations to explain trace gas uptake and reaction.

Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins

Clathrin-coated structures must assemble on cell membranes to internalize receptors. These structures can grow surprisingly large, containing over 20 clathrin seemingly locked in a hexagonal lattice, yet often fail to form productive vesicles. We show that these abortive events happen spontaneously when adaptor availability is low, despite abundant clathrin. Here, we combine recent in vitro kinetic measurements with stochastic reaction-diffusion simulations and theory to differentiate mechanisms of stable vs unstable clathrin assembly on membranes.

Computational Design of Photocured Polymers Using Stochastic Reaction–Diffusion Simulation

AbstractPhotopolymerization is widely used for creating materials for biomedical applications, lithography, and fast 3D fabrication. The resin composition and curing protocol during polymerization define a high‐dimensional parameter space that dictates the reaction kinetics, network structures, and physical properties of polymerized materials. But a quantitative map from the input parameter space to the transient and final properties does not exist. Here, a computational method is presented to simulate network growth as a stochastic process using reaction‐diffusion master equations. The evolving diffusivity of reacting species during polymerization is modeled using effective medium method, thereby making the rates of reaction and fluctuation events sensitive to the spatial configuration of a growing network. The numerical studies show that this method can correlate fabrication parameters to polymer properties like crosslink heterogeneity, relaxation time, phonon density of states, and bulk modulus, in silico. Hence, stochastic simulation of polymerization is broadly applicable to accelerate the design of polymeric materials with targeted physical properties for specific applications.

Efficient Green's Function Reaction Dynamics (GFRD) simulations for diffusion-limited, reversible reactions

In diffusion controlled reversible bimolecular reactions in three dimensions, a dissociation step is typically followed by multiple, rapid re-association steps slowing down the simulations of such systems. In order to improve the efficiency, we first derive an exact Green's function describing the rate at which an isolated pair of particles undergoing reversible bimolecular reactions and unimolecular decay separates beyond an arbitrarily chosen distance. Then the Green's function is used in an algorithm for particle-based stochastic reaction–diffusion simulations for prediction of the dynamics of biochemical networks. The accuracy and efficiency of the algorithm are evaluated using a reversible reaction and a push–pull chemical network. The computational work is independent of the rates of the re-associations.

Parallel STEPS: Large Scale Stochastic Spatial Reaction-Diffusion Simulation with High Performance Computers.

Stochastic, spatial reaction-diffusion simulations have been widely used in systems biology and computational neuroscience. However, the increasing scale and complexity of models and morphologies have exceeded the capacity of any serial implementation. This led to the development of parallel solutions that benefit from the boost in performance of modern supercomputers. In this paper, we describe an MPI-based, parallel operator-splitting implementation for stochastic spatial reaction-diffusion simulations with irregular tetrahedral meshes. The performance of our implementation is first examined and analyzed with simulations of a simple model. We then demonstrate its application to real-world research by simulating the reaction-diffusion components of a published calcium burst model in both Purkinje neuron sub-branch and full dendrite morphologies. Simulation results indicate that our implementation is capable of achieving super-linear speedup for balanced loading simulations with reasonable molecule density and mesh quality. In the best scenario, a parallel simulation with 2,000 processes runs more than 3,600 times faster than its serial SSA counterpart, and achieves more than 20-fold speedup relative to parallel simulation with 100 processes. In a more realistic scenario with dynamic calcium influx and data recording, the parallel simulation with 1,000 processes and no load balancing is still 500 times faster than the conventional serial SSA simulation.

Stochastic reaction–diffusion algorithms for macromolecular crowding

Compartment-based (lattice-based) reaction–diffusion algorithms are often used for studying complex stochastic spatio-temporal processes inside cells. In this paper the influence of macromolecular crowding on stochastic reaction–diffusion simulations is investigated. Reaction–diffusion processes are considered on two different kinds of compartmental lattice, a cubic lattice and a hexagonal close packed lattice, and solved using two different algorithms, the stochastic simulation algorithm and the spatiocyte algorithm (Arjunan and Tomita 2010 Syst. Synth. Biol. 4, 35–53). Obstacles (modelling macromolecular crowding) are shown to have substantial effects on the mean squared displacement and average number of molecules in the domain but the nature of these effects is dependent on the choice of lattice, with the cubic lattice being more susceptible to the effects of the obstacles. Finally, improvements for both algorithms are presented.

