Kinetic Parameters Estimation in the Polymerase Chain Reaction

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Kinetic Parameters Estimation in the Polymerase Chain Reaction Process Using the Genetic Algorithm Lanting Li,† Chao Wang,‡ Bo Song,† Lijuan Mi,*,† and Jun Hu† †

Laboratory of Physical Biology, Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, China Department of Biomedical Engineering, Oregon Health & Science University, Beaverton, Oregon 97006, United States

‡

S Supporting Information *

ABSTRACT: Analyzing the polymerase chain reaction (PCR) process by mathematical modelling is of importance in terms of uncovering the dynamic mechanism of PCR and predicting DNA amplification performance. Construction of an ideal PCR model, however, requires sufficient and accurate kinetic parameters, which cannot be easily obtained directly from experimental data. In this work, a genetic algorithm-based approach was introduced for optimizing kinetic parameters of PCR such as the rate constant of polymerase catalyst reaction. The fitted model agrees well with experimental data and predicts the DNA amplification yields as a function of cycle number. The kinetic parameters are not dependent on initial concentrations or the fragment types of DNA amplification. The model also allows us to predict the threshold cycle in real-time PCR, which is helpful for estimating the initial amount of DNA template and determining the optimal PCR reaction conditions. a number of fields due to its advantage over other deterministic optimization methods, especially the nonlinear parameter estimation.22−24 Its global and stochastic search ability and its nature of independence of initial estimates make the solutions converge to the global minimal largely and find the real values of the model.21 It has been successfully applied to solve various problems, especially in bioprocesses. For example, an intracellular signal transduction pathway in the neuronal cell was used as a model system to implement the proposed parameter estimate procedure, and from the results a mathematical model was established to analyze the interacting network quantitatively and understand the dynamic mechanism.20 The GA has been used in biomanufacturing processes, such as a fed batch fermentor for penicillin production,25 the process of plant cell suspension culture of catharanthus roseus and Nicotiana tabacum (grown in a stirred bioreactor),26 the photosynthesis,27 and so on. These studies show that the GA-based estimation methods can efficiently obtain globally optimal solutions, while nonlinear fitting methods are more sensitive to the initial values. Using GA, one is able to estimate parameters of a nonlinear system with satisfactory results. In this work, we used the multiobjective GA-based approach to optimize the rate constants of PCR which are unknown in the conventional models. Furthermore, the combination of fitness function, correlation coefficient (R2), and average relative error determine the optimal solution. GA is helpful to develop an accurate and predictive PCR model. The calculation results agreed well with the experimental results in the literature. The model is allowed to predict the DNA amplification yields as a function of cycle number.

1. INTRODUCTION The polymerase chain reaction (PCR), an important bimolecular technique to amplify special DNA fragments in vitro,1 has been widely used in many fields.2−4 The procedure of PCR includes denaturation, annealing, and extension, but the reaction conditions differ from one PCR process to another because of DNA fragments of various lengths or from diverse sources across all disciplines of diagnostic pathology and research, hence the need for optimizing experimental conditions. Traditionally, the univariate approach is used in a usual PCR, by changing just one factor at a time while keeping the others unchanged, at the expenses of time and cost. For selecting appropriate operating conditions, however, PCR involves so many variables of the reactants and reaction processes that it is impractical to investigate all possible combinations of the reaction conditions. Especially for PCR reactions with additives, it became very difficult to evaluate the optimal conditions because of the complicated mechanisms.5−9 Therefore, researchers have been developing kinetic models to predict or investigate the mechanisms of some bioprocess,10−18 such as mathematical models of PCR, to calculate the kinetic progression of target DNA concentration with the cycle number.16−18 Ideally, a good model for PCR optimization shall consider all individual steps in the process. But most of the models available omitted some important steps of the PCR.16,17 For example, the conformational change step before phosphoryl transfer in the catalyst pathway of polymerase may be rate-limiting.19 The parameters during the process cannot be easily obtained directly from experimental data. When only a few kinetic parameters are available to implement a model, one may resort to a theoretical estimate, by optimizing the unknown parameters of a model to achieve the best agreement between the simulated and experimental data.20 The genetic algorithm (GA) is an effective, stochastic, and global search algorithm that is inspired by the evolutionary features of biological systems.21 This algorithm is widely used in © 2012 American Chemical Society

