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Current address: Division of Basic Science, Fred Hutchinson Cancer Research Center, Seattle, Washington, United States of America

Conceived and designed the experiments: SSA. Performed the experiments: SSA. Analyzed the data: SSA. Wrote the paper: SSA RB. Wrote the software: SSA NJA. Provided support and guidance: RB APA.

The authors have declared that no competing interests exist.

Most cellular processes depend on intracellular locations and random collisions of individual protein molecules. To model these processes, we developed algorithms to simulate the diffusion, membrane interactions, and reactions of individual molecules, and implemented these in the Smoldyn program. Compared to the popular MCell and ChemCell simulators, we found that Smoldyn was in many cases more accurate, more computationally efficient, and easier to use. Using Smoldyn, we modeled pheromone response system signaling among yeast cells of opposite mating type. This model showed that secreted Bar1 protease might help a cell identify the fittest mating partner by sharpening the pheromone concentration gradient. This model involved about 200,000 protein molecules, about 7000 cubic microns of volume, and about 75 minutes of simulated time; it took about 10 hours to run. Over the next several years, as faster computers become available, Smoldyn will allow researchers to model and explore systems the size of entire bacterial and smaller eukaryotic cells.

We developed a general-purpose biochemical simulation program, called Smoldyn. It represents proteins and other molecules of interest with point-like particles that diffuse, interact with surfaces, and react, all in continuous space. This high level of detail allows users to investigate spatial organization within cells and natural stochastic variability. Although similar to the MCell and ChemCell programs, Smoldyn is more accurate and runs faster. Smoldyn also supports many unique features, such as commands that a “virtual experimenter” can execute during simulations and automatic reaction network expansion for simulating protein complexes. We illustrate Smoldyn's capabilities with a model of signaling between yeast cells of opposite mating type. It investigates the role of the secreted protease Bar1, which inactivates mating pheromone. Intuitively, it might seem that inactivating most of the pheromone would make a cell less able to detect the local pheromone concentration gradient. In contrast, we found that Bar1 secretion improves pheromone gradient detectability: the local gradient is sharpened because pheromone is progressively inactivated as it diffuses through a cloud of Bar1. This result helps interpret experiments that showed that Bar1 secretion helped cells distinguish between potential mates, and suggests that Bar1 helps yeast cells identify the fittest mating partners.

One hurdle to the computational modeling of cellular systems is the lack of adequate tools. If one assumes that molecules inside cells are well-mixed, and that they behave deterministically, then one can model the chemical reactions that cells use to operate with differential equations (recently reviewed by Alves and coworkers

Computational biologists have pursued four main approaches to simulating biochemical systems with spatial and stochastic detail. These differ in how they represent space, time, and molecules (

Simulation method | Simulation programs | Space representation | Time treatment | Molecule representation |

Spatial Gillespie | MesoRD |
coarse lattice | event-based | populations |

SmartCell |
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GMP |
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Fine lattice | GridCell |
fine lattice | fixed steps | individuals |

Spatiocyte |
||||

Particle-based | ChemCell |
continuous | fixed steps | individuals |

MCell |
||||

Cell++ |
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Smoldyn (this work) | ||||

GFRD | E-Cell |
continuous | event-based | individuals |

GMP is event-based for reactions and uses fixed time steps for diffusion.

MCell uses an event-based scheduling system in which short steps are used for fast processes and long steps for slow processes.

Cell++ represents small molecules, such as metabolites, as concentrations on a coarse lattice and large molecules, such as enzymes, as individual particles in continuous space.

GFRD is in process of being added to E-Cell.

