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Open AcceSoftwareBiochemical Network Stochastic Simulator (BioNetS): software for stochastic modeling of biochemical networksDavid Adalsteinsson*1, David McMillen2 and Timothy C Elston1

Address: 1Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA and 2Department of Chemical and Physical Sciences, University of Toronto at Mississauga, Mississauga, ON L5L 1C6, Canada

Email: David Adalsteinsson* - david@amath.unc.edu; David McMillen - mcmillen@utm.utoronto.ca; Timothy C Elston - telston@amath.unc.edu

* Corresponding author

AbstractBackground: Intrinsic fluctuations due to the stochastic nature of biochemical reactions can have largeeffects on the response of biochemical networks. This is particularly true for pathways that involvetranscriptional regulation, where generally there are two copies of each gene and the number ofmessenger RNA (mRNA) molecules can be small. Therefore, there is a need for computational tools fordeveloping and investigating stochastic models of biochemical networks.

Results: We have developed the software package Biochemical Network Stochastic Simulator (BioNetS)for efficiently and accurately simulating stochastic models of biochemical networks. BioNetS has a graphicaluser interface that allows models to be entered in a straightforward manner, and allows the user to specifythe type of random variable (discrete or continuous) for each chemical species in the network. Thediscrete variables are simulated using an efficient implementation of the Gillespie algorithm. For thecontinuous random variables, BioNetS constructs and numerically solves the appropriate chemicalLangevin equations. The software package has been developed to scale efficiently with network size,thereby allowing large systems to be studied. BioNetS runs as a BioSpice agent and can be downloadedfrom http://www.biospice.org. BioNetS also can be run as a stand alone package. All the required files areaccessible from http://x.amath.unc.edu/BioNetS.

Conclusions: We have developed BioNetS to be a reliable tool for studying the stochastic dynamics oflarge biochemical networks. Important features of BioNetS are its ability to handle hybrid models thatconsist of both continuous and discrete random variables and its ability to model cell growth and division.We have verified the accuracy and efficiency of the numerical methods by considering several test systems.

BackgroundMathematical modeling of complex biological networkshas a lengthy history [1-5]. In the past, the standardapproach for modeling these systems has been to deriveordinary differential equations (ODEs) based on the lawof mass action for the concentrations of the biochemicalspecies involved in the network [6-16]. Experimentalstudies [17-19] have demonstrated, however, that sto-

chastic effects can be significant in cellular reactions, par-ticularly in the case of transcriptional regulation, wheregenerally there are two copies of each gene and thenumber of messenger RNA (mRNA) molecules can besmall. A number of recent experimental and modelingstudies have addressed the role of fluctuations in geneexpression [20-31]. Many modeling studies haveemployed the well-established Gillespie Monte Carlo

Published: 08 March 2004

BMC Bioinformatics 2004, 5:24

Received: 07 November 2003Accepted: 08 March 2004

This article is available from: http://www.biomedcentral.com/1471-2105/5/24

© 2004 Adalsteinsson et al; licensee BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL.

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algorithm [32] or one of its more recent variants [33,34].These algorithms offer an exact solution to the stochasticevolution of chemical systems, but they are computation-ally very expensive. A much more efficient approach is toapproximate the species as continuous variables and for-mulate the problem in terms of stochastic differentialequations (SDEs), often referred to as chemical Langevinequations [24,28,35]. This approximation works remark-ably well for many cases, even when the number of parti-cles involved is as small as ten, and the resultingsimulations can run orders of magnitude more quicklythan the discrete Monte Carlo approach. In other cases,when some or all of the particle numbers are very small,the system may need to be modeled using the discreteapproach, or a hybrid method in which some species aretreated discretely while others are evolved using the con-tinuum approximation. With the increasing interest informulating accurate models of large biochemical net-works, there is a need for reliable software packages thatcorrectly incorporate stochastic effects, yet are fast enoughto simulate large interconnected sets of reacting species(as found, for example, in signaling cascades or geneticregulatory networks). We have developed the BIOchemi-cal NETwork Stochastic Simulator, "BioNetS," to meet thisneed. BioNetS is capable of performing full discrete simu-lations using an efficient implementation of the Gillespiealgorithm. It is also able to set up and solve the chemicalLangevin equations, which are a good approximation tothe discrete dynamics in the limit of large abundances.Finally, BioNetS can handle hybrid models in whichchemical species that are present in low abundances aretreated discretely, whereas those present at high abun-dances are handled continuously. Thus, the user can pickthe simulation method that is best suited to their needs.All aspects of the software are highly optimized forefficiency.

The remainder of this manuscript is arranged in the fol-lowing way. In the Implementation section, the mathe-matical background for the Gillespie method, chemicalLangevin equations and hybrid models is presented,along with a discussion of the numerical algorithms usedin BioNetS. Under Results and Discussion we provide abrief introduction to BioNetS along with several exam-ples. The examples serve two purposes: 1) to illustratehow to use the software and 2) to verify its efficiency andaccuracy. More complete documentation can be found athttp://x.amath.unc.edu/BioNetS, and in the documenta-tion included with the package.

ImplementationWe first develop the mathematical methodology onwhich BioNetS is built. Readers interested in using Bio-NetS without going into its underlying structure can pro-ceed directly to the Results and discussion section.

Discrete reactions and the gillespie algorithmBioNetS makes use of elementary reactions (zeroth, firstand second order). The following examples illustrateseach type of reaction:

In the above reactions, the calligraphic letters denote asingle molecule of a chemical species. The number of mol-ecules of a particular species in the system at time t isdenoted with uppercase letters (e.g., A(t), B(t), A_B(t),and V(t)). All the rate constants, γ, δ, and k1-k6, have unitsof per time. Eq. 1 represents a process in which a moleculeA is produced when the reaction proceeds in the forwarddirection and is degraded in the reverse direction. In theforward direction the reaction is zeroth order and pro-ceeds with an average rate of γ. In the backward direction,the reaction is first order, and the average rate ofdegradation is δA(t). The forward reaction in Eq. 2 repre-sents a process in which chemical species A is converted tospecies B. In this case A and B might represent two differ-ent conformations of the same molecule. In Eq. 2 both theforward and backward reactions are first order because thereaction rates are proportional to the respective concentra-tions. The forward reaction given in Eq. 3 is a second orderreaction in which an A molecule and a B molecule cometogether to form the complex A_B. The average rate for thereaction is k1A(t)B(t). The backward reaction is a firstorder reaction in which A_B dissociates at an average rateof k2A_B(t). In Eq. 4 the forward reaction produces a mol-ecule V. The difference between this reaction and the for-ward reaction in Eq. 1 is that the average rate is k3V(t). Thisleads to exponential growth of V(t). This reaction is par-ticularly useful if V(t) is interpreted as the cell volume. Inthe backward reaction, two V molecules come togetherand degrade one of the V molecules. The average rate forthis reaction is k4V(t)(V(t) - 1). The V(t) - 1 term arisesbecause two of V(t) molecules must be chosen to react.This type of term also arises in reactions that producehomodimers. This reaction eventually stops the exponen-tial growth of V. The net effect of these two reactions is toproduce logistic growth. The total average reaction rate forthe set of reactions given in Eqs. 1–4 is

∅ γδ� ( )1

� �k

k1

22( )

� � � �+ k

k3

43_ ( )

� � �k

k5

64+ ( )

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where Fi and Bi are the average forward and backwardrates, respectively, for the ith reaction.

For the rest of this section, we assume that the volume ofthe cell is not changing and only consider Eqs. 1–3. In theExamples we consider a case in which the volume ischanging. If A(t), B(t) and A_B(t) are present in largenumbers, then the law of mass action can be applied toderive equations that govern the concentrations [A]= A(t)/V, [B]= B(t)/V and [AB]= A_B(t)/V, where V is the cell vol-ume. These equations are

The primed rate constants indicate that they have beenappropriately scaled by the volume (i.e, k'3= k3V and γ' =γ/V), and, therefore, have units of either per time per con-centration or concentration per time. Note that to convertto units of molar, we also have to appropriately scale therate constants by Avagadro's number. Eqs. 6–8 represent amacroscopic description of the process, because theyignore fluctuations in the concentration that arise fromthe stochastic nature of chemical reactions.

In general, A(t), B(t) and A_B(t) are random variables thattake on any nonnegative integer value. The Gillespie algo-rithm [32] can be used to generate sample paths of theprocess. This algorithm assumes that the random time∆Ti, between the ith and i + 1 reaction, is exponentiallydistributed. For the simple example given by Eqs. 1–3, themean waiting time between reactions, which characterizesthe exponential distribution, is µ∆Ti = γ + δ A(ti) + k1A(ti) +k2B(ti) + k3A(ti)B(ti) + k4A_B(ti), where ti is the time atwhich the ith reaction occurred. Therefore, ti+1 = ti + ∆Ti.Once the time at which the next reaction occurs has beendetermined, the following probabilities are used to deter-mine which reaction occurred:

Once the reaction has been determined, the chemical spe-cies are updated accordingly. As discussed in the Numeri-cal methods section, BioNetS uses an efficientimplementation of the Gillespie algorithm [33].

Another description of discrete stochastic processes isachieved through use of the master equation that governshow the probabilities of the various random variables inthe process evolve in time. Let pa, b,a_b(t) = Pr [A(t) = a, B(t)= b, A_B(t) = a_b], then Pa,b,a_b(t) satisfies the masterequation

The master equation is the starting point for deriving var-ious approximate schemes for describing the system [28].In the next section, we discuss an approximate schemethat is valid in the limit of large, but finite molecule num-bers. The simplest approximation scheme is achieved byconsidering the first moments of the process. We will useover bars to denote averaging. For example,

. Eq. 15 can be used to derive

equations that govern the time evolution of all the firstmoments. Because of the second order reaction in Eq. 3,the equations for the means are coupled to the secondmoments. In fact, the nth moment equations containterms that involve the n+ l moments. Thus, there is no clo-sure to the system. The simplest closure scheme is to

assume that all moments factorize (e.g., ). Thisrepresents the macroscopic limit in which fluctuations areignored. In this limit, we recover Eqs. 6–8 from the masterequation.

µ γ δ( ) ( ) ( ) ( )

( ) ( ) _ ( ) ( ) (

t A t k A t k B t

k A t B t k A B t k V t k V

= + + + ++ + +1 2

3 4 5 6 tt V t

F Bii

i

)( ( ) )

( ) ( )

−

= +=∑

1

51

4

d A

dtk B k A k AB k B

[ ]( [ ] )[ ] [ ] [ ] ( )= − ′ + + + + + ′3 1 4 2 6δ γ

d B

dtk A k B k A k AB

[ ]( [ ] )[ ] [ ] [ ] ( )= − ′ + + +3 2 1 4 7

d AB

dtk A B k AB

[ ][ ][ ] [ ] ( )= ′ −3 4 8

Pr[ ] ( )∅ =γ γµ

�

∆Ti

9

Pr[ ]( )

( )�δ δ

µ ∅ =

A ti

Ti∆10

Pr[ ]( )

( )� �k i

T

k A t

i

1 1 11 =µ∆

Pr[ ]( )

( )� �k i

T

k B t

i

2 2 12=µ∆

Pr[ _ ]( ) ( )

( )� � � �+ =k i i

T

k A t B t

i

3 3 13µ∆

Pr[ _ ]_ ( )

( )� � � �k i

T

k A B t

i

4 4 14+ =µ∆

dp

dtk a k b k ab k a b p pa b a b

a b a b a b a, , _

, , _ , , _[ ( ) _ ]= − + + + + + + −γ δ γ1 2 3 4 1 bb a b a b

a b a b a b a

a p

k a p k b p

+ +

+ + + +

+

+ − − +

δ( )

( ) ( )

, , _

, , _ , , _

1

1 1

1

1 1 1 2 1 1 bb

a b a b a b a bk a b p k a b p+ + + + ++ + − − − +3 1 1 1 4 1 1 11 1 1 15( )( ) ( _ ) ( ), , _ , , _

A t ap ta b a ba b a b( ) ( ), , _, , _

= ∑

A A2 2=

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The diffusion limit and the chemical langevin equationsThe general form of the master equation for a system con-sisting of N chemical species and M reactions is

where n is a N-dimensional vector of species numbers, Fiand Bi are the backward and forward rates for the ith reac-tion, and the vectors δi contain the stoichiometric con-stants for the ith reaction. For the simple model given byEqs. 1–3, N = 3, M = 3, and pn(t) = Pr[A(t) = n1, B(t) = n2,and A_B(t) = n3]. The forward and backward rates are F1 =γ, B1 = δn1, F2 = k1n1, B2 = k2n2, F3 = k3n1n2, and B3 = k4n3.The δi vectors are the rows of the stoichiometric matrix

The (i,j) element in the above matrix represents thechange in the jth chemical species when the ith reactionproceeds in the forward direction.

