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Abdullah et. al.the searching capability of the CSA method. Theevolutionary operations of the DE method are commonlyutilized to enhance the proliferation process in the CSAmethod, thereby substantially utilizing the informationregarding the adjoining clones [15-16].In this work, the research that is relevant to theimprovement of the searching capability of standardCSA method is extended by using the evolutionaryoperations of DE method. In this variant of the CSAmethod, the crossover and mutation operation areimplemented to exploit the information regardingdifferent antibodies in the population. Simultaneously,the antibodies providing insignificant solutions arerelocated randomly to enhance the fitness values. Bydoing so, the method can efficiently improve thesearching quality as well as utilizing the computationaltime. The effectiveness of the proposed method is testedto estimate parameter values in a biological model andthe statistical results are then compared with thestandard CSA, PSO and GA methods. The rest of thepaper is organized as follows. Sections II introduce thestandard CSA, standard DE methods, and the proposedDifferential Clonal Evolution (DICE) method.Subsequently, Section III presents the experimentalresults. Section IV discusses the contribution of thework and Section V presents the conclusion and futureworks.II. METHODSA. Standard Clonal Selection Algorithm (CSA) MethodThe clonal selection principle [21] describes thereaction of immune system to pathogens and the processof improving the capability to identify these unintendedagents. In particular, the theory illustrates that a numberof immune cells that identify the pathogens willproliferate. Some of them will become the effector cellswhile the others maintain their role as memory cells [18].In general, the CSA method employs three main phases:cloning, mutation and selection. The method starts witha population of d-dimensional search vectors, calledantibodies. The ith antibody, X, of the whole populationat a specific generation t is given by:In CSA method, the fitness value of each antigen isrepresented as affinity, which implies the goodness of theantibody to generate antigen for the specific pathogen.Initially, the population of antibody is initiatedrandomly and the affinity of each antibody is evaluated.The antibodies that produce good affinity values areselected to undergo cloning phase. As a result, a new setof population is created. Next, the mutation process is(1)performed to every clone, based on the mutationconstant. Hence, the mutated clones are formed withnew components and the affinity values are then beenevaluated to measure the fitness. In the last phase, themutated clones are selected to replace the originalantibodies. Eventually, the population is built with thenew improved antibodies. The overall procedure of thestandard CSA method is outlined in Figure 1:1: Begin2: Initiate population, X3: // evaluate fitness of each antibody4: While max number of generation is not met5: // Select m best antibodies6: For i = 1 to m antibody7: // cloning selected antibodies8: // mutate clones9: // select improved clones to10: replace old antibodies11: End For12: //include improved best13: antibodies to population14: End While15: End BeginFig. 1. The standard CSAB. Standard Differential Evolution (DE) MethodThe DE method is also a stochastic populationbasedoptimization method. The method is proposedbased on the evolutionary operations of the GA method[13]. Compare to GA, this method employs a mutationoperation to produce a trivial chromosome from theoriginal chromosome. Then, this trivial chromosome iscrossovered with its original counterpart to generate anoffspring chromosome. A simple selection operation isperformed to select the chromosome with a better fitnessvalue. In each generation, a range of search space isspecified to find a good solution. Thus, at initialgeneration or t = 0, each chromosome is initialized, witha lower and an upper bound, and respectively [13]:where R is a random number generated between 0 and 1and j is the dimension size.In order to produce the trivial chromosome, V i , themutation operation is executed according to thedifferentiation of neighborhood chromosomes, given asfollowing:(3)where x best(t) denotes the current best chromosome, F isthe scaling factor, while x r1(t) and x r2(t) are randomlychosen chromosomes [13]. Using this chromosome, an(2)314 | P a g e