The dominant particle based simulators are ChemCell

Feature | ChemCell | MCell3 | Smoldyn 2.1 |

Simulation methods | ODE, Gillespie, particle | particle | particle |

Time steps | fixed | adaptive | fixed |

System dimensionality | 3 | 3 | 1, 2, 3 |

System boundaries | reflective, periodic | reflective, absorbing, transparent | reflective, absorbing, periodic, transparent |

Geometry primitives | triangles, spheres, boxes, planes, cylinders | triangles | triangles, rectangles, spheres, cylinders, hemispheres, disks |

Surface molecule states | transmembrane | integral states: top-front, top-back | integral, peripheral states: up, down, front, back |

Accuracy of diffusion | volume: exact | volume: exact | volume: exact |

surface: approx | surface: approx. | surface: approx. | |

Accuracy of reactions in solution | order 0: none | order 0: none | order 0: exact |

order 1: exact | order 1: exact | order 1: exact | |

order 2: approx. | order 2: approx. | order 2: exact | |

Accuracy of reactions on surfaces | order 0: none | order 0: none | order 0: exact |

order 1: exact | order 1: exact | order 1: exact | |

order 2: not quantitative | order 2: approx. | order 2: not quantitative | |

Dissociation reaction product placement | at reactants, not quantitative | stochastic, for microscopic reversibility | fixed separation, for accurate reaction rates |

User can fix molecular concentrations | no | near surfaces | on surfaces, in compartments |

Location-specific reactions | no | surfaces | surfaces, compartments |

Surface interactions | adsorb: not quantitative | adsorb: not quantitative | adsorb: exact |

desorb: exact | desorb: exact | desorb: exact | |

permeable: not quantitative | permeable: not quantitative | permeable: exact | |

Parallel processing | MPI | MPI | POSIX threads |

Graphics | post-run with pizza.py | post-run with DReAMM | during simulation |

Source code | open, GPL license | closed | open, GPL license |

Benchmark run time | 99 s | 120 s | 47 s |

Computer systems | Mac, Linux | Mac, Linux, Windows | Mac, Linux, Windows |

This article focuses on the latest version of Smoldyn, Smoldyn 2.1. Smoldyn 1.0 embodied several algorithms that were based on Smoluchowski reaction dynamics

(A) Model of

The algorithms that Smoldyn uses, and the program's name, derive from a biophysical description of space and chemical reactions that von Smoluchowski defined in 1917

Smoldyn reads a configuration file that describes the system or cellular process under study. This file lists the system dimensionality, initial numbers of molecules, membrane locations, chemical reactions, and the rules for molecule-surface interactions. It also contains directives for a virtual experimenter, a software agent under the direction of the human researcher, which can measure and manipulate the system during its simulation. For example, at any given point during the simulation run, the virtual experimenter can count the number of particular molecules or add a new surface to represent intrusion of a membrane vesicle into the simulated space. After calculating simulation parameters, Smoldyn performs the simulation with fixed-length time steps, typically set by the researcher to be shorter than characteristic reaction or diffusion timescales (0.1 ms time steps often work well

Smoldyn represents

Surfaces, which might represent biological membranes or the sides of a reaction vessel, are modeled as being infinitely thin and locally smooth. Each surface is composed of

We summarize the core algorithms here and in

The bottom-right panel is a key. The front and back sides of surfaces are noted with ‘F’ and ‘B’, respectively. (A) Diffusion for solution and surface-bound molecules; note that there is no excluded volume. (B) From left to right, interactions between molecules and surfaces are: reflection, absorption, transmission, adsorption, desorption, and surface-state conversion. (C) Zeroth order chemical reaction. (D) First order chemical reaction. (E) In these sequential association and dissociation reactions, _{b}_{u}_{c}

(

(

(

(

(

(^{2}s^{−1} diffusion coefficients and a 10^{6} M^{−1}s^{−1} reaction rate constant have a binding radius of 3.4 nm when a 0.1 ms time step is used). Tournier

(

(

Taken together, this set of algorithms allows Smoldyn to represent most biochemical processes that take place among proteins and small molecules, in 1, 2, or 3 dimensional space and on surfaces and membranes. For example, this capability will bring most cell signaling systems within reach of particle-based modeling. Currently absent, however, from Smoldyn and nearly all comparable simulators are algorithms that specifically address moving or distorting membranes, and the dynamics of biopolymers (including microtubules, actin filaments, and most conspicuously, DNA).