If the molecule numbers are large as compared to 1, thenthe master equation Eq. 16 can be approximated by thecontinuous process [28,35]

where

This result can be derived in several ways. One method isto note that Eq. 15 represents a second order finite differ-encing of Eq. 18, with a grid size of 1. Another method isto make use of the shift operator

where f(n) is an arbitrary smooth function and for ourpurposes k is an integer. If the shift operator is used in Eq.15, the diffusion limit is achieved when the Taylor seriesexpansion given in Eq. 21 is truncated at j = 2.

Sample paths consistent with Eq. 18 can be generatedusing the following set of SDEs

where the wk(t) are independent Gaussian white noiseprocesses. These equations are often referred to as thechemical Langevin equations. For Eqs. 1 – 3, the explicitform of the SDEs are

BioNetS generates numerical solutions to the SDEs givenby Eq. 22 using either an explicit or semi-implicit Eulermethod. The form of these methods is

where ε = 0 for the explicit method and ε = 1 for the semi-implicit method and the Zk(t) are independent standardnormal random variables. The advantage of using thechemical Langevin equations is that in the appropriateparameter regime, numerical solutions to the set of SDEsgiven by Eq. 22 can be generated much more efficientlythan using the Gillespie algorithm. We expand upon thispoint in the Examples section. Higher order numericalalgorithms for SDEs are available [36], but the noise struc-ture of the chemical Langevin equations makes theseschemes very cumbersome to implement. In the Exam-ples, we verify that the Euler methods given by Eq. 26 aresufficient to produce reliable results. We note that the ∆matrix is generally sparse, and BioNetS takes advantage ofthis sparseness to optimize the efficiency of the two Eulermethods (see Numerical Methods, below).

Hybrid schemesIt is often desirable to allow some of the chemical speciesto be treated as continuous random variables and some tobe treated discretely. This is particularly true for the case oftranscriptional regulation by transcription factors. In thissituation there can be as few as one DNA/transcriptionfactor binding site and mRNA abundances can be as small

dp

dtF B p F p B pi i

i

M

ii

M

ii

M

i in

n n n= + + +=

−=

+=

∑ ∑ ∑( ) ( )1 1 1

16δ δ

∆∆ = −− −

1 0 0

1 1 0

1 1 1

17( )

∂∂

= − ∂∂

+ ∂∂ ∂

ρ ρ ρ( , )( ) ( , ) ( ) ( , ) ( ),

,

nn n n n

t

t nA t D t

jj

n nj k

j k

N

j k

12

182

∑∑∑j

N

A F Bj j i i ii

M( ) ( ) ( ),n = −∑∆ 19

D F Bj k i j i k i ii

M

, , ,( ) ( ) ( )n = +∑∆ ∆ 20

f n k kn

f nk

j nf n

j j

jj

( ) exp( ) ( )!

( ) ( )+ = ∂∂

= ∂∂=

∞∑ 21

0

dN

dtA F B w t

jj k j k k k

k

M= + +

=∑( ) ( ) ( ) ( ) ( ),N N N∆ 22

1

dA

dtk k B A k B k A B A w t

k A k Bw t k AB

= − + + + + + ′ + +

− + −

( ) _ ( )

( )

1 3 2 4 1

1 2 2 3

δ γ δ γ

++ k A Bw t4 3 23_ ( ) ( )

dB

dtk k A B k A k A B k A k Bw t

k AB k A Bw t

= − + + + + +

− +

( ) _ ( )

_ ( ) ( )

2 3 1 4 1 2 2

3 4 3 24

dA B

dtk A B k AB k AB k A Bw t

__ _ ( ) ( )= − + + +4 3 3 4 2 25

N t t N t tA t t

t F t B t Z t

j j j

k j k k k

( ) ( ) ( ( ))

( ( )) ( ( )) ( ),

+ = + +

+ +

∆ ∆ ∆

∆ ∆

N

N N

ε

(( )261k

M

=∑

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as 10 or fewer. In contrast, protein abundances can be inthe thousands. The technical difficulty with implementinghybrid schemes that include both discrete and continuousrandom variables is that the Gillespie method requiresconstant transition rates between reactions. This may notbe the case, if some of the chemical species are evolvingcontinuously in time. BioNetS overcomes this problem inone of two ways.

Let Nd <N be the number of discrete chemical species andMd ≤ M the number of reactions that produce a change inone of the Nd chemical species. The overall reaction rate attime tj for the discrete set of chemical species is

If the time step ∆t for the SDEs is small enough such that

then pt is approximately the probability of a transition in∆t. In the above equation ε is a user specified tolerance.The probability of two discrete transitions in ∆t is propor-tional to (∆t)2. Choosing ε < 0.1, which means the proba-bility of two reactions in ∆t is less than 0.01, generallyproduces good results. However, this should be verifiedon a case by case basis. At each time step, BioNetS checksto verify that Ineq. 28 is satisfied for the specified ε. If so,a uniform random number R is generated and comparedagainst pt. If R < pt, then a transition occurred and the con-ditional probability R/pt is used to determine which of thediscrete transitions occurred. If pt > ε, then the discretereactions determine the fastest time scale in the system. Inthis case the Gillespie algorithm is used to update the dis-crete reactions, and the random time step ∆tj is used toupdate the SDEs.

A pseudo-code description of the above algorithm is dis-played in Table 3:

Numerical methodsBioNetS generates code that is tailored to efficiently simu-late biochemical reactions. The optimization techniquesused by BioNetS allows the software to simulate large sys-tems in reasonable times without requiring high-endcomputational hardware.

