Page 1

A Computational Approach to Increase Time Scales

in Brownian Dynamics–Based

Reaction-Diffusion Modeling

ZACHARY FRAZIER and FRANK ALBER

ABSTRACT

Particle-based Brownian dynamics simulations offer the opportunity to not only simulate

diffusion of particles but also the reactions between them. They therefore provide an oppor-

tunity to integrate varied biological data into spatially explicit models of biological processes,

such as signal transduction or mitosis. However, particle based reaction-diffusion methods

often are hampered by the relatively small time step needed for accurate description of the

reaction-diffusion framework. Such small time steps often prevent simulation times that are

relevant for biological processes. It is therefore of great importance to develop reaction-

diffusion methods that tolerate larger time steps while maintaining relatively high accuracy.

Here, we provide an algorithm, which detects potential particle collisions prior to a BD-based

particle displacement and at the same time rigorously obeys the detailed balance rule of

equilibrium reactions. We can show that for reaction-diffusion processes of particles mim-

icking proteins, the method can increase the typical BD time step by an order of magnitude

while maintaining similar accuracy in the reaction diffusion modelling.

Key words: Brownian dynamics, protein-protein interactions, reaction-diffusion.

1. INTRODUCTION

B

Due to advances in experimental technology (including fluorescence imaging, quantitative mass spectros-

copy and cryo electron tomography), quantitative information is increasingly available about the kinetics in

reaction networks of cellular components and alsoabout the molecular organization of living cells. There is a

pressing need to integrate these varied data into spatially explicit, predictive models of biological processes

such as signal transduction, genome separation, and mitosis.

Mathematical and computational modeling has been critical for predicting the systems level behavior of

reaction networks. Typically, mathematical modeling treats the entire system or parts of it as well mixed

solutions with a spatially homogeneous environment, which can be modeled by ordinary or stochastic dif-

ferential equations. However, the cellular environment is highly inhomogeneous (Beck et al., 2011) due to

iological processes, such as the selective nucleo-cytoplasmic protein transport or gene

expression regulation typically involve the intricate relationship of hundreds of cellular components.

Program in Molecular and Computational Biology, University of Southern California, Los Angeles, California.

JOURNAL OF COMPUTATIONAL BIOLOGY

Volume 19, Number 6, 2012

# Mary Ann Liebert, Inc.

Pp. 606–618

DOI: 10.1089/cmb.2012.0027

606

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spatial gradients in the distribution of biomolecules, crowding effects (Ando and Skolnick, 2010; Kim and

Yethiraj, 2010, 2011; Minton, 2001, 2006), and cellular compartmentalization (Delon et al., 2006). Many

cellular processes, such as cell division and nucleo-cytoplasmic transport, are either spatially constrained or

segregated (Terry et al., 2007). Moreover, when biomolecules are present in relatively low copy numbers, their

local concentrations can fluctuate widely, which can cause stochastic effects in reaction processes (Dobrzynski

et al., 2007; Turner et al., 2004). The effective behavior of such molecules may be very different from their

behavior under a constant distribution, which can significantly influence gene regulation (Cai et al., 2006;

Raser and O’Shea, 2005), signal transduction (Kollmann et al., 2005), and many other processes.

Particle based simulations naturally incorporate the concepts of space, crowding, and stochasticity

(Dobrzynski et al., 2007; Dugosz and Trylska, 2011). Those methods treat proteins or other reactants

explicitly, and the time-evolution of particle positions is sampled at discrete time intervals by Brownian

dynamics (BD) simulations (Ermak and McCammon, 1978; Northrup and Erickson, 1992). In BD, the net

force experienced by a particle contains a random element in addition to contributions from interactions with

other particles. The random element is an explicit approximation to the statistical properties of Brownian

forces, due to the effects of collisions with solvent molecules, which are not explicitly modeled. More

specifically, the particles are displaced from their position at each time interval by a random vector whose

norm is chosen from a probability distribution function that is a solution to the Einstein diffusion equation.

