Generalized binomial τ -leap method for biochemical kinetics incorporating both delayand intrinsic noiseAndré Leier, Tatiana T. Marquez-Lago, and Kevin Burrage Citation: The Journal of Chemical Physics 128, 205107 (2008); doi: 10.1063/1.2919124 View online: http://dx.doi.org/10.1063/1.2919124 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear biochemical signal processing via noise propagation J. Chem. Phys. 139, 144108 (2013); 10.1063/1.4822103 An effective method for computing the noise in biochemical networks J. Chem. Phys. 138, 084106 (2013); 10.1063/1.4792444 Improved delay-leaping simulation algorithm for biochemical reaction systems with delays J. Chem. Phys. 136, 144108 (2012); 10.1063/1.3702433 A data-integrated method for analyzing stochastic biochemical networks J. Chem. Phys. 135, 214110 (2011); 10.1063/1.3664126 Intrinsic biochemical noise in crowded intracellular conditions J. Chem. Phys. 132, 185102 (2010); 10.1063/1.3427244

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Generalized binomial �-leap method for biochemical kinetics incorporatingboth delay and intrinsic noise

André Leier,1,a� Tatiana T. Marquez-Lago,1,b� and Kevin Burrage1,2,3,c�

1Advanced Computational Modelling Centre, The University of Queensland, Brisbane, Queensland 4072,Australia2Institute for Molecular Biology, The University of Queensland, Brisbane, Queensland 4072, Australia3COMLAB, University of Oxford, Oxford OX1 3QD, United Kingdom

�Received 3 August 2007; accepted 11 April 2008; published online 30 May 2008�

The delay stochastic simulation algorithm �DSSA� by Barrio et al. �Plos Comput. Biol. 2, 117�E��2006�� was developed to simulate delayed processes in cell biology in the presence of intrinsicnoise, that is, when there are small-to-moderate numbers of certain key molecules present in achemical reaction system. These delayed processes can faithfully represent complex interactions andmechanisms that imply a number of spatiotemporal processes often not explicitly modeled such astranscription and translation, basic in the modeling of cell signaling pathways. However, for systemswith widely varying reaction rate constants or large numbers of molecules, the simulation time stepsof both the stochastic simulation algorithm �SSA� and the DSSA can become very small causingconsiderable computational overheads. In order to overcome the limit of small step sizes, various�-leap strategies have been suggested for improving computational performance of the SSA. In thispaper, we present a binomial �-DSSA method that extends the �-leap idea to the delay setting andavoids drawing insufficient numbers of reactions, a common shortcoming of existing binomial�-leap methods that becomes evident when dealing with complex chemical interactions. Theresulting inaccuracies are most evident in the delayed case, even when considering reaction productsas potential reactants within the same time step in which they are produced. Moreover, we extendthe framework to account for multicellular systems with different degrees of intercellularcommunication. We apply these ideas to two important genetic regulatory models, namely, the hes1gene, implicated as a molecular clock, and a Her1/Her 7 model for coupled oscillating cells. © 2008American Institute of Physics. �DOI: 10.1063/1.2919124�

I. INTRODUCTION

Stochasticity and discreteness are two important aspectswhen modeling biochemical systems that involve small-to-moderate numbers of certain key molecules. In this setting,the stochastic simulation algorithm �SSA� has been widelyused as an appropriate tool for studying intrinsic noise. TheSSA is an essentially exact procedure for simulating stochas-tic kinetics in a well-mixed chemical reaction system anddescribes the evolution of the system as a discrete nonlinearMarkov process. Since its introduction by Gillespie1 �the di-rect method and first reaction method�, other SSA methodshave been proposed by Cao et al.2 �the optimized directmethod�, Gibson and Bruck3 �the next reaction method�, andMcCollum et al.4 �the sorting direct method�. Moreover, anefficient version of a spatial SSA has been implemented byMarquez-Lago and Burrage5 using a modification of the nextreaction method by Ehrenberg and co-workers.6,7

The basic idea of the SSA is that a state vector x�t�= �X1�t� , . . . ,XN�t�� representing numbers of molecules Xi�t�of N different species Si, i=1, . . . ,N, is evolved by an updatestep x�t+��=x�t�+� j. Here � is an exponentially distributed

waiting time to the next reaction, � j is the jth stoichiometricvector associated with reaction Rj, and j=1, . . . ,M, that ischosen based on the relative sizes of the propensity functionsthat describe the probabilities of a particular reactionoccurring.

Furthermore, delays constitute an important aspect whenmodeling cell signaling pathways and genetic regulation.8

They are intrinsic to slow biochemical processes that do notoccur instantaneously and are often affected by spatial inho-mogeneities. Delays are associated with transcription andtranslation, two key mechanisms that imply a number of spa-tiotemporal processes often not explicitly modeled. In eu-karyotes, these include diffusion and translocation into andout of the nucleus, RNA polymerase activation, splicing, pro-tein synthesis, and protein folding. By incorporating delaysinto the temporal model, we can capture essential informa-tion on a macroscopic level, the delay itself accounting for amultitude of biochemical processes and events happening ona microscopic time scale.

In order to take proper account of time delays along withdiscreteness and intrinsic noise, Barrio et al.9 developed adelay stochastic simulation algorithm �DSSA� that general-izes the SSA for models with delayed reactions. Indepen-dently, Bratsun et al.10 developed a SSA without consideringwaiting times for delayed reactions while only nonconsum-ing reactions can be specified to be delayed. More recently,

a�Author to whom correspondence should be addressed. Tel.: �61-7-3365-7445. FAX: �61-7-3365-6136. Electronic mail: leier@maths.uq.edu.au.

b�Electronic mail: tmarquez@maths.uq.edu.au.c�Electronic mail: kevin.burrage@comlab.ox.ac.uk.

THE JOURNAL OF CHEMICAL PHYSICS 128, 205107 �2008�

0021-9606/2008/128�20�/205107/14/$23.00 © 2008 American Institute of Physics128, 205107-1

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Cai11 introduced a direct DSSA method and showed that bothapproaches, the DSSA by Barrio et al. and the direct method,are exact for chemical reaction systems with delays. Sincethe DSSA is a delay algorithm, there are similar issues toinitialization as there are when solving a delay differentialequation numerically. Basically, one can proceed in twoways to assign initial conditions.

�1� Consider initial molecular concentrations that are rel-evant to the biological problem in question assumingthere are no initial delayed reactions.

�2� The second way is to perform a prerun according to �1�for a certain period of time, and then use the outcomeof this prerun as an initial condition for the simulation,including the history of delayed reactions to beupdated.

For all simulations in this paper we used the first optionfor assigning initial conditions.

