Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 1 of 10. Towards the Inference of Genetic Networks

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1 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page of 0 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inverse Problems: Towards the Inference of Genetic Networks Daisuke Tominaga, Masahiro Okamoto,2,Yukihiro Maki 3, Shoji Watanabe 3 and Yukihiro Eguchi 3 Department of Biochemical Engineering & Science, Kyushu Institute of Technology 680, Kawazu, Iizukacity Fukuoka , Japan 2 Phone: Fax: okahon@bse.kyutech.ac.jp 3 Research Institute, Mitsui Knowledge Industry Co., Ltd 27 Higashinakano, Nakanoku Tokyo 68555, Japan ABSTRACT Nonlinear systems, such as metabolic pathways and genetic networks have richly complex structures and the details of the mechanism at the molecular level that govern interactions among system components are generally not well known. Estimation of the interaction mechanisms among system components via experimentally observed dynamic responses (timecourses) of some of the system components is generally an inverse problem. The Ssystem, which belongs to powerlaw formalism, is one of the best representations to solve this type of inverse problem: the Ssystem is rich enough in structure to capture all relevant dynamics. In the present paper, for the purpose of solving the inverse problem, we apply the Genetic Algorithm and propose an efficient procedure for the estimation of large numbers of parameters in the Ssystem formalism. The proposed procedure enables genetic network architecture to be reconstructed based on experimentally observed time courses in gene expression. Keywords : Nonlinear system, Numerical optimization, Gene Expression Network, Genetic Algorithm, Powerlaw formalism, inverse problem INTRODUCTION Organizationally complex systems such as gene expression networks and metabolic path ways are comprised of numerous, richly interacting components. In the case where the details of the molecular mechanism that govern interactions among system components (state variables) are not well known, most of these processes are nonlinear. How then do we mathematically model such complex processes? Description of these processes requires a representation that is general enough to capture the essence of the experimentally observed response. One of the best approaches that satisfies this requirement is the Ssystem [ Savageau, 976; Voit, 99; Tominaga and Okamoto, 998] which is a type of to powerlaw formalism because it is based on a particular type of ordinary differential equation in which the component processes are characterized by powerlaw functions () where n is the number of state variables or reactants (X i ) and i and j (< i, j < n) are suffixes of state variables. The terms g i j are the interactive effectivity of X j to X i. The first term represents all influences that increase X i, whereas the second term represents all influences that decrease X i. In a biochemical engineering context, the nonnegative parameters i and ß i are called rate constants, and realvalued exponents g i j are referred to as kinetic orders. The Ssystem is rich enough in

