Splitting Strategy for Simulating Genetic Regulatory Networks · ResearchArticle Splitting Strategy...

Research ArticleSplitting Strategy for Simulating Genetic Regulatory Networks

Xiong You Xueping Liu and Ibrahim Hussein Musa

Department of Applied Mathematics Nanjing Agricultural University Nanjing 210095 China

Correspondence should be addressed to Xiong You youxnjaueducn

Received 30 May 2013 Accepted 24 October 2013 Published 2 February 2014

Academic Editor Damien Hall

Copyright copy 2014 Xiong You et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The splitting approach is developed for the numerical simulation of genetic regulatory networks with a stable steady-state structureThe numerical results of the simulation of a one-gene network a two-gene network and a p53-mdm2 network show that the newsplitting methods constructed in this paper are remarkably more effective and more suitable for long-term computation with largesteps than the traditional general-purpose Runge-Kutta methods The new methods have no restriction on the choice of stepsizedue to their infinitely large stability regions

1 Introduction

The exploration of mechanisms of gene expression and regu-lation has become one of the central themes in medicine andbiological sciences such as cell biology molecular biologyand systems biology [1 2] For example it has been acknowl-edged that the p53 tumor suppressor plays key regulatoryroles in various fundamental biological processes includingdevelopment ageing and cell differentiation It can regulateits downstream genes through their signal pathways andfurther implement cell cycle arrest and cell apoptosis [3ndash6]The qualitative analysis as well as numerical simulation hasbecome an important route in the investigation of differentialequations of genetic regulatory networks (GRNs) in thepast few years [7ndash10] Up till now algorithms used in thesimulation of GRNs have primarily been classical Runge-Kutta (RK) methods (typically of order four) or Runge-Kutta-Fehlberg embedded pairs as employed in the scientificcomputing software MATLAB [11ndash13] However if we arerequired to achieve a very high accuracy we have to takevery small stepsize Moreover the traditional Runge-Kuttatype methods often fail to retain some important qualitativeproperties of the system of interest This prevents us fromacquiring correct knowledge of the dynamics of geneticregulatory networks

Geometric numerical integration aims at solving differ-ential equations effectively while preserving the geometricproperties of the exact flow [14] Recently You et al [15]

develop a family of trigonometrically fitted Scheifele two-step(TFSTS) methods derive a set of necessary and sufficientconditions for TFSTS methods to be of up to order fivebased on the linear operator theory and construct twopracticalmethods of algebraic four andfive respectively Veryrecently You [16] develops a new family of phase-fitted andamplification methods of Runge-Kutta type which have beenproved very effective for genetic regulatory networks with alimit-cycle structure

Splitting is one of the effective techniques in geometricintegration For example Blanes and Moan [17] constructa symmetric fourth- and sixth-order symplectic partitionedRunge-Kutta and Runge-Kutta-Nystrom methods and showthat these methods have an optimized efficiency For asystematic presentation of the splitting technique the readeris referred to Hairer et al [14] The purpose of this paperis to develop the splitting methods for GRNs In Section 2we present the system of differential equations governing theGRNs and basic assumptions for the system In Section 3 wedescribe the idea and formation of the approach of splittingstrategy which intends to simulate exactly the characteristicpart of the system Section 4 gives the simulation results ofthe new splitting methods and the traditional Runge-Kuttamethods when they are applied to a one-gene network a two-gene network and a p53-mdm2 network We compare theiraccuracy and computational efficiency Section 5 is devoted toconclusive remarks Section 6 is for discussions In Appendixthe linear stability of the new splitting methods is analyzed

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014 Article ID 683235 9 pageshttpdxdoiorg1011552014683235

2 Computational and Mathematical Methods in Medicine

2 Materials

21 mRNA-ProteinNetworks An119873-gene regulatory networkcan be described by the following system of ordinary differ-ential equations

119903 (119905) = minusΓ119903 (119905) + 119865 (119901 (119905))

(119905) = minus119872119901 (119905) + 119870119903 (119905)

(1)

where 119903(119905) = (1199031(119905) 119903

119873(119905)) and 119901(119905) = (119901

1(119905) 1199012(119905)

119901119873(119905)) are 119873-dimensional vectors representing the concen-

trations of mRNAs and proteins at time 119905 respectively and119865(119901(119905)) = (119865

1(119901(119905)) 119865

119873(119901(119905))) Γ = diag(120574

1 120574

119873)

119872 = diag(1205831 120583

119873) and119870 = diag(120581

1 120581

119873) are diagonal

matrices The system (1) can be written in components as

119903119894(119905) = minus120574

119894119903119894(119905) + 119865

119894(119901 (119905))

119894(119905) = minus120583

119894119901119894(119905) + 120581

119894119903119894(119905)

(2)

where for 119894 = 1 2 119873 119903119894(119905) and 119901

119894(119905) isin R+ are the

concentrations of mRNA 119894 and the corresponding protein 119894

at time 119905 respectively 120574119894and 120583

119894 positive constants are the

degradation rates of mRNA 119894 and protein 119894 respectively 120581119894is

a positive constant and119865119894(sdot) the regulatory function of gene 119894

is a nonlinear andmonotonic function in each of its variablesIf gene 119894 is activated by protein 119895 120597119865

119894120597119901119895gt 0 and if gene 119894 is

inhibited by protein 119895 120597119865119894120597119901119895lt 0 If protein 119895 has no action

on gene 119894 119901119895will not appear in the expression of 119865

119894

In particular we are concerned in this paper with thefollowing two simple models

(I) The first model is a one-gene regulatory networkwhich can be written as

119903 (119905) = minus120574119903 (119905) + 119891 (119901 (119905))

(119905) = minus120583119901 (119905) + 120581119903 (119905)

(3)

where 119891(119901(119905)) = 120572(1 + 119901(119905)21205792) represents the action of an

inhibitory protein that acts as a dimer and 120574 120583 120581 120572 and 120579 arepositive constants This model with delays can be found inXiao and Cao [18]

(II) The second model is a two-gene cross-regulatorynetwork [7 19]

1199031= 1205821ℎ+(1199012 1205792 1198992) minus 12057411199031

1199032= 1205822ℎminus(1199011 1205791 1198991) minus 12057421199032

1= 12058111199031minus 12057511199011

2= 12058121199032minus 12057521199012

(4)

where 1199031and 1199032are the concentrations ofmRNA 1 andmRNA

2 respectively 1199011and 119901

2are the concentrations of their

corresponding products protein 1 and protein 2 respectively1205821 and 120582

2represent the maximal transcription rates of gene

1 and gene 2 respectively 1205741and 1205742are the degradation rates

of mRNA 1 and mRNA 2 respectively 1205751and 120575

2are the

degradation rates of protein 1 and protein 2 respectively

ℎ+(1199012 1205792 1198992) =

1199011198992

2

1199011198992

2+ 1205791198992

2

ℎminus(1199011 1205791 1198991) =

1205791198991

1

1199011198991

1+ 1205791198991

1

(5)

are the Hill functions for activation and repression respec-tively 119899

1and 1198992are the Hill coefficients and 120579

1and 1205792are the

thresholds It is easy to see that the activation function ℎ+ is

increasing in 1199012and the repression function ℎ

minus is decreasingin 1199011

22 A p53-mdm2Regulatory Pathway Anothermodelwe areinterested in is for the damped oscillation of the p53-mdm2regulatory pathway which is given by (see [20])