Techniques used to optimize the Gillespie method are:

• For the discrete variables, the program uses data struc-tures that allow only the chemical species and reactionrates that are affected by the current reaction to beupdated.

• A bisection search is used to determine which reactionoccurred.

The code has both an explicit and a semi-implicit solver,for simulating the chemical Langevin equations. The userspecifies at runtime which method to use. By default thesemi-implicit solver will be used. The semi-implicit solveruses Newton's method to solve the implicit equations,and for that the program needs to compute the Jacobianand solve a linear system at each iteration. For updatingthe chemical Langevin equations and hybrid models opti-mization techniques include:

• The sparse nature of the stoichiometric matrix is used toefficiently store and per form matrix operations.

• After every reaction, only the species and reaction ratesaffected by that reaction are updated. This can be seen inthe Rates.cpp file, where all the different cases have beenworked out and written for optimal execution speed.

• The Jacobian is sparse, and the code takes full advantageof this fact. The program solves and factorizes the Jacobianusing sparse methods. Before the code generation, Bio-NetS computes the entries in the Jacobian symbolicallyand finds a permutation that decreases the number of fill-ins during the LU factorization. As a result, no zero entriesare saved, and the sparse structure is fully exploited. Thesparse structure is then used in the LU solve. In the code,no pivots are visible, and no if-statements are left.

Results and discussionIn this section we present several examples which serve asillustrations of how to use BioNetS and test the accuracyand efficiency of the numerical methods. One particularconcern is the accuracy of the Euler methods. While these

methods are only of order , we show that when theapproximations that lead to the chemical Langevin equa-tions are valid, the difference between the numerical solu-tions of the SDEs and the exact discrete Gillespie methodare negligible. Currently, the graphical user interface toBioNetS runs on the Macintosh OS X operating system,though the software will generate portable C/C++ codethat can be compiled and run in any computing environ-ment. The files needed to install and run BioNetS can bedownloaded from http://x.amath.unc.edu/BioNetS. Thefollowing examples illustrate the way in which models areentered and run in BioNetS. More detailed documenta-tion is available with the software package.

DimerizationWe begin with a simple system that consists of the follow-ing two reactions:

µt i j i ji

M

j

d

F t B t= +=∑[ ( ) ( )] ( )27

1

p tt t j= < <<µ ε∆ 1 28( )

∆t

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In this system, monomer molecules M are produced at anaverage rate γ and degraded at an average rate δmM(t). Twomonomers can then bind to form a dimer molecule D.The average forward and backward rates for the this reac-tion are k1M(t)(M(t) - 1) and k2D(t), respectively. Thedimers are degraded at a rate δd. We will treat two cases. Inthe first case the cell volume is assumed to be constant,and in the second case the cell is allowed to grow anddivide. To model cell growth, the cell volume Vc is treatedas a random variable Vc = αV, where V is a non-negativediscrete random variable and α represents a unit of vol-ume. The random variable V is governed by the reaction

The above reaction causes V to grow exponentially fastwith an average rate of k3. Note that logistic growth is pro-duced when the backward reaction in Eq. 32 is included.

Constant volumeWe start by considering the simple case in which the vol-ume of the cell remains constant. To use BioNetS followthese steps. Copy BioNetS onto your machine, and doubleclick to launch. Help is included as part of the program,and accessed from the Help menu. The Help documentwill walk you through all the steps needed to enter reac-tions and run the simulator.

The user interface asks you to enter the reaction and corre-sponding rate constants in the top part of the script win-dow as shown in Fig. 1a. In the bottom part of the scriptwindow, you can toggle between panels. The Speciespanel is shown in Fig. 1a and allows the user to specifyhow the simulator treats each chemical species, discrete orcontinuous. The Constants panel lists the order in whichthe rate constants are referenced. The Output panel allowsthe user to specify the ouput type. There are two ways togenerate program output, either binary or ASCII. Binaryoutput is based on MATLAB binary files, so it is possibleto drive the program with MATLAB and use MATLAB'splotting routines to view the output. It is also possible togenerate time series and histograms of the data fromwithin BioNetS. Using ASCII files for I/O allows the sim-ulator to be run through shell scripts. The Executablepanel allows the user to generate either an executable fileor source code. BioNetS generates portable C/C++ code

that can be compiled and run in any computing environ-ment. BioNetS can directly compile the C/C++ code.However, this requires the Developer tools, included onall recent Apple machines and available directly fromhttp://developer.apple.com for free. The compiled codecan then be run from within BioNetS. The Commentspanel is available for the user to enter descriptive com-ments about the model.

To run BioNetS as a BioSpice agent, you need to move thesource directory onto a OAA-supported system. Oncethere, open up the MakeOAA file and specify the locationsof your oaalib folder. Then just type "make -f MakeOAA"(without the quotes) to create the agent.

Figures 2A and 2B show plots of time series for the mon-omer number generated by BioNetS. The parameter valuesused to generate these figures are given in the caption. Fig-ure 2A is the result obtained when M and D are treated asdiscrete variables. Figure 2B is the result from the chemicalLangevin equations. The solid line shown in both panelsis the result from the following equations for the firstmoments:

Figure 3A shows the probability density function (PDF) ofthe dimer concentration at various times for the discreteand continuous case. Notice that for all times, the agree-ment between the two different methods is very good. Atthe final time, t = 200 s, the system has reached steadystate. These figures indicate that the chemical Langevinequations are accurately capturing the dynamics andsteady-state behavior of the discrete system.