Implementation of parallel spatial stochastic reaction-diffusion simulation in STEPS

Spatial stochastic reaction-diffusion simulation has been recognized as an essential modeling tool in computational neuroscience studies of signaling pathways, as shown in an increasing number of recent studies such as our previous work on the stochastic effects of calcium dynamics in Purkinje cells [1]. A critical performance issue arises when attempting to model large pathway models with complex morphologies, for example the one proposed in Human Brain Project [2], due to the serial nature of Gillespie's direct method [3], the fundamental algorithm of many spatial stochastic reaction-diffusion simulators including STEPS [4]. Various solutions have been proposed to improve the computational efficiency of the Gillespie method, including the tau-leaping approximation [5], most of which, however, remain serial implementations. The need of parallel implementation of stochastic reaction-diffusion simulators has become urgent, as the scale and complexity of model being studied surpasses the speedup gained from hardware upgrade and algorithm improvement. However, such a task is not trivial as the original Gillespie SSA is known to be extremely serial. In CNS2014 we proposed a parallel solution to approximate diffusion events in STEPS [6], which significantly improves the performance while maintaining high accuracy. We now further improve this solution by introducing a multinomial algorithm for fast diffusion direction selection of multiple molecules in a single subvolume. We also combine this diffusion approximation with a new operator splitting solution for reaction events in the SSA system. The combined solution is implemented in STEPS as its first parallel solver named TetOpSplit. Current implementation of TetOpSplit uses MPI as its parallel protocol and aims to provide solutions for large scale simulations such as whole cell reaction-diffusion models, in modern supercomputers like Blue Gene. In this poster we discuss the difficulties we encountered during the transformation from the serial TetOpSplit algorithm to its parallel counterpart, as well as our solutions. We also provide performance results of the new solver via different examples, and compare them with the results gained from our original serial SSA implementation.

Convergence of methods for coupling of microscopic and mesoscopic reaction–diffusion simulations

In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction–diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment–placement method (CPM). The third method that is introduced and analysed in this paper is called the ghost cell method (GCM), since it works by constructing a “ghost cell” in which molecules can disappear and jump into the compartment-based simulation. Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step Δt (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter h, the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered:(i)Δt→0 and h is fixed;(ii)Δt→0 and h→0 such that Δt/h is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model.

Accurate approximation and MPI parallelization of spatial stochastic reaction-diffusion in STEPS

Spatial stochastic reaction-diffusion simulations have become an important component of molecular modeling in Computational Neuroscience, as shown in a growing number of recent studies including our previous work in which we show that stochastic effects, in particular stochastic calcium dynamics, contribute to Purkinje cell calcium burst variability [1]. However, the computational cost of exact, serial algorithms such as Gillespie’s direct method as implemented in STEPS [2] in which every reaction event and cross-subvolume diffusion event is simulated, impose severe restrictions on model complexity. Serial solutions limit the number of molecular species and their reaction channels that can be represented, as well as imposing restrictive upper-limits on the number of subvolumes in the discretized space. It is desirable to provide a solution that allows investigation of stochastic effects in larger systems, with complex meshes of many hundreds of thousands or millions of subvolumes representing complex neuronal morphology, in projects such as the Human Brain project [3], which is not possible with serial solutions. Therefore, parallel solutions for stochastic reaction-diffusion is vital for the future of this field. There are common challenges of parallelization for reaction-diffusion simulators as well as some unique challenges for STEPS, in which space is discretized into irregular tetrahedrons and not cubes that are commonly used by other reaction-diffusion simulators. Exact solutions are not beneficial, where stochastic diffusion across geometry partitions result in frequent conflict between nodes, requiring regular costly rollbacks. Therefore it is essential that an approximation algorithm is applied. Such an approximation must minimise loss of accuracy whilst maximising performance gain. Existing solutions such as the Gillespie-Multi-Particle method (GMP) [4,5] have been designed for cubic subvolumes, which have the advantage of regularity (yet restrict morphological accuracy [2]). We review such approaches from a theoretical standpoint and find that a systematic diffusion slowing error exists with naive application to irregular subvolumes, which have local scaled diffusion coefficients and therefore localized expected diffusion times. We present a method that is tailored for irregular subvolumes, with a defined theoretical upper limit to the communication time-step. The method could, however, be generalised for different subvolume configurations where adaptable step-sizes could be beneficial. We further expand our method to allow multiple particle leaps, which for some systems this can greatly increase the minimum acceptable step-size and thus benefit performance whilst still ensuring acceptable accuracy. We demonstrate accuracy of our approximation by comparing to analytical solutions on a number of test cases. Our method is implemented on an MPI framework, and we present performance gains under two scenariosglobal application of the method, and application only at partition boundaries. We discuss the potential of this method for future large-scale parallel simulations in STEPS.

Python-based geometry preparation and simulation visualization toolkits for STEPS.