To incorporate reactions within a BD framework, reactions occur upon collisions of particles according

to specific probabilities, which are chosen to reproduce the correct reaction kinetics (Morelli and ten

Wolde, 2008). A number of reaction-diffusion algorithms incorporating Brownian dynamics have been

developed with various levels of detail. Some include atomic level details (Gabdoulline and Wade, 1997,

2002), while others have various degrees of coarseness (Andrews, 2009; Andrews and Bray, 2004;

Barenbrug et al., 2002; Boulianne et al., 2008; Byrne et al., 2010; Ermak and McCammon, 1978; Morelli

and ten Wolde, 2008; Northrup and Erickson, 1992; Ridgway et al., 2008; Strating, 1999; Sun and

Weinstein, 2007; van Zon and ten Wolde, 2005). For instance, Morelli and ten Wolde (2008) introduced a

coarse-grained Brownian dynamics algorithm for simulating reaction-diffusion systems that rigorously

obeys detailed balance for equilibrium reactions to omit systematic errors in the simulation.

The disadvantage of particle methods is that often they require relatively small time steps in order to

accurately simulate the dynamics of diffusion and reaction kinetics. The reason lies partly in the ap-

proximations used to derive reaction event probabilities and the incomplete detection of particle collisions,

which prohibit the use of larger time steps. The use of small time steps often prevents reaching simulation

times that are relevant for biological processes. Biological processes occur on a wide range of time scales.

Some proteins may encounter each other in fractions of a millisecond, while others take hours. For instance,

DNA transcription and translation are completed in a few minutes, but these processes are made up of

thousands of small diffusion-limited reactions. It is an ongoing challenge to develop particle-based sim-

ulations that can cover a wide range of time scales while accurately reproducing the properties of diffusion

and reaction networks.

Several methods have been developed to increase simulation time steps. Event-driven simulations, such

as the Green’s function reaction dynamics (GFRD) scheme (Takahashi et al., 2005; van Zon and ten Wolde,

2005), use Green’s function exact expression for one and two particle reaction-diffusion systems to take

large steps in time when the particles are far apart from each other (Takahashi et al., 2005; van Zon and ten

Wolde, 2005). When particle concentrations are low, GFRD is significantly more efficient than traditional

Brownian Dynamics (Takahashi et al., 2005). However, for reactions near surfaces or in crowded cellular

environments, the benefits of GRFD vanish as these methods also require relatively short time steps to omit

unphysical particle overlaps to accurately simulate particle diffusion (van Zon and ten Wolde, 2005).

Here, we describe a Brownian Dynamic algorithm that tolerates increased time steps in comparison to

traditional Brownian dynamics algorithms while maintaining relatively high accuracy in the reaction-

diffusing modeling. Our method builds upon a more accurate description of particle collisions in com-

parison to the traditional Brownian dynamics scheme. One of the causes of simulation errors when using

increasingly larger time steps lies in the treatment of collisions during a BD time step. Typically particle

collisions are considered when particle overlaps are detected after the displacement of the particles in a

time step. Such a procedure underestimates the number of collisions per time steps, as not all potential

collisions of the particles during a time step are considered. With increasing length of the time step this

underestimation becomes more pronounced, which leads to inflated reaction probabilities when parame-

terizing the microscopic reaction rates. Previously, Barenbrug et al. (2002) proposed a method that

BROWNIAN DYNAMICS–BASED REACTION-DIFFUSION MODELING607

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corrected the traditional BD step by augmenting the collision detection with a correction step for non-

overlapping particles based on an analytical term for missed collisions. However, the method did not satisfy

the detailed balance rule for reversible reactions. In contrast, we provide a method, which not only detects

potential collisions prior to a BD move but also determines reaction probabilities consistent with the

detailed balance rule for reversible reactions. We can show that our Reaction Before Move (RBM) method

allows an increase of BD time steps by an order of magnitude while maintaining relatively high accuracy

with respect to analytical solutions of a reaction system.

2. METHODS

To study processes at biologically relevant time scales (in the microsecond to second time range) it is

required to simulate reaction-diffusion systems with relatively large time steps. Here, we describe a

Brownian Dynamic algorithm that allows increased time steps in comparison to traditional Brownian

dynamics algorithms while maintaining minimal impact on accuracy. We begin with an introduction of BD.

2.1. Brownian dynamics

The motion of a particle undergoing diffusion can be described by Einstein’s diffusion equation (Kim

and Shin, 1999),

q

qtP(r‚tjr0‚t0)=D=2P(r‚tjr0‚t0)(1)

where P(r, tjr0, t0) is the probability that the particle will be at position r at time t given the particle was

initially at position r0at time t0. The rate of diffusion is given by the parameter D. This problem can be

solved with the addition of an initial condition (the particle starts at r0) and a boundary condition.