Cell growth and divisionIn this section we describe how cell growth and divisioncan be modeled using BioNetS. We will assume that thecell is experiencing exponential growth up until the timeit divides. As discussed above, the cell volume Vc = αV istreated as a random variable. In this model cell divisionoccurs when V exceeds a threshold value Vmax. Note thatthe choice of Vmax influences the degree of variabilityobserved in the cell division times: cells growing from V =1 to Vmax = 2 will have a large amount of variability in theirdivision times, while those growing from V = 100 to Vmax= 200 will have less variable times, and those rangingfrom V = 1000 to Vmax = 2000 will be still less variable.Changing the range of V in this way requires rescaling therelationship of V to the cell volume by adjusting the value

∅ γδm

� ( )29

� � �+ k

k1

230( )

�δd → ∅ ( )31

� � �k3 32 → + ( )

dM

dtk M M k Dm= − − + +2 2 331

22δ γ ( )

dD

dtk D k M Dd= − + −2 1

2 34δ ( )

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of α. When cell division occurs the volume is halved, andthe proteins are randomly divided between the two cellsusing a binomial distribution. Only one of the daughtercells is tracked. Because second order reactions requiretwo molecules to collide, the rate constants for these reac-tions should scale like k1 = k'1/Vc. We also assume that theproduction rate of monomers scales as γ = γ'Vc. This is areasonable assumption, because as the cell grows the tran-scription and translation machinery increases. These

assumptions produce the following rate equations for theconcentrations

A) A screen shot of BioNetS that illustrates how the dimerization example with constant volume is enteredFigure 1A) A screen shot of BioNetS that illustrates how the dimerization example with constant volume is entered. The tabs at the bottom of the screen allow the user to enter various options. B) A screen shot that illustrates how the dimerization example with cell growth and division is entered into BioNetS. In each screen shot, the area to the right of the reaction entry interface is a a slide-out testing panel that allows the user to enter rate constants and initial conditions, then view the results of a run from within BioNetS.

A

B

d M

dtk M k D k Mm

[ ][ ] [ ] ( )[ ] ( )= − ′ + + ′ − +2 2 351

22 3γ δ

d D

dtk M k D k Dd

[ ][ ] [ ] ( )[ ] ( )= ′ − − +1

22 3 36δ

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The terms in Eqs. 35 and 36 that involve k3 arise becauseof dilution due to cell growth. We use the same parametervalues as in the constant volume case except δm = 1 and δd= 0. The cell growth rate is k3 = 0.02 (assuming a scaling of1 time unit to one minute, this yields an average cell divi-sion time of ln 2/k3 � 35 minutes, typical for bacteria), thescale factor for the cell volume is α = 1 (for simplicity),and Vmax = 100. With these choices of parameter values,Eqs. 35 and 36 are identical with Eqs. 33 and 34, and we

expect the average behavior of this system to be similar tothat of the constant volume case.

The screen shot shown in Fig. 1B illustrates how thismodel is entered into BioNetS. Figure 4A shows the timeseries for the volume and monomer number treating eachvariable discretely. The results for the continuous case arevirtually identical. In Fig. 4B the concentration is plottedas a function of time. This figure should be compared withFig. 2A. The solid line in Fig. 4B is the result from solvingEqs. 35–37. Figure 3B shows the PDFs for the dimer con-centration at various times. Both the discrete and contin-uous results are shown. By comparing Fig. 3A with Fig. 3B,

A) A single realization of M(t) for the discrete processFigure 2A) A single realization of M(t) for the discrete process. B) A single realization of M(t) produced by the chemical Langevin equa-tions. The solid line in both panels is the result produced from Eqs. 33 and 34. The parameter values used to generate these fig-ures were γ = 50, δm = 1.02, δd = 0.02, k1 = 0.01, and k2 = 0.1. The initial conditions used were M(0) = 0 and D(0) = 0 and a time step of 0.01 was used for the continuous case.

time

Continuous

Count

0 10 20 30 40 50 60 70 80 90 100

0

20

40

60

time

DiscreteCount

0 10 20 30 40 50 60 70 80 90 100

0

20

40

60

A

B

dV

dtk Vc

c= 3 37( )

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we see not surprisingly that for this simple example themain effect of volume growth is to act as an additionalnoise source and increase the variability of thedistributions.

A chemical oscillatorWe next use BioNetS to simulate a two gene system thathas been studied in the literature [37]. In this system, theprotein A coded for by gene a acts as an activator for genea and gene r, by binding to the promoter regions, Pa and

Pr, of the respective gene. This increases the rate of mRNAaand mRNAr production by a factor αaand αr, respectively.The protein R, acts as a represser for both genes by bindingto A to form the inactive complex A_R. All gene products,mRNA and protein, are actively degraded. However, theheterodimer A_R protects the R subunit from degradation.The system consists of 9 chemical species and the follow-ing 14 biochemical reactions:

PDFs for the dimer concentration at various timesFigure 3PDFs for the dimer concentration at various times. The staircase plots are the results for the discrete case and the continuous lines are the results for the continuous case. Panel A is for the case of constant volume and Panel B is the varying volume case. The parameter values and initial conditions are the same as in Figs. 2 and 4, for the constant volume case and varying volume case, respectively. Each PDF consists of 10, 000 realizations of the process.

Varying Volume

[AA]

t = 200t = 20

t = 20

PD

F

4 5 6 7 8 9 10 11 12 13 14 15 16

0

0.5

1

1.5

Constant Volume

[AA]

t = 200t = 20

t = 20

PD

F

4 5 6 7 8 9 10 11 12 13 14 15 16

0

0.5

1

1.5

A

B

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A) Time series for the volume and monomer number for the case of cell growth and divisionFigure 4A) Time series for the volume and monomer number for the case of cell growth and division. The parameter values used to generate these figures were γ = 50, δm = 1.0, δd = 0.0, k1 = 0.01, k2 = 0.1, k3 = 0.02, and Vmax = 100. The initial conditions used were M(0) = 0, D(0) = 0 and V(0) = 50. B) The monomer concentration M(t)/V(t) as a function of time. The solid line is the result from Eqs. 35–36.

time

DiscreteCount

0 10 20 30 40 50 60 70 80 90 100

0

20

40

60

80

100

time

Discrete

Concentration

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

A

B

� � ���ak

a am1 38+ ( )

� � � � ���ak

a aa m_ _ ( )α 1 39+

� � ���rk

r rm2 40+ ( )

� � � � ���rk

r rr m_ _ ( )α 2 41+

m mak

a��� ��� �3 42+ ( )

m mrk

r��� ��� �4 43+ ( )

� � � �+ k

k5

644_ ( )

� � � �ak

k a+ 7

845_ ( )

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Figure 5 shows the way this system is entered intoBioNetS.

An interesting feature of the system is that it is capable ofproducing sustained oscillations [37]. Figure 6A shows atimes series for the repressor protein number when all thechemical species are treated as discrete random variables.The values of the rate constants used to generate this figureare listed in Fig. 5.

