- Research article
- Open Access
Parameter estimation for stiff equations of biosystems using radial basis function networks
© Matsubara et al; licensee BioMed Central Ltd. 2006
- Received: 05 June 2005
- Accepted: 27 April 2006
- Published: 27 April 2006
The modeling of dynamic systems requires estimating kinetic parameters from experimentally measured time-courses. Conventional global optimization methods used for parameter estimation, e.g. genetic algorithms (GA), consume enormous computational time because they require iterative numerical integrations for differential equations. When the target model is stiff, the computational time for reaching a solution increases further.
In an attempt to solve this problem, we explored a learning technique that uses radial basis function networks (RBFN) to achieve a parameter estimation for biochemical models. RBFN reduce the number of numerical integrations by replacing derivatives with slopes derived from the distribution of searching points. To introduce a slight search bias, we implemented additional data selection using a GA that searches data-sparse areas at low computational cost. In addition, we adopted logarithmic transformation that smoothes the fitness surface to obtain a solution simply. We conducted numerical experiments to validate our methods and compared the results with those obtained by GA. We found that the calculation time decreased by more than 50% and the convergence rate increased from 60% to 90%.
In this work, our RBFN technique was effective for parameter optimization of stiff biochemical models.
- Genetic Algorithm
- Logarithmic Transformation
- Radial Basis Function Network
- Stiff System
- Biochemical Model
The application of mathematical expressions for biochemical systems is useful for understanding complex biological phenomena. Reactions are expressed as rate equations, and the numerical integration of variables is applied to simulate these reactions. The simulation of biochemical models requires the identification of all kinetic parameters of each rate equation. However, almost all biochemical models, and large-scale models in particular, involve parameters that are extremely difficult to measure in vivo or in vitro [1–3] and the exclusive use of results from wet-bench experiments may be insufficient for the identification of all parameters. Therefore, to establish precise models, it is essential to determine unknown kinetic parameters from the time-courses of concentrations.
Various optimization algorithms such as the Levenberg-Marquardt method, genetic programming, simulated annealing, and evolutionary algorithms have been applied to infer parameters or equations in biochemical models [4–9]. The genetic algorithm (GA)  is commonly applied to these problems because of its global optimization ability [11–15].
However, the GA, including the real-coded GA , relies on stochastic optimization algorithms and requires vast numerical integrations to evaluate estimated parameters. In addition, the computational cost increases further when the target model has the stiffness that often appears in equations for biochemical systems because each solution requires small integration steps [17–19]. The application of the stochastic method to large-scale or stiff models involves enormously expensive computational time and superior hardware performance. In addition, use of GA elicits the sampling bias phenomenon [20, 21]. This becomes an inherent problem that may result in failure in cases where an optimal parameter exists around the boundary of the parameter space. In particular, parameters of stiff systems tend to be allocated around the boundary of the search space because they involve different-order parameters.
The radial basis function network (RBFN) [22, 23], a type of artificial neural network (ANN) , is one of the artificial learning methods that describe complex non-linear relationships between inputs and outputs. The RBFN can solve parabolic, hyperbolic, and elliptic partial differential equations numerically  and is able to approximate non-linear time-courses effectively [26–28]. Due to the increased accumulation of biological information, RBFN are now applied in the biochemical field [29–33].
In the work presented here, we adopted RBFN to parameter estimation for local and global searches. RBFN enable to reduce the computational cost by omitting numerical integrations, resulting in solution of the above problems. In applying RBFN to biochemical modeling, we implemented subsidiary 2 improvements: (1) Since it is difficult to model highly nonlinear biochemical systems, we adopted a logarithmic transformation to both the input and output layer of the RBFN, thereby facilitating parameter-optimization. (2) RBFN require the selection of appropriate additional learning data to obtain a global minimum. Here we propose additional data selection using a GA for selecting additional learning data sets. This method adopts elemental GA to search data-sparse areas in the parameter space. We applied the proposed method to parameter estimation in a stiff biosystem. Our results demonstrate that compared to the GA, our method facilitates the acquisition of equivalent or more accurate parameters at half the calculation time and yields a 50% increase in the optimization success rate.
