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
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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
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