Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2A7, Canada

Department of Biology, Carleton University, Ottawa, ON, K1S 5B6, Canada

Abstract

Background

Realistic biochemical simulators aim to improve our understanding of many biological processes that would be otherwise very difficult to monitor in experimental studies. Increasingly accurate simulators may provide insights into the regulation of biological processes due to stochastic or spatial effects.

Results

We have developed GridCell as a three-dimensional simulation environment for investigating the behaviour of biochemical networks under a variety of spatial influences including crowding, recruitment and localization. GridCell enables the tracking and characterization of individual particles, leading to insights on the behaviour of low copy number molecules participating in signaling networks. The simulation space is divided into a discrete 3D grid that provides ideal support for particle collisions without distance calculation and particle search. SBML support enables existing networks to be simulated and visualized. The user interface provides intuitive navigation that facilitates insights into species behaviour across spatial and temporal dimensions. We demonstrate the effect of crowing on a Michaelis-Menten system.

Conclusion

GridCell is an effective stochastic particle simulator designed to track the progress of individual particles in a three-dimensional space in which spatial influences such as crowding, co-localization and recruitment may be investigated.

Background

One of the main goals of computational cell biology aims to accurately simulate large biological systems at molecular resolution. Stochastic effects and spatial constraints are increasingly being recognized as important factors in the normal functioning of molecular networks

We developed GridCell to simulate biological models with specific consideration for stochasticity, locality, and collision. GridCell is based on a simplified model for molecular movement and interaction. It uses a discrete three-dimensional cubic grid based on the D3Q27 model often used in the application of the Lattice-Boltzmann Method (LBM)

D3Q27 cubic grid structure

**D3Q27 cubic grid structure**. The 27 possible locations of a D3Q27 cubic grid that a given voxel can access.

The simulation space is visualized via a 3D interface and 2D graphs, and surface plots summarize molecule concentrations over space and time. GridCell supports models specified by the Systems Biology Markup Language (SBML) (please see Availability & requirements for further information). SBML models can be obtained from public repositories such as EBI's Biomodels database (please see Availability & requirements for further information) or designed using software such as SBMLeditor or JDesigner (please see Availability & requirements for further information).

Implementation

Algorithm

The simulation employs a two-phase process in which particles (1) attempt to move and then (2) attempt to react every turn.

Movement Phase

A particle can move at most once per timestep. Since a particle only has access to its immediate surrounding, a particle can only move in one of 27 nearest locations, including the current location. The selection of the movement direction is made randomly; therefore the particles follow a Brownian random walk. Figure

Random Brownian walk

**Random Brownian walk**. Random walks of 4 different particles in GridCell after 1000 timesteps.

Diffusion

Particles following a Brownian random walk should also follow the well-known Einstein-Smoluchowski equation

^{2}> = 2

where <^{2}> is the mean-square displacement, ^{2}> (averaged over 1000 iterations) versus the number of timesteps _{ts }when the probability of movement of the particles at every timestep is equal to 1. As expected for an uncrowded case, the mean-square displacement increases linearly with the number of timesteps. This leads to the following relation

Diffusion in GridCell

**Diffusion in GridCell**. Mean-square distance (over 1000 iterations) of particles versus the number of timesteps in GridCell.

^{2}> = _{ts},

where _{vox }is the length of the sides of the voxels in meters and _{ts }is the length of the timestep in seconds, we get

Since the probability of movement at each timestep of the particle is equal to 1, _{max }that GridCell can support for a given timestep and voxel size. This upper limit on diffusion speed is caused by the design decision of restraining particle movement to its immediate neighbourhood (the D3Q27 grid). By calculating the slope of the graph and setting the dimensionality _{max }can be calculated as

_{max }= 0.335_{vox}^{2}/_{s}.

Smaller diffusion speeds are simulated by applying a different probability of movement such that

_{m}_{max},

where _{m }is the probability of movement of a particle at every timestep. As long as the diffusion speeds of the particles are smaller than _{max}, diffusion will be modeled correctly. If a larger diffusion speed is needed, one can reduce the timestep or increase the size of the voxels.

Reaction Phase

A particle may react only once per turn and only with its immediate surrounding. The reaction phase is completely independent from the movement phase, therefore it does not matter if a particle previously moved or collided with another particle. Common interactions include aggregation events such as molecular complex formation/dissolution or conversion events such as chemical reactions. Only the simplest reactions involving 3 or less particles are directly supported. Complex reactions involving more than 3 particles are decomposed into several elementary reactions. The probability of reaction per timestep is derived from the overall rate of reaction and is very similar to the approach taken by ChemCell

Both reactions have a forward rate of reaction ^{-1 }and timestep is ^{-kt}). Considering each particle individually, each particle has a probability equal to 1 - ^{-kt }to react during each timestep. In our stochastic model, a uniform random number _{n }between 0 and 1 is generated for each particle, and the reaction occurs if _{n }< 1 - ^{-kt}. In a reaction with only 1 reactant and 1 product, the reactant is simply replaced by the product. In a reaction with 1 reactant and 2 products, a search is first conducted in the surrounding area. If there is at least 1 free voxel in the surrounding area of the particle, the reaction takes place, and the second product is positioned in that free location while the first product is placed at the position of the initial reactant. The reaction is blocked if no free position is found. This limitation only modifies the overall reaction rate of the reaction in a situation where the whole cell is completely filled which would prevent any movement and reaction to take place.