The chemical species Pa, Pr, Pr_A, and Pr_A are binary ran-dom variables: they can only take on the values 0 or 1.

Therefore, these species can not be approximated as con-tinuous random variables. All the other chemical speciesappear in sufficient quantities to justify the continuumapproximation. Figure 6B shows a time series correspond-ing to Fig. 6A using the hybrid model. The hybrid modelwas run using the semi-implicit Euler method, and forthese parameter values, runs 3 times faster than fullmodel. Visually, the agreement between the two methodsappears good. To test the accuracy of the Euler method, we

� � � �rk

k r+ 9

1046_ ( )

�k11 47∅ ( )

�k12 48∅ ( )

m ak

��� 13 49∅ ( )

m rk

��� 14 50∅ ( )

� � �_ ( )k15 51

A screen shot of the BioNetS user interface for the chemical oscillator exampleFigure 5A screen shot of the BioNetS user interface for the chemical oscillator example. The parameter values shown in the figure are the ones used to produce the results shown in Figs. 6, 7, 8. The area to the right of the reaction entry interface is a a testing panel provided by BioNetS to allow the user to enter rate constants and initial conditions, then examine the results of a run.

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used BioNetS to construct 2-D histograms of R versusmRN Ar. The results for the discrete and hybrid models areshown in Figs. 7B and 7B. To construct these histograms10, 000 oscillations were used. Excellent agreementbetween the discrete and hybrid model is seen. This indi-cates that the hybrid model is accurately sampling thesteady-state distribution. To verify that the hybrid model

faithfully captures the dynamics of the system, we com-puted the power spectra of both models. The results areshown in Figs. 8A and 8B. Again, excellent agreement isseen between the discrete and hybrid model.

Sample paths for the repressor protein numberFigure 6Sample paths for the repressor protein number. Panel A is the fully discrete case and Panel B is the hybrid model.

Time(s)

R(t)

Discrete

0 50 100 150 200 250 300 350 400

0

2000

4000

6000

8000

10000

Time(s)

R(t)

Hybrid

0 50 100 150 200 250 300 350 4000

2000

4000

6000

8000

10000

A

B

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An engineered promoter systemUsing standard techniques in modern molecular biology,it is possible to design novel systems of promoter-genepairs, such that virtually any desired regulatory networkarchitecture may be instantiated; such networks are oftencalled "synthetic gene networks." Recent implementa-

tions have included direct negative [22] and positive [23]feedback, a bistable switch [12], an oscillator [11], anintercellular communication system [38], and a bimodalself-activating system [39].

2-D histograms of repressor mRNA number versus repressor protein numberFigure 72-D histograms of repressor mRNA number versus repressor protein number. Red indicates regions of large frequency (i.e., where the system spends a lot of time) and blue indicates regions of low frequency. Panel A is for the discrete system and Panel B is the hybrid model. Roughly 10, 000 oscillations were used to produce both plots.

Hybrid method

mR

NA

R

R10

30 1 2 3 4 5 6 7 8 9 10 11 12

0

10

20

30

40

50

60

70

Discrete methodm

RN

AR

R10

30 1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

60

70A

B

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In this example, we use BioNetS to implement a model ofa simple, open-loop network based around a novel engi-neered promoter, which has been designed andconstructed by N. Guido and J. J. Collins at Boston Uni-versity. The promoter, called OROlac, combines the Olac,OR1, and OR2 operator sites, so that it is repressed by thelac repressor protein (LacI) and activated by the lambdarepressor protein (CI); see Fig. 9. Experiments have beenconducted in which the promoter, along with other sitesto produce the activator and repressor proteins, is inte-grated into a high copy number plasmid and inserted intoa strain of Escherichia coli. The promoter's activity is

observed using a fluorescent reporter, Green FluorescentProtein (GFP). A detailed modeling study with directcomparisons to experimental results has been carried outusing a fully discrete stochastic approach, and will bereported elsewhere (McMillen et al., manuscript in prepa-ration). Our goal here is to provide a reasonably complextest case to evaluate the performance of BioNetS.

The processes to be captured by the model are: transcrip-tion and degradation of mRNA strands; translation ofmRNA into protein; degradation of protein; formation ofprotein multimers (dimers in the case of CI, tetramers in

Power spectra for the repressor protein numberFigure 8Power spectra for the repressor protein number. Panel A is the discrete case and B is the hybrid model.

Hybrid

Frequency

Power

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

50

100

Discrete

Frequency

Power

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

50

100

A

B

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the case of LacI); LacI binding to isopropyl-β-D-thioga-lactopyranoside (IPTG), a chemical inducer that reducesLacI's binding affinity for Olac; and protein-DNA bindingat the OROlac promoter's operator sites. We define the fol-lowing chemical species: G, GFP; Mg, mRNA coding forGFP; X, CI monomer; X2, CI dimer; Mx, mRNA coding forCI; Dx, the arabinose-inducible pBAD promoter siteproducing CI; Y, LacI monomer; Y2, LacI dimer; Y4, LacItetramer; I0, IPTG (present in massive excess and thustaken to be constant); YI, LacI tetramer bound to IPTG; My,mRNA coding for LacI; and Dy, the PLtetO1 site constitu-tively producing LacI. In addition to these, we define spe-cies D0 through D8, representing the various permutationsof proteins bound to the three operator sites in the OROlacpromoter (see Table for a list). There are twelve combina-torial possibilities, but we eliminate three of them on thebasis that CI (X2) binding OR2 but notOR1 is unlikely,because of the low binding affinitity of CI for OR2 com-pared toOR1. Table also lists the effect on the basal rate ofproduction when the promoter is in each state. Thisreflects the regulatory effect of the proteins; for example,CI bound to OR2 leads to a 10-fold increase in transcrip-tion rate, while LacI bound to Olac halts transcription com-pletely (note that we assume in the event of simultaneous

binding of activator and repressor, repression "wins" andtranscription is halted).