However, a major drawback of both the SSA and DSSAis that they entail high computational costs when the simu-lated time steps become very small due to large numbers ofmolecules or widely varying rate constants. In order to re-duce the computational load and overcome the limit of smallstep sizes, the so-called �-leap methods have been suggested.The general idea behind these methods is to advance thesimulation in time leaps while updating the system state ac-cording to a reasonably good approximation for the accumu-

lated number of reactions within the time leap, hence coarsegraining the simulation in time. Thus, the update formulatakes the form x�t+��=x�t�+� j=1

M � j� j, where the � j representsthe likely number of occurrences of reaction Rj in the timeinterval �t , t+��.

Initially, Gillespie12 proposed the Poisson �-leap methodand the midpoint �-leap method in which the number of re-actions in each �-leap is sampled from a Poisson distribution.Following these methods, further improvements were madeby, for instance, Gillespie and Petzold,13 Rathinam et al.,14

and Cao et al.15,16

Since sample values from a Poisson distribution rangefrom zero to infinity, negative numbers of molecules canoccur when larger step sizes are used. In order to avoid this,Tianhai and Burrage17 and later Chatterjee et al.18 proposedthe binomial �-leap method where the number of reactions ina leap are drawn from a binomial distribution. Recently,Peng et al.19 developed a modified binomial leap method thatestimates the number of reaction products within a �-leapstep allowing them to participate in additional reactions inthe same leap. A different implementation of the binomial�-leap methods are the so-called R-leap methods20 in whichthe total number of reactions in a step is first selected and the� j are then assigned through a selection based on a binomialdistribution.

In this paper, we review the binomial �-leap methods byTianhai and Burrage and Peng et al. and present a new ver-sion, which overcomes some of the difficulties that are asso-ciated with these methods. In order to design models of bio-chemical processes incorporating delays, we extended thesemethods to include delays. We applied all three methods tomodels of molecular clocks9,21,22 that incorporate both delays

TABLE I. Some simple reactions Rj and their corresponding propensities aj, stoichiometric coefficients v j, and maximum number of potential reaction eventsNj.

Reaction Rj Propensity aj Nj Stoichiometric coefficients

First order Sk ——→cj

Slaj =cjXk Xk � j,k=−1, � j,l=1

Heterodimeric Sk+Sl ——→cj

Smaj =cjXkXl min�Xk ,Xj� � j,k=� j,l=−1� j,m=1

Homodimeric Sk+Sk ——→cj

Slaj +cjXk�Xk−1� /2 �Xk

2� � j,k=−2, � j,l=1

Hill fct. type Sk ——→cjf

Sk+Sl aj =cjf�Xk� where f�Xk�t��=1−1

1+ �Xk�t� /X0�h

�activation� or f�Xk�t��=1

1+ �Xk�t� /X0�h �inhibition�

with Hill coefficient h and X0 such that f�X0�=1 /2

Const, Nj �1�see text�

� j,l=1

FIG. 1. Schematic representation of the DSSA implementation �Ref. 23�.Here, �i refers to the waiting time until the next reaction Ri is scheduled and�i is the delay time of reaction Ri. The dotted arrows point to the time lineindicating when a reaction is updated. The reaction is specified below thearrow. Ru denotes a nondelayed and Rd a delayed reaction. Nondelayedreactions are updated when triggered. In case of a delayed consuming reac-tion, its reactants and products are separately updated. This is marked byRd,r and Rd,p, respectively. If time steps and reactions are drawn but thenignored, they are crossed out. Those steps are marked as gray dashed linesand the steps replacing an ignored step are marked as gray solid lines.

FIG. 2. Illustration of hes1 autoinhibition.

205107-2 Leier, Marquez-Lago, and Burrage J. Chem. Phys. 128, 205107 �2008�

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and intrinsic noise and found that the first two methods werenot always able to capture the correct oscillatory dynamics.However, our generated binomial �-DSSA method is able toefficiently yield the dynamics of models with �and without�delayed reactions with high accuracy.

II. DELAY STOCHASTIC SIMULATION ALGORITHM

The DSSA generalizes the SSA for chemical kineticswith delayed reactions.9,23 The discrete nonlinear Markovprocess described by the SSA has a probability density func-tion that is the solution of the so-called chemical masterequation �CME�, a discrete partial differential equation. Byenumerating the state space, the CME leads to a system ofODEs that describes the deterministic behavior of the meanassociated with an infinite number of SSA runs. A CME forthe DSSA, namely, a DCME, has been derived from firstprinciples.9 By consequence, the DSSA has a correspondingrepresentation as a system of delay differential equations�DDEs� when the molecular numbers become large. Analo-gously to the SSA case, any trajectory of the DSSA is anexact solution of the DCME. Thus there are very naturalgeneralizations from the nondelayed setting to the delayedsetting via a CME approach.

The implementation by Barrio et al. is illustrated in Fig.1. The DSSA differs from the regular SSA by making a cleardistinction between a waiting time and a reaction delay.Simulation proceeds by drawing reactions and their waitingtimes. If a nondelayed reaction is selected, then the state isupdated in the standard way �SSA�, but if it is a delayedreaction that is selected we have to distinguish between con-suming and nonconsuming reactions. In case of nonconsum-ing reactions, the corresponding reactants and products arenot updated until the appropriate time point would be passedby another simulation step. The last drawn reaction thatpassed the time point when the delayed reaction was sched-uled is ignored and instead the state is updated according tothe delayed reaction. Simulation continues at the delayed re-action time point. On the other hand, if the reaction is con-suming, reactants and products of delayed consuming reac-tions must be updated separately, namely, when the delayedreaction is selected and when it is completed.

III. �-LEAP METHODS

In this section, we describe in detail the binomial �-leapmethods by Tianhai and Burrage �method 1�, Peng et al.�method 2�, and our proposed new method, B�-�D�SSA�method 3�, which accounts for both delayed and nondelayedreaction dynamics. We assume that the chemical reactionsystem consists of N molecular species �S1 , . . . ,SN� thatchemically interact through M reactions �R1 , . . . ,RM� in awell stirred solution of fixed volume and at constant tem-perature. We denote by Xi�t� the number of molecules ofspecies Si at time t, aj�x� the propensity of reaction Rj,a�x�= �a1�x� , . . . ,aM�x��T the vector of propensities, a0�x�=� j=1

M aj�x�, and Nj�x� the maximum number of potential re-action events of type Rj, given the state x�t�= �X1�t� , . . . ,XN�t��T. See Table I on how to calculate aj�x�and Nj�x� for specific types of reactions. Calculating theNj�x� is straightforward for elementary reactions.17 However,for Hill-type reactions, which are often used to model coop-erative binding and regulation involved in transcription, it isnot obvious how to do this and we will discuss this issue.