Kinetic equations used in the model. Reactions 1, 2, and 3 are reversible Henri-Michaelis-Menten reactions with one substrate and one product. Reaction 4 is an irreversible Henri-Michaelis-Menten reaction with one substrate and one product. Reaction 5 is a mass action reaction with 2 substrates and one product.
Kinetic parameters used in the model. These values were determined artificially. Details of the kinetic parameters are shown in Table 1. The search ranges were K i ∈ [0.1,1000].
K11 = 10
K31 = 1000
K12 = 0.1
K32 = 1
K13 = 4
K33 = 0.1
K14 = 3
K34 = 1
K21 = 1000
K41 = 2
K22 = 1
K42 = 1
K23 = 0.1
K51 = 1000
K24 = 3
K52 = 0.1
Initial conditions of training data sets from No.1 to No.10 and test data sets from No.11 to No.20. Examples of time-courses using initial conditions No.3 and No.5 are shown in Figure 3. Concentrations are non-unit.
Parameters applied in this work. The top column shows the parameters of RBFN, the bottom column the parameters of ADSGA using SPX as the recombination- and ranking selection as the selection procedure.
parameter of RBFN
number of near individual
number of far individual
neighborhood control parameter
generalization control parameter
parameter of ADSGA
number of individual
max of generation
initial mutation ratio
selection control parameter
selection control parameter
Performance of ADSGA
Second, we inferred all parameters in the Figure 1 model. Table 6 shows that we succeeded in obtaining desirable results in terms of generalization ability using ADSGA and k NN with k = 4. k NN with the other k values attained lower achievements compared with our algorithm. Although ADSGA employed GA for determining additional data, the optimization speed was raised compared with k NN.
Performance of logarithmic transformation
Table 6 shows the results using parameter transformation and fitness transformation in the inference of all parameters in the Figure 3 model. Log indicates the application of the logarithmic transformation of the fitness. The logarithmic transformation of the parameters applies to all cases. We found that the logarithmic transformation reduced the learning time by 23% while maintaining a high convergence rate. However, the test error increased by 16%. Optimization becomes simple with logarithmic transformation in cases where the view of the fitness space could be transformed to gently sloping. On the other hand, the test error increased slightly since it also changes the neighbor view of the answer. Since the parameters of power-law formalisms affect the parts of exponential factors, the perturbation leads to large changes in the variables and the individual fitness during their optimizations. The fitness values tend to become 0 in most of the search space since the results of numerical integrations become infinite. The smoothing of the error surface by the logarithmic transformation enables to simplify the search surface to optimize. The transformation achieved 1.2-fold faster optimization.
RBFN application to stiff systems
Parameters for GA. SPX is employed as the recombination method, ranking selection as the selection method.
number of individual
initial mutation ratio
selection control parameter
selection control parameter
Validation of two subsidiary improvements and comparative experiments between GA and RBFN including above-mentioned improvements. The comparison between k NN and ADSGA are shown in the columns headed k NN and the second columns from right headed ADSGA. The results acquired with the conventional k NN method and our proposed ADSGA are shown in the columns headed k NN and ADSGA. The improved results obtained with the fitness transformation are demonstrated in the 2 columns headed ADSGA and ADSGA+Log. The validation of main improvement that is the application of RBFN is presented in the leftmost GA column and the rightmost ADSGA+Log column. The proposed method yielded equivalent or more accurate results compared to the parameters obtained with GA at half the calculation time and a 50% increase in the optimization success rate.
k = 2
k = 4
k = 8
k = 16
Convergence rate (%)
Processing time (min)
Test error (%)
10.3 ± 2.5
12.9 ± 3.9
10.4 ± 2.0
11.0 ± 2.3
11.8 ± 3.4
9.3 ± 2.2
10.8 ± 2.1
The calculation time was reduced to less than half compared with GA. The 1.6-fold and 2.1-fold increases in the optimization speed are ascribable to RBFN and the combination of RBFN and the logarithmic transformation, respectively. RBFN learning omit the calculation of numerical integration because it learns the relationship between a parameter and its evaluation space. When problems are examined that require a long calculation time for each integration, the difference in the processing time between RBFN and GA increases further.