P(r‚t0jr0‚t0)=d(r-r0)

lim

r!1P(r‚tjr0‚t0)=0

The solution that describes the position of the particle P(r, t) at a given time is known as Green’s

function, and for a single isolated particle is the Gaussian distribution associated with a continuous random

walker (Agmon and Szabo, 1990; Rice, 1985).

P(r‚t+Dtjr0‚t)=(4pDDt)-3=2exp

-jr-r0j2

4DDt

!

(2)

When the diffusive motions of multiple particles are simulated, in a traditional Brownian dynamics

algorithm the distribution of particle displacements P(r, t) is sampled for each particle every time step. To

prevent unphysical overlaps between particles, typically individual moves are rejected that would result in

overlaps of the excluded volume of the particles. Because collisions are only determined at the end of each

time step, the length of the time step is naturally limited. The size must be small enough to prevent particles

passing through each other without detecting a collision in a time step propagation.

2.2. Reaction-diffusion modeling

Brownian dynamics simulations cannot only be used to simulate particle diffusion but also the reactions

between them (Morelli and ten Wolde, 2008). Within the Brownian dynamics framework, reactions can be

simulated as follows. A second order reaction, where two species A and B form a product C (e.g. the

formation of a protein complex), can be formulated as:

A+BÐ

kon

koffC(3)

where the konand koffare the macroscopic reaction rates, which are related to the equilibrium(Keq) and the

concentrations of the species by

608 FRAZIER AND ALBER

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Keq=kon

koff

=[A][B]

[C]

(4)

where [A] is the concentration of the species A. This reaction can be decomposed into two events (Agmon

and Szabo, 1990; Morelli and ten Wolde, 2008).

A+B Ð

kD

k-DA ? BÐ

ka

kdC(5)

The first event is the formation of the encounter complex (A$B), when the particles come into contact. In

a typical Brownian dynamics framework, the formation of an encounter complex is naturally simulated

when two particles are close to each other and a trial displacement of one of the two leads to an overlap.

The formation of an encounter complex occurs at the diffusion limited rate kD. For spherical particles, this

rate can be solved analytically (Agmon and Szabo, 1990; Rice, 1985) and is the diffusion limited Smo-

luchowski rate kD= 4pRD, where D is the sum of the diffusion rates, and R is the sum of their radii (Rice,

1985).

In the second step, the encounter complex (A$B) can either advance to form the reaction product C or the

encounter complex can remain dissociated as the individual components A and B. The formation of the

product from the encounter complex occurs by an intrinsic microscopic reaction rate ka. The correct

calibration of this reaction probability accounts for the many forces and rotational motions involved in

progressing from the encounter complex to the reaction products. This reaction rate reflects the time spent

by the two reaction partners achieving an orientation that allows the reaction to progress. When dealing

with proteins, the actual value depends on steering effects, due to electrostatic interactions from charged

residues or dipole moment orientations, but also reflects the size and other physical properties of the

binding sites (Gabdoulline and Wade, 1997, 2002). kamust be appropriately calibrated based on experi-

ments so that physically meaningful reaction rates can be simulated.

The reverse first order reaction (e.g., the dissociation of a protein complex) can also be formalized in a

two-step process. First, an interaction dissociates at an intrinsic microscopic dissociation rate of kd, re-

sulting in the re-formation of the encounter complex of the two unbound proteins still positioned at contact

distance R. In a second step, the individual proteins in the newly formed encounter complex can diffuse into

the bulk solution.

The key point in accurate reaction-diffusion modeling is to correctly relate the microscopic rate constants

to the probabilities that a particle collision leads to a reaction or a reaction product leads to particle

dissociation during a time step. Following Morelli and ten Wolde (2008), the ratio of the microscopic

reaction rates for forward and reverse reactions must be equal to the products of the corresponding reaction

probabilities.

Keq=kon

koff

=ka

kd

=Pcol(r)Pacc

Psep(r)Pdis

(6)

where the individual event probabilities are defined as follows: Pcol(r) is the collision probability. It is the

probability that particles initially separated by r will form an encounter complex during the time step of

length Dt. Paccis the acceptance probability. It is the probability that the reaction will occur to form C given

that the reactants are in an encounter complex. Pdisis the probability that the reaction product C dissociates

to form an encounter complex of particles A$B. The lifetime of a stable complex is modeled with a Poisson

distribution of waiting times and is defined by a dissociation probability such that Pdis= 1 - exp

(-kdDt) & kdDt when kdDt/1 (kdbeing the intrinsic dissociation rate) (Morelli and ten Wolde, 2008).