Abdullah et. al.Mackey [22] introduced a mathematical model for theregulation of induction process in the lac operon thatconsidered the dynamics of the permease enabling theinternalization of several biomolecules such as lactoseand β-galactosidase. The model is important for theobservation of the conversion of lactose to allolactose,glucose and galactose; the allolactose interactions withthe lac repressor; and the mRNA [22]. The model isformed through the following equations:where A, B and L are the concentrations of allolactose, β-galactosidase and lactose, respectively; M is the mRNAtranslation; is time; , and are the productionrate constants; and are the loss rate constants; isthe dilution rate constant; and are the equilibriumconstants of allolactose and lactose, respectively [22].Thus, in this work, the values of , , , , ,and parameters are tended to be estimated. Theexperimental values of these parameters are given inTable 1 [22].TABLE IEXPERIMENTAL VALUES OF THE REGULATION MODELParameterExperimental Value1.76 × 10 4 min -11.66 × 10 -2 min -12.15 × 10 4 min -15.20 × 10 -1 min -18.33 × 10 -4 min -12.26 × 10 -2 min -11.95 × 10 -3 M9.70 × 10 -4 MIn this work, the experimental data is obtained insilico by generating noisy and sparse version of themodel data. Firstly, the model is simulated and the valuesat several randomly chosen time points are evaluated.Then, the Gaussian noise is added to the values so that itwill simulate the measurement noise [23]. The modeldata and the generated noisy and sparse experimentaldata of β-galactosidase and allolactose are illustrated inFigure 3 and Figure 4, respectively.B. Parameter EstimationGenerally, the parameter estimation problem isformulated in the following way. Suppose that a systemis formed by the d-dimensional state variable, x, at time t,which is the distinctive solution of the initial valueproblem:(6)(7)where p is the parameters [24]. So, let y signify theobservation of experimental value, i, corresponding tothe measurement, j, and represented by the followingequation:where σ ij > 0 and ɛ ij is a Gaussian distributed randomvariable [24]. Thus, the parameter estimation problem ofa biological system consists of finding the optimalparameter p such that the difference of the experimentaldata and the simulated data is minimized:(8)(9)(10)where is the trajectory at time t, n is the totalnumber of parameters and m is the total number ofobserved values [24].The results obtained from the proposed method arecompared with those from the standard CSA, PSO andGA methods. For each method, a population size of 50particles or chromosomes is initiated and the maximumnumber of generations is set to 200. Furthermore, eachmethod is executed 100 times independently to observeits reliability and consistency. Table 2 shows the averagefitness values and the corresponding standard deviationfor each method. In general, the proposed DICE methodhas outperformed the standard methods. Hence, theaccuracy of the proposed method is better compared tothe other methods, as the overall fitness value obtained isthe lowest among those from the other methods.TABLE IIACCURACY AND SPEED PERFORMANCEMethod GA PSO CSA DICEAverage 3.72×10 -3 3.56×10 -3 4.64×10 -4 1.93×10 -9StandardDeviationAverageSpeed(second)3.07×10 -3 3.00×10 -3 7.94×10 -4 4.15×10 -90.358 6.240 0.483 0.452316 | P a g e

Hybrid Evolutionary Clonal Selection for Parameter Estimation of Biological ModelFig. 3. Comparison of the model data and the experimental data for concentration of β-galactosidaseFig. 4. Comparison of the model data and the experimental data concentration of allolactoseTo address the performance of the proposedmethod in terms of convergence speed, Figure 5illustrates the graph of convergence for all methods.Obviously, the standard GA and PSO methodsconverged prematurely while the standard CSA methodsuccessfully finds better fitness values compared to theGA and PSO methods. However, the method waseventually trapped in one of the local optima starting atthe 165th generation. This problem has been effectivelysolved by the proposed method as the values are keptdecreasing until the maximum number of generations isreached.In addition, a statistical analysis of the observedmeasurements and the fitted data produced by theproposed DICE method is conducted. In this analysis,confidence interval estimates using chi-squared (χ 2 )distribution is used. The result of this analysis ispresented in Table 3. The result shows that the proposedmethod is reliable for the estimation of the parametervalues as the mean error is substantially small for bothcomponents of the model. Moreover, the variance pointlies between the interval estimates. Thus it is confirmedthat the estimate obtained using the DICE method canbe generally considered as valid.317 | P a g e