The convergence rate of RBFN was raised from 60% to 90% compared with that of GA. The RBFN application, not 2 subsidiary improvements, achieved to raise the convergence rate. Ref  reported a peculiar problem that pertains to the sampling bias phenomenon when the optimal solution exists at the edge of a parameter space. GA was unable to find the optimal solution because some parameters existed at the edge of a parameter space. As RBFN are only biased to the area near the estimated optimal solution, the optimal solution can be searched regardless of its location in a parameter space. The test error of RBFN did not decrease compared with that of GA. RBFN, which employ logarithmic transformation of fitness, are slightly inferior to GA in terms of local searches, resulting in a decline in the generalization ability compared with GA. If the logarithmic transformation is not applied in RBFN, the generalization ability of the 2 algorithms is equivalent and high convergence and fast optimization are retained.
GA involves the sampling bias phenomenon because the search region of unimodal normal-distribution crossover , parent-centric crossover , or SPX is biased to the interior of a parameter space. The phenomenon is bound to occur more frequently as the parameters become larger. To solve this problem, extrapolation-directed crossover  has been proposed; it is biased toward extrapolative crossover. In addition, boundary extension by mirroring  and toroidal search space conversion  have been proposed; they extend the search space arbitrarily. However, these techniques raise computational costs rather than conquering the sampling bias phenomenon.
The performance of RBFN was superior to that of GA in the parameter inference of stiff systems since it enables to reduce the number of numerical integrations.
For further improvement, the optimization algorithm is switched to the gradient method which is deterministic and can search local minima quickly at the end of the learning stage. A decrease in test error can be expected when a combination of algorithms is applied. It is considered that the additional improvement by the numerical integration methods such as Gear algorithm  is harmonious with our parameter estimation algorithm. The hybrid method is also one of tasks to be confirmed in the future. In this paper, we tested the case of stiff equations, and it is considered that non-stiff equations could also show the advantage of our method because of the simple dynamics.
We obtained solutions with low generalization ability when only a few time-course sets were given as input data. Predicted solutions with high generalization ability are likely to be biologically meaningful parameters. Careful selection of the number of sampling points  and application of the data smoothing method [19, 45] are necessary when our method is applied to real problems because the most appropriate combination with our method is dependent on the data and the system in question.
RBFN are optimized by adding a new function O(p2) and then adding a new data set O(m2+ mp + p2), where p is the number of learned data and m is the number of the kernel function. The inference of numerous parameters causes a serious problem. Biological restriction as well as the deletion of several learning data in overcrowded areas can lead to a reduction in computational costs. In addition, problem decomposition strategies [15, 46] are useful to reduce the number of parameters that should be inferred simultaneously.
We applied RBFN to the parameter estimation of biochemical models, especially stiff systems. RBFN method could reduce the number of numerical integrations. Especially, it leaded to the fast optimization in stiff systems of biochmical models. It yielded equivalent or more accurate results compared to the parameters obtained with GA at half the calculation time and a 50% increase in the optimization success rate. Also we adopted 2 secondary learning techniques to RBFN. One is the fitness transformation that changes a fitness function into a gently sloping function and reduces the calculation time 0.8-fold. The other is the new selection method of learning data, ADSGA. ADSGA was able to quickly obtain optimal solutions with high generalization ability, almost equivalent to k NN with optimal k. In this paper, it was shown that our RBFN technique attained the parameter optimization for stiff biochemical models from measured time-courses only.
Overview of RBFN optimization method
Parameters are generated and uniform random numbers are assigned to each parameter within the search space.
Each parameter is set to the kinetic model to calculate the relative squared errors E between calculated and given time-courses. The fitness F is the reciprocal of E, i.e. F = 1/E). If F is high, the calculated time-course resembles the given time-course. The goal is to search for the parameter space that maximizes F.
where l is the number of time-course sets, n the number of experimentally given state variables, the numerically calculated concentration at time t of a state variable X i , and represents the given concentration at time t of a state variable X i . For expedience, when is 0, the denominator of E is set to 0.1.
(c) RBFN learning
RBFN learn the relationship between parameters and fitness values. The output of RBFN f(x) defines the view of the fitness space in the parameter estimation. The value is calculated from
where w j is the j th weight, x the learning data from parameter space, m the number of learning data, and h j is the j th kernel function, e.g.,
where c j is the center of the j-th kernel and r is the radial of kernels. The optimal weight of each kernel is immediately obtained by the matrix operation. Various methods determining internal parameters of RBFN, e.g. generalized cross-validation  or calculation of the marginal likelihood of the data , can adjust the centers of kernels . In this study, we assume that the inputs of the data set are used as the fixed centers since adjustment involves high computational costs. For details of the supervised training scheme, refer to .