Consider the following reaction with 2 reactants:

with a rate constant ^{-1 }and a timestep between each iteration of _{a }particle of type _{b }particle of type _{v}, then the total number of reactions _{r }in a well-mixed system is given by

On average, the desired number of reactions in our system should be equivalent to the result of the above equation. In our system, particles can only react with their immediate surrounding locations. In a well-mixed system, the number of _{a}_{b}_{c}/_{c }is the volume of the cube containing the 26 "neighbouring" voxels and _{r }= _{r }= _{a}_{b}_{v}

The formula is independent of _{a }and _{b }as expected. Also, for a given rate constant _{c }such that _{c}) has to be selected. A random number _{n }is generated. If _{n }<

When 2 reactants of the same species form a product such as

the individual rate of reaction of particle A needs to be modified to ensure that the overall rate of reaction is respected since each reactant will attempt to react with the other one. Assuming

More complex reactions are implemented by creating a cascade of several elementary equations. This process, done automatically by the software, will break the complex reactions into a series of simpler reactions by introducing "temporary" species. For example, consider the following reaction with 1 reactant and 5 products.

where ^{-1}. For each product exceeding 2, a temporary species is created. In this case, 3 temporary species are created. It follows that the reaction is broken down into:

_{1 }→

_{2 }→

_{3 }→ _{1}

_{2 }+ _{3}

where _{1}, _{2 }and _{3 }are respectively the first, second and third temporary species. By setting the rate of reaction of equation 17 equal to

Shown below is the case where more than 2 reactants merge into a single product:

The procedure is similar to the previous case, 1 temporary species is created for each reactant above 2.

_{1},

_{2},

_{1 }→ _{3},

_{2 }+ _{3 }→

where _{1}, _{2 }and _{3 }are respectively the first, second and third temporary species. In order to obtain the same overall probability of reaction and to reduce the impact of the temporary species on the system to a minimum, the probability of reaction of any reaction containing temporary species on the reactant side (equation 21 and 22) is set to 1. Assuming that _{1 }and _{2 }are the probability of the first and second simple reactions _{1 }and _{2 }then, we set _{1}_{2}. We also set _{1 }= _{2}. Equating the 2 equations gives _{1 }= _{2 }=

In general, the probability of the simple reactions _{n }containing no temporary species is equal to

where _{reactants }is the number of reactants of the initial reaction. Each temporary particle has a parameter

Reversible reactions are handled by creating 2 different separate reactions, 1 for the forward reaction with the forward reaction rate and 1 for the backward reaction with the corresponding backward reaction rate. Assuming the following reaction

with forward reaction rate _{f }and backward reaction rate _{b}. This reversible reaction is then split into

with a reaction rate _{f }and

with reaction rate _{b}.

Temporary particles involved in a reversible reaction have a flag mentioning if they are participating in a forward or backward reaction such that they can revert back to the proper reactants when their lifetime reaches zero.

Performance analysis

Preliminary tests have been conducted to determine how the software reacts to different system sizes. The tests have been executed on a stand-alone microprocessor: a 3.2 GHz P4 with 2GB of RAM. The current algorithm is computed serially. As it can be shown in Table ^{7}. Table

GridCell performance versus system size

Number of Voxels

1e3

1e4

1e5

1e6

Number of Particles

3e2

3e3

3e4

3e5

Time (s)

1.62e-4

1.58e-3

1.6e-2

1.7e-1

Time required to compute a timestep versus the size of a simulation.

GridCell performance versus number of voxels

Number of Voxels

1e3

1e4

1e5

1e6

Time (s)

1.6e-5

1.6e-4

2.14e-3

2.06e-2

Time required to compute a timestep versus the number of voxels. No particles in the system

GridCell performance versus number of particles

Number of Particles

1e3

1e4

1e5

5e5

Time (s)

21.3e-2

26.4e-2

68.1e-2

22.8e-1

Time required to compute a timestep versus the number of particles. The system contains 1e6 voxels.

GridCell performance versus the average number of reactions

Average Number of Reactions

0

16.5e3

29.5e3

39.5e3

47.5e3

Time (s)

7.4e-2

7.3e-2

7.25e-2

7.2e-2

7.1e-2

Time required to compute a timestep versus the average number of reactions occuring at each timestep. The system contains 1e6 voxels and 1e5 particles.