Psep(r) is the distribution describing the separation distance between the particles A and B after the

dissociation of the reaction product C. To omit systematic errors, the detailed balance in the reaction system

must be satisfied, which states that a reverse move must be generated according to a probability distribution

Psep(r) that is the same as that by which the forward move is generated (Pcol(r)), but properly normalized to

a total probability of 1 (Morelli and ten Wolde, 2008). The normalization factor is the integral of Pcol(r)

over all possible locations r.

Psep(r)=

Pcol(r)

R

jrj>RPcol(r)dr=

Pcol(r)

RPcol(r)r2dr

4pR1

(7)

BROWNIAN DYNAMICS–BASED REACTION-DIFFUSION MODELING609

Page 5

Then the probability of accepting a reaction Paccupon particle collision can be written as

Pacc=ka

kd

Psep(r)Pdis

Pcol(r)

=

kaDt

4pR1

Rr2Pcol(r)dr

(8)

where kais the microscopic association rate, r is the distance between two particles A and B, and R is the

sum of the radii of the two particles.

The key step for accurate reaction modeling is to analytically determine the Pcol(r). The analytical

solution depends on the Brownian dynamics framework used. Morelli and ten Wolde (2008) derived the

solution for the classic Brownian dynamics scheme. At each time step, a trial move for a particle is chosen

based on the Einstein diffusion equation and a potential particle collision detected if the trial move leads to

an overlap with another particle. In this scheme, the collision probability Pcol(r) is the analytically de-

termined probability that two particles initially separated by a distance r will be placed after the time step

Dt at positions that lead to a particle overlap.

However, a substantial number of actual collisions will not be detected by this criterion, as two particles

can also experience collisions during the time step, even if their final positions do not overlap. The method

therefore underestimates the number of collisions. This underestimation increases with the time step, and

results in a simulation error proportional to the square root of the time step.

2.3. Increasing the time step by applying the RBM method

Here, we introduce an alternative Brownian dynamics scheme and determine all event probabilities in

such a way that detailed balance in the reaction process is considered. Instead of checking for overlaps after

a trial move of one of the two particles (i.e., at the end of the time step), we determine the probability of two

particles colliding analytically prior to the particle displacement. Given the positions of the two proteins at

the beginning of a time step, we derive an analytical solution for the probability that they collided during

the time step without explicitly resolving their path. This probability distribution can be derived in analogy

to the analytical solution of particle diffusion around an absorbing sphere (Barenbrug et al., 2002).

The probability of two particles coming into contact during a time step is given by the solution of the

Green’s function for the two-body problem (Kim and Shin, 1999). For two particles A and B diffusing with

a combined diffusion rate of D = DA+ DB, and separated by distance r, there exists a closed form ex-

pression for the probability of the particles coming into contact during a time step of length Dt (Agmon and

Szabo, 1990; Rice, 1985). The probability of collision is

Pcol(r)=R

rerfc

r -R

ffiffiffiffiffiffiffiffiffiffiffi

4DDt

p

??

(9)

where R is the sum of the two particle’s radii and D the sum of the diffusion coefficients for both particles.

The function erfc is the complementary error function which can be defined in terms of the error function

erf:

erfc(x)=1-erf(x)=

2ffiffiffi

p

p

Z1

x

e-t2dt:

(10)

With an expression for Pcol(r), we can now calculate the normalizing factor found in the denominator of

Pacc, which we call N, in order to derive the remaining reaction probabilities:

Z1

N =4p

R

Pcol(r)r2dr =4p RDDt+2R2

ffiffiffiffiffiffiffiffiffi

p

DDt

r

!

:

(11)

The acceptance probability Pacc(r) is now determined:

Pacc(r)=kaDt

N

=

kaDt

4p RDDt+2R2

ffiffiffiffiffiffi

DDt

p

q

??:

(12)

Correspondingly the new distribution with which dissociated particles must be placed while satisfying

the detailed balance condition is calculated as

610FRAZIER AND ALBER

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Psep(r)=R

rNerfc

r-R

ffiffiffiffiffiffiffiffiffiffiffi

4DDt

p

??