119868= 119904119901+ 119895119886119875119860minus (119889119901+ 119896119886119878 (119905)) 119875

119868minus 119896119888119875119868119872+ 119895

119888119862

= 1199041198980

+1199041198981119875119868+ 1199041198982119875119860

119875119868+ 119875119860+ 119870119898

+ 119896119906119862 + 119895119888119862 minus (119889

119898+ 119896119888119875119868)119872

= 119896119888119875119868119872minus (119895

119888+ 119896119906) 119862

119860= 119896119886119878 (119905) 119875

119868minus (119895119886+ 119889119901) 119875119860

(6)

where 119875119868represents the concentration of the p53 tumour

suppressor119872 (mdm2) is the concentration of the p53rsquos mainnegative regulator 119862 is the concentration of the p53-mdm2complex 119875

119860is the concentration of an active form of p53

that is resistant against mdm2-mediated degradation 119878(119905) is atransient stress stimulus which has the form 119878(119905) = minus119890

119888119904119905 119888119904=

120574119896119906 119904lowast(lowast = 11990111989801198981) are de novo synthesis rates 119896

lowast(lowast =

119886 119888 119906) are production rates 119895lowast(lowast = 119886 119888) are reverse reactions

(eg dephosphorylation) 119889119901is the degradation rate of active

p53 and119870119898is the saturation coefficient

3 Methods

31 Runge-Kutta Methods Either the mRNA-protein net-work (1) or the p53-mdm2 regulatory pathway (6) can beregarded as a special form of a system of ordinary differentialequations (ODEs)

1199111015840= 119892 (119911) (7)

where 119911 = 119911(119905) isin 119877119889 and the function 119892 R119889 rarr R119889

is smooth enough as required The system (7) together withinitial value 119911(0) = 119911

0is called an initial value problem (IVP)

Throughout this paper we make the following assumptions

(i) The system (7) has a unique positive steady state 119911lowastthat is there is a unique 119911

lowast= (119911lowast

1 119911lowast

2 119911

lowast

119889) with

119911lowast

119894gt 0 for 119894 = 1 2 119889 such that 119892(119911lowast) = 0

(ii) The steady state 119911lowast is asymptotically stable that is forany solution 119911(119905) of the system (7) through a positiveinitial point 119911

0 lim119905rarr+infin

119911(119905) = 119911lowast

Computational and Mathematical Methods in Medicine 3

The most frequently used algorithms for the system (7)are the so-called Runge-Kutta methods which read

119885119894= 119911119899+ ℎ

119904

sum

119895=1

119886119894119895119892 (119885119895) 119894 = 1 119904

119911119899+1

= 119911119899+ ℎ

119904

sum

119894=1

119887119894119892 (119885119894)

(8)

where 119911119899is an approximation of the solution 119911(119905) at 119905

119899 119899 =

0 1 119886119894119895 119887119894 119894 119895 = 1 119904 are real numbers 119904 is the number

of internal stages119885119894 and ℎ is the step sizeThe scheme (8) can

be represented by the Butcher tableau

119888 119860

119887119879 =

1198881

11988611

1198861119904

d

119888119904

1198861199041

119886119904119904

1198871

119887119904

(9)

or simply by (119888 119860 119887(])) where 119888119894= sum119904

119895=1119886119894119895for 119894 = 1 119904

Two of the most famous fourth-order RK methods have thetableaux as follows (see [13])

0

12 12

12 12

1 0 0 1

16 26 26 16

0

13 13

23 1

11 1

18 38 38 18

minus1

minus12 (10)

which we denote as RK4 and RK38 respectively

32 Splitting Methods Splitting methods have been provedto be an effective approach to solve ODEs The main ideais to split the vector field into two or more integrable partsand treat them separately For a concise account of splittingmethods see Chapter II of Hairer et al [14]

Suppose that the vector field 119892 of the system (7) has a splitstructure

1199111015840= 119892[1]

(119911) + 119892[2]

(119911) (11)

Assume also that both systems 1199111015840 = 119892[1](119911) and 119911

1015840= 119892[2](119911)

can be solved in closed form or are accurately integrated andtheir exact flows are denoted by 120593[1]

ℎand 120593[2]

ℎ respectively

Definition 1 (i) The method defined by

Ψℎ= 120593[2]

ℎ∘ 120593[1]

ℎ

Φℎ= 120593[1]

ℎ∘ 120593[2]

ℎ

(12)

is the simplest splitting method for the system (7) based onthe decomposition (11) (see [13])

(ii) The Strang splitting is the following symmetric ver-sion (see [21])

Ψℎ= 120593[1]

ℎ2∘ 120593[2]

ℎ∘ 120593[1]

ℎ2 (13)

(iii) The general splitting method has the form

Ψℎ= 120593[2]

119887119898ℎ∘ 120593[1]

119886119898ℎ∘ sdot sdot sdot ∘ 120593

[2]

1198872ℎ∘ 120593[1]

1198862ℎ∘ 120593[2]

1198871ℎ∘ 120593[1]

1198861ℎ (14)

where 1198861 1198871 1198862 1198872 119886

119898 119887119898are positive constants satisfying

some appropriate conditions See for example [22ndash25]

Theorem 56 in ChapterII of Hairer et al [14] gives theconditions for the splitting method (14) to be of order 119901

However in most occasions the exact flows 120593[1]ℎ

and 120593[2]

ℎ

for 119892[1] and 119892[2] in Definition 1 are not available Hence we

have to use instead some approximations or numerical flowswhich are denoted by 120595

1and 120595

2

33 Splitting Methods for Genetic Regulatory Networks Basedon Their Characteristic Structure For a given genetic reg-ulatory network different ways of decomposition of thevector field 119891 may produce different results of computationThus a question arises as follows which decomposition ismore appropriate or more effective In the following we takethe system (1) for example The analysis of the p53-mdm2pathway (6) is similar Denote 119911(119905) = (119903(119905) 119901(119905))

119879 Thenthe 119873-gene regulatory network (1) has a natural form ofdecomposition

119892[1]

(119911) = (minusΓ 0

119870 minus119872)119911 119892

[2](119911) = (

119865 (119901 (119905))

0) (15)

Unfortunately it has been checked through practical test thatthe splitting methods based on this decomposition cannotlead to effective results To find a way out we first observethat a coordinate transform 119909(119905) = 119903(119905) minus 119903

lowast 119910(119905) = 119901(119905)minus119901

lowast

translates the steady state (119903lowast 119901lowast) of the system (1) to the

origin and yields

(119905) = minusΓ119909 (119905) + 1198651015840(119901lowast) 119910 (119905) + 119866 (119910 (119905))

119910 (119905) = 119870119909 (119905) minus 119872119910 (119905)

(16)

where 1198651015840(119901lowast) is the Jacobian matrix of 119865(119901) at point 119901lowast

and 119866(119910(119905)) = 119865(119901lowast+ 119910(119905)) minus 119865

1015840(119901lowast)119910(119905) minus 119865(119901

lowast)


We employ this special structure of the system (16) to reachthe decomposition of the vector field

119892[1]

(119911) = (minusΓ 119865

1015840(119901lowast)

119870 minus119872)119911 119892

[2](119911) = (

119866 (119910 (119905))

0)

(17)

where 119911(119905) = (119909(119905) 119910(119905))119879 The system = 119892

[1](119911) here is

called the linearization of the system (1) at the steady state(119903lowast 119901lowast) 119892[1] in (17) is the linear principal part of the vector

field 119892 which has the exact flow 120593[1]