Abdullah et. al.Fig. 5. Convergence behaviours of each methodParameter estimation of complex biological modelsis usually presented as an optimization problem [23, 24].The approximation of the parameter values is alwayshindered by the noise and incompleteness of theexperimental data. Thus, optimization methods such asthe GA and PSO methods have always been consideredfor this problem because they are capable of fitting theexperimental data with the model prediction effectively.However, a substantial number of studies have shownthat these methods are frequently trapped in one of thelocal optima [6]. Moreover, these methods alwaysinvolve a huge search space that requires a large amountof computational time. Hence, a significant number ofstudies have been conducted to merge several methods toovercome this challenge [6-11]. Nevertheless, thisapproach shows potential in improving the accuracy andspeed of the standard methods.TABLE IIISTATISTICAL ANALYSIS OF FITTED DATA BY DICE METHODComponent β-galactosidase AllolactoseError 0.21% 0.40%Variance Point 4.65×10 -8 2.49×10 -1Variance Interval [3.74×10 -8 ,[2.00×10 -1 ,3.35×10 -1 ]6.26×10 -8 ]Real Variance 4.64×10 -8 2.48×10 -1χ 2 TestPassIV. DISCUSSIONIn this work, the proposed DICE method haspresented another prospective alternative for enhancingthe quality of the parameter estimation results. As shownin Table 2, the method has outperformed all thecompetitive methods efficiently, in terms of both,accuracy and speed performance. The accuracyperformance of the proposed DICE method has shownremarkable improvement compared to the resultsproduced by the other methods. This is because of thetwo main reasons. Firstly, the DICE method employsevolutionary operations to the antibodies that yieldpotentially good fitness values. As the operations areperformed to these antibodies, the fitness values areimproved significantly at each generation as theinformation regarding different antibodies is utilized toproduce more significant fitness values. Secondly, theantibodies that produced insignificant fitness values aresubjected to undergo randomization operation. By doingso, the method can enhance the fitness values of theseantibodies, thus allowing the method to escape the localoptima more effectively. This is shown by theconvergence behavior of the DICE method in Figure 4.Nonetheless, there is only a small difference ofspeed performance between the proposed method and itsstandard counterpart. This is due to the fact that theproposed method uses the computational timeextensively for each antibody to exchange informationbetween its neighbors. Even though finding the possiblebest values can be achieved more effectively, thisrequires numerous runtimes to execute the evolutionaryoperation on every antibody in the population. Hence, thescalability of the problem dimension may affect thespeed performance of the method. However, statisticalanalysis performed on the results produced by theproposed DICE method show that the method is capableof estimating the parameter values accurately. Themethod passed the χ2 test, indicating that the valuesestimated by the proposed method are very close to theactual values.V. CONCLUSIONGlobal optimization problems present a majorchallenge in both scientific and industrial fields. Thus, asignificant number of optimization methods have been318 | P a g e

Hybrid Evolutionary Clonal Selection for Parameter Estimation of Biological Modeldeveloped to overcome these problems. In most cases,global optimization methods are always chosen due tothe capability to handle nonlinearity of the problems.However, these methods are usually hampered by somelimitations including huge computational timeconsumption and getting stuck in one of the local optima.This led to the development of hybrid optimizationmethods, which mainly tends to combine severaldifferent methods to improve the limitations by utilizingthe advantages of the combined methods.This paper presented a new hybrid optimizationmethod based on the CSA method and the evolutionaryoperations adopted from the DE method. Theeffectiveness of the new method is tested using noisy andincomplete experimental data of a bacterial lactoseproduction model. The results are compared to thestandard CSA, PSO and GA methods. The comparisonsuggests that the accuracy and the speed performance ofthe proposed method are better than that can be obtainedfrom other methods. Despite of this achievement, thereare several limitations which need to be addressed. Thecomputational time constitutes one such limitation.Hence, research is needed to overcome this challenge.The future research work may involve the improvementof the proposed DICE method through a use of localoptimization approach and adaptive features. In addition,this study only considered one nonlinear model, whichmay ponder the restriction of the actual performance ofthe proposed method. Therefore, in the future, theperformance of the method will be verified by using anumber of different models to show the reliability androbustness of the method.REFERENCES[1] N. Noman and H. Iba, ―Accelerating differential evolutionusing an adaptive local search,‖ IEEE Transactions onEvolutionary Computation, pp.107-125, volume 12, no. 1, 2008.[2] J. Kennedy and R. 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Khammash, "Parameter estimation andmodel selection in computational biology," PLoSComputational Biology, e1000696, volume 6, no.3, 2010.[24] E. Balsa-Canto, M, Peifer, J.R. Banga, J. Timmer and C. Fleck:,―Hybrid optimization method with general switching strategyfor parameter estimation, ― BMC Systems Biology, pp.26-35,volume 2, no. 1, 2008.319 | P a g e