(13)

The main outcome of this section is the formulation of new event probabilities (Pcol(r), Psep(r), and Pacc)

that are consistent with the detailed balance rule but incorporate a more accurate description of the collision

frequency in comparison to the traditional BD scheme. As we will demonstrate in the results section, this

formulation will allow an increase in time step length by an order of magnitude while maintaining high

accuracy in the reaction-diffusion kinetics.

2.4. The RBM algorithm

With the derivation of all event probabilities, we can now formulate the modified algorithm for reaction-

diffusion simulations. Our algorithm can be divided into two main parts: First, the detection of particle

collisions and reactions prior to the BD particle movement; and second, the actual particle displacement

according to the diffusion equation post reaction detection (Fig. 1).

2.4.1. Detection of potential reactions for a given configuration.

between particles located within a certain cutoff distance are detected. For these pairs of particles the

collision probability Pcol(r) and acceptance probability Paccof the potential reaction is determined.

First, all potential reactions

2.4.2. Acceptance of potential reactions for a given configuration.

or rejected according to the reaction probabilities. For second-order reactions (A + B/C), the reaction

probability is the product of the collision and acceptance probability (Pcol(r)Pacc) for the corresponding

pairs of particles. For dissociation reactions and other first order reactions, the reaction is accepted with

probability Pdis.

Next, reactions are accepted

2.4.3. Resolving conflicting reactions.

time step. If a particle participates in multiple accepted reactions, only one of these reactions is randomly

chosen, while all other reactions are rejected.

A particle can only be involved in at most one reaction per

2.4.4. Placement of reaction products.

mined. For second-order reactions, (A + B/C), both reactant particles are removed and the product

particle is placed at the location of one of the randomly chosen reactant particles. If the placement is

rejected because of hard-sphere overlap, the reaction is not carried out, and both reactant particles are

restored to their positions at the beginning of the time step. For dissociation reactions (A/B + C), the

reactant particle is removed, and one of the product particles is placed at the reactant particles position.

Then the second particle is placed at a distance from the first product particle according to the probability

distribution Psep(r) so that the detailed balance relationship is fulfilled. If overlap with other particles are

encountered this placement is repeated several times. If after several attempts no location for the second

particle is found, the reaction is rejected, and the reactant is restored, while the product particles are

removed.

Finally, new positions of the reaction products are deter-

2.4.5. Displacement of all non-reacting particles according to the BD scheme.

acting particles have been moved all other particles that did not participate in a reaction or whose

reaction was rejected are displaced according to the Einstein’s diffusion equation. Each particle move

is considered in a random order. Several trial moves are performed if a move resulted in particle

overlaps. If after several attempts, a new position has not been found, the particle is left at its current

position.

After all re-

3. RESULTS

After establishing our approach, we now assess its accuracy and compare its performance with the

traditional reaction-diffusion BD scheme, particularly in view of larger simulation time steps. Examples are

shown for particle radii and diffusion constants typically observed for proteins.

BROWNIAN DYNAMICS–BASED REACTION-DIFFUSION MODELING611

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3.1. Collision probability

We first focus on the differences in the collision probabilities of two particles observed in the traditional

BD and the just discussed RBM approach. When particle collisions are detected during the time step, the

collision probability is significantly increased in comparison to the traditional BD scheme. For instance,

when two equally sized particles are initially placed close to their contact distance (i.e., the sum of their

FIG. 1.

Flowchart of the RBM method.

612FRAZIER AND ALBER

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radii), the collision probability in our approach is more than twice as large as in traditional BD (Fig. 2).

This observation is true also when the initial particle distance is larger then the collision distance. The

increased collision probability has important consequences for calculating the acceptance probability Pacc

and the expected error in simulations with larger time steps. First, an increased collision probability leads to

an increase in the normalization factor N, which in turn influences the acceptance probability Pacc. With

increasing time step Dt, the behavior of N differs dramatically between the traditional BD and the RBM

approach. Whereas in the traditional method the function NBD(Dt) reaches a plateau at already relatively

small time steps, in our approach NRBM(Dt) increases almost linearly with increasing time steps, leading to

a significantly larger slope for the function (Fig. 3A). For example, for particle sizes and diffusion constants

typically observed in proteins, NRBMis about 10 times larger than NBDwhen a time step of 1ns is used (Fig.