ℎ= exp(ℎ ( minusΓ 1198651015840(119901lowast)

119870 minus119872))

However it is not easy or impossible to obtain the exactsolution of 119892[2](119909

119894(119905) 119910119894(119905)) due to its nonlinearity So we have

to use an approximation flow 120595[2]

ℎand form the splitting

method

Ψℎ= 120595[2]

ℎ∘ 120593[1]

ℎ (18)

When 120595[2] is taken as an RK method then the resulting

splitting method is denoted by Split(ExactRK) Hence wewrite Split(ExactRK4) and Split(ExactRK38) for the split-ting methods with120595[2] taken as RK4 and RK38 respectively

4 Results

In order to examine the numerical behavior of the newsplitting methods Split(ExactRK4) and Split(ExactRK38)we apply them to the three models presented in Section 2Their corresponding RK methods RK4 and RK38 are alsoused for comparison We will carry out two observationseffectiveness and efficiency For effectiveness we first findthe errors produced by each method with different values ofstepsize We also solve each problem with a fixed stepsize ondifferent lengths of time intervals

41 The One-Gene Network Table 1 gives the parametervalues which are provided by Xiao and Cao [18] This systemhas a unique steady state (119903

lowast 119901lowast) = (06 2) where the

Jacobianmatrix has eigenvalues1205961= minus12500+29767119894 120596

2=

minus12500 minus 29767119894 where 119894 is the imaginary unit satisfying1198942= minus1 Since the two eigenvalues both have negative real

parts the steady state is asymptotically stableIn order to apply the splitting methods Split(ExactRK4)

and Split(ExactRK38) the vector field of the system (3) isdecomposed in the way of (16) as

119892[1]

(119911 (119905)) = (minus120574 minus

2120572119901lowast1205792

(1 + 119901lowast21205792)2

120581 minus120583

)(119909 (119905)

119910 (119905))

119892[2]

(119911 (119905))

= (

120572

1 + (119901lowast + 119910 (119905))2

1205792+

2120572119901lowast1205792

(1 + 119901lowast21205792)2119910 (119905)

0

)

(19)

The system is solved on the time interval [0 100] withinitial values ofmRNAand protein 119903(0) = 06 119901(0) = 08 andwith different stepsizes The numerical results are presentedin Table 2

Then we solve the problem with a fixed stepsize ℎ = 2

on several lengths of time intervalsThe numerical results aregiven in Table 3

42 The Two-Gene Network Table 4 gives the parametervalues which can be found in Widder et al [19] Thissystem has a unique steady state (119903

lowast

1 119903lowast

2 119901lowast

1 119901lowast

2) =

(0814713 0614032 0814713 0614032) Since the foureigenvalues 120596

1= minus19611 + 09611119894 120596

2= minus19611minus

09611119894 1205963= minus00389 + 09611119894 and 120596

4= minus00389 minus 09611119894

of the Jacobian matrix at the steady state (119903lowast 119901lowast) all have

negative real parts the steady state is asymptotically stableFor this system the decomposition (16) becomes

119892[1]

(119911 (119905))

=(((

(

minus1205741

0 0119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)2

0 minus1205742minus119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)2

0

1205811

0 minus1205751

0

0 1205812

0 minus1205752

)))

)

times(

1199091(119905)

1199092(119905)

1199101(119905)

1199102(119905)

)

119892[2]

(119911 (119905))

=(((

(

1205821(1199102(119905) + 119901

lowast

2)1198992

(1199102(119905) + 119901

lowast

2)1198992

+ 1205791198992

2

minus119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)21199102(119905)

12058221205791198991

1

(1199101(119905) + 119901

lowast

1)1198991

+ 1205791198991

1

+119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)21199101(119905)

0

0

)))

)

(20)

The system is solved on the time interval [0 100] withthe initial values 119903

1(0) = 06 119903

2(0) = 08 119901

1(0) = 06 and

1199012(0) = 08 andwith different stepsizesThenumerical results

are presented in Table 5Then we solve the problem with a fixed stepsize ℎ = 2


43 The p53-mdm2 Network Table 7 gives the parame-ter values which are used by van Leeuwen et al [20]For simplicity we take the small function 119878(119905) equiv 0


Table 1 Parameter values for the one-gene network

120572 = 3 120574 = 1 120583 = 15 120581 = 5

This system has a unique steady state (119875lowast

119868119872lowast 119862lowast 119875lowast

119860) =

(942094 00372868 349529 0) Since the eigenvalues 1205961=

minus384766 1205962= minus00028 + 00220119894 120596

3= minus00028 minus 00220119894

and 1205964= minus02002 of the Jacobian matrix at the steady state

all have negative real parts the steady state is asymptoticallystable

For this system decomposition (16) becomes

119892[1]

(119911 (119905)) = 119869(

1199111(119905)

1199112(119905)

1199113(119905)

1199114(119905)

)

119892[2]

(119911 (119905)) = (

minus1198961198881199111(119905) 1199112(119905)

120592 (119905)

1198961198881199111(119905) 1199112(119905)

0

)

(21)

where

119869 =(((

(

minus119896119888119872lowastminus 119889119901

minus119896119888119875lowast

119868119895119888

119895119886

1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

minus 119896119888119872lowast

minus119889119898minus 119896119888119875lowast

119868119896119906+ 119895119888

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

119896119888119872lowast

119896119888119875lowast

119868minus (119895119888+ 119896119906) 0

0 0 0 minus (119895119886+ 119889119901)

)))

)

120592 (119905) = 1199041198980

+1199041198981

(1199111(119905) + 119875

lowast

119868) + 1199041198982

(1199114(119905) + 119875

lowast

119860)

1199111(119905) + 119875

lowast

119868+ 1199114(119905) + 119875

lowast

119860+ 119870119898

minus 119889119898119872lowast+ 1198961198881199111(119905) 1199112(119905) + 2119896

1198881199111(119905)119872lowast

minus 119896119888119875lowast

119868119872lowastminus1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199111(119905) minus

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199114(119905)

+ (119896119906+ 119895119888) 119862lowast

(22)

The system is solved on the time interval [0 100] withinitial values 119875

119868(0) = 942 nM 119862(0) = 0037 nM 119872(0) =

349 nM and 119875119860(0) = 0 nM and with different stepsizes The

numerical results are presented in Table 8Then we solve the problem with a fixed stepsize ℎ = 2


5 Conclusions

In this paper we have developed a new type of splittingalgorithms for the simulation of genetic regulatory networksThe splitting technique has taken into account the specialstructure of the linearizing decomposition of the vectorfield From the results of numerical simulation of Tables2 5 and 8 we can see that the new splitting methodsSplit(ExactRK4) and Split(ExactRK38) are much moreaccurate than the traditional Runge-Kutta methods RK4 andRK38 For large steps when RK4 and RK38 completely loseeffect Split(ExactRK4) and Split(ExactRK38) continue towork verywell On the other hand Tables 3 6 and 9 show thatfor comparatively large steps RK4 and RK38 can solve theproblem only on short time intervals while Split(ExactRK4)and Split(ExactRK38) work for very long time intervals

We conclude that for genetic regulatory networkswith an asymptotically stable steady state compared with

the traditional Runge-Kutta the new splitting methods havetwo advantages

(a) They are extremely accurate for large steps Thispromises high efficiency for solving large-scale sys-tems (complex networks containing a large numberof distinct proteins) in a long-term simulation