STEPS is a stochastic reaction-diffusion simulation engine that implements a spatial extension of Gillespie's Stochastic Simulation Algorithm (SSA) in complex tetrahedral geometries. An extensive Python-based interface is provided to STEPS so that it can interact with the large number of scientific packages in Python. However, a gap existed between the interfaces of these packages and the STEPS user interface, where supporting toolkits could reduce the amount of scripting required for research projects. This paper introduces two new supporting toolkits that support geometry preparation and visualization for STEPS simulations.

Editorial: Special Issue on Stochastic Modelling of Reaction–Diffusion Processes in Biology

Editorial: Special Issue on Stochastic Modelling of Reaction–Diffusion Processes in Biology

Analysis of the Two-Regime Method on Square Meshes

The two-regime method (TRM) has been recently developed for optimizing stochastic reaction-diffusion simulations [M. Flegg, J. Chapman, and R. Erban, J. Roy. Soc. Interface, 9 (2012), pp. 859--868]. It is a multiscale (hybrid) algorithm which uses stochastic reaction-diffusion models with different levels of detail in different parts of the computational domain. The coupling condition on the interface between different modeling regimes of the TRM was previously derived for one-dimensional models. In this paper, the TRM is generalized to higher dimensional reaction-diffusion systems. Coupling Brownian dynamics models with compartment-based models on regular (square) two-dimensional lattices is studied in detail. In this case, the interface between different modeling regimes contains either flat parts or right-angle corners. Both cases are studied in the paper. For flat interfaces, it is shown that the one-dimensional theory can be used along the line perpendicular to the TRM interface. In the direction tan...

Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release

The intracellular release of calcium from the endoplasmic reticulum is controlled by ion channels. The resulting calcium signals exhibit a rich spatio-temporal signature, which originates at least partly from microscopic fluctuations. While stochasticity in the gating transition of ion channels has been incorporated into many models, the distribution of calcium is usually described by deterministic reaction-diffusion equations. Here we test the validity of the latter modeling approach by using two different models to calculate the frequency of localized calcium signals (calcium puffs) from clustered IP3 receptor channels. The complexity of the full calcium system is here limited to the basic opening mechanism of the ion channels and, in the mathematical reduction simplifies to the calculation of a first passage time. Two models are then studied: (i) a hybrid model, where channel gating is treated stochastically, while calcium concentration is deterministic and (ii) a fully stochastic model with noisy channel gating and Brownian calcium ion motion. The second model utilises the recently developed two-regime method [M. B. Flegg, S. J. Chapman, and R. Erban, "The two-regime method for optimizing stochastic reaction-diffusion simulations," J. R. Soc., Interface 9, 859-868 (2012)] in order to simulate a large domain with precision required only near the Ca(2+) absorbing channels. The expected time for a first channel opening that results in a calcium puff event is calculated. It is found that for a large diffusion constant, predictions of the interpuff time are significantly overestimated using the model (i) with a deterministic non-spatial calcium variable. It is thus demonstrated that the presence of diffusive noise in local concentrations of intracellular Ca(2+) ions can substantially influence the occurrence of calcium signals. The presented approach and results may also be relevant for other cell-physiological first-passage time problems with small ligand concentration and high cooperativity.

Improved spatial direct method with gradient-based diffusion to retain full diffusive fluctuations.

The spatial direct method with gradient-based diffusion is an accelerated stochastic reaction-diffusion simulation algorithm that treats diffusive transfers between neighboring subvolumes based on concentration gradients. This recent method achieved a marked improvement in simulation speed and reduction in the number of time-steps required to complete a simulation run, compared with the exact algorithm, by sampling only the net diffusion events, instead of sampling all diffusion events. Although the spatial direct method with gradient-based diffusion gives accurate means of simulation ensembles, its gradient-based diffusion strategy results in reduced fluctuations in populations of diffusive species. In this paper, we present a new improved algorithm that is able to anticipate all possible microscopic fluctuations due to diffusive transfers in the system and incorporate this information to retain the same degree of fluctuations in populations of diffusing species as the exact algorithm. The new algorithm also provides a capability to set the desired level of fluctuation per diffusing species, which facilitates adjusting the balance between the degree of exactness in simulation results and the simulation speed. We present numerical results that illustrate the recovery of fluctuations together with the accuracy and efficiency of the new algorithm.