3A). As a consequence the resulting acceptance probability PRBM

significantly smaller in our approach (Fig. 3B). Indeed, the function Pacc(Dt) differs dramatically between

the traditional BD approach and our RBM method (Fig. 3B). Most importantly, the values of PRBM

at relatively low values over a wide range of time steps, while PBD

dramatically with increasing time step (Fig. 3B). For instance, for relatively fast reaction rates, PBD

at significantly faster rates and reaches levels above PRBM

acc(Dt) as a function of the time step Dt is

accremain

accin the traditional BD scheme increases

accgrows

acc>0:1 already at time steps of Dt & 0.3. In

1.0

0.8

0.6

0.2

0.4

0.0

1.0 1.2

1.4

r [nm]

1.6

1.8 2.0

RBM

BD

r

Collision probability during a time step

P (r) [a.u.]

col

FIG. 2.

Pcol(r) as a function of initial particle distance between

our RBM and the traditional BD scheme. Pcol(r) is the

probability of observing a collision during a time step

between two particles initially located at a distance r.

(Green line) Pcol(r) in the RBM scheme, where particle

collisions are considered that occur during the time step

and (blue line) the traditional BD scheme, where par-

ticle collisions are detected only at the end of each time

step. The specific example is shown for two particles

with typical parameters of proteins, each with a radius

of 0:5nm and diffusion rate D=0:5 nm2/ns, and a time

step of 0:1ns.

Comparison of the collision probabilities

4

4

3

1

2

0

2.0

1.5

0.5

1.0

0.0

N

[ns]

N( ) [a.u.]

RBM

BD

0.3

0.1

0.2

0

0.4

0.5

Probability of reaction upon contact

2.0

1.5

0.5

1.0

0.0

[ns]

RBM

BD

AB

Pacc

[a.u.]

( )

Normalization factor

FIG. 3.

tor N(Dt) as a function of the simula-

tion time step Dt. (Green line)N as

observed in the RBM method and

(blue line) N as observed in the tradi-

tional BD scheme. (B) The probability

of accepting a reaction upon the colli-

sion between two particles Pacc(Dt) as

a function of the time step (Dt) used in

the simulation. The function Pacc(Dt)

differs dramatically between the tra-

ditional BD approach and our RBM

method. Most importantly, the values

of Paccremain at relatively low values

over a wide range of time steps, while

Paccin the traditional BD scheme in-

creaseswiththelengthofthetimestep.

Simulation parameters are the same as

Figure 2.

(A) The normalization fac-

BROWNIAN DYNAMICS–BASED REACTION-DIFFUSION MODELING613

Page 9

contrast PRBM

(Fig. 3B).

Importantly, due to approximations used in the derivation of the reaction-diffusion formulas, it is

generally considered that the time step Dt must be selected so that Pacc< 0.1 (Morelli and ten Wolde,

2008). Because PRBM

accis sufficiently lower even at larger time steps in comparison to the traditional method,

larger time steps can be tolerated while maintaining high accuracy in the reaction kinetic modeling (Fig.

3B). For instance, for reacting particles with radii of 5nm and D = 1nm2/ns, using Pacc< 0.1 as a limit,

traditional BD schemes would require a time step Dt < 0.03ns, while time steps can be as large as Dt = 2ns

in our approach with sufficiently small Pacc.

acc

reaches a plateau and remains below PRBM

acc<0:05 even at time steps as large as Dt = 2ns

3.2. Assessment of the RBM

Next, we assess the accuracy of our method by comparing our simulations with known analytical

solutions. In the following section, we first focus on the radial distribution functions of two reacting

particles. The integral of the radial distribution function will provide the survival probability of the reaction

partners for a given elapsed simulation time (i.e., the fraction of particles not reacted). Analytical solutions

for both, the radial distribution function and survival probability can be determined for the case of two

reacting particles (Kim and Shin, 1999).

Radial distribution function. The radial distribution function describes the probability of distances be-

tween two reacting particles after an elapsed simulation time. The two particles are initially separated by

their contact distance R and diffuse for an elapsed time. If particles react during this time period, the

simulation is terminated. If after the simulation time the particles have not reacted yet, their final separation

is measured. The distribution of the final separation distances is measured from 30,000 independent

simulations. Comparison of the radial distribution function with the analytical solution (Kim and Shin,

1999) therefore allows assessment of both, the reaction and diffusion of the particles.