(b) They work effectively for very long time intervalsThis makes it possible for us to explore the long-run behavior of genetic regulatory network which isimportant in the research of gene repair and genetherapy

The special structure of the new splitting methodsand their remarkable stability property (see Appendix) areresponsible for the previous two advantages

6 Discussions

The splitting methods designated in this paper have openeda novel approach to effective simulation of the complexdynamical behaviors of genetic regulatory network with acharacteristic structure It is still possible to enhance theeffectiveness of the new splitting methods For examplehigher-order splitting methods can be obtained by recursivecomposition (14) or by employing higher order Runge-Kuttamethods see II5 of [13] Another possibility is to consider


Table 2 One-gene network average errors for different stepsizes

Stepsize RK4 RK38 Split(ExactRK4) Split(ExactRK38)12 17733 times 10

062404 times 10

minus113910 times 10


minus3

15 32171 times 1017

12337 times 101

11862 times 10minus3


20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062



Table 3 One-gene network average errors for fixed stepsize ℎ = 2 on different time intervals

Time interval RK4 RK38 Split(ExactRK4) Split(ExactRK38)[0 100] 33619 times 10

5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240



[0 1000] NaN NaN 55903 times 10minus5




Table 4 Parameter values for the two-gene network

1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3

Table 5 Two-gene network average errors for different stepsizes





minus4

12 25157 times 10minus1




15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068



Table 6 Two-gene network average errors for fixed stepsize ℎ = 2 on different time intervals


015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095







Table 7 Parameter values for the p53-mdm2 pathway

1199041198980

= 2 times 10minus3 nMminminus1 119896

119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981

= 015 nMminminus1 119896119888= 4minminus1 nMminus1 119895

119888= 2 times 10

minus3minminus1

1199041198982

= 02 nMminminus1 119896119906= 04min minus1 119889

119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1

Table 8 p53-mdm2 network average errors for different stepsizes





minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100



012 NaN NaN 34986 times 10minus6





Table 9 p53-mdm2 network average errors for fixed stepsize ℎ = 10 on different time intervals

Time interval RK4 RK38 Split(ExactRK4) Split(ExactRK38)[0 100] NaN NaN 68810 times 10


minus2









0

Stability region of RK4

minus10 minus5 0 5 10minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)

Figure 1 (a) Stability region of RK4 (left) and (b) stability region of RK38 (right)


minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v

Stability region of Split(ExactRK4)

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)

Figure 2 (a) Stability region of Split(ExactRK4) (left) and (b) stability region of Split(ExactRK38) (right)

embedded pairs of two splitting methods which can improvethe efficiency see II4 of [13]

The genetic regulatory networks considered in thispaper are nonstiff For stiff systems (whose Jacobian pos-sesses eigenvalues with large negative real parts or with

purely imaginary eigenvalues of large modulus) the pre-vious techniques suggested by the reviewer are appli-cable Moreover the error control technique which canincrease the efficiency of the methods is an interesting themefor future work


There are more qualitative properties of the geneticregulatory networks that can be taken into account in thedesignation of simulation algorithms For example oscil-lation in protein levels is observed in most regulatorynetworks Symmetric and symplectic methods have beenshown to have excellent numerical behavior in the long-term integration of oscillatory systems even if they are notHamiltonian systems A brief account of symmetric andsymplectic extended Runge-Kutta-Nystrom (ERKN) meth-ods for oscillatory Hamiltonian systems and two-step ERKNmethods can be found for instance in Yang et al [26] Chenet al [27] Li et al [28] and You et al [29]

Finally a problem related to this work remains openWe observe that in Tables 3 and 9 for the p53-mdm2pathway as the time interval extends the error producedby Split(ExactRK4) and Split(ExactRK38) becomes evensmaller This phenomenon is yet to be explained

Appendix

Stability Analysis of Runge-Kutta Methods andSplitting Methods

Stability analysis is a necessary step for a new numericalmethod before it is put into practice Numerically unstablemethods are completely useless In this appendix we examinethe linear stability of the new splitting method constructedin Section 3 To this end we consider the linear scalar testequation

119910 = 120582119910 + 120598119910 (A1)

where 120582 is a test rate which is an estimate of the principalrate of a scalar problem and 120598 is the error of the estimationApplying the splitting method Φ

ℎ(18) to the test equation

(A1) we obtain the difference equation

119910119899+1

= 119877 (119906 V) 119910119899 (A2)

where 119877(119906 V) is called the stability function with 119906 = 120582ℎ andV = 120598ℎ

Definition A1 The region in the 119906-V plane

R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)

is called the stability region of the method and the curvedefined by |119877(119906 V)| le 1 is call the boundary of stability region

By simple computation we obtain the stability functionof an RK method

119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)

where 119890 = (1 1 1)119879 119868 is the 119904 times 119904 unit matrix and the

stability function 119877(119906 V) a splitting method Split(ExactRK)

119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)

The stability regions of RK4 and RK38 are pre-sented in Figure 1 and those of Split(ExactRK4) and

Split(ExactRK38) are presented in Figure 2 It is seen thatSplit(ExactRK4) and Split(ExactRK38) have much largerstability regions than RK4 and RK38 Moreover the infin-ity area of the stability regions of Split(ExactRK4) andSplit(ExactRK38) means that these two methods have nolimitation to the stepsize ℎ for the stability reason but forthe consideration of accuracy while RK4 and RK38 willcompletely lose effect when the stepsize becomes large tosome extent This has been verified in Section 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees for theirinvaluable comments and constructive suggestions whichhelp greatly to improve this paper This work was partiallysupported by NSFC (Grant no 11171155) and the Fundamen-tal Research Fund for the Central Universities (Grant noY0201100265)

References

[1] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008

[2] F Ren and J Cao ldquoAsymptotic and robust stability of geneticregulatory networks with time-varying delaysrdquo Neurocomput-ing vol 71 no 4ndash6 pp 834ndash842 2008

[3] J J Chou H Li G S Salvesen J Yuan and G Wagner ldquoSolu-tion structure of BID an intracellular amplifier of apoptoticsignalingrdquo Cell vol 96 no 5 pp 615ndash624 1999

[4] C A Perez and J A Purdy ldquoTreatment planning in radiationoncology and impact on outcome of therapyrdquo Rays vol 23 no3 pp 385ndash426 1998

[5] B Vogelstein D Lane and A J Levine ldquoSurfing the p53networkrdquo Nature vol 408 no 6810 pp 307ndash310 2000

[6] J P Qi S H Shao D D Li and G P Zhou ldquoA dynamicmodel for the p53 stress response networks under ion radiationrdquoAmino Acids vol 33 no 1 pp 75ndash83 2007

[7] A Polynikis S J Hogan and M di Bernardo ldquoComparingdifferent ODE modelling approaches for gene regulatory net-worksrdquo Journal ofTheoretical Biology vol 261 no 4 pp 511ndash5302009

[8] A Goldbeter ldquoA model for circadian oscillations in theDrosophila period protein (PER)rdquo Proceedings of the RoyalSociety B vol 261 no 1362 pp 319ndash324 1995

[9] C Gerard D Gonze and A Goldbeter ldquoDependence of theperiod on the rate of protein degradation inminimalmodels forcircadian oscillationsrdquo Philosophical Transactions of the RoyalSociety A vol 367 no 1908 pp 4665ndash4683 2009