We define Nj�x ,��, with ����1 , . . . ,�M� and � j �Nj�x�,as the maximal number of potential reaction events of typeRj when �1 , . . . ,�M reactions of R1 , . . . ,RM occur in the�-step. In method 3, when calculating Nj�x ,��, we set� j , . . . ,�M =0 since only the already sampled reaction num-bers �1 , . . . ,� j−1 are considered.

Let �ij ,ij 0 �1� i�N, 1� j�M� be the number ofreactant and product molecules of species Si in reaction Rj,respectively, then, �=−�+ is the stoichiometric matrix with�ij determining the change in the number of molecules ofspecies Si due to reaction Rj. For example, for the system ofMichaelis–Menten reactions,

R1: E + S → ES ,

R2: ES → E + S ,

R3: ES → S + P ,

� = − 1 1 1

− 1 1 0

1 − 1 − 1 where S1 = S,S2 = E,S3 = ES .

FIG. 3. Delta-notch signaling in a one-dimensional multicellular setting.

FIG. 4. Mean trajectory of the decaying-dimerizing model obtained from104 SSA runs.

205107-3 Generalized binomial �-leap method J. Chem. Phys. 128, 205107 �2008�

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An additional issue with computing the Nj�x� in bino-mial �-leap methods is the reaction network. A set of reactionchannels constitutes a network G= �V ,E� with vertices V= �R1 , . . . ,RM� and edges E= ��Ri ,Rj� , ∃k ,1�k�N ,�ki ,�kj

�0�. This implies that two reactions, i.e., two vertices, areconnected by an edge if and only if they have one or morecommon reactant species. The connected subnetworks G� ofG divide the set of reactions into mutually exclusive subsets.In method 1, this information is used to sample the total

number of reactions of each subnetwork from a binomialrandom variable. Unlike method 1, method 3 samples reac-tion numbers for each reaction by considering only those

reactions it shares reactant species with, namely, its direct

neighbors in G. Method 2 also considers potential reaction

products that might occur within in �t , t+��, generally lead-ing to a higher estimate for the available number of mol-

ecules P� i for each species Si as follows:

P� i =� xi if �ij � 0, ∀ j

xi + max�0,� ��ij�0

�ijaj�x� + ��� ��ij�0

�ij2 aj�x� otherwise, �1�

with control parameter �� �−1,1�. Based on P� i, bounds P� ij

for the maximal number of potentially available Si moleculesfor reaction Rj are sampled. These essentially distribute the

P� i molecules of species Si amongst those reactions that de-pend on Si molecules. The actual sampling of reaction num-bers uses the maximum number of potential reaction events

Nj�P�1j , . . . , P�Nj� that are calculated on the basis of P� ij for allspecies Si.

One of the most important aspects associated with �-leapmethods is the choice of a step size strategy since this canprevent the occurrence of obtaining negative numbers ofmolecules as well as control the accuracy of the simulation.These step size strategies must be used in conjunction with aglobal error control parameter � that makes sure that thepropensity functions do not change too much in any givenstep. This is done by attempting to estimate the mean andcovariance matrix occurring in a Taylor series expansion ofthe discrete approximation for the chemical kinetics. For theselection of � in methods 1–3, we use one of the followingstep-size selection formulas:

�a� � = min� �a0�x�max

1�j�M�� j�x���

,��a0�x��2

max1�j�M

�� j2�x�� ,

�b� � = min1�j�M

�max��aj�x�,cj�� j�x��

,�max��aj�x�,cj��2

� j2�x� ,

�c� � = min1�i�N

1�j�M

�ij�0

�max��xi/gi, ��ij��� � i�x��

,�max��xi/gi, ��ij���2

�� i2�x� ,

with Aij�x�=�ai�x� /�xj, F=A�, =Fa, � =�a, �2= F̃a, and

��2= �̃a, where F̃ij =Fij2 and �̃ij =�ij

2 , ∀1� i�N, 1� j�M,and gi=� /�i where the �i are chosen such that the relativechanges in all propensities will be bounded by a prespecifiederror control parameter � ,0���1 �see Ref. 16�. Formulas�b� and �c� were originally developed for use in a step-sizeselection procedure for the Poisson �-leap method in order toreduce the likelihood of producing negative molecular num-bers when a larger step size is used.16 The parameter � al-lows us to adjust the accuracy and performance of a �-leapmethod, so that a smaller � results in a smaller � and, hence,a smaller number of reactions per �-step but at the expense oflarger computation time. However, if � is too small, that is,less than K /a0 for a prespecified K, with 1�K�10, it isgenerally more efficient to perform just a standard �D�SSAstep. This modification is considered in both methods 1 and 3

TABLE II. The error statistics are based on 104 simulations for the SSA and all three �-leap algorithms using the �-selection formula �a� with �=0.01. Themaximum absolute errors and the maximum relative errors �per species� are calculated from the mean trajectories with 300 equidistant time points in �0,30�.The maximum absolute errors occur at t=0.4 in the molecular numbers of species S3 for method 1 and at t=0.3 in all other cases. The maximal relative errors�relative to the expected molecular numbers� occur at t=0.2 in the molecular numbers of species S3 for all three methods and at t=0.3 otherwise. The tablealso presents the mean error �the deviation from the SSA mean trajectory� at time point t=30. For the SSA, mean and standard deviations of the molecularnumbers of �S1 ,S2 ,S3� at t=30 are �2441.86,12 101.07,31 329.83� and �49.69,95.84,102.74�, respectively. The smallest errors are highlighted. The maximalrelative error values are rounded to the fourth decimal place; all other values are rounded to the second decimal place.

Method

Maximum absolute error Maximum relative error Mean error at t=30

S1 S2 S3 S1 S2 S3 S1 S2 S3 Sum

1 7.20 17.79 12.51 0.0015 0.0004 0.0315 0.33 1.19 0.97 2.502 7.07 18.36 13.23 0.0015 0.0004 0.0333 0.27 1.19 2.35 3.803 6.57 7.65 12.80 0.0014 0.0004 0.0328 0.96 1.12 1.55 3.62

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but not in the original implementation of method 2. In orderto make the algorithms compatible, we included this modifi-cation in method 2 as well. If not stated otherwise, we choseK=1 in our simulations.

We now give a brief overview of the binomial �-leapmethods of Tianhai and Burrage and Peng et al. �methods 1and 2, respectively�. We then present a more generalizedapproach �method 3� in which complicated reaction networksare treated in a more transparent manner. We then give somecomments on these methods and then show how delays canbe incorporated into these frameworks.

A. Method 1 „binomial �-leap by Tianhai and Burrage…

Prestep. Identify the subnetworks Gi of G. For a givenGi, let �Ril

, . . . ,Ris� be the set of corresponding reactions with

propensities ai1�x� , . . . ,ais

�x� and maximal numbers ofreaction events �Ni1

, . . . ,Nis�.