We have calculated the radial distribution functions based on simulations with 6 different time steps,

which differ over 6 orders of magnitude (Fig. 4A). We demonstrate that the determined radial distribution

functions and the survival probabilities show excellent agreement with theory, even for the simulations

with relatively very large time steps (see time step Dt = 10) (Fig. 4A). The reaction method we have

described is exact for the two particle case, so all of the curves are very close to the theoretical curve. The

sources of error are mainly due to the placement of particles and can be rationalized as follows. The RBM

method places particles randomly, with equal probability also near other potentially reacting particles. This

situation deviates from the theoretical distribution, which will have potentially reacting particles being

located with a lower probability near other potentially reacting particles, because some fraction of these

particles will have undergone the reaction. Although the reaction calculations are exact, this error in

particle placement is more pronounced when using relatively small time steps, because more time steps are

needed to reach the simulation time. With an increase in the number of time steps, the simulation error

accumulates to larger values. For larger time steps, the error is lower, since a smaller number of time steps

is needed.

Survival probabilities. The integral of the radial distribution function is the survival probability of the

reacting particles. We have plotted the survival probabilities for each time step length and compared it with

the analytical solution (Fig. 4B). The errors in the survival probability are relatively small and range

between <0.1% and <3% (Fig. 4B,C). Interestingly, in the RBM method the errors with respect to the

analytical solution decreases significantly with increasing time steps Dt (Fig. 4C). This behavior is opposite

to the one observed in the BD approach and demonstrate the good performance of the RBM method for

larger time steps.

3.3. Annihilation experiment

Next, we perform a simulation of a diffusion-controlled second order reaction. Consider a reaction

system A + A/; where particles that come into contact instantly react and annihilate each other. As-

suming the particles are distributed at a steady state, there is a closed form solution for the number of

particles that have survived (Barenbrug et al., 2002). We chose 105particles with radius 1nm, which

initially are randomly placed into a box with periodic boundary conditions and diffuse at a rate of

D = 1nm2/ns (Fig. 5). Any collision will lead to a reaction, resulting that both particles are removed from

the simulation. At each time step in the simulation, the number of particles that have survived is calculated.

614FRAZIER AND ALBER

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We now compare the analytical solution of this annihilation reaction with the simulations from the tra-

ditional BD and RBM methods. More specifically, we perform the simulations at three different time steps

of variable length.

Interestingly, for the RBM method the total error between simulation and theory remains almost constant

with increasing time steps (Fig. 5). In contrast, the traditional BD scheme performs better at very small time

steps, however with increasing time steps the error in the traditional BD scheme increases dramatically

20 155 100 2530

r [nm]

Radial distribution functions

0.04

0.00

g(r) [a.u.]

0.02

0.06

0.08

A

2015

5

10

0 0

elapsed time [ns]

Survival probability [a.u.]

0.88

0.92

0.96

1.00

Survival probability with

respect to elapsed time

B

C

0.0001 0.001 0.01

0.1

1.0

10

time step [ns]

% error rate

0

0.5

1.0

1.5

2.0

2.5

3.0

Error in survival probability

as a function of time step

FIG. 4.

particle system generated from simulations from ana-

lytical solution. Simulations are performed for 6 dif-

ferent time steps with length spaning 6 orders of

magnitude. Particle radii and diffusion constant are

chosen as they are observed for typical proteins. Parti-

cles have a radius 0:5nm, and diffusion rate of 0.5nm2/

ns, and react with the intrinsic rate, ka=0:16kD, where

kDis the diffusion limited rate, and corresponds to re-

action upon any contact. The reaction rate is chosen

close to the diffusion-limited rate in order to accent any

deviation from theory. At the beginning of each simu-

lation the particles are placed at contact distance, with

centers separated by 1nm. The elapsed total simulation

time is 20ns. (B) Survival probabilities with respect for

the elapsed simulation time for simulations with 6 dif-

ferent time steps. (C) Relative error rate in the survival

probabilities with respect to the analytical solution

shown for the 6 different simulations and calculated at

the elapsed simulation time of 20ns.

(A) The radial distribution functions for a two

BROWNIAN DYNAMICS–BASED REACTION-DIFFUSION MODELING615

Page 11

leading to errors that are an order of magnitude larger in comparison to the RBM method even at relatively

modest time steps (Fig. 5).