Received: Revised: Accepted: Published: 13268

February 10, 2012 August 27, 2012 September 13, 2012 September 13, 2012 dx.doi.org/10.1021/ie3003717 | Ind. Eng. Chem. Res. 2012, 51, 13268−13273

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2. METHODS 2.1. Development of Model. Figure 1 shows schematics of the PCR pathway. A PCR process consists of three stages at

according the literature published previously. The deactivation rate constant of polymerase of 1.9 × 10−4/s is from ref 16. In the annealing stage, the association rate constant of template and primer of 106/M·s is from ref 28.28 The nonspecific association rate constant of template and primer is the same as the specific one according to the experimental results in ref 29, because kinetics measurements involving the mismatched duplexes show that association rates are almost the same as those for the complementary cases.29According to ref 16, all the forward rate constants between the primers are 104/M·s, which is 100-fold lower than the specific annealing rate. The dissociation rate constants are decided based on calculation of Gibb’s free energy of chemical reaction from the primer design software (primer premier 5). The possible reactions occurring in this stage include the formation of matched product, mismatched product, cross-dimer, and self-dimers. The Gibbs’s free energy can be determined by ΔGT0 = −RT ln(K), where R = 1.9872 cal·mol−1·K−1 is the gas constant, T is the temperature, and K is the equilibrium constant at a defined temperature. So the reverse rate constants are calculated by the division of forward rate constants and equilibrium constants. For the reaction of template−primer complex and polymerase, the equilibrium dissociation constant for the DNA−polymerase is reported as 103 nM at 70 °C for Thermus aquaticus (Taq) polymerase.30 So the forward rate constant for enzyme−DNA binding is 108/M·s, and the dissociation rate constant of the DNA−enzyme complex is therefore 10/s. Experimental results in ref 31 indicate that the introduction of the mismatched base pairs at the different positions of the 3′-end of the primer decreases the association rate constants and increases the dissociation rate constants by several times, resulting in a significant decrease in the binding affinity of polymerase to the DNA template−primer complex. So the association rate constant of nonspecific template−primer and polymerase is sixfold lower than the specific case, and the dissociation rate constant is eightfold higher than the specific case.31 Detailed description about section 2.2 is shown in Table S2. 2.3. Algorithm To Estimate Kinetic Parameters in Extending Stage. The theory of GA is based on the concept of the fittest survival to find the best solution from a wide search range. First, the optimization solutions for a problem are encoded by a population of strings. The population is generated randomly and covers the entire range of possible solutions. The evolution starts from a population of randomly generated individuals. Second, the fitness of each member of a starting population is evaluated in each generation. The fitness function, which is always problem-dependent, is defined over the genetic representation and measures the quality of the represented solution. Once the genetic representation and the fitness function are defined, the operators of crossover, mutation, and selection are used to generate the next population. During each successive generation, a proportion of the existing population which has fitter solutions is more likely to be selected to breed a new generation. The next generation population is different from the initial generation by this procedure, and generally the average fitness will decrease, since only the best organisms from the first generation are selected for breeding. The operators of crossover, mutation, and selection effectively modify the locations in which the search is implemented. At last, this repetitive iteration is carried out until a termination condition has been reached, such as a maximum number of generations produced, or a satisfactory fitness level has been reached for the population.21

Figure 1. Schematic view of the PCR process.