The following irreversible reactions represent the proc-esses of transcription, translation, and degradation:

Schematic of the OROlac engineered promoterFigure 9Schematic of the OROlac engineered promoter. The promoter fuses three operator sites, one of which (Olac) yields repression when the LacI tetramer is bound to it, while binding of the CI dimer to OR2 yields approximately ten-fold activation (the OR1 site assists activation by cooperatively enhancing CI binding to OR2 when CI is bound to OR1). In the experimental system, CI and LacI are produced in the cell by external promoters, not shown here. The Green Fluorescent Protein (GFP) product is then used to monitor the output of the promoter by flow cytometry.

Olac

OR1

OR2

LacI CI

GFP

� � �0 0 52βg

g → + ( )

� � �1 1 53βg

g → + ( )

� � �210

2 54βg

g → + ( )

� � �y y yyβ

→ + ( )55

� � �x x xxβ → + ( )56

� � �g gTβ → + ( )57

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As in previous reactions, the caligraphic letters representindividual molecules of each species. We scale all timesand rates by the cell division time.

Experimental measurements generally provide equilib-rium rather than rate constants, and thus when writingreversible reactions we use the following notational con-vention: a reaction with equilibrium constant K has for-ward rate constant KR and backward rate constant R,where R is a scaling factor which sets the speed at whichthe reaction approaches equilibrium (we will considerthree values of R: 1, 10, and 100). Using this notation, werepresent protein-protein binding with the following setof reactions

Finally, protein-DNA binding is given by:

In all, the system consists of 21 species, participating in 34reactions. The reactions are entered into BioNetS using thesame method described in the previous examples. We useBioNetS' ability to represent individual species as eitherdiscrete or continuous to formulate three versions of themodel: fully discrete, fully continuous, and a hybrid ver-sion in which the DNA species D0 through D8 are discretewhile all other species are continuous. We vary the valueof R, the scaling factor for reversible reactions, and keepall other parameters fixed at the following nondimension-alized values: βg = 0.1, βy = 1, βx = 0.5, βT = 10, γmrna = 3.5,γprot = 0.7, Ky = 0.01, Ky2 = 0.1, KyI = 2 × 10-6, Kx = 0.05, K1= 0.3, K2 = 2K1,K3 = 0.008, K4 = 1.4 × l0-4K3, I0 = 1 × 106.

To evaluate the steady-state probability distributions pro-duced by the reaction system, simulations 250000 cellcycles in length were used to accumulate histograms (abuilt-in feature of BioNetS) of the number of molecules ofGFP (species G), for each of the three versions of themodel. As Fig. 10A shows, the resulting distributions areessentially identical, indicating that the continuumapproximations used in the fully continuous and hybridforms of the model were valid. Not all species in the sys-

� � �y yTβ → + ( )58

� � �x xTβ → + ( )59

�gmrnaγ → ∅ ( )60

�ymrnaγ → ∅ ( )61

�xmrnaγ → ∅ ( )62

�γprot → ∅ ( )63

�γprot → ∅ ( )64

�γprot → ∅ ( )65

� � �+Ky

2 66( )

� � �2 2 42

67+Ky

( )

� �4 0 68+ IK

IyI

( )

� � �+ Kx2 69( )

� � �0 2 11 70+ K

( )

� � �1 2 22 71+ K

( )

� � �0 4 33 72+ K

( )

� � �1 4 43 73+ K

( )

� � �3 2 41 74+ K

( )

� � �2 4 53 75+ K

( )

� � �4 2 52 76+ K

( )

� � �0 64 77+ I

K( )

� � �1 74 78+ I

K( )

� � �6 2 71 79+ K

( )

� � �2 84 80+ I

K( )

� � �7 2 82 81+ K

( )

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tem are well approximated as continuous variables: Fig.10B shows the continuous probability distribution forspecies D8, representing a promoter fully populated withtwo CI dimers and a LacI tetramer-IPTG complex. This sit-uation is very rare in our parameter regime, and the sys-tem spends essentially all of its time with D8 = 0. The fullycontinuous model, however, fluctuates into negative val-ues, indicating that the continuum approximation hasbroken down. This does not significantly affect the distri-bution for GFP because the other, more common DNAstates dominate the system's behavior; note, however, thatif we were considering genomic DNA rather than a highcopy number plasmid, we would not be able to employ afully continuous model. The hybrid model, by treating theDNA species as continuous, eliminates the fluctuationsinto negative values. In general, the appropriate approxi-mations will depend on both the system and the variablesof interest: in the present example, if we were interested inthe behavior of the operator sites themselves, we wouldnot be able to use the fully continuous version of themodel, but as a model solely of GFP expression theapproximation suffices. Comparisons between types ofmodels should be made to test the underlying assump-tions, and BioNetS facilitates this process.

We used simulations 200 cell cycles in length to test thespeed at which the three model versions ran. In each case,200 simulations were run using a consistent set of 200 dif-ferent random seeds; all runs were started with identicalinitial conditions. For the fully continuous and hybridsystems, the semi-implicit scheme was numerically stableand yielded consistent histograms for all time step sizesbetween dt = 0.001 and dt = 0.5, but the latter correspondsto just two time points per cell division cycle (recall thatall times are scaled by the cell division time), and wechose instead to sample 20 points per cycle and set dt =0.05. As shown in Table 2, the fully continuous methodwas always fastest, with the degree of improvement overthe exact, fully discrete method depending strongly on thevalue of R, the scaling factor for the reversible reactionrates. For R = 1, the fully continuous method was only 1.4-fold faster than the fully discrete method, but as R isincreased this speed advantage increases to over 4-fold atR = 10, then to over 30-fold at R = 100. (Note that thespeed advantage of the fully continuous over the fully dis-crete method increases with the abundances of the chem-ical species. Shifting parameters to generate higher proteinnumbers can yield cases in which the continuum approx-imation is hundreds of times faster than the discreteapproach; runs not shown here.) Use of a hybrid discrete/continuous method did not, for this particular model sys-tem, offer any speed gain over the fully discrete approach;the increased time involved in computing the Jacobian forthe semi-implicit method is more time-consuming thansimply simulating the reactions directly. Optimizing effi-

ciency requires testing various potential approaches, andBioNetS makes this a simple process.