[10] J C Leloup and A Goldbeter ldquoToward a detailed computa-tionalmodel for themammalian circadian clockrdquoProceedings ofthe National Academy of Sciences of the United States of Americavol 100 no 12 pp 7051ndash7056 2003

[11] J C Butcher Numerical Methods for Ordinary DifferentialEquations John Wiley amp Sons 2nd edition 2008


[12] J C Butcher and G Wanner ldquoRunge-Kutta methods somehistorical notesrdquo Applied Numerical Mathematics vol 22 no 1ndash3 pp 113ndash151 1996

[13] E Hairer S P Noslashrsett and G Wanner Ordinary DifferentialEquations I Nonstiff Problems Springer Berlin Germany 1993

[14] E Hairer C Lubich and GWanner Solving Geometric Numer-ical Integration Structure-Preserving Algorithms Springer 2ndedition 2006

[15] X You Y Zhang and J Zhao ldquoTrigonometrically-fittedScheifele two-step methods for perturbed oscillatorsrdquo Com-puter Physics Communications vol 182 no 7 pp 1481ndash14902011

[16] X You ldquoLimit-cycle-preserving simulation of gene regulatoryoscillatorsrdquo Discrete Dynamics in Nature and Society vol 2012Article ID 673296 22 pages 2012

[17] S Blanes and P C Moan ldquoPractical symplectic partitionedRunge-Kutta and Runge-Kutta-Nystrom methodsrdquo Journal ofComputational andAppliedMathematics vol 142 no 2 pp 313ndash330 2002

[18] M Xiao and J Cao ldquoGenetic oscillation deduced from Hopfbifurcation in a genetic regulatory network with delaysrdquoMath-ematical Biosciences vol 215 no 1 pp 55ndash63 2008

[19] S Widder J Schicho and P Schuster ldquoDynamic patternsof gene regulation I simple two-gene systemsrdquo Journal ofTheoretical Biology vol 246 no 3 pp 395ndash419 2007

[20] I M M van Leeuwen I Sanders O Staples S Lain andA J Munro ldquoNumerical and experimental analysis of thep53-mdm2 regulatory pathwayrdquo in Digital Ecosystems F ABasile Colugnati L C Rodrigues Lopes and S F AlmeidaBarretto Eds vol 67 of Lecture Notes of the Institute forComputer Sciences Social Informatics and TelecommunicationsEngineering pp 266ndash284 2010

[21] G Strang ldquoOn the construction and comparison of differenceschemesrdquo SIAM Journal on Numerical Analysis vol 5 pp 506ndash517 1968

[22] R D Ruth ldquoCanonical integration techniquerdquo IEEE Transac-tions on Nuclear Science vol NS-30 no 4 pp 2669ndash2671 1983

[23] M Suzuki ldquoFractal decomposition of exponential operatorswith applications to many-body theories and Monte Carlosimulationsrdquo Physics Letters A vol 146 no 6 pp 319ndash323 1990

[24] M Suzuki ldquoGeneral theory of higher-order decompositionof exponential operators and symplectic integratorsrdquo PhysicsLetters A vol 165 no 5-6 pp 387ndash395 1992

[25] H Yoshida ldquoConstruction of higher order symplectic integra-torsrdquo Physics Letters A vol 150 no 5ndash7 pp 262ndash268 1990

[26] H Yang X Wu X You and Y Fang ldquoExtended RKN-typemethods for numerical integration of perturbed oscillatorsrdquoComputer Physics Communications vol 180 no 10 pp 1777ndash1794 2009

[27] Z Chen X YouW Shi and Z Liu ldquoSymmetric and symplecticERKNmethods for oscillatoryHamiltonian systemsrdquoComputerPhysics Communications vol 183 no 1 pp 86ndash98 2012

[28] J Li B Wang X You and X Wu ldquoTwo-step extended RKNmethods for oscillatory systemsrdquo Computer Physics Communi-cations vol 182 no 12 pp 2486ndash2507 2011

[29] X You Z Chen and Y Fang ldquoNew explicit adapted Numerovmethods for second-order oscillatory differential equationsrdquoApplied Mathematics and Computation vol 219 pp 6241ndash62552013


2 Materials

21 mRNA-ProteinNetworks An119873-gene regulatory networkcan be described by the following system of ordinary differ-ential equations

119903 (119905) = minusΓ119903 (119905) + 119865 (119901 (119905))

(119905) = minus119872119901 (119905) + 119870119903 (119905)

(1)

where 119903(119905) = (1199031(119905) 119903

119873(119905)) and 119901(119905) = (119901

1(119905) 1199012(119905)

119901119873(119905)) are 119873-dimensional vectors representing the concen-

trations of mRNAs and proteins at time 119905 respectively and119865(119901(119905)) = (119865

1(119901(119905)) 119865

119873(119901(119905))) Γ = diag(120574

1 120574

119873)

119872 = diag(1205831 120583

119873) and119870 = diag(120581

1 120581

119873) are diagonal

matrices The system (1) can be written in components as

119903119894(119905) = minus120574

119894119903119894(119905) + 119865

119894(119901 (119905))

119894(119905) = minus120583

119894119901119894(119905) + 120581

119894119903119894(119905)

(2)

where for 119894 = 1 2 119873 119903119894(119905) and 119901

119894(119905) isin R+ are the

concentrations of mRNA 119894 and the corresponding protein 119894

at time 119905 respectively 120574119894and 120583

119894 positive constants are the

degradation rates of mRNA 119894 and protein 119894 respectively 120581119894is

a positive constant and119865119894(sdot) the regulatory function of gene 119894

is a nonlinear andmonotonic function in each of its variablesIf gene 119894 is activated by protein 119895 120597119865

119894120597119901119895gt 0 and if gene 119894 is

inhibited by protein 119895 120597119865119894120597119901119895lt 0 If protein 119895 has no action

on gene 119894 119901119895will not appear in the expression of 119865

119894

In particular we are concerned in this paper with thefollowing two simple models

(I) The first model is a one-gene regulatory networkwhich can be written as

119903 (119905) = minus120574119903 (119905) + 119891 (119901 (119905))

(119905) = minus120583119901 (119905) + 120581119903 (119905)

(3)

where 119891(119901(119905)) = 120572(1 + 119901(119905)21205792) represents the action of an

inhibitory protein that acts as a dimer and 120574 120583 120581 120572 and 120579 arepositive constants This model with delays can be found inXiao and Cao [18]

(II) The second model is a two-gene cross-regulatorynetwork [7 19]

1199031= 1205821ℎ+(1199012 1205792 1198992) minus 12057411199031

1199032= 1205822ℎminus(1199011 1205791 1198991) minus 12057421199032

1= 12058111199031minus 12057511199011

2= 12058121199032minus 12057521199012

(4)

where 1199031and 1199032are the concentrations ofmRNA 1 andmRNA

2 respectively 1199011and 119901

2are the concentrations of their

corresponding products protein 1 and protein 2 respectively1205821 and 120582

2represent the maximal transcription rates of gene

1 and gene 2 respectively 1205741and 1205742are the degradation rates

of mRNA 1 and mRNA 2 respectively 1205751and 120575

2are the

degradation rates of protein 1 and protein 2 respectively

ℎ+(1199012 1205792 1198992) =

1199011198992

2

1199011198992

2+ 1205791198992

2

ℎminus(1199011 1205791 1198991) =

1205791198991

1

1199011198991

1+ 1205791198991

1

(5)

are the Hill functions for activation and repression respec-tively 119899

1and 1198992are the Hill coefficients and 120579

1and 1205792are the

thresholds It is easy to see that the activation function ℎ+ is

increasing in 1199012and the repression function ℎ

minus is decreasingin 1199011

22 A p53-mdm2Regulatory Pathway Anothermodelwe areinterested in is for the damped oscillation of the p53-mdm2regulatory pathway which is given by (see [20])