3.4. Diffusion in crowded environments using large time steps

Finally, we investigate the errors in diffusion rates when relatively large time steps are used. The most

challenging scenario is the diffusion of particles under crowded conditions (Dugosz and Trylska, 2011;

Kim and Yethiraj, 2010, 2011; Minton, 2001, 2006). More specifically, we have placed crowding particles

inside a box and measured the time for a central particle to escape a given volume defined by a sphere

(Fig. 6). Simulations are performed for 4 different time steps, of varying length and also at different

crowding levels, ranging from 0% to 20% of total amount of volume occupied by crowding particles (Fig.

6A,B). For each time step 5000 independent simulations are performed each time with a random config-

uration of the crowding particles. From the 5000 simulations, we measure the average time required for a

central particle to escape the crowded environment from its starting location in the center of the crowding

box. Results are then compared for different time steps and crowding levels (Fig. 6B).

As expected, the escape times increase with the crowding level (Banks and Fradin, 2005; Kozer and

Schreiber, 2004; Sun and Weinstein, 2007). Generally, with increasing time steps, the escape time in-

creases. We observe that, for crowding levels between 0% and 15% volume occupancy, increasing the time

step by two orders of magnitude does have only a modest effect on the escape times and hence the motion

of the particles. However, at the highest crowding level of 20% volume occupancy, a more significant

increase of escape times is observed, indicating that errors in particle diffusion must be considered.

FIG. 6.

the diffusion rates of particles in

crowded environments. Shown are

the average elapsed times needed for

a particle of radius 1 nm and a dif-

fusion rate of0:25 nm2=ns to reach a

distance of 10 nm away from its ini-

tialstartinglocation.The escapetime

is measured as the average over 5000

simulations, each time containing

random configurations of crowding

particles at the given crowding level.

Simulations are performed at 6 dif-

ferent crowding levels, which is de-

fined as the total volume fraction

occupied by the crowding particles.

The effect of time step on

200150

50 1000

250 300 350

RBMBD

= 1.00

= 0.10

= 0.01

Theory

10

3

10

4

10

5

Number of particles

Number of time steps

FIG. 5.

react upon collision. Shown are the results of simula-

tions for three different time steps for the traditional BD

method and our RBM method. The analytical solution

for the annihilation reaction is also shown. The tests are

performed with the following parameters: 105particles

with radius 1nm, are placed into a cubic box and pe-

riodic boundary conditions are imposed. The particles

diffuse at a rate of 1.0 nm2/ns.

Annihilation reaction of 10,000 particles that

616 FRAZIER AND ALBER

Page 12

4. CONCLUSION

We have introduced a particle-based reaction-diffusion method for the use of larger time steps in

comparison to traditional BD while maintaining similar accuracy for reaction-diffusion events. Our method

builds upon a more accurate description of particle collisions in comparison to traditional BD. Moreover,

the method obeys the detailed balance rule for equilibrium reactions.

In our method, particle collisions are detected analytically prior to the trial displacements of particles in

BD, which allows the detection of collisions during a time step, which are otherwise missed in the

traditional BD scheme. In particular for longer time steps, this approach allows a more accurate detection of

particle collisions. This procedure therefore increases the collision probability between particles in com-

parison to the traditional BD scheme, which in turn naturally reduces the acceptance probabilities for

reaction events. Because for accurate simulations the time steps must be chosen so that the acceptance

probabilities remain below a cutoff value, our approach naturally increases the range of allowable time

steps for accurate reaction-diffusion modeling. The testing of our approach confirms its applicability. Our

approach therefore provides a step towards the goal of increasing time scales in reaction-diffusion modeling

of biological processes.

ACKNOWLEDGMENTS

We would like to acknowledge Dr. Harianto Tjong and Dr. M.S. Madhusudhan for useful discussions

and Dr. Harianto Tjong for help with numerical integration. This work is supported by the Human Frontier

Science Program (grant RGY0079/2009-C to F.A.), Alfred P. Sloan Research Foundation (grant to F.A.),

NIH (grants 1R01GM096089 and 2U54RR022220 to F.A.), and NSF CAREER grant 1150287 (to F.A.).

F.A. is a Pew Scholar in Biomedical Sciences, supported by the Pew Charitable Trusts.

DISCLOSURE STATEMENT

No competing financial interests exist.

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Address correspondence to:

Dr. Frank Alber

Program in Molecular and Computational Biology

University of Southern California

1050 Childs Way RRI 201B

Los Angeles, CA 90089

E-mail: alber@usc.edu

618FRAZIER AND ALBER