ConclusionsWe have developed BioNetS to be a reliable tool for stud-ying the stochastic dynamics of large chemical networks.The software allows the user to specify which of the chem-ical species in the network should be treated as discreterandom variables and which can be approximated ascontinuous random variables. The software is highly opti-mized for speed and should be be able to simulate net-works consisting of hundreds of chemical species. Wehave verified the accuracy of the numerical methods byconsidering several test systems (a dimerization reaction,a chemical oscillator, and an engineered promoter), eachof which shows excellent agreement between the fully dis-crete version and the fully or partially continuous ver-sions. Our hope is that BioNetS, by providing a simple,user-friendly interface, will allow biological experimental-ists to formulate biochemical reaction models of their sys-tems quickly and easily, ideally increasing the number ofsystems in which direct comparisons are availablebetween models and experimental results. Clearly, notevery possible biological system can be captured in thecurrent version of BioNetS, and its capabilities will con-tinue to grow in the future. We wish to encourage users, orpotential users, to contact us regarding which additionalfeatures would be most helpful to them.

Availability and requirements• Project name: BlOchemical NETwork Stochastic Simu-lator (BioNetS)

• Project home page: http://x.amath.unc.edu/BioNetS

• Operating system:

* User interface: Macintosh OS X, version 10.2 or above.

* Generated source code: Ability to compile portable C++code. Makefiles included for OS X and Linux.

• Programming language: C++.

• Other requirements: None.

• License: BSD license.

• Restrictions on use by non-academics: None.

Authors' contributionsDA wrote the BioNetS code in its entirety, providing theuser interface, the numerical optimizations and the codegenerator. TE provided the mathematical derivations, car-ried out the dimer and oscillator examples, wrote an ini-

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tial draft of the paper, and worked with DA on thealgorithms and numerical methods. DM revised and final-ized the paper, provided the engineered promoter exam-ple, and worked extensively with DA on testing anddebugging of the software. All authors read, edited, andapproved the final version of the paper.

A) Probability densities of GFP molecules, for three versions of the OROlac promoter modelFigure 10A) Probability densities of GFP molecules, for three versions of the OROlac promoter model. Densities were generated by accu-mulating statistics in runs 250000 cell cycles in duration. (Solid line) Fully discrete model. (Dashed line) Fully continuous model. (Dash-dotted line) Hybrid model: the DNA binding states are represented by discrete variables, while all other species are con-tinuous. All three methods produce virtually identical probability distributions. B) Probability density for species D8, for the continuous model. The hybrid and fully discrete methods produce identical distributions in which D8 spends 99.996% of the time at zero. This histogram shows that the continuous model artificially allows negative numbers (see inset time series).

D8

PD

F

-6 -5 -4 -3 -2 -1 0 1 2 3 4

0

0.5

1

1.5

Discrete

Continuous

Hybrid

GFP

PD

F

0 100 200 300

0

5

10

15

20

D8 vs time

20 40 60 80 100

-1

0

1

A

B

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Table 1: List of DNA-binding states in the OROlac engineered promoter example.

Species Olac OR2 OR1 Production Rate

D0 - - - 1

D1 - - X2 1

D2 - X2 X2 10

D3 Y4 - - 0

D4 Y4 - X2 0

D5 Y4 X2 X2 0

D6 YI - - 0

D7 YI - X2 0

D8 YI X2 X2 0

Column Olac represents the lac operator site, while columns OR1 and OR2 represent the two operator sites in the prm region of bacteriophage λ. Entries in these columns indicate which regulatory proteins are bound to the operator sites in each state. The production rate column indicates how quickly transcription from the promoter proceeds in each state: basal expression occurs in states D0 and D1; enhanced expression occurs in state D2; and the remaining states display full repression (no expression).

Table 2: Execution times for three versions of the OROlac promoter model.

Fully continuous Fully discrete HybridR Mean (s) Std (s) Mean (s) Std (s) Mean (s) Std (s)

1 0.26 [1] 0.01 0.36 [1.4] 0.01 0.69 [2.7] 0.0110 0.27 [1] 0.01 1.15 [4.3] 0.01 6.91 [27] 0.12100 0.27 [1] 0.01 9.11 [34] 0.28 69.43 [260] 1.13

The mean values represent the number of seconds of CPU time required to run a simulation of 200 cell cycles on an otherwise unloaded 700 MHz PowerPC G4 processor, averaged over a set of 200 different random seeds, identical for each version of the model. "Std" indicates the standard deviation of this same set of 200 runs. Three values of parameter R, a scaling factor giving the relative speed of the reversible reactions in the system, have been used. In each row, the values in square brackets are normalized by the shortest execution time. The fully continuous version runs substantially faster than the other methods, and its execution time does not depend on R. The fully continuous version produces identical histograms to the fully discrete and hybrid methods for GFP (see Fig. 10A), but for the low-number species such as D8 it produces spurious negative values (see Fig. 10B). If accuracy is required for the small-number states, the fully discrete method should be used; note that for this system, the hybrid approach is consistently slower than the fully discrete method.

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Table 3

• Initialize all species and rate constants• Compute all reaction rates• Loop:

* Set µ = sum of rates for the discrete reactions* if (pt = µ∆t > ε), use Gillespie algorithm:

* R = a uniform random number in (0,1)* Set timeStep = -log(R)/µ* Find which reaction occurred, update the species involved

* else, use small ∆t approximation:* R = a uniform random number in [0,1]* timeStep = continuousTimeStep* if (R <pt = µ × timeStep), discrete transition has occurred:

• Determine which discrete transition occurred:

• Find the first value of k for which

• If , the forward reaction occurred, otherwise the backward reaction occurred

* else, no discrete transition:• No discrete reaction occurs, update is entirely due to continuous reactions (below)

* end if (small ∆t method, determination if discrete transition occurred)* end if (selection of Gillespie or small ∆t method for discrete reactions)* Update the continuous species using the Langevin equation, with step size timeStep (where timeStep is either equal to continuousTimeStep or to the step size found by the Gillespie algorithm), using a semi-implicit numerical method* Update any rates that have been changed by the continuous reactions and the single discrete reaction* Break when user-defined total simulation time is reached

• end loop

[ ]F B Ri iik + ≥=∑ µ

1

[ ]F B F Ri i kik + + ≥=

−∑ µ11

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AcknowledgementsThis work was supported by DARPA grant F30602-01-2-0579. D. Adal-steinsson acknowledges support by the Alfred P. Sloan Foundation.

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