(d) Capturing the RBFN shape by GA
GA is used to search for the optimal parameters for the maximum F based on the function of parameters (input) and the fitness value (output) from the previous step. We employed real-coded GA; it could attain a high convergence success rate, fast optimization, and high accuracy compared with conventional binary GA . We then adopted the GA for searching the optimal solution of the network at the present stage. The calculation cost of this procedure is very small since the GA does not require numerical integration in the evaluation phase.
(e) Additional data selection
If the solution obtained in the previous phase is not satisfactory, additional learning parameters are used to improve fitness. Two types of learning parameters are employed for relearning : One is selected from the neighborhood of the current optimal parameters to add local information near the optimal point, the other is selected from a sparse area in the parameter space to add global information. Iterative learning using 2 types of data enables a local and global search for the optimal parameters.
(f) Repeat (b)-(e)
If the current optimal solution is unsatisfactory, learning is repeated from (b).
RBFN capture the function of parameters and the fitness value by exercising feed-forward control of the optimal parameters. Additional learning is performed to search parameters with higher fitness. Parameters for learning are carefully selected from 2 areas in order to consider global and local optimizations. In one area learning data are few because a whole view of fitness function can be obtained by adding such data. The k-nearest-neighbor method (k NN) [52, 53] is a clustering technique to find the sparse area. The density function of a data set x q is described as f(x q ) = dist(x q , y), where y is a set of k data near x q in all the data D, and dist(x q ,y) represents the Euclidean distance of x q and y. k NN defines x, which has the maximum value of f(x q ), as the most sparse data set. Optimal k changes with the distribution of data learned early, the dimension of the parameter space, and the number of groups or units of data. Therefore, searching the optimal parameters involves the computational costs for obtaining the optimal k.
We propose a method for selecting additional data for RBFN learning and capturing the rough surface using elemental GA. Note that GA was not applied to the parameter-estimation phase. We employ the simple GA to accelerate finding the next data. This is indicated in paragraph (d) of the previous section. This procedure is called additional data selection using GA (ADSGA). The rule for selecting data in ADSGA is as follows. First, the density function of the learned data is created by RBFN, where [1 1 ··· 1]⊤ is applied as F. Next, the minimum value of the function is searched by simply using GA. ADSGA searches the minimum value of the function as the sparsest data set. Since ADSGA does not require control parameters such as k, it can create the density function without pilot studies.
The other additional learning data are selected from the area near the current optimal solution to obtain more detailed information. The size of this neighborhood is controlled by , where represents the parameters for learning in the next repetition, is the current optimal solution, and rand(A, B) generates a vector consisting of uniform random numbers ∈ [A, B]. When the estimated optimal solution varies greatly with every learning session, the search continues to be defined as an early stage. When the solution seldom changes even with repeated learning sessions, the search is defined as a finishing stage. We recommend decreasing the L value as the search advances.
Two conditions are then defined to change the value of L: (1) the current estimated optimal solution is near the preceding estimated optimal solution , and (2) the fitness of is better than that of . If conditions (1) and (2) are fulfilled, L – α is substituted for the value of L. If only condition (1) is fulfilled, L0, which is the initial value of L, is substituted for the value of L. If neither (1) nor (2) is fulfilled, the value of L remains unchanged.
The simultaneous inclusion of global and local data can prevent convergence into a local minimum solution, and can obtain a better solution near the current optimal solution.
We thank Dr. Koji Tsuda, Max Planck Institute for Biological Cybernetics, Germany, for discussions of the RBFN application. This work was supported by a grant from the Ministry of Education, Culture, Sports, Science and Technology as COE program, NEDO and JST.