�1.1� Calculate a1�x� , . . . ,aM�x� ,a0, and N1�x� , . . . ,NM�x�.�1.2� Choose a �-selection procedure and update the corre-

sponding variables.�1.3� �Check of step-size conditions�:

For each subnetwork Gi of G:

�1.3.1� Calculate Ni�=min�Ni1, . . . ,Nis

� and ai��x�=�k=1

s aik�x�.

�1.3.2� If �Ni��0� and �ai��x�� /Ni��1�, then�=Ni� /ai��x�.

�1.4� If ��K /a0, perform a normal �D�SSA step, otherwisego to �1.5�.

�1.5� �Sampling�:For each subnetwork Gi of G with Ni��0:

�1.5.1� Generate a sample value �total for the total reactionnumber of �Ri1

, . . . ,Ris�:

�total = B�Ni�,Pi� with Pi = ai��x��/Ni�.

�1.5.2� For 1�r�s−1, generate a sample value �ir,

for reaction Rir:

�ir= B��total − �

k=1

r−1

�ik,air

�x�

ai��x� � .

�1.5.3� Calculate the number �isof reaction events of

type Ris:

�is= �total − �

k=1

s−1

�ik.

�1.6� Update the system state:

x�t + �� = x�t� + �j=1

M

� j� j where � j is the

reaction number of Rj calculated in �1.5� .

B. Method 2 „modified binomial �-leaping by Penget al.…

Prestep. For each species Si, identify the reactions whereSi is a consumed, reacting species; let �Rj1

, . . . ,Rjs� be the set

of these reactions with corresponding stoichiometric matrixentries �ij1

, . . . ,�ijs�with �ij1

�0� and propensitiesaj1

�x� , . . . ,ajs�x�.

�2.1� Calculate a1�x� , . . . ,aM�x�, a0.�2.2� Choose a �-selection procedure �Peng et al. suggest ei-

ther �b� or �c��; update the corresponding variables.�2.3� If ��K /a0, perform a normal �D�SSA step, otherwise

go to �2.4�.�2.4� �Sampling I�:

For each species Si �i=1, . . . ,N�:

�2.4.1� Calculate P� i �an estimate for the available numberof Si molecules�.

�2.4.2� Distribute the P� i molecules among �Rj1, . . . ,Rjs

�:Set k=1, �=0; calculate a�i�x�=�l=1

s ��ijl�ajl

�x�.While ��� P� i� and �k�s�:

P� ijk= B�P� i − �,

��ijk�ajk

�x�

a�i�x� − �l=1k−1��ijl

�ajl�x�� ,

� = � + P� ijk; k = k + 1.

�2.5� �Sampling II�:For each reaction Rj �j=1, . . . ,M�:

�2.5.1� Calculate Nj�=Nj�P�1j , . . . , P�Nj�.

�2.5.2� Generate a sample value � j for the reaction num-ber of type Rj:

If Nj��0, then � j =B�Nj� , Pj� with Pj =aj�x� � /Nj�, else� j =0.

�2.6� Update the system state:

x�t + �� = x�t� + �j=1

M

� j� j where � j is the

reaction number of Rj calculated in �2.5� .

TABLE III. The statistics are obtained as described in Table II, using the �-selection formula �a� with�=0.05.

Method

Maximum absolute error Maximum relative error Error at t=30

S1 S2 S3 S1 S2 S3 S1 S2 S3 Sum

1 62.23 132.44 89.48 0.0128 0.0028 0.1647 0.65 3.53 5.44 9.622 37.15 298.32 269.91 0.0077 0.0064 0.2945 1.17 11.87 23.91 36.953 58.49 128.02 87.12 0.0121 0.0027 0.1603 0.20 5.34 3.34 8.87

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C. Method 3 „generalized binomial �-leaping…

Prestep. Identify for each reaction Rj the reactions itshares reacting species with; let Ij be the set of indices ofthese reactions.

�3.1� Calculate a1�x� , . . . ,aM�x�, a0, and N1�x� , . . . ,NM�x�.�3.2� Choose a �-selection procedure; update corresponding

variables.�3.3� �Check of step-size conditions�:

For each reaction Rj:

�3.3.1� Calculate Nj�=min�Ni�x� , i� Ij� and aj��x�=�i�Ij

ai�x�.�3.3.2� If �Nj��0� and �aj��x�� /Nj��1�, then �

=Nj� /aj��x�.

�3.4� If ��K /a0, perform a normal �D�SSA step, otherwisego to �3.5�.

�3.5� �Sampling�:

� � ��1, . . . ,�M� = 0;

For each reaction Rj �j=1, . . . ,M�:

�3.5.1� Calculate Nj�=Nj�x ,��.�3.5.2� Generate a sample value � j for the reaction num-

ber of type Rj:If Nj��0, � j =B�Nj� , Pj� with Pj =aj�x�� /Nj�,else � j =0.

�3.6� Update the system state:

x�t + �� = x�t� + �j=1

M

� j� j .

D. Notes on methods 1–3

�1� Method 2 had to be modified in order to take properaccount of Hill-type reactions and reactions of type A→A+B. In contrast to the original implementation, the

set of reactions �Rj1, . . . ,Rjs

� is determined �in theprestep� by the reactant matrix entries �ij1

, . . . ,�ijs�with

�ijl�0�. The corresponding stoichiometric matrix entry

is zero and therefore not suitable. Together with this

modification, in step �2.4.2� a�i�x� and P� ijkare calculated

based on �ijl, and not on ��ijl

�.�2� The size of � plays an important role in the computa-

tional performance and in reproducing the appropriatedynamics. If � is too large then, if in methods 1 and 3,� does not satisfy the step-size conditions and it is set toa smaller value automatically. If � is too small the al-gorithm will perform �D�SSA steps instead of �-steps.

�3� We noticed that for large �, method 2 is likely to pro-duce negative molecular numbers for certain models.This is due to unreasonably large reaction numbers instep �2.4.1� that result from overestimates of the avail-

able number of certain molecules P� i in Eq. �1�. Fine-tuning the control parameter � is possible but that mayimpact on the computational performance.

�4� Chemically, reactions of type A→c

A+B are often usedto model synthesis processes such as transcription andtranslation with corresponding synthesis rate constantc. For transcription reactions, species A may representeither the DNA or a transcription factor and for trans-lation reactions A represents the mRNA. If A is a tran-scription factor, the regulatory effect on the chemicalkinetics is often described by a Hill function f resultingin the propensity a=cf�A�. Calculating the Nj�x� forsuch reactions involves some subtlety. For unregulated�non-Hill-type� reactions �see Sec. IV B for an ex-ample�, we decided to choose Nj�x�=Xj as this matcheswith the �D�SSA approach where the propensity is pro-portional to the number of molecules of A. For Hill-type reactions, we define Nj�x�=C where C is constant.In our simulations, we found that unless C is too small��10� it has no noticeable effect on the simulationoutcome.