Exact on-lattice stochastic reaction-diffusion simulations using partial-propensity methods

Stochastic reaction-diffusion systems frequently exhibit behavior that is not predicted by deterministic simulation models. Stochastic simulation methods, however, are computationally expensive. We present a more efficient stochastic reaction-diffusion simulation algorithm that samples realizations from the exact solution of the reaction-diffusion master equation. The present algorithm, called partial-propensity stochastic reaction-diffusion (PSRD) method, uses an on-lattice discretization of the reaction-diffusion system and relies on partial-propensity methods for computational efficiency. We describe the algorithm in detail, provide a theoretical analysis of its computational cost, and demonstrate its computational performance in benchmarks. We then illustrate the application of PSRD to two- and three-dimensional pattern-forming Gray-Scott systems, highlighting the role of intrinsic noise in these systems.

The two-regime method for optimizing stochastic reaction–diffusion simulations

Spatial organization and noise play an important role in molecular systems biology. In recent years, a number of software packages have been developed for stochastic spatio-temporal simulation, ranging from detailed molecular-based approaches to less detailed compartment-based simulations. Compartment-based approaches yield quick and accurate mesoscopic results, but lack the level of detail that is characteristic of the computationally intensive molecular-based models. Often microscopic detail is only required in a small region (e.g. close to the cell membrane). Currently, the best way to achieve microscopic detail is to use a resource-intensive simulation over the whole domain. We develop the two-regime method (TRM) in which a molecular-based algorithm is used where desired and a compartment-based approach is used elsewhere. We present easy-to-implement coupling conditions which ensure that the TRM results have the same accuracy as a detailed molecular-based model in the whole simulation domain. Therefore, the TRM combines strengths of previously developed stochastic reaction-diffusion software to efficiently explore the behaviour of biological models. Illustrative examples and the mathematical justification of the TRM are also presented.

Tetrahedral mesh generation and visualization for stochastic reaction-diffusion simulation

The generation of high quality three-dimensional meshes is important in voxel-based computational modeling and simulation of stochastic reaction-diffusion systems. Compared with cubical meshes used in some reaction-diffusion simulators such as MesoRD [1], tetrahedral meshes are more flexible in representing complex geometry boundaries, for instance, the spherical head of a dendritic spine. We describe the interaction between tetrahedral mesh generators (CUBIT [2], TetGen [3]) and our STochastic Engine for Pathway Simulation (STEPS) [4], including the generation and importing of meshes, as well as the visualization of the simulations in a 3D graphical environment. STEPS is a GNU-licensed simulation platform that uses an extension of Gillespie's SSA [5] to deal with reactions and diffusion of molecules in 3D reconstructions of neuronal morphology and tissue [4]. In order to support a range of research goals, it provides a flexible Python frontend together with an efficient C++ backend. In STEPS, the diffusion of molecules is simulated as diffusive fluxes between tetrahedral elements in the mesh, represented by a series of first-order reactions. Since many mesh generating methods and software packages are available in public, instead of implementing our own mesh generator, STEPS provides a generic, extendable Python-based import interface for different tetrahedral mesh formats. Thus the first step is to create a suitable mesh, which can be challenging. In our research [6] it involves the reconstruction of anatomical/morphological structures from combinations of simple geometry primitives like cylinders and spheres. Once the mesh file is generated, it can be imported to the simulations using either pre-implemented import functions (currently supporting Abaqus and TetGen mesh formats) or user-developed functions by adapting the STEPS element proxy interface. The element proxy interface performs automatic mapping between mesh elements in external files and the ones used by the STEPS simulation. Monitoring mesh quality is possible via mesh generators or the STEPS internal quality metric. This metric calculates the Radius-Edge Ratio (RER) for each tetrahedron, given by dividing the radius of the tetrahedron's circumsphere with the length of the shortest edge. The smaller this value, the more regular the tetrahedron. The minimum value of this metric is given by computing the RER for a fully regular tetrahedron, that is, sqrt(6)/4. Although the visualization of stochastic reaction-diffusion simulations with 3D meshes is optional due to the high number of reactions/diffusions that take place during a simulation, it is sometimes required for demonstration and education. Animations of such a simulation can be made by a script-controlled 3D graphical environment, for example the python frontend of CUBIT, with user-specified data extracted from the STEPS simulation. Alternatively, STEPS provides a preliminary Python-based visual toolkit for general purposes of displaying meshes and simple reaction-diffusion simulations. Overall, the collaborative workflow between STEPS and external mesh generators discussed here has shown its flexibility and will facilitate a broad range of applications in computational pathway modeling and simulation.