- Isaacs FJ, Hasty J, Cantor CR, Collins JJ: Prediction and measurement of an autoregulatory genetic module. Proc Natl Acad Sci U S A 2003, 100: 7714–7719. 10.1073/pnas.1332628100PubMed CentralView ArticlePubMedGoogle Scholar
- Gardner TS, di Bernardo D, Lorenz D, Collins JJ: Inferring genetic networks and identifying compound mode of action via expression profiling. Science 2003, 301: 102–105. 10.1126/science.1081900View ArticlePubMedGoogle Scholar
- Kalir S, Alon U: Using a quantitative blueprint to reprogram the dynamics of the flagella gene network. Cell 2004, 117: 713–720. 10.1016/j.cell.2004.05.010View ArticlePubMedGoogle Scholar
- Mendes P, Kell D: Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation. Bioinformatics 1998, 14: 869–883. 10.1093/bioinformatics/14.10.869View ArticlePubMedGoogle Scholar
- Moles CG, Mendes P, Banga JR: Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res 2003, 13: 2467–2474. 10.1101/gr.1262503PubMed CentralView ArticlePubMedGoogle Scholar
- Swameye I, Muller TG, Timmer J, Sandra O, Klingmuller U: Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. Proc Natl Acad Sci USA 2003, 100: 1028–1033. 10.1073/pnas.0237333100PubMed CentralView ArticlePubMedGoogle Scholar
- Bentele M, Lavrik I, Ulrich M, Stosser S, Heermann DW, Kalthoff H, Krammer PH, Eils R: Mathematical modeling reveals threshold mechanism in CD95-induced apoptosis. J Cell Biol 2004, 166: 839–851. 10.1083/jcb.200404158PubMed CentralView ArticlePubMedGoogle Scholar
- Tsai KY, Wang FS: Evolutionary optimization with data collocation for reverse engineering of biological networks. Bioinformatics 2005, 21: 1180–1188. 10.1093/bioinformatics/bti099View ArticlePubMedGoogle Scholar
- Sugimoto M, Kikuchi S, Tomita M: Reverse engineering of biochemical equations from time-course data by means of genetic programming. Biosystems 2005, 80: 155–164. 10.1016/j.biosystems.2004.11.003View ArticlePubMedGoogle Scholar
- Goldberg DE: Genetic algorithms in search, optimization and machine learning. Edited by: Boston MA. USA: Addison-Wesley Longman Publishing Co., Inc.; 1989.Google Scholar
- Park LJ, Park CH, Park C, Lee T: Application of genetic algorithms to parameter estimation of bioprocesses. Med Biol Eng Comput 1997, 35: 47–49.View ArticlePubMedGoogle Scholar
- Pinchuk RJ, Brown WA, Hughes SM, Cooper DG: Modeling of biological processes using self-cycling fermentation and genetic algorithms. Biotechnol Bioeng 2000, 67: 19–24. 10.1002/(SICI)1097-0290(20000105)67:1<19::AID-BIT3>3.0.CO;2-CView ArticlePubMedGoogle Scholar
- Hatakeyama M, Kimura S, Naka T, Kawasaki T, Yumoto N, Ichikawa M, Kim JH, Saito K, Saeki M, Shirouzu M, Yokoyama S, Konagaya A: A computational model on the modulation of mitogen-activated protein kinase (MAPK) and Akt pathways in heregulin-induced ErbB signalling. Biochem J 2003, 373: 451–463. 10.1042/BJ20021824PubMed CentralView ArticlePubMedGoogle Scholar
- Kikuchi S, Tominaga D, Arita M, Takahashi K, Tomita M: Dynamic modeling of genetic networks using genetic algorithm and S-system. Bioinformatics 2003, 19: 643–650. 10.1093/bioinformatics/btg027View ArticlePubMedGoogle Scholar
- Kimura S, Ide K, Kashihara A, Kano M, Hatakeyama M, Masui R, Nakagawa N, Yokoyama S, Kuramitsu S, Konagaya A: Inference of S-system models of genetic networks using a cooperative coevolutionary algorithm. Bioinformatics 2005, 21: 1154–1163. 10.1093/bioinformatics/bti071View ArticlePubMedGoogle Scholar
- Jonikow CZ, Michalewicz Z: An experimental comparison of binary and floating point representations in genetic algorithms. Proceedings of International Conference on Genetic Algorithms 1991, 31–38.Google Scholar