User Interface Features

The rendering is implemented in OpenGL, and most user-interface functions are written using the PLIB library, which is available online

GridCell user interface

**GridCell user interface**. GridCell user interface with (a) menu, (b) 3D space, (c) species panel, (d) 2D plot of concentration versus time, and (e) 2D surface plot of concentration versus space. Simulated model involves the translocation of particles through a membrane with embedded enzymes.

The menu system (Figure

A key feature of the GridCell user interface is the ability to interact with the three-dimensional simulation volume (Figure

The species panel (Figure

Finally, two plots to summarize particle concentrations with respect to time (Figure

Results and discussion

Michaelis-Menten reaction

The Michaelis-Menten equations are used to describe most enzymatic reactions. Its kinetics is given by the following equation:

The enzyme _{1}. _{2}, or reverts back to its original form _{r}.

Crowding

One of the main differences between GridCell and other simulators is its ability to simulate particle crowding. Molecular crowding occurs when particle density affects movement and reactivity. Crowding is typically ignored in most models since kinetics are often based on controlled, in vitro conditions that are not crowded. In addition, simulators do not typically support this feature since it is computationally expensive to keep track of all particle positions and their excluded volume, and to implement collision-detection algorithms. Some simulators (e.g. Smoldyn ^{-20 }litres, this amounts to approximately 30000 inert particles per step of 10%.

Simulation parameters

Volume (litres)

10^{-14}

Number of

3000

Number of

1000

_{1 }(^{-1}^{-1})

10^{7}

_{2 }(^{-1})

1

_{r }(^{-1})

1

Simulation time (s)

10

Timestep (s)

10^{-3}

Parameters of the Michaeles-Menten simulation.

Effect of crowding on Michaelis-Menten product formation using GridCell

**Effect of crowding on Michaelis-Menten product formation using GridCell**. Effect of increasing the number of inert particles on product formation of a Michaelis-Menten system using GridCell. The mean has been calculated over 20 iterations. Percentage of voxels occupied by inert particles.

Interestingly, the maximum rate of reaction is obtained when the inert particles occupy 20% of the volume, which agrees with the fact that macromolecular crowding may enhance reaction rates, as the particles have to search a smaller volume to find each other

Related Work

GridCell is related to a family of Monte Carlo (MC) simulators (Table

Spatial simulators

GridCell

SmartCell

MesoRD

Cell++

MCell

Smoldyn

ChemCell

Molecule representation

Particle

Population

Population

Large particles and populations of small particles

Particle

Particle

Particle

Stochastic

Yes

Yes

Yes

Large particles only

Yes

Yes

Yes

Space

Discretized

Discretized

Discretized

Continuous (large particles) and discretized (small particles)

Continuous

Continuous

Continuous

Particle-collision support

Yes

No

No

No

No

No

No

Diffusion support

Yes

Yes

Yes

Yes

Yes

Yes

Yes

SBML support

Yes

Yes

Yes

No

No

No

No

Web availability

Yes

Yes

Yes

Yes

Yes

Yes

No

Comparison and description of various simulators.

Future Directions

GridCell performance is tightly linked to the number of voxels in the simulation space. The simulator can currently support a maximum of 10^{7 }voxels/particles which is not enough to simulate at a molecular resolution structures as complex as a complete cell, the long term goal of GridCell. However, the simple and regular algorithm of GridCell, which does not require any searches or complex operations, is a prime candidate for acceleration by parallelization to achieve performance speedup and simulate large-scale systems.

Conclusion

GridCell is a stochastic simulator that uses a 3D grid and accounts for locality, very low concentration stochastic effects and particle collisions. Its user-interface makes it easy to use while providing several tools to analyze the system. GridCell reproduces the results obtained with ODEs and the Stochastic Simulation Algorithm (SSA) for simple systems when crowding and locality do not affect the system. We also show that particle collisions can have a significant impact on the speed of reaction and that the well-mixed assumption and dimensionless particles can induce a significantly different response in a biological system. The discrete 3D grid and the nearest-neighbour interactions remove the need to do any distance calculation, particle search and floating-point arithmetic. The regularity and simplicity of the algorithm makes it a good candidate for acceleration with a parallel architecture which will open the door to the simulation of even more complex systems.

Availability and Requirements

The software is available at

Authors' contributions

LB designed and programmed the GridCell simulator. SAA contributed to the programming of the GridCell simulator and its graphical user interface. MD and WJG contributed to the conception and design of the GridCell simulator. LB and SAA drafted the manuscript and MD and WJG revised the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors gratefully acknowledge funding provided by the Natural Sciences and Engineering Research Council of Canada.