TABLE IV. The statistics are obtained as described in the caption of Table II, using the �-selection formula �b�with �=0.01 for all three �-leap methods.

Method

Maximum absolute error Maximum relative error Error at t=30

S1 S2 S3 S1 S2 S3 S1 S2 S3 Sum

1 7.78 5.47 3.09 0.0012 0.0002 0.0087 0.32 1.98 1.11 3.412 7.00 7.08 3.05 0.0014 0.0002 0.0086 0.28 1.19 1.86 3.333 8.37 6.87 3.14 0.0013 0.0002 0.0088 0.09 2.42 1.17 3.68

TABLE V. The statistics are obtained as described in the caption of Table II, using the �-selection formula �b�with �=0.05 for all three �-leap methods.

Method

Maximum absolute error Maximum relative error Error at t=30

S1 S2 S3 S1 S2 S3 S1 S2 S3 Sum

1 35.68 85.34 65.62 0.0057 0.0018 0.1208 0.80 5.38 8.68 14.862 37.26 84.01 64.56 0.0060 0.0018 0.1186 0.88 3.07 6.53 10.483 36.13 84.15 64.39 0.0058 0.0018 0.1185 0.10 5.52 8.86 14.48

205107-6 Leier, Marquez-Lago, and Burrage J. Chem. Phys. 128, 205107 �2008�

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E. �-leaps with delayed reactions

In order to account for delayed reactions we have toadjust the update step, step �i.6� in method i=1,2 ,3. In whatfollows, the subscripts ndj and dj represent the jth nonde-layed and delayed reactions, respectively, while M� denotesthe number of nondelayed reactions.

Modifications for systems with delayed reactions:�i.6� Update the system state:

�i.6.1� For each delayed reaction Rdjwith delay � j,

record �djrandom update time points t+� j +uk� with uk

�U�0,1�, k=1, . . . ,�dj.

If Rdjis a consuming, delayed reaction, update x�t+��

=x�t+��+�dj.�i.6.2� Update for all nondelayed reactions Rndj

withreaction number �ndj

as calculated in �i.5�:

x�t + �� = x�t� + �j=1

M�

�ndj�ndj

.

�i.6.3� Update for all delayed reactions Rdjscheduled

within �t , t+��:

x�t + �� = x�t + ��

+ djfor consuming, delayed reactions,

x�t + �� = x�t + �� + �dj, otherwise.

In order to obtain a better estimate of the maximumnumber of potential reaction events in �t , t+�� we can con-sider the products of all delayed reactions that occur withinthe �-step. In this case, the update of the delayed reactionshas to happen prior to the �first� sampling step. However, formethod 3, there is no indication that this could lead to asignificantly better accuracy.

TABLE VI. Parameters used for the Hes1-dimer model.

Parameter Description Value Reference

c1 Hes1 mRNA degradation rate 0.03 min−1 23c2 Hes1 degradation rate 0.06 min−1 Biological plausible

valuesc3 Hes1 dimer degradation rate 0.03 min−1 Biological plausible

valuesc4 Association rate 0.001

min−1Biological plausible

valuesc5 Dissociation rate 0.01 min−1 Biological plausible

valuesc6 Translation rate 1 min−1 8c7 Maximum transcription rate 1 min−1 normalized; 8X0 Critical number of Hes1 protein

�Hill function parameter�50 22 and 8

h Hill cooperativity factor�Hill function parameter�

5 22 and 8

� Total delay�Transcription, translation,translocation�

20 min 8

FIG. 5. Single run of method 1 using �-selection formula �a� with �=0.01.

FIG. 6. Single runs of methods 2 and 3 using �-selection formulas �b� and�a� with �=0.05, respectively. The plots show the differences in the resultingdynamics in the numbers of monomeric Hes1.

205107-7 Generalized binomial �-leap method J. Chem. Phys. 128, 205107 �2008�

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IV. NUMERICAL SIMULATIONS

We present the simulation results of three different mod-els for all three �-leap �D�SSA methods and the �D�SSA�implementation by Barrio et al.9�. The first model is asimple four-reaction decaying-dimerizing nondelayedmodel.17,19

The other two models are models of genes that act asmolecular clocks that regulate coordinated somite segmenta-tion. They involve genetic regulation modeled by delayedHill-type reactions. The second model �Fig. 2� is a model ofhes1 autoinhibition by dimerized Hes1 proteins in mousewhere the expression levels for Hes1 mRNA and Hes1 pro-tein oscillate in a 2 h cycle with a phase lag of approximately15 min between the oscillatory profiles.8,9,22,24 The thirdmodel �Fig. 3� describes the Delta-Notch dependent synchro-nization of Her1 and Her7 protein levels in an array of fivecells in zebrafish.21,22 The two linked genes her1 and her7 are

autorepressed by their own gene products and positivelyregulated by Delta-Notch signaling that leads to oscillatorygene expression with a period of about 30 min, generatingregular patterns of somites, that is, future segments of thevertebrae. In both models, the transcriptional and transla-tional delays are responsible for the oscillatory behavior.

All algorithms are implemented and run in MATLAB 7.However, instead of using MATLAB standard binomial ran-dom number generator �binornd�, we use the faster randomnumber generator �randbinom� from the lightspeed toolboxby Minka,25 which is available online.

A. Simple four-reaction model without delays

The first test system has been used to benchmark thePoisson �-leap SSA, the original binomial �-leap by Tianhaiand Burrage,17 and the �-leap method by Peng et al.19 Itcontains three reactant species and four reaction channels:

FIG. 7. Mean molecular number of monomeric Hes1 obtained from 103 runsof method 2 using �-selection formula �a� for different �.

FIG. 8. Mean molecular number of monomeric Hes1 obtained from 103 runsof method 2 using �-selection formula �a� for different �. Single DSSA runswere not allowed.

FIG. 9. Mean molecular number of monomeric Hes1 obtained from 103 runsof method 2 using �-selection formula �b� for different �.

FIG. 10. Mean molecular number of monomeric Hes1 obtained from 103

runs of method 2 using �-selection formula �b� for different �. Single DSSAruns were not allowed.