119868= 119904119901+ 119895119886119875119860minus (119889119901+ 119896119886119878 (119905)) 119875

119868minus 119896119888119875119868119872+ 119895

119888119862

= 1199041198980

+1199041198981119875119868+ 1199041198982119875119860

119875119868+ 119875119860+ 119870119898

+ 119896119906119862 + 119895119888119862 minus (119889

119898+ 119896119888119875119868)119872

= 119896119888119875119868119872minus (119895

119888+ 119896119906) 119862

119860= 119896119886119878 (119905) 119875

119868minus (119895119886+ 119889119901) 119875119860

(6)

where 119875119868represents the concentration of the p53 tumour

suppressor119872 (mdm2) is the concentration of the p53rsquos mainnegative regulator 119862 is the concentration of the p53-mdm2complex 119875

119860is the concentration of an active form of p53

that is resistant against mdm2-mediated degradation 119878(119905) is atransient stress stimulus which has the form 119878(119905) = minus119890

119888119904119905 119888119904=

120574119896119906 119904lowast(lowast = 11990111989801198981) are de novo synthesis rates 119896

lowast(lowast =

119886 119888 119906) are production rates 119895lowast(lowast = 119886 119888) are reverse reactions

(eg dephosphorylation) 119889119901is the degradation rate of active

p53 and119870119898is the saturation coefficient

3 Methods

31 Runge-Kutta Methods Either the mRNA-protein net-work (1) or the p53-mdm2 regulatory pathway (6) can beregarded as a special form of a system of ordinary differentialequations (ODEs)

1199111015840= 119892 (119911) (7)

where 119911 = 119911(119905) isin 119877119889 and the function 119892 R119889 rarr R119889

is smooth enough as required The system (7) together withinitial value 119911(0) = 119911

0is called an initial value problem (IVP)

Throughout this paper we make the following assumptions

(i) The system (7) has a unique positive steady state 119911lowastthat is there is a unique 119911

lowast= (119911lowast

1 119911lowast

2 119911

lowast

119889) with

119911lowast

119894gt 0 for 119894 = 1 2 119889 such that 119892(119911lowast) = 0

(ii) The steady state 119911lowast is asymptotically stable that is forany solution 119911(119905) of the system (7) through a positiveinitial point 119911

0 lim119905rarr+infin

119911(119905) = 119911lowast



119885119894= 119911119899+ ℎ

119904

sum

119895=1

119886119894119895119892 (119885119895) 119894 = 1 119904

119911119899+1

= 119911119899+ ℎ

119904

sum

119894=1

119887119894119892 (119885119894)

(8)


119899 119899 =




119888 119860

119887119879 =

1198881

11988611

1198861119904

d

119888119904

1198861199041

119886119904119904

1198871

119887119904

(9)


119895=1119886119894119895for 119894 = 1 119904


0

12 12

12 12

1 0 0 1

16 26 26 16

0

13 13

23 1

11 1

18 38 38 18

minus1

minus12 (10)




1199111015840= 119892[1]

(119911) + 119892[2]

(119911) (11)


1015840= 119892[2](119911)


ℎand 120593[2]

ℎ respectively


Ψℎ= 120593[2]

ℎ∘ 120593[1]

ℎ

Φℎ= 120593[1]

ℎ∘ 120593[2]

ℎ

(12)



Ψℎ= 120593[1]

ℎ2∘ 120593[2]

ℎ∘ 120593[1]

ℎ2 (13)


Ψℎ= 120593[2]

119887119898ℎ∘ 120593[1]

119886119898ℎ∘ sdot sdot sdot ∘ 120593

[2]

1198872ℎ∘ 120593[1]

1198862ℎ∘ 120593[2]

1198871ℎ∘ 120593[1]

1198861ℎ (14)

where 1198861 1198871 1198862 1198872 119886





and 120593[2]

ℎ



1and 120595

2



119892[1]

(119911) = (minusΓ 0

119870 minus119872)119911 119892

[2](119911) = (

119865 (119901 (119905))

0) (15)


lowast 119910(119905) = 119901(119905)minus119901

lowast


origin and yields

(119905) = minusΓ119909 (119905) + 1198651015840(119901lowast) 119910 (119905) + 119866 (119910 (119905))

119910 (119905) = 119870119909 (119905) minus 119872119910 (119905)

(16)


and 119866(119910(119905)) = 119865(119901lowast+ 119910(119905)) minus 119865

1015840(119901lowast)119910(119905) minus 119865(119901

lowast)



119892[1]

(119911) = (minusΓ 119865

1015840(119901lowast)

119870 minus119872)119911 119892

[2](119911) = (

119866 (119910 (119905))

0)

(17)

where 119911(119905) = (119909(119905) 119910(119905))119879 The system = 119892

[1](119911) here is




119870 minus119872))





method

Ψℎ= 120595[2]

ℎ∘ 120593[1]

ℎ (18)



4 Results





2=




119892[1]

(119911 (119905)) = (minus120574 minus

2120572119901lowast1205792

(1 + 119901lowast21205792)2

120581 minus120583

)(119909 (119905)

119910 (119905))

119892[2]

(119911 (119905))

= (

120572

1 + (119901lowast + 119910 (119905))2

1205792+

2120572119901lowast1205792

(1 + 119901lowast21205792)2119910 (119905)

0

)

(19)





lowast

1 119903lowast

2 119901lowast

1 119901lowast

2) =


1= minus19611 + 09611119894 120596

2= minus19611minus

09611119894 1205963= minus00389 + 09611119894 and 120596

4= minus00389 minus 09611119894



119892[1]

(119911 (119905))

=(((

(

minus1205741

0 0119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)2

0 minus1205742minus119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)2

0

1205811

0 minus1205751

0

0 1205812

0 minus1205752

)))

)

times(

1199091(119905)

1199092(119905)

1199101(119905)

1199102(119905)

)

119892[2]

(119911 (119905))

=(((

(

1205821(1199102(119905) + 119901

lowast

2)1198992

(1199102(119905) + 119901

lowast

2)1198992

+ 1205791198992

2

minus119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)21199102(119905)

12058221205791198991

1

(1199101(119905) + 119901

lowast

1)1198991

+ 1205791198991

1

+119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)21199101(119905)

0

0

)))

)

(20)


1(0) = 06 119903

2(0) = 08 119901

1(0) = 06 and







120572 = 3 120574 = 1 120583 = 15 120581 = 5



119860) =


minus384766 1205962= minus00028 + 00220119894 120596

3= minus00028 minus 00220119894




119892[1]

(119911 (119905)) = 119869(

1199111(119905)

1199112(119905)

1199113(119905)

1199114(119905)

)

119892[2]

(119911 (119905)) = (

minus1198961198881199111(119905) 1199112(119905)

120592 (119905)

1198961198881199111(119905) 1199112(119905)

0

)

(21)

where

119869 =(((

(


minus119896119888119875lowast

119868119895119888

119895119886

1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

minus 119896119888119872lowast


119868119896119906+ 119895119888

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

119896119888119872lowast

119896119888119875lowast

119868minus (119895119888+ 119896119906) 0

0 0 0 minus (119895119886+ 119889119901)