- Michalski R, Rode W, Les A: DerivFit: a program for rate equation parameter fitting using derivatives. Comput Biomed Res 1998, 31: 71–89. 10.1006/cbmr.1998.1469View ArticlePubMedGoogle Scholar
- Meng TC, Somani S, Dhar P: Modeling and simulation of biological systems with stochasticity. Silico Biol 2004, 4: 293–309.Google Scholar
- Voit EO, Almeida J: Decoupling dynamical systems for pathway identification from metabolic profiles. Bioinformatics 2004, 20: 1670–1681. 10.1093/bioinformatics/bth140View ArticlePubMedGoogle Scholar
- Larry JE, Keith EM, Schaffer JD: Crossover operator biases. Proceedings of International Conference on Genetic Algorithms 1997, 354–361.Google Scholar
- Tsutsui S, Goldberg DE: Search space boundary extension method in real-coded genetic algorithms. Inf Sci 2001, 133: 229–247. 10.1016/S0020-0255(01)00087-1View ArticleGoogle Scholar
- Broomhead DS, Lowe D: Multivariable functional interpolation and adaptive networks. Complex Syst 1988, 2: 321–335.Google Scholar
- Musavi MT, Ahmed W, Chen KH, Faris KB, Hummels DM: On the training of radial basis function classifiers. Neural Netw 1992, 5: 595–603. 10.1016/S0893-6080(05)80038-3View ArticleGoogle Scholar
- Rumelhart DE, Hinton GE, Williams RJ: Learning representations by back-propagating errors. Nature 1986, 323: 533–536. 10.1038/323533a0View ArticleGoogle Scholar
- Jianyu L, Siwei L, Yingjian Q, Yaping H: Numerical solution of elliptic partial differential equation using radial basis function neural networks. Neural Netw 2003, 16: 729–734. 10.1016/S0893-6080(03)00083-2View ArticlePubMedGoogle Scholar
- X H, SA B: Dual-orthogonal radial basis function networks for nonlinear time series prediction. Neural Netw 1998, 11: 479–493. 10.1016/S0893-6080(97)00132-9View ArticleGoogle Scholar
- Cowper MR, Mulgrew B, Unsworth CP: Nonlinear prediction of chaotic signals using a normalised radial basis function network. Signal Process 2002, 82: 775–789. 10.1016/S0165-1684(02)00155-XView ArticleGoogle Scholar
- Rank E: Application of Bayesian trained RBF networks to nonlinear time-series modeling. Signal Process 2003, 83: 1393–1410. 10.1016/S0165-1684(03)00088-4View ArticleGoogle Scholar
- Agatonovic-Kustrin S, Glass BD, Wisch MH, Alany RG: Prediction of a stable microemulsion formulation for the oral delivery of a combination of antitubercular drugs using ANN methodology. Pharm Res 2003, 20: 1760–1765. 10.1023/B:PHAM.0000003372.56993.39View ArticlePubMedGoogle Scholar
- Agatonovic-Kustrin S, Evans A, Alany RG: Prediction of corneal permeability using artificial neural networks. Pharmazie 2003, 58: 725–729.PubMedGoogle Scholar
- Germani M, Magni P, De Nicolao G, Poggesi I, Marsiglio A, Ballinari D, Rocchetti M: In vitro cell growth pharmacodynamic studies: a new nonparametric approach to determining the relative importance of drug concentration and treatment time. Cancer Chemother Pharmacol 2003, 52: 507–513. 10.1007/s00280-003-0688-7View ArticlePubMedGoogle Scholar
- Ji W, Naguib RNG, Ghoneim MA: Neural network-based assessment of prognostic markers and outcome prediction in bilharziasis-associated bladder cancer. IEEE Trans Inf Technol Biomed 2003, 7: 218–224. 10.1109/TITB.2003.813796View ArticlePubMedGoogle Scholar
- Niwa T: Prediction of biological targets using probabilistic neural networks and atom-type descriptors. J Med Chem 2004, 47: 2645–2650. 10.1021/jm0302795View ArticlePubMedGoogle Scholar
- Takahashi K, Kaizu K, Hu B, Tomita M: A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics 2004, 20: 538–546. 10.1093/bioinformatics/btg442View ArticlePubMedGoogle Scholar
- Kimura S, Kawasaki T, Hatakeyama M, Naka T, Konishi F, Konagaya A: OBIYagns: a grid-based biochemical simulator with a parameter estimator. Bioinformatics 2004, 20: 1646–1648. 10.1093/bioinformatics/bth122View ArticlePubMedGoogle Scholar