different temperatures: denaturation (94−98 °C), primer annealing (50−60 °C), and primer extension (72 °C). We assume the outputs in each step will be the reactants in the following step during repeating 30−40 experimental cycles. Usually, double-stranded DNA (dsDNA) first denatures into single-stranded DNA (ssDNA) (step 1), each primer hybridizes ssDNA in the annealing stage, and then the primer−ssDNA junction is extended from the 3′ hydroxy end of the primer (steps 2 and 3). Nonspecific amplifications, such as false priming complexes of template and primer, SPns (steps 4 and 5), cross-dimer, RC (step 6), and self-dimers, R and R′ (Step 7 and 8), take place in a similar way. In the extension stage, seven basic reversible reactions which are catalyzed by the polymerase take place.19 For instance, step 9 is the binding of polymerase to DNA substrate, while step 10 involves binding of deoxyribonucleoside triphosphate (dNTP) to the polymerase/DNA complex, forming E:DNA:dNTP. Step 11 represents a conformational change process, forming the product which is denoted by E*:DNA:dNTP. Then, phosphoryl is transferred (step 12), turning into E*·DNAn+1·PPi. After the nucleotide is covalently attached to the extending primer, there is a conformational change step in the polymerase that relaxes from the E*·DNAn+1·PPi to the E·DNAn+1·PPi (step 13). Following this step, pyrophosphoric acid (PPi) is released as the first product from the polymerase, forming E·DNAn+1 (step 14). At last, the polymerase is dissociated from the E·DNAn+1 to renew polymerization on another substrate, and the desired product DNAn+1 is released (step 15).19 It is worth noting that the nonspecific complexes are also extended by polymerase from DNAns, and the complex DNAns here includes SPns, RC, R, and R′. The reactions are listed in Table S1 (Supporting Information). According to the literature or the calculation, we obtained part parameters (steps 1−9) required in model construction (see Table S2). Other parameters (steps 10−15) were estimated using GA. Table S1 shows the key reactions in steps 10−15. According to the reactions, we listed a set of ordinary differential equations based on the law of mass action (see Table S3). 2.2. Determination of Kinetic Parameters According to the Literature Published Previously. We set the kinetic parameters of denaturation process and annealing stage 13269

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3. RESULTS AND DISCUSSION 3.1. Kinetic Parameters Estimated Using GA. Here we investigate the effect of the initial DNA concentrations and DNA template sources on the concreted kinetic parameters using GA in real-time PCR amplification. Experimental data of different initial DNA concentrations and different DNA sources are from ref 34 and ref 35, respectively. The values are all identical to that reported in the literature. First, the search procedures of a given DNA template with different initial concentrations are carried out. The DNA template is a genetically modified cotton variety, herbicidetolerance Mon1445.34 The experimental conditions are as follows: the initial reactant concentrations of polymerase, primer, and dNTP are 2.5 nM, 0.4 μM and 0.2 mM, respectively. The DNA concentrations are 103, 104, 105, 106, and 107 copies per microliter; the time periods of denaturation, annealing, and extension are 1, 10, and 30 s, respectively; and the cycle number is 50. Second, we investigate the influence of different DNA templates on the parameters estimation. Three kinds of DNA fragment are chosen to implement the simulation, including Porcine teschovirus (PTV), Porcine enterovirus type 8 (PEV 8), and Porcine enterovirus type 9 (PEV 9), respectively, which belong to Porcine enterovirus (PEV) cytopathic effect (CPE) groups I, II, and III.35 The DNA concentration is chosen as 105 copy/μL, while the concentrations of the polymerase, primer, and dNTP are 1 nM, 0.2 μM, and 0.2 mM, respectively; the time periods of denaturation, annealing, and extension are 15, 15, and 30 s, respectively, and the cycle number is 45. We find that the values of the corresponding kinetic parameters are basically in the same order of magnitude in the different initial DNA concentrations and different DNA sources, though the estimated values of some parameters have a small variation across the solution groups. The results indicate that the values optimized are basically not affected by the factors mentioned above (see Table S5). We obtained the values which were simultaneously estimated by multiobjective GA. The result is shown in Table 1.

The standard GA procedure is as follows: Step 1: Construct fitness function, which is set as the sum of the error where the error is the difference between the experimental data and the calculated data for the parameter set. Step 2: Initialize a population containing N individuals, and a population of the possible solutions is randomly generated in binary form. Step 3: Evaluate the fitness of each individual in that population; sum of errors is calculated. Step 4: Select the best-fit individuals for reproduction. Step 5: Breed new individuals through crossover and mutation operations to give birth offspring. 5.1 Selection: select parent chromosomes from a population according to their fitness. 5.2 Crossover: a new offspring over the parents is formed with crossover probability. 5.3 Mutation: new offspring at each locus is mutated with a mutation probability. 5.4 Accepting: new offspring is accepted in a new population. Step 6: Evaluate the individual fitness of new individuals. Step 7: Replace least-fit population with new individuals. Step 8: Repeating steps 2−7 until the stop criteria are met, and the final solution is found. In our optimization, the model consists of 71 ordinary differential equations and 12 unknown parameters. The stochastic uniform is used as a selection function, and scattering is used as a crossover function. The crossover fraction and mutation fraction are 0.8 and 0.2, respectively. The numerical solutions of the differential equations are calculated using a 2−3 order Runge−Kutta method. The sum of the “distance” between the predicted time course ( f(pred)) of the concentration of the DNA product and that effectively measured by the experiments of the same variable of PCR (f(exp)) is used to evaluate each group’s fitness function (eq 1). The final fitness function is obtained by multiplying each objective function by a weighing factor and summing up all weighted objective functions; the equations are shown below:

Table 1. Kinetic Parameters Estimated by Multiobjective GA

n

di =

∑ |f

(pred)

j=1

( j) − f

(exp)

( j) | (1)

m

d=

∑ ωidi i=1

(2)

where j is a data point, n is the number of experimental values considered for comparison, i is a data point, m is the number of fitness function, and ω is the weigh factor, which is set to the same value of 0.125 for 8 groups. At the same time, the optimization terminated when the average change in the fitness value is less than 10−6. 2.4. Determination of Search Range. Searching a correct parameter range is important to implement the optimization so as to save the time. According to kinetic data of the polymerases which have been studied so far, such as Escherichia coli polymerases (pol) I exo- (KF-),32 pol II exo- (pol II-),32 bacteriophage polymerase T7 exo- (T7-), and HIV-1 reverse transcriptase (RT),33 the GA search range of Taq for the reaction orders are set encompassing the reported values in the literature (see Table S4).

a

parameter

definition (step)a

k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12

kforward (10) kreverse (10) kforward (11) kreverse (11) kforward (12) kreverse (12) kforward (13) kreverse (13) kforward (14) kreverse (14) kforward (15) kreverse (15)

values 0.6 0.4 0.5 0.3 0.4 0.2 0.1 1.0 0.1 0.3 0.9 0.1

× × × × × × × × × × × ×

109 104 103 10−1 104 10−1 104 10−3 105 108 101 108

k, rate constant; step, refer to Figure 1.

Comparing Table 1 and Table S5, it was found that the parameters according the multiobjective GA are basically at the same order as the values estimated solely. Therefore, we selected the parameters resulting from multiobjective GA at the minimum fitness value to develop the model. In addition, when several different parameter value combinations give almost the same fitness value, two new criteria, the correlation coefficient 13270

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not exceed 0.1 for the five groups with amounts of DNA. Figure 2b indicates that the optimal solution is obtained for no more than 10 generations. The fitness values which stand for the error between the experimental and calculated data do not exceed 0.2 in three different DNA fragment groups. However, in the case of the PEV9 group, the slower converging rate of fitness value may be a result of DNA fragments, the matching of DNA template and primer, and so on. We also plotted the mean fitness values versus the iteration generation for estimating parameters simultaneously by multiobjective GA. In Figure 2c, we found the optimal solution converging more slowly than the single case. The iteration generation is more than 40 at the optimal solution. The optimization system is more complicated when the multiobjective GA is performed. 3.3. Model Verification. The threshold cycle (Ct) is a basic and essential parameter in real-time PCR, which is defined as the fractional PCR cycle number at which the reporter fluorescence is greater than the minimal detection level (i.e., the threshold). The plot of the log of initial DNA copy numbers versus Ct is a straight line.36,37 The model’s capability of reproducing or predicting experimental results has been tested with refs 38 and 39. Real-time PCR of human cytomegalovirus (HCMV) DNA was performed with serial dilutions (103−107) of a plasmid containing the major immediate early (MIE) gene region of HCMV in 20 μL solutions, with 3 × 10−9 M polymerase, 0.8 μM primer, and 0.8 mM for each of the four dNTPs. The process contained 45 cycles, including 10 s of denaturation, 10 s of annealing, and 30 s of extension.38 Figure 3a shows the calculated DNA concentration plotted against cycle number for the amplification of HCMV fragment. The vertical axis represents the normalized product concentration; the horizontal axis represents the cycle number. Another simulation was performed to reproduce the amplification of Hepatitis B virus (HBV) fragment (102−106) according to the literature. Mutations at the active site of DNA polymerase of HBV, and tyrosine−methionine−aspartate− aspartate (YMDD) motif, made the infected patients resistant

(R2) and the average relative error, are calculated to help us determine the optimal parameters. 3.2. Fitness Function Evaluation. The GA code is executed on a Gateway TM profile computer with 4 GB of RAM and a Pentium 4TM 3 GHz processor. The code requires 3−4 days to execute. We run it 10 times to get statistically meaningful values for the time taken. The fitness function is responsible for the GA evaluation and returning fitness value for that solution, which reflects whether the solution fits. In our model, the fitness function is set as the difference between the experimental data and the calculated data. The time evolutions of the fitness values are investigated for data of the eight groups (see Figure 2). A fast decrease in the fitness value is seen in the