205107-8 Leier, Marquez-Lago, and Burrage J. Chem. Phys. 128, 205107 �2008�

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R1: S1 ——→c1

� ,

R2: S1 + S1 ——→c2

S2,

R3: S2 ——→c3

S1 + S1,

R4: S2 ——→c4

S3,

with rate constants c1=1, c2=0.002, c3=0.5, and c4=0.04.When applied to this model, all three �-leap methods

show basically the same dynamics as the SSA solution �Fig.4� for a reasonable value of � �e.g., �=0.01�. We simulatedthe system in the time interval �0,30� with initial state x0

= �105 ,0 ,0� using the �-selection formulas �a� and �b� for allthree �-leap methods. For method 2, we choose �=0. Peng etal. did not specify the value of �, so we are unable to directlycompare our results. We calculated the means and standarddeviations of the molecular numbers for 104 simulation runsat 300 equidistant time points for the SSA and each of thethree �-leap methods. The resulting error statistics are givenin Tables II–V. This example demonstrates two aspects.First, for this particular model, method 3 is as good as any ofthe other �-leap methods since all three methods capture theSSA dynamics with high accuracy with �=0.01 the maximalrelative error is less than 0.04% �in the number of S3 mol-ecules� of the corresponding mean SSA molecular number.Second, our simulations do not confirm the conclusions byPeng et al. in which method 2 performs more accurately thanmethod 1 on this particular model. Their conclusion is basedon the histogram distance errors for different values of �.However, we believe that this approach might not properlyreflect the accuracies of the �-leap methods as the compari-son is based only on the molecular number of a single spe-cies, S3, at a single time point, t=4, as opposed to all mo-lecular species at all times. As a counterexample, at timepoint t=30, with �=0.01, methods 1 and 3 with �-selectionformula �a� yield a smaller error than method 2 with�-selection formula �b�.

B. The Hes1-dimer model

The Hes1-dimer model consists of three species, Hes1mRNA �S1�, monomeric Hes1 �S2� and dimeric Hes1 �S3�and the following seven reactions:

R1: S1 ——→c1

� ,

R2: S2 ——→c2

� ,

R3: S3 ——→c3

� ,

R4: S2 + S2 ——→c4

S3,

R5: S3 ——→c5

S2 + S2,

R6: S1 ——→c6

S1 + S2,

R7: S3 ——→c7

S1 + S3,

where R1, R2, and R3 are degradations, R4 and R5 are theassociation and dissociation of Hes1 and Hes1-dimer, respec-tively, R6 represents the translation of Hes1 mRNA into Hes1protein, and R7 the Hes1-dimer regulated transcription of thehes1 gene. Both the transcriptional and the translational timedelays are taken into account by a delayed Hill function de-scribing the inhibitory regulation of reaction R7 �see Table I�.The model parameters used in our simulations are shown inTable VI all of which are biological reasonable.

The model is similar to the original model of Monk,8

apart from the addition of dimerization. If not stated other-wise, we simulated the system on the time interval �0,500�with initial state x0= �X1�0� ,X2�0� ,X3�0��= �100,3 ,0� usingthe �-selection formulas �a� and �b� for all three �-leap meth-ods and with K=1 �allowing single DSSA steps to be per-formed�. Again, for method 2, we choose �=0. For theDSSA and the three �-leap methods, we performed 103 simu-lations, each.

As can be observed from Figs. 5 and 6, methods 1 and 2do not properly capture the oscillatory dynamics in all threespecies concentrations resulting from the DSSA simulationof the hes1 regulatory network. In method 1, the numbers of

TABLE VII. Statistics of method 2: mean values over 100 simulation runs.

�

No. of DSSAsteps �nondelayed�

No. of DSSAsteps

�delayed�

No. of reactions in�-leap steps�nondelayed�

No. of reactions in�-leap steps

�delayed�

�-leapsteps�%�

Simulationswith �-selectionformula �1a�

0.1 1 0 9 450 136 99.970.01 1 139 65 12 015 27 80.820.005 3 514 119 9 949 4 52.070.003 10 762 165 38 0 0.14

Simulations 0.1 0 0 13 191 84 100.00with �-selection 0.03 2 489 100 11 293 21 60.12formula �1b� 0.01 10 692 164 0 0 0.00

205107-9 Generalized binomial �-leap method J. Chem. Phys. 128, 205107 �2008�

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molecules increase monotonically and, in the case of Hes1monomers, they even increase exponentially �Fig. 5�. Due tothe large concentration of molecules, we simulated the sys-tem for 350 min only. The dynamics does not change signifi-cantly with varying � unless � is below a certain thresholdvalue so that the �-leap method becomes the DSSA. Thisthreshold is about 0.003 for �-selection formula �a�.

On the other hand, method 2 yields oscillatory dynamicsbut with a much longer cycle time, approximately 250 min,than the DSSA simulations of approximately 150 min �Fig.6�. The averaged dynamics over 103 simulations for different�- and �-selection formulas �Figs. 7–10� show that this is notmerely the result of stochasticity in a single simulation. Ofcourse, the range of values for which � is actually effectivedepends on the selection formula. Although Figs. 7 and 9suggest that for �=0.003 and �=0.01, method 2 can yieldsimilar oscillatory behaviors to the DSSA, what may not to

be noticed is that the method performs almost only DSSAsteps. This becomes obvious in Figs. 8 and 10, which showthe mean simulation results when the single DSSA stepmechanism is disabled. Table VII presents statistics takenfrom 100 simulation runs for different � when the singleDSSA step mechanism is enabled. Apparently, not enoughnumbers of delayed reactions are drawn, and these are thereactions that drive the molecular clock. According to DSSAsimulations, we would expect about 160–170 delayed reac-tions within 500 min but, instead, we count less than 140delayed reactions and, in the case of selection formula �b�with �=0.1, there are less than 90 delayed reactions. In ad-dition, we made several simulations with different values of�� �−1,1� in Eq. �1� but this did not noticeably affect theoutcome.

Method 3 performs best for all parameter settings. Itcaptures well the oscillatory dynamics resulting from DSSA

FIG. 11. Mean molecular number of monomeric Hes1 obtained from 103

runs of method 3 using �-selection formula �a� for different �.

FIG. 12. Mean molecular number of monomeric Hes1 obtained from 103

runs of method 3 using �-selection formula �a� for different �. Single DSSAruns were not allowed.

FIG. 13. Mean molecular number of monomeric Hes1 obtained from 103

runs of method 3 using �-selection formula �b� for different �.

FIG. 14. Mean molecular number of monomeric Hes1 obtained from 103

runs of method 3 using �-selection formula �b� for different �. Single DSSAruns were not allowed.

205107-10 Leier, Marquez-Lago, and Burrage J. Chem. Phys. 128, 205107 �2008�

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simulations in single runs as well as in the mean of 103

independent simulations �Figs. 6 and 11–14�. In order to il-lustrate any differences in the simulation outcomes fordifferent � values, we plot the number of Hes1 monomers inFigs. 11–14 only for the time interval �300,500�. Thedifferences are marginal. See Table VIII for correspondingstatistics obtained from 100 simulation runs containingexpected numbers of performed delayed reactions for differ-ent �.