- Dhar P, Meng TC, Somani S, Ye L, Sairam A, Chitre M, Hao Z, Sakharkar K: Cellware-a multi-algorithmic software for computational systems biology. Bioinformatics 2004, 20: 1319–1321. 10.1093/bioinformatics/bth067View ArticlePubMedGoogle Scholar
- Tsutsui S, Yamamura M, Higuchi T: Multi-parent recombination with simplex crossover in real coded genetic algorithms. Proceedings of Genetic and Evolutionary Computation Conference 1999, 73–80.Google Scholar
- Ono I, Kobayashi S: A real-coded genetic algorithm for function optimization using unimodal normal distribution crossover. Proceedings of International Conference on Genetic Algorithms 1997, 246–253.Google Scholar
- Deb K, Anand A, Joshi D: A computationally efficient evolutionary algorithm for real-parameter optimization. Evol Comput 2002, 10: 371–395. 10.1162/106365602760972767View ArticlePubMedGoogle Scholar
- Sakuma H, Yamamura M: Extrapolation-directed crossover for real-coded GA:overcoming deceptive phenomena by extrapolative search. Proceedings of Congress on Evolutionary Computation 2001, 553–560.Google Scholar
- Tsutsui S: Multi-parent Recombination in Genetic Algorithms with Search Space Boundary Extension by Mirroring. Proceedings of the International Conference on Parallel Problem Solving from Nature 1998, 428–437.Google Scholar
- Someya H, Yamamura M: Robust Evolutionary Algorithms With Toroidal Search Space Conversion For Function Optimization. Proceedings of the Genetic and Evolutionary Computation Conference, San Francisco 2002, 553–560.Google Scholar
- Gear CW: Numerical initial value problems in ordinary differential equations. Upper Saddle River, NJ, USA: Prentice Hall PTR; 1971.Google Scholar
- Kutalik Z, Cho KH, Wolkenhauer O: Optimal sampling time selection for parameter estimation in dynamic pathway modeling. Biosystems 2004, 75: 43–55. 10.1016/j.biosystems.2004.03.007View ArticlePubMedGoogle Scholar
- Chen L, Bernard O, Bastin G, Angelov P: Hybrid modeling of biotechnological processes using neural networks. Control Eng Pract 2000, 8: 821–827. 10.1016/S0967-0661(00)00036-8View ArticleGoogle Scholar
- Maki Y, Ueda T, Okamoto M, Uematsu N, Inamura Y, Eguchi Y: Inference of genetic network using the expression profile time course data of mouse P19 cells. Genome Inform 2002, 13: 382–383.Google Scholar
- Golub G, Heath M, Wahba G: Generalised cross-validation as a method for choosing a good ridge parameter. Thchnometrics 1979, 21: 215–223. 10.2307/1268518View ArticleGoogle Scholar
- MacKay D: Bayesian interpolation. Neural Comput 1992, 4: 415–447.View ArticleGoogle Scholar
- Orr M: Regularisation in the selection of radial basis function centres. Neural Comput 1995, 7: 606–623.View ArticleGoogle Scholar
- Haykin S: Neural networks: A comprehensive foundation. Upper Saddle River, NJ, USA: Prentice Hall PTR; 1994.Google Scholar
- Nakayama H, Arakawa M, Sasaki R: A Computational Intelligence Approach to Optimization with Unknown Objective Functions. Proceedings of the International Conference on Artificial Neural Networks 2001, 73–80.Google Scholar
- Cover T, Hart P: Nearest neighbor pattern classifiation. IEEE Trans Information Theory 1967, 13: 21–27. 10.1109/TIT.1967.1053964View ArticleGoogle Scholar
- Dasarathy BV: Nearest-neighbor pattern classification techniques. Edited by: Los Alomitos. CA: Computer Society Press; 1991.Google Scholar
- Oyman AI, Beyer HG, Schwefel HP: Analysis of the (1, lambda)-ES on the parabolic ridge. Evol Comput 2000, 8: 249–265. 10.1162/106365600750078772View ArticlePubMedGoogle Scholar
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