Figure 2. Curves of the fitness function versus the iteration generation for different initial concentrations (a) and kinds of template (b) and simultaneously estimated by multiobjective GA (c).

early generations, and then it remains stable in the following generations. These calculations are not terminated until the fitness value has stabilized. The curves represent the mean fitness values versus the iteration generation. Figure 2a shows that the fitness value does

Figure 3. Real-time PCR curves of HCMV (a) and HBV (b); the comparisons between the experimental and calculated results for HCMV (c) and HBV (d). 13271

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to antiviral drug therapy,39 with 3 × 10−9 M polymerase, 0.5 μM primer, and 0.2 mM dNTP. The time periods of denaturation, annealing, and extension were 15, 15, and 30 s, respectively, and the cycle number was 35. Real-time PCR reactions were carried out with a fluorometric thermal cycler (Rotor-Gene 3000; Corbett Research, Australia) in final volumes of 20 μL. The fluorescent signal was detected once per cycle upon completion of the extension step. Figure 3b shows the calculated result for the amplification of HBV fragment; here the vertical axis represents the log of normalized product concentration, and the horizontal axis represents the cycle number. The comparisons between the simulated data and the experimental results are given in Figure 3c and 3d, where the thresholds are the same as those in the literature. A straight line (with solid circle) represents the results of calculation according to our model. The solid squares indicate the experimental data from ref 38 and ref 39. The square regression coefficients (r2) of calculated results and experimental ones are 0.9993 and 0.9969 in HCMV real-time PCR (Figure 3c), while they are 0.9966 and 0.9996 in HBV real-time PCR (Figure 3d). In addition, the R2 and the average relative errors for two groups are shown in Figure 4. The R2 and average relative

amplification, investigate the possible mechanisms of PCR, and provide some help to the molecular biologist.

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ASSOCIATED CONTENT

S Supporting Information *

Tables S1−S5 as described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: 086-21-39194184. Fax: 086-21-59552394. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Shanghai Municipal Commission for Science and Technology (09395811700), National Basic Research Program of China (2007CB936000), National Natural Science Foundation (No.21073222), and the Chinese Academy of Sciences (No.KJCX2-EW-N03) for financial support.

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REFERENCES

(1) Saiki, R. K.; Scharf, S.; Faloona, F.; Mullis, K. B.; Horn, G. T.; Erlich, H. A.; Arnheim, N. Enzymatic amplification of beta-globin genomic sequences and restriction site analysis for diagnosis of sickcell anemia. Science 1985, 230, 1350−1354. (2) Zhang, L.; Cui, X. F.; Schmitt, K.; Hubert, R.; Navidi, W.; Arnheim, N. Whole genome amplification from a single cellimplications for genetic-analysis. Proc. Natl. Acad. Sci.U.S.A. 1992, 89, 5847−5851. (3) Lee, M. S.; Chang, P. C.; Shien, J. H.; Cheng, M. C.; Shieh, H. K. Identification and subtyping of avian influenza viruses by reverse transcription-PCR. J. Virol. Methods 2001, 97, 13−22. (4) Gilbert, L. L.; Dakhama, A.; Bone, B. M.; Thomas, E. E.; Hegele, R. G. Diagnosis of viral respiratory tract infections in children using a reverse transcription-PCR panel. J. Clin. Microbiol. 1996, 34, 140−143. (5) Fan, C. H.; Mi, L. J.; Wen, Y. Q.; Pan, D.; Wang, Y. H.; Hu, J. Modulation of DNA polymerases with gold nanoparticles and their applications in hot-start PCR. Small 2009, 5, 2597−2600. (6) Li, H. K.; Huang, J. H.; Lv, J. H.; An, H. J.; Zhang, X. D.; Zhang, Z. Z.; Fan, C. H.; Hu, J. Nanoparticle PCR: Nanogold-assisted PCR with enhanced specificity. Angew. Chem., Int. Ed. 2005, 44, 5100−5103. (7) Li, M.; Lin, Y. C.; Wu, C. C.; Liu, H. S. Enhancing the efficiency of a PCR using gold nanoparticles. Nucleic Acids Res. 2005, 33, e184. (8) Willson, R. C.; Vu, B. V.; Litvinov, D. Gold nanoparticle effects in polymerase chain reaction: Favoring of smaller products by polymerase adsorption. Anal. Chem. 2008, 80, 5462−5467. (9) Wan, W. J.; Yeow, J. T. W. The effects of gold nanoparticles with different sizes on polymerase chain reaction efficiency. Nanotechnology 2009, 20, 325702. (10) Van Hecke, W.; Bhagwat, A.; Ludwig, R.; Dewulf, J.; Haltrich, D.; Van Langehove, H. Kinetic modeling of a bi-enzymatic system for efficient conversion of lactose to lactobionic acid. Biotechnol. Bioeng. 2009, 102, 1475−1482. (11) Kapil, A.; Wilson, K.; Lee, A. F.; Sadhukhan, J. Kinetic modeling studies of heterogeneously catalyzed biodiesel synthesis reactions. Ind. Eng. Chem. Res. 2011, 50, 4818−4830. (12) Romero, A.; Santos, A.; Ruiz, G.; Simon, E. Phenol production kinetic model in the cyclohexanol dehydrogenation process. Ind. Eng. Chem. Res. 2011, 50, 8498−8504. (13) Wang, Y. H.; Li, Y.; Li, Y. H.; Ma, X. H.; Yang, S. L.; Yang, L. Investigations into the analysis and modeling of the cytochrome P450 cycle. J. Phys. Chem. B 2006, 110, 10139−10143.