^{2} test, we found no statistically significant differences between data and theory (see

Each panel presents simulation results with points and theoretical results for the same parameters with solid lines. Where present, different shape points represent simulation data for different rates. Inset panels present fluctuations of the same simulation data shown in the main panels, with parameters that are reduced so as to highlight the fluctuations and provide meaningful comparisons between data sets. Here, solid lines represent theoretical expectation values and dashed lines are drawn one standard deviation, analytically calculated from theory, above and below the expectation values. Configuration files and analytic calculations are included with ^{−1} simulation data.

Smoldyn simulation run times scale linearly with the number of simulated molecules (

We compared the run times for ChemCell, MCell, and Smoldyn using identical models of a Michaelis-Menten reaction. These models comprised 10,000 molecules (10% enzyme, 90% substrate initially) and ran for 10 s of simulated time in 1 ms time steps. As

We used Smoldyn's capabilities to explore a simple model. When haploid

(A) A snapshot of the system shown at steady state and with a target cell release of about 4×10^{4} α-factor molecules per second. It is surrounded by a triangulated spherical boundary which absorbs molecules with Smoldyn's “unbounded-emitter” method. The central sphere is a ^{+} and Bar1^{–} cells (see legend). These data fit well to Hill functions with unit cooperativity but with a 5-fold difference in their EC_{50}s. (C) Average gradient of GPCR-α complexes across the surface of the

_{50} by about a factor of 5 (

We simulated systems at each of nine α-factor secretion rates. During the first 100 s of each secretion rate, the systems equilibrated to a nearly steady state, which we assessed using time-dependent concentrations and concentration gradients of all simulated molecular species. Every 2 s for the next 400 s, Smoldyn recorded the number and the mean position of receptor molecules bound to α-factor. We defined the vector that pointed from the

Many features of Smoldyn facilitated the above investigation. The simulations used nearly diffusion-limited reaction rates for the Bar1 protease reaction, which Smoldyn handles accurately; they tracked up to 190,000 molecules at a time, which required high computational efficiency; and they used system boundaries that absorbed molecules so that the distribution of molecules in the bounded system roughly matched the distribution if space extended indefinitely

Many aspects of the biochemical reactions that animate cellular processes are inherently spatial. These include diffusion in complex spaces, sub-cellular localization, and transient membrane associations. Additionally, important cellular processes often rely on molecular species present in low numbers. Computer models that ignore these spatial and stochastic aspects of biological function clearly cannot offer insights into them. For example, non-spatial, non-stochastic models of the

We devised Smoldyn 2.1 to help address the need for accurate and efficient spatial stochastic simulation software. We designed it to simulate molecules and membranes, and events including diffusion, chemical reactions, adsorption, and desorption. We demonstrated the capabilities of Smoldyn with a model of signaling between yeast cells through a diffusible pheromone. The model suggested that, by degrading pheromone, the protease secreted by

On a contemporary laptop computer (2006 MacBook Pro) Smoldyn can perform useful simulations involving assemblages of more 100,000 molecules with relative ease. This power is sufficient to investigate many biochemical systems, including the

However, many challenges to simulations of entire cells and populations remain. First, neither Smoluchowski dynamics nor Green's Function Reaction Dynamics are wholly adequate. For that reason, researchers will need to develop new physical theories for reactions and diffusion in crowded cytoplasms, the mechanical interactions between cytoskeletal filaments and cell membranes, and the functions of extended macromolecular complexes. These theories, which may be partially empirical, need to isolate the essential behaviors of these processes so that they can be modeled. Second, these theories will need to be embodied in algorithms so that modelers can account for the corresponding processes in their cellular models. Third, not all aspects of cellular processes require attention to spatial and stochastic detail (for example, there are likely to be over a million ATP molecules in an

We wrote the core portion of Smoldyn in the C programming language. This core is linked to the OpenGL library for graphics, the libtiff library for saving tiff format images, the libmoleculizer library for rule-based reaction network generation

Additional information for

(0.53 MB PDF)

Details for

(0.61 MB PDF)

Runtime scales linearly with the number of molecules

(0.12 MB PDF)

Bar1 model details

(0.15 MB PDF)

We thank Karen Lipkow, Shahid Khan, and Aki Kusumi for files used for