)))

)

120592 (119905) = 1199041198980

+1199041198981

(1199111(119905) + 119875

lowast

119868) + 1199041198982

(1199114(119905) + 119875

lowast

119860)

1199111(119905) + 119875

lowast

119868+ 1199114(119905) + 119875

lowast

119860+ 119870119898

minus 119889119898119872lowast+ 1198961198881199111(119905) 1199112(119905) + 2119896

1198881199111(119905)119872lowast

minus 119896119888119875lowast

119868119872lowastminus1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199111(119905) minus

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199114(119905)

+ (119896119906+ 119895119888) 119862lowast

(22)


119868(0) = 942 nM 119862(0) = 0037 nM 119872(0) =




5 Conclusions







6 Discussions





062404 times 10



minus3

15 32171 times 1017

12337 times 101



20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062





5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240








1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3






minus4





15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068





015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095








1199041198980


119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981


119888= 2 times 10

minus3minminus1

1199041198982


119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1






minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100











minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References

































119885119894= 119911119899+ ℎ

119904

sum

119895=1

119886119894119895119892 (119885119895) 119894 = 1 119904

119911119899+1

= 119911119899+ ℎ

119904

sum

119894=1

119887119894119892 (119885119894)

(8)


119899 119899 =




119888 119860

119887119879 =

1198881

11988611

1198861119904

d

119888119904

1198861199041

119886119904119904

1198871

119887119904

(9)


119895=1119886119894119895for 119894 = 1 119904


0

12 12

12 12

1 0 0 1

16 26 26 16

0

13 13

23 1

11 1

18 38 38 18

minus1

minus12 (10)




1199111015840= 119892[1]

(119911) + 119892[2]

(119911) (11)


1015840= 119892[2](119911)


ℎand 120593[2]

ℎ respectively


Ψℎ= 120593[2]

ℎ∘ 120593[1]

ℎ

Φℎ= 120593[1]

ℎ∘ 120593[2]

ℎ

(12)



Ψℎ= 120593[1]

ℎ2∘ 120593[2]

ℎ∘ 120593[1]

ℎ2 (13)


Ψℎ= 120593[2]

119887119898ℎ∘ 120593[1]

119886119898ℎ∘ sdot sdot sdot ∘ 120593

[2]

1198872ℎ∘ 120593[1]

1198862ℎ∘ 120593[2]

1198871ℎ∘ 120593[1]

1198861ℎ (14)

where 1198861 1198871 1198862 1198872 119886





and 120593[2]

ℎ



1and 120595

2



119892[1]

(119911) = (minusΓ 0

119870 minus119872)119911 119892

[2](119911) = (

119865 (119901 (119905))

0) (15)


lowast 119910(119905) = 119901(119905)minus119901

lowast


origin and yields

(119905) = minusΓ119909 (119905) + 1198651015840(119901lowast) 119910 (119905) + 119866 (119910 (119905))

119910 (119905) = 119870119909 (119905) minus 119872119910 (119905)

(16)


and 119866(119910(119905)) = 119865(119901lowast+ 119910(119905)) minus 119865

1015840(119901lowast)119910(119905) minus 119865(119901

lowast)



119892[1]

(119911) = (minusΓ 119865

1015840(119901lowast)

119870 minus119872)119911 119892

[2](119911) = (

119866 (119910 (119905))

0)

(17)

where 119911(119905) = (119909(119905) 119910(119905))119879 The system = 119892

[1](119911) here is




119870 minus119872))





method

Ψℎ= 120595[2]

ℎ∘ 120593[1]

ℎ (18)



4 Results





2=




119892[1]

(119911 (119905)) = (minus120574 minus

2120572119901lowast1205792

(1 + 119901lowast21205792)2

120581 minus120583

)(119909 (119905)

119910 (119905))

119892[2]

(119911 (119905))

= (

120572

1 + (119901lowast + 119910 (119905))2

1205792+

2120572119901lowast1205792

(1 + 119901lowast21205792)2119910 (119905)

0

)

(19)





lowast

1 119903lowast

2 119901lowast

1 119901lowast

2) =


1= minus19611 + 09611119894 120596

2= minus19611minus

09611119894 1205963= minus00389 + 09611119894 and 120596

4= minus00389 minus 09611119894



119892[1]

(119911 (119905))

=(((

(

minus1205741

0 0119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)2

0 minus1205742minus119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)2

0

1205811

0 minus1205751

0

0 1205812

0 minus1205752

)))

)

times(

1199091(119905)

1199092(119905)

1199101(119905)

1199102(119905)

)

119892[2]

(119911 (119905))

=(((

(

1205821(1199102(119905) + 119901

lowast

2)1198992

(1199102(119905) + 119901

lowast

2)1198992

+ 1205791198992

2

minus119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)21199102(119905)

12058221205791198991

1

(1199101(119905) + 119901

lowast

1)1198991

+ 1205791198991

1

+119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)21199101(119905)

0

0

)))

)

(20)


1(0) = 06 119903

2(0) = 08 119901

1(0) = 06 and







120572 = 3 120574 = 1 120583 = 15 120581 = 5



119860) =


minus384766 1205962= minus00028 + 00220119894 120596

3= minus00028 minus 00220119894




119892[1]

(119911 (119905)) = 119869(

1199111(119905)

1199112(119905)

1199113(119905)

1199114(119905)

)

119892[2]

(119911 (119905)) = (

minus1198961198881199111(119905) 1199112(119905)

120592 (119905)

1198961198881199111(119905) 1199112(119905)

0

)

(21)

where

119869 =(((

(


minus119896119888119875lowast

119868119895119888

119895119886

1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

minus 119896119888119872lowast


119868119896119906+ 119895119888

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

119896119888119872lowast

119896119888119875lowast

119868minus (119895119888+ 119896119906) 0

0 0 0 minus (119895119886+ 119889119901)

)))

)

120592 (119905) = 1199041198980

+1199041198981

(1199111(119905) + 119875

lowast

119868) + 1199041198982

(1199114(119905) + 119875

lowast

119860)

1199111(119905) + 119875

lowast

119868+ 1199114(119905) + 119875

lowast

119860+ 119870119898

minus 119889119898119872lowast+ 1198961198881199111(119905) 1199112(119905) + 2119896

1198881199111(119905)119872lowast

minus 119896119888119875lowast

119868119872lowastminus1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199111(119905) minus

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199114(119905)

+ (119896119906+ 119895119888) 119862lowast

(22)


119868(0) = 942 nM 119862(0) = 0037 nM 119872(0) =




5 Conclusions







6 Discussions





062404 times 10



minus3

15 32171 times 1017

12337 times 101



20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062





5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240








1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3






minus4





15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068





015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095








1199041198980


119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981


119888= 2 times 10

minus3minminus1

1199041198982


119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1






minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100











minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References

































119892[1]

(119911) = (minusΓ 119865

1015840(119901lowast)

119870 minus119872)119911 119892

[2](119911) = (

119866 (119910 (119905))

0)

(17)

where 119911(119905) = (119909(119905) 119910(119905))119879 The system = 119892

[1](119911) here is




119870 minus119872))





method

Ψℎ= 120595[2]

ℎ∘ 120593[1]

ℎ (18)



4 Results





2=




119892[1]

(119911 (119905)) = (minus120574 minus

2120572119901lowast1205792

(1 + 119901lowast21205792)2

120581 minus120583

)(119909 (119905)