Figure 4. Correlation coefficient and average relative error between experimental and simulated data for HCMV (a) and HBV (b).

errors of HCMV and HBV are 0.9985, 0.0978 and 0.9985, 0.0281, respectively. These results indicate that the parameters estimation by multi-GA is reliable, and the comparison between the experiments and simulation indicate that they fit well. However, some factors, such as the magnesium concentration, PCR instruments, and detective methods, are not considered in our model due to the complicated interactions among the variables, which have influence on the PCR amplification and the experimental data used to estimate parameters. Sometimes, some optimization may converge slowly due to different system, such as PEV9. These factors may result in the deviation between the experiments and simulation data. Actually, it provides a pathway to study the kinetic mechanism of PCR and should be useful in wide applications.

4. CONCLUSION The lack of kinetic rates in reliable in vitro experiments is the major limitation to the creation of a model with accurate prediction of PCR. In this work, we have used the GA method to estimate the values of the parameters in mathematical modelling of PCR. The 12 reaction rate constants were successfully fitted by the GA. It showed that the values of the estimation using GA are independent of the initial conditions and DNA sources. The model results fit well with the experimental data. Therefore, the model can predict the DNA 13272

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teschoviruses and enteroviruses by LightCycler real-time PCR. J. Virol. Methods 2003, 113, 51−63. (36) Arya, M.; Shergill, I. S.; Williamson, M.; Gommersall, L.; Arya, N.; Patel, H. R. H. Basic principles of real-time quantitative PCR. Expert Rev. Mol. Diagn. 2005, 5, 209−219. (37) Valasek, M. A.; Repa, J. J. The power of real-time PCR. Adv. Physiol. Educ. 2005, 29, 151−159. (38) Nitsche, A.; Steuer, N.; Schmidt, C. A.; Landt, O.; Siegert, W. Different real-time PCR formats compared for the quantitative detection of human cytomegalovirus DNA. Clin. Chem. 1999, 45, 1932−1937. (39) Chieochansin, T.; Chutinimitkul, S.; Payungporn, S.; Theamboonlers, A.; Tangkijvanich, P.; Komolmit, P.; Poovorawan, Y. Rapid detection of lamivudine-resistant hepatitis B virus mutations by PCR-based methods. Tohoku J. Exp. Med. 2006, 210, 67−78.