Reverse-engineering of biochemical reaction networks from spatio-temporal correlations of fluorescence fluctuations

Recent developments of fluorescence labeling and highly advanced microscopy techniques have enabled observations of activities of biosignaling molecules in living cells. The high spatial and temporal resolutions of these video microscopy experiments allow detection of fluorescence fluctuations at the timescales approaching those of enzymatic reactions. Such fluorescence fluctuation patterns may contain information about the complex reaction–diffusion system driving the dynamics of the labeled molecule. Here, we have developed a method of identifying the reaction–diffusion system of fluorescently labeled signaling molecules in the cell, by combining spatio-temporal correlation function analysis of fluctuating fluorescent patterns, stochastic reaction–diffusion simulations, and an iterative system identification technique using a simulated annealing algorithm. In this report, we discuss the validity and usability of spatio-temporal correlation functions in characterizing the reaction–diffusion dynamics of biomolecules, and demonstrate application of our reaction–diffusion system identification method to a simple conceptual model for small GTPase activation.

Anchoring proteins and phosphodiesterases: A stochastic reaction diffusion simulation of cAMP microdomains

Event Abstract Back to Event Anchoring proteins and phosphodiesterases: A stochastic reaction diffusion simulation of cAMP microdomains Rodrigo F. Oliveira1*, Myungsook Kim2, Kim T. Blackwell2 and Manuela Zaccolo3 1 George Mason University, United States 2 Krasnow Institute for Advanced Studies, United States 3 Faculty of Biomedical & Life Sciences, United Kingdom Cyclic AMP (cAMP) is an important second messenger molecule responsible for regulation of many aspects of cell function such as long term synaptic plasticity in neurons. Though cAMP diffuses throughout a cell relatively rapidly, specificity in cAMP activation of downstream targets is achieved by mechanisms which produce spatial gradients and microdomains of cAMP concentration. Recent experiments suggest that compartmentalized phosphodiesterases (PDE) act as localized 'sinks', producing gradients of cAMP. To test the role of compartmentalized PDEs in producing spatial gradients, we develop a computational model of cAMP production, PKA activation and compartmentalized PDE activity using the Rapid, Approximate, Stochastic reaction-diffusion (RASrd) algorithm. In the model cell, adenylyl cyclase, PKA and PDE4B are anchored to the submembrane compartment, PKA and PDE4D are anchored in the cytosol, and cAMP freely diffuses. To precisely compare simulated results with FRET imaging data, we calculate the theoretical FRET signal from simulated concentrations of cAMP bound and free Epac or PKA, and include contamination terms due to overlap of emission and excitation spectra. The model reproduces the cAMP gradient (both free cAMP and the calculated FRET signal) between submembrane and cytosolic compartments. Simulations results with reduced PDE activity and different cAMP diffusion constants also show the dependence of the gradient on both PDE4D cytosoloic anchoring and cAMP diffusion suggesting that reduced cAMP diffusivity may also play a role in compartmentalization. Furthermore, our simulations demonstrate the utility of the new stochastic reaction-diffusion algorithm for exploring signaling pathways in spatially complex structures such as neurons. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Oliveira RF, Kim M, Blackwell KT and Zaccolo M (2008). Anchoring proteins and phosphodiesterases: A stochastic reaction diffusion simulation of cAMP microdomains. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.048 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Rodrigo F Oliveira, George Mason University, Fairfax, United States, roliveir@gmu.edu Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Rodrigo F Oliveira Myungsook Kim Kim T Blackwell Manuela Zaccolo Google Rodrigo F Oliveira Myungsook Kim Kim T Blackwell Manuela Zaccolo Google Scholar Rodrigo F Oliveira Myungsook Kim Kim T Blackwell Manuela Zaccolo PubMed Rodrigo F Oliveira Myungsook Kim Kim T Blackwell Manuela Zaccolo Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Stochastic reaction–diffusion simulation of enzyme compartmentalization reveals improved catalytic efficiency for a synthetic metabolic pathway

We have demonstrated the accuracy of a spatial stochastic model of Escherichia coli central carbon metabolism using the next subvolume method (NSM), an efficient implementation of the Gillespie direct method of stochastic simulation. Using this model, we demonstrate that compartmentalization of the enzymes comprising an engineered pathway for biosynthesis of R-1,2-propanediol leads to improved kinetic properties for the pathway enzymes, especially when substrate diffusivities are low. Our results suggest that enzyme compartmentalization is a powerful approach for improving the catalytic turnover of a channeled carbon substrate and should be particularly useful when applied to synthetic metabolic pathways that suffer from poor translation efficiency, are present in highly variable copy numbers, and have low turnover for new substrates. Furthermore, this approach represents a generic modeling framework for simultaneously analyzing spatial and stochastic events in cellular metabolism and should enable quantitative evaluation of the effect of enzyme compartmentalization on virtually any recombinant pathway.