119910 (119905))

119892[2]

(119911 (119905))

= (

120572

1 + (119901lowast + 119910 (119905))2

1205792+

2120572119901lowast1205792

(1 + 119901lowast21205792)2119910 (119905)

0

)

(19)





lowast

1 119903lowast

2 119901lowast

1 119901lowast

2) =


1= minus19611 + 09611119894 120596

2= minus19611minus

09611119894 1205963= minus00389 + 09611119894 and 120596

4= minus00389 minus 09611119894



119892[1]

(119911 (119905))

=(((

(

minus1205741

0 0119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)2

0 minus1205742minus119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)2

0

1205811

0 minus1205751

0

0 1205812

0 minus1205752

)))

)

times(

1199091(119905)

1199092(119905)

1199101(119905)

1199102(119905)

)

119892[2]

(119911 (119905))

=(((

(

1205821(1199102(119905) + 119901

lowast

2)1198992

(1199102(119905) + 119901

lowast

2)1198992

+ 1205791198992

2

minus119899212058211205791198992

2119901lowast

2

1198992minus1

(119901lowast

2

1198992 + 1205791198992

2)21199102(119905)

12058221205791198991

1

(1199101(119905) + 119901

lowast

1)1198991

+ 1205791198991

1

+119899112058221205791198991

1119901lowast

1

1198991minus1

(119901lowast

1

1198991 + 1205791198991

1)21199101(119905)

0

0

)))

)

(20)


1(0) = 06 119903

2(0) = 08 119901

1(0) = 06 and







120572 = 3 120574 = 1 120583 = 15 120581 = 5



119860) =


minus384766 1205962= minus00028 + 00220119894 120596

3= minus00028 minus 00220119894




119892[1]

(119911 (119905)) = 119869(

1199111(119905)

1199112(119905)

1199113(119905)

1199114(119905)

)

119892[2]

(119911 (119905)) = (

minus1198961198881199111(119905) 1199112(119905)

120592 (119905)

1198961198881199111(119905) 1199112(119905)

0

)

(21)

where

119869 =(((

(


minus119896119888119875lowast

119868119895119888

119895119886

1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

minus 119896119888119872lowast


119868119896119906+ 119895119888

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

119896119888119872lowast

119896119888119875lowast

119868minus (119895119888+ 119896119906) 0

0 0 0 minus (119895119886+ 119889119901)

)))

)

120592 (119905) = 1199041198980

+1199041198981

(1199111(119905) + 119875

lowast

119868) + 1199041198982

(1199114(119905) + 119875

lowast

119860)

1199111(119905) + 119875

lowast

119868+ 1199114(119905) + 119875

lowast

119860+ 119870119898

minus 119889119898119872lowast+ 1198961198881199111(119905) 1199112(119905) + 2119896

1198881199111(119905)119872lowast

minus 119896119888119875lowast

119868119872lowastminus1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199111(119905) minus

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199114(119905)

+ (119896119906+ 119895119888) 119862lowast

(22)


119868(0) = 942 nM 119862(0) = 0037 nM 119872(0) =




5 Conclusions







6 Discussions





062404 times 10



minus3

15 32171 times 1017

12337 times 101



20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062





5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240








1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3






minus4





15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068





015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095








1199041198980


119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981


119888= 2 times 10

minus3minminus1

1199041198982


119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1






minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100











minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References

































120572 = 3 120574 = 1 120583 = 15 120581 = 5



119860) =


minus384766 1205962= minus00028 + 00220119894 120596

3= minus00028 minus 00220119894




119892[1]

(119911 (119905)) = 119869(

1199111(119905)

1199112(119905)

1199113(119905)

1199114(119905)

)

119892[2]

(119911 (119905)) = (

minus1198961198881199111(119905) 1199112(119905)

120592 (119905)

1198961198881199111(119905) 1199112(119905)

0

)

(21)

where

119869 =(((

(


minus119896119888119875lowast

119868119895119888

119895119886

1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

minus 119896119888119872lowast


119868119896119906+ 119895119888

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

119896119888119872lowast

119896119888119875lowast

119868minus (119895119888+ 119896119906) 0

0 0 0 minus (119895119886+ 119889119901)

)))

)

120592 (119905) = 1199041198980

+1199041198981

(1199111(119905) + 119875

lowast

119868) + 1199041198982

(1199114(119905) + 119875

lowast

119860)

1199111(119905) + 119875

lowast

119868+ 1199114(119905) + 119875

lowast

119860+ 119870119898

minus 119889119898119872lowast+ 1198961198881199111(119905) 1199112(119905) + 2119896

1198881199111(119905)119872lowast

minus 119896119888119875lowast

119868119872lowastminus1199041198981

(119875lowast

119860+ 119870119898) minus 1199041198982119875lowast

119860

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199111(119905) minus

1199041198982

(119875lowast

119868+ 119870119898) minus 1199041198981119875lowast

119868

(119875lowast

119868+ 119875lowast

119860+ 119870119898)2

1199114(119905)

+ (119896119906+ 119895119888) 119862lowast

(22)


119868(0) = 942 nM 119862(0) = 0037 nM 119872(0) =




5 Conclusions







6 Discussions





062404 times 10



minus3

15 32171 times 1017

12337 times 101



20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062





5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240








1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3






minus4





15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068





015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095








1199041198980


119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981


119888= 2 times 10

minus3minminus1

1199041198982


119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1






minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100











minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References


































062404 times 10



minus3

15 32171 times 1017

12337 times 101



20 33619 times 1059

60455 times 1047



100 87073 times 1062

76862 times 1062





5960455 times 10

4754917 times 10


minus4

[0 500] 45549 times 10299

20782 times 10240








1205821= 18 120574

1= 1 120581

1= 1 120575

1= 1 120579

1= 06542 119899

1= 3

1205822= 18 120574

2= 1 120581

2= 1 120575

2= 1 120579

1= 06542 119899

2= 3






minus4





15 43953 times 100

29958 times 100



20 41821 times 1024

75238 times 1017



50 19692 times 1069

32225 times 1068





015342 times 10

019352 times 10


minus3

[0 1000] 44396 times 100

10172 times 100



[0 1500] NaN 23844 times 1095








1199041198980


119886= 20minminus1 119895

119886= 02minminus1 120574 = 25

1199041198981


119888= 2 times 10

minus3minminus1

1199041198982


119898= 04minminus1

119904119901= 14 nMminminus1 119870

119898= 100 nM 119889

119901= 2 times 10

minus4minminus1






minus6

008 46118 times 100

45057 times 100



010 NaN 53550 times 100











minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References



































minus2









0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)



minus8

minus6

minus4

minus2

0

2

4

6

8

10

0

u

v


(a)

0



minus8

minus6

minus4

minus2

0

2

4

6

8

10

u

v

(b)








Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References


































Appendix



119910 = 120582119910 + 120598119910 (A1)




119910119899+1

= 119877 (119906 V) 119910119899 (A2)



R119904= (119906 V) | |119877 (119906 V)| le 1 (A3)



119877 (119906 V) = 1 + (119906 + V) 119887119879(119868 minus (119906 + V) 119860)minus1119890 (A4)



119877 (119906 V) = exp (V) (1 + 119906119887119879(119868 minus 119906119860)

minus1119890) (A5)





Acknowledgments


References































Splitting Strategy for Simulating Genetic Regulatory Networks · ResearchArticle Splitting Strategy...

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