(14) Zhou, W.; Li, Y.; Wang, X.; Wu, L.; Wang, Y. MiR-206-mediated dynamic mechanism of the mammalian circadian clock. BMC Syst. Biol. 2011, 5, 141. (15) Wang, X.; Li, Y.; Xu, X.; Wang, Y.-h. Toward a system-level understanding of microRNA pathway via mathematical modeling. Biosystems 2010, 100, 31−38. (16) Mehra, S.; Hu, W. S. A kinetic model of quantitative real-time polymerase chain reaction. Biotechnol. Bioeng. 2005, 91, 848−60. (17) Gevertz, J. L.; Dunn, S. M.; Roth, C. M. Mathematical model of real-time PCR kinetics. Biotechnol. Bioeng. 2005, 92, 346−55. (18) Hsu, J. T.; Das, S.; Mohapatra, S. Polymerase chain reaction engineering. Biotechnol. Bioeng. 1997, 55, 359−366. (19) Berdis, A. J. Mechanisms of DNA polymerases. Chem. Rev. 2009, 109 (7), 2862−2879. (20) Arisi, I.; Cattaneo, A.; Rosato, V. Parameter estimate of signal transduction pathways. BMC Neurosci. 2006, 7, s6. (21) Holland, J. H. Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence; University of Michigan Press: Ann Arbor, MI, 1975; pp viii, 183. (22) Harris, S. D.; Elliott, L.; Ingham, D. B.; Pourkashanian, M.; Wilson, C. W. The optimization of reaction rate parameters for chemical kinetic modelling of combustion using genetic algorithms. Comput. Method Appl. Mech. Eng. 2000, 190, 1065−1090. (23) Katare, S.; Bhan, A.; Caruthers, J. M.; Delgass, W. N.; Venkatasubramanian, V. A hybrid genetic algorithm for efficient parameter estimation of large kinetic models. Comput. Chem. Eng. 2004, 28, 2569−2581. (24) Park, S. J.; Bhargava, S.; Chase, G. G. Fitting of kinetic parameters of NO reduction by CO in fibrous media using a genetic algorithm. Comput. Chem. Eng. 2010, 34, 485−490. (25) Ranganath, M.; Renganathan, S.; Gokulnath, C. Identification of bioprocesses using genetic algorithm. Bioprocess Engin. 1999, 21, 123− 127. (26) Park, L. J.; Park, C. H.; Park, C.; Lee, T. Application of genetic algorithms to parameter estimation of bioprocesses. Med. Biol. Eng. Comput. 1997, 35, 47−49. (27) Su, Y.; Zhu, G.; Miao, Z.; Feng, Q.; Chang, Z. Estimation of parameters of a biochemically based model of photosynthesis using a genetic algorithm. Plant, Cell Environ. 2009, 32, 1710−1723. (28) Plum, G. E.; Breslauer, K. J.; Richard, W. R. Thermodynamics and kinetics of nucleic acid association/dissociation and folding processes. Compr. Nat. Prod. Chem. 1999, 7, 15−33. (29) Wang, S. H.; Friedman, A. E.; Kool, E. T. Origins of high sequence selectivityA stopped-flow kinetics study of DNA/RNA hybridization by duplex-forming and triplex-forming oligonucleotides. Biochemistry 1995, 34, 9774−9784. (30) Datta, K.; LiCata, V. J. Thermodynamics of the binding of Thermus aquaticus DNA polymerase to primed-template DNA. Nucleic Acids Res. 2003, 31, 5590−5597. (31) Tsoi, P. Y.; Yang, M. S. Kinetic study of various binding modes between human DNA polymerase beta and different DNA substrates by surface-plasmon-resonance biosensor. Biochem. J. 2002, 361, 317− 325. (32) Lowe, L. G.; Guengerich, F. P. Steady-state and pre-steady-state kinetic analysis of dNTP insertion opposite 8-oxo-7,8-dihydroguanine by Escherichia coli polymerases I exo(-) and II exo(-). Biochemistry 1996, 35, 9840−9849. (33) Furge, L. L.; Guengerich, F. P. Analysis of nucleotide insertion and extension at 8-oxo-7,8-dihydroguanine by replicative T7 polymerase exo(-) and human immunodeficiency virus-1 reverse transcriptase using steady-state and pre-steady-state kinetics. Biochemistry 1997, 36, 6475−6487. (34) Yang, L. T.; Pan, A. H.; Zhang, K. W.; Yin, C. S.; Qian, B. J.; Chen, J. X.; Huang, C.; Zhang, D. B. Qualitative and quantitative PCR methods for event-specific detection of genetically modified cotton Mon1445 and Mon531. Transgenic Res. 2005, 14, 817−831. (35) Krumbholz, A.; Wurm, R.; Scheck, O.; Birch-Hirschfeld, E.; Egerer, R.; Henke, A.; Wutzler, P.; Zell, R. Detection of porcine 13273

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