Table IX lists simulation run times for different �- andthe �-selection formulas �a� and �b� for methods 2 and 3.Method 3 performs better than method 2, which is mainlydue to the overhead produced by the additional sampling stepin the latter. Since the MATLAB code for both methods isidentical except for those parts that are specific for the re-spective methods, we believe that a comparison based onCPU time is reasonable. However, method 2 is not capableof capturing the oscillatory dynamics of the system and inthis sense the corresponding CPU times are less relevant. For�=0.1 and using �-selection formula �a�, method 3 takesabout 0.33 s and it is therefore approximately as fast as asingle DSSA simulation �about 0.3 s�. This is because of theadditional calculations involved in the �-leap approach. Asindicated by the following example, we can expect this over-head to become increasingly negligible for larger models.

Before concluding this section, we make a few remarksabout accuracy issues. Just as with the SSA and �-leap meth-ods, the accuracy of the delay �-leap methods is comparedwith that of the DSSA. This can be done either by computingappropriate moments or by analysis �human eye, Fourier, andspectral� of the dynamics of the trajectories. This then gives

a very good idea of suitable ranges of �. Of course, thisrange is problem dependent but nevertheless a robust rangecan be determined. Another approach would be to comparethe master equation of the DSSA with that of the delay �-leapmethods as a function of �. As far as we are aware, this hasnot even been done in the nondelayed case, but if it can bedone in that setting it will be possible to do this in the de-layed setting. This will be the subject of future research.

C. The Her1/Her7 model for five coupled cells

Our model is based on the chemical reaction models byboth Lewis22 and Horikawa et al.21 Lewis modeled a singlecell and two coupled cells. His work is generalized byHorikawa et al. to a one-dimensional array of cells. Webriefly described the model for five cells, which we use as athird benchmark model for �-leap method 3 showing the gainin simulation run time while preserving an adequateaccuracy.

For each cell, we simulate the dynamics of 6 differentspecies controlled by 12 reactions. Denote by Mh1i

, Mh7i,

Mdi, Ph1i

, Ph7i, and Pdi

the species Her1 mRNA, Her7mRNA, DeltaC mRNA, Her1 protein, Her7 protein, andDeltaC protein in a particular cell i. For each of the speciesS=Mh1i

,Mh7i,Mdi

, Ph1i, Ph7i

, Pdi, the model contains a degra-

dation reaction

S→c

�

with associated rate constant c=ch1 ,ch7 ,cd ,bh1 ,bh7 ,bd. Thethree different proteins Ph1i

, Ph7i, and Pdi

are synthesizedwith translational delays �h1p, �h7p, and �dp, respectively. Thecorresponding reactions are

S1→c

S1 + S2

with �S1 ,S2�= �Mh1i, Ph1i

� , �Mh7i, Ph7i

� , �Mdi, Pdi

� and associ-ated reaction rate constants c=ah1 ,ah7 ,ad. The transcriptionsof Mh1i

, Mh7i, and Mdi

are regulated reactions with transcrip-tional delays �h1m, �h7m, and �dm, respectively. The reactionsare

S1 ——→cf

S1 + S2

with �S1 ,S2�= �Ph1i,Mh1i

� or �Ph7i,Mh7i

� and associated reac-tion rate constants c=kh1 ,kh7 and

TABLE VIII. Statistics of method 3: mean values over 100 simulation runs.

�

No. of DSSAsteps �nondelayed�

No. of DSSAsteps

�delayed�

No. of reactions in�-leap steps�nondelayed�

No. of reactions in�-leap steps

�delayed�

�-leapsteps�%�

Simulationswith �-selectionformula �1a�

0.1 15 4 10 654 166 98.980.01 2 147 132 8 543 33 58.690.005 4 774 160 5 955 7 33.250.003 10 770 164 46 0 0.18

Simulations 0.01 13 3 10 824 164 99.29with �-selection 0.03 3 350 133 7 496 31 44.68formula �1b� 0.01 10 879 167 0 0 0.00

TABLE IX. Mean simulation runtimes �averaged over 100 simulation runs�of methods 2 and 3. A single DSSA simulation takes about 0.3 s.

Meanruntimes in�s� � Method 2 Method 3

Simulationswith �-selectionformula �1a�

0.1 0.15 0.330.01 3.09 1.780.005 6.63 3.470.003 6.87 2.85

Simulations 0.1 0.77 0.64with �-selection 0.03 6.06 3.43formula �1b� 0.01 6.85 3.13

205107-11 Generalized binomial �-leap method J. Chem. Phys. 128, 205107 �2008�

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S1 ——→cg

S1 + S2

with �S1 ,S2�= �Pdi,Mdt

� and c=kd. As described in detail inRef. 21 the individual negative and positive regulations aremodeled using specific Hill functions f and g. For cells i=2, 3, and 4, the Hill function f is defined by

f�Ph1i,Ph7i

,Pdi−1,Pdt+1

�

= rh1

1 + Ph1iPh7i

/P02

+ rhd1

1 + Ph1iPh7i

/P02

Pdi−1+ Pdi+1

2D0 + Pdi−1+ Pdi+1

,

and for cells 1 and 5, it is

f�Ph11,Ph71

,Pd1� =

1

1 + Ph1iPh7i

/P02

Pd1/D0

1 + Pd1/D0

and

f�Ph15,Ph75

� =1

1 + Ph15Ph75

/P02

1

1 + D0/500,

respectively. The parameters rh and rhd are weight param-eters that determine the balance of internal and external con-tribution of oscillating molecules. In our model, we assume100% coupling, i.e., rhd=1. For all cells, the Hill function gthat describes the inhibition of DeltaC mRNA synthesis byHer1 and Her7 is given by

g�Ph1i,Ph7i

� =1

1 + Ph1iPh7i

/P02 .

See Table VII to Table X for the full list of model param-eters. For all simulations, the system is initialized with statex0= �0, . . . ,0�.

When applied to this model, the generalized binomial�-leap approach �method 3� reveals its computational powercombined with high accuracy, when compared with theDSSA. Figures 15 and 16 present the dynamics of Delta�mRNA and Protein� and Her1 �mRNA and Protein�, respec-tively, for cell 3. The figures result from a single DSSA run

and a single B�-DSSA simulation using method 3. For theB�-DSSA, we used �-selection formula �a� and set �=0.1and Nj�x�=105 for all Hill-type reactions.

Single DSSA simulations took about 80 min, whileB�-DSSA takes only about 70 s for �=0.01 and about 50 sfor �=0.1, which is approximately 70–100 times faster thanthe original DSSA approach. Despite the speedup, a highaccuracy is still preserved: the frequencies of oscillationsmatch almost perfectly for each species in each cell and theamplitudes fluctuate in an expected spectrum. Methods 1 and2 failed to yield the correct dynamics.

As already mentioned, for Hill-type reactions Rj, wechose Nj�x�=const. We investigated the effect of Nj�x� bysimulating model 3 with method 3 for three different values�Nj�x�=101 ,105 ,109� without the option of single DSSAsteps. Table XI shows that the differences in the number andsize of �-steps and the number of delayed and nondelayedreactions for Nj�x�=105 and Nj�x�=109 are marginal. As ex-pected, for Nj�x�=101, the step size is lower but, at the sametime, the number of reactions per �-steps is higher. In fact,the total number of reactions is slightly higher than for largervalues of Nj�x�. Since the number of �-steps is larger forNj�x�=101, the CPU time of the simulation is also larger.

TABLE X. Parameters for the five-cell Her1-Her7 model. Parameter values are taken from Horikawa et al.�Ref. 21�.

Parameter Description Value

bh1 ,bh7 ,bd Her1/Her7/DeltaC protein degradation rate 0.23 min−1

ch1 ,ch7 ,cd Her1/Her7/DeltaC mRNA degradation rate 0.23 min−1

ah1 ,ah7 ,ad Her1/Her7/DeltaC protein synthesis rate �max.� 4.5 min−1

kh1 ,kh7 ,kd Her1/Her7/DeltaC mRNA synthesis rate �max.� 33 min−1

P0 Critical number of Her1+Her7 protein/cell 40D0 Critical number of Delta protein/cell 1000�h1m ,�h7m ,�dm Time to produce a single Her1/Her7/DeltaC

mRNA molecule12.0, 7.1, and 16.0 min

�h1p ,�h7p ,�dp Time to produce a single Her1/Her7/DeltaCprotein

2.8, 1.7, and 20.5 min

FIG. 15. DSSA and �-DSSA �method 3� dynamics of Delta mRNA andDelta protein in cell 3. The numbers of proteins are scaled by 0.05.

205107-12 Leier, Marquez-Lago, and Burrage J. Chem. Phys. 128, 205107 �2008�

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Figure 17 shows the values of � for all steps in two indi-vidual simulations with Nj�x�=101 and Nj�x�=105. Figure 18shows that for both values of Nj�x� the dynamics are almostidentical.

V. DISCUSSIONS AND CONCLUSIONS

In this paper, we have presented a new generalized bi-nomial �-leap simulation method. This method improves onsome of the difficulties associated with previous binomialleap implementations by Tianhai and Burrage and Peng et al.The original difficulties arose from subtleties when comput-ing the Nj needed in the simulation of the binomial randomvariables B�Nj , P� in the case of complex interaction net-works. This new approach also seems more robust and effi-cient than the other approaches when dealing with chemicalreactions associated with Hill-type propensities. Nor are sys-tems containing multiple reactions sharing the same reactantsa constraint on the efficiency of our new approach. In par-ticular, we note that method 3 considers only the directneighbors in the reaction network G, while method 1 consid-ers the entire subnetworks. Consequently, the Nj� are usuallysmaller in method 1 than in method 3 since in the former wecalculate the Nj� as the minimum over a larger set of reac-

tions. By consequence, obtaining a smaller � is more likelyto happen in method 1. Additionally, the Nj� in sampling step�1.5� are usually smaller than the Nj�x ,�� considered in step�3.5�. These two factors together result in method 3 samplingslightly larger numbers of reactions, allowing for the consid-eration of delayed reactions that are known to play a majorrole in driving molecular clocks and would otherwise be ne-glected by a more restrictive time step choice. We also notethat by choosing binomial �-leap as a basis for modelingdelayed reactions we are not drawing any conclusions aboutthe relative efficiencies with other �-leap approaches.

The second aspect of this paper is concerned with theintroduction of delayed processes into our binomial �-leapframework. This is done in a relatively simple way using theexisting framework by dividing the reactions into nonde-layed and delayed and using some of the ideas from Barrio etal. in their implementation of DSSA. The new method�method 3� appears superior to the other implementations byappearing to sample enough of the delayed reactions, whichare crucial in the molecular clock.

Our numerical simulations reveal that, unlike previousbinomial �-leap methods, the B�-DSSA approach is able tocapture the oscillatory dynamics of molecular clock models

TABLE XI. Effect of Nj for all Hill-type reactions in the Her1/Her7 model on � and the number of reactions �delayed/nondelayed� for method 2 with�-selection formula �a�. Single DSSA steps are not allowed. Data are based on a single simulation over 500 min. As a point of reference, from a single DSSAthere are 693 411 nondelayed and 708 017 delayed reactions for the same simulation time.

Nj�x� � No. of �-leap steps Average � Maximum �

No. of reactions in�-leap steps�nondelayed�

No. of reactions in�-leap steps

�delayed�

101 0.1 12 389 0.0404 0.1922 682 963 696 0150.01 14 003 0.0357 0.1940 666 583 682 259

105 0.1 2 851 0.1754 0.2114 651 827 665 2630.01 5 040 0.0992 0.2114 662 595 678 959

109 0.1 2 717 0.1841 0.2114 665 511 679 6680.01 4 891 0.1023 0.2114 615 221 630 213

FIG. 16. DSSA and �-DSSA �method 3� dynamics of Her1 mRNA and Her1protein in cell 3. The numbers of proteins are scaled by 0.05.

FIG. 17. Sizes of �-steps over time for single �-DSSA runs �method 3� fortwo different values of Nj�x�. The single DSSA step mechanism wasdisabled.

205107-13 Generalized binomial �-leap method J. Chem. Phys. 128, 205107 �2008�

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such as Hes1 autoinhibition in a single cell, and Her1/7-Delta-Notch signaling in a multicellular setting, when num-bers of delayed reactions play a crucial role in maintainingthe cyclic behavior. Our findings show that the B�-DSSA cansimulate complex models with large numbers of species andreactions, both accurately and significantly faster than theDSSA. Specifically, we simulated the Her1/7-Delta-Notchsignaling pathway in five coupled cells 70–100 times fasterthan the DSSA by Barrio et al. Our method is also applicableto reaction systems without delays. In such cases, it performsequally well in terms of accuracy compared to other bino-mial �-leap methods.

ACKNOWLEDGMENTS

K.B. would like to thank the Australian Research Coun-cil for funding through the Federation Fellowship program.

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FIG. 18. Her1 protein dynamics in cell 3 obtained from single �-DSSA runs�method 3� for two different values of Nj�x�. The single DSSA step mecha-nism was disabled.

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