2 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 2 of 0 structure to capture all relevant dynamics; an observed response (dynamic response) may be monotone or oscillatory, it may contain limit cycles or exhibit deterministic chaos. As long as it can be formulated as a system of ordinary differential equations, the Ssystem can be formulated as a canonical model. Furthermore, the simple homogeneous structure of the Ssystem has a great advantage in terms of system analysis and control design, because the structure allows analytical and computational methods to be customized specifically for this structure [ Irvine and Savageau, 990]. However, the Ssystem formalism has a major disadvantage in that this formalism includes a large number of parameters that must be estimated ( i, ß i, g i j ). The estimation of these parameters oftern causes a bottleneck, and matching the model to the experimentally observed responses (time courses of relevant state variables or reactants) is almost never straightforward and is almost always difficult. The number of estimated parameters in Ssystem formalism is 2n(n + ), where n is the number of state variables (X i ). In the present paper, we propose an algorithm and procedures for the estimation (optimization) of a large number of parameters [ Okamoto et al., 997; Tominaga and Okamoto, 999]. The basic idea is as follows: the Genetic Algorithm (GA) [ Baker, 985; Goldberg, 989; Davis, 99] is introduced as a nonlinear optimization method which is much less likely to be stranded in local minima. Furthermore, in order to find the skeletal structure (smallsized system) of Ssystem formalism that matches the experimentally observed responses, some of the parameters (g i j ), the absolute values of which are less than a given threshold value, are to be removed (reset to 0) during optimization procedures. By introducing this algorithm, referred to as structure skeletalizing [ Tominaga and Okamoto, 998; Tominaga and Okamoto, 999], optimized essential Ssystem model matching to the experimentally observed responses should be possible. In the last section of the paper, we show that the proposed procedures are applicable to the inference of genetic networks from gene expression patterns produced by gene disruptions and over expressions. OPTIMIZATION PROCEDURES Since the Ssystem is a formalism of ordinary nonlinear differential equations, the system can easily be solved numerically by using a suitable numerical calculation program, such the RungeKuttaGill method. However, when an adequate timecourse of relevant state variables is given, a set of parameter values i, ß i, g i j, in many cases, will not be uniquely determined, because it is highly possible that the other sets of parameter values will also show a similar timecourse. Therefore, even if one set of parameter values that matches the observed timecourse is obtained, this set is still only one of the best candidates that explains the observed timecourse. Our strategy is to explore and exploit these candidates within the immense searching space of parameter values. In this optimization problem, each set of parameter values to be estimated is evaluated using the following procedure. Suppose that X i;cal;t is the numerically calculated timecourse at time t of state variable X i and that X i;exp;t represents the experimentally observed timecourse at time t of X i. Sum the relative error between X i;cal;t and X i;exp;t to get the total error f (2) where N is the number of experimentally observable state variables and T is the number of sampling points of the experimental data. The problem is to find a set of parameters that minimizes f. The proposed method is based on the simple GA, and the structure of the genome (design code) of each individual (each set of parameter values) is shown in Figure. Figure : Design code of an individual. Two n vectors of i and ß i, and two n * n matrices of n * n g i j form an n *

3 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 3 of 0 (2n + 2) matrix. This matrix represents one Ssystem model. A genome (corresponds to one individual) contains a set of Ssystem parameters ( n i s and ß i s, and n*n g i j s s) which form an n*(2n+2) matrix. An individual represents one Ssystem model. Each small square in Figure corresponds to each parameter that has a real value. We introduced the following coding for representing real numbers: a 32bit unsigned integer format within a given searching region, that is, each dimensional region to be searched is divided into 2 32 discrete points and is numbered using an unsigned integer. A real value within a searching region is represented by scaling a unsigned integer using an offset. Genetic Algorithm The optimization procedure in GA is as follows: (0) Prepare a set of experimentally observed timecourse data of n state variables. The number of sampling points (T) is common for each state variable. Determine the number of individuals P, maximum limit of generation G max, search regions for i (ß i ) and g i j (h i j ), initial mutation rate m 0 and the threshold of structure skeletalizing S for g i j. () P initial guesses (P sets of i, ß i, g i j ) are randomly created. Each matrix element of an individual (Figure ) has a real number and is randomly set within a given searching region. (2) Evaluate each individual. Solve a simultaneous differential equation (Equation ) for each individual, and calculate total error f using Equation 2. The fitness of each individual is calculated as the reciprocal of total error f. Within the group of P individuals in every generation, select the individual having the largest fitness (elite individual) and check whether the total error f of this individual is less than a given threshold value of convergence. If the largest fitness converges or the number of generations is beyond the specified G max, or if the fitness of the elite individual does not change during G n (< G max ) generations, exit the loop and terminate. The elite individual is reported as the best matched model. (3) Otherwise, continue searching by introducing genetic operations (crossover and muta tion), as shown in Figure 2. Only one elite individual is selected from among the present and past generations as an elite and the design code of this elite individual is kept as it is (is not submitted to the genetic operations) and is succeeded to the next generation. Figure 2: Crossover, mutation and structure skeletalizing. In order to create a child individual for

4 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page of 0 the next generation, select two individuals from the individual group according to the probabilities proportional to their fitness. Genes of a child are created from the parents' genes by crossover. For each gene (Ssystem parameter), () Pick the gene from either parent. (2)(3) Determine which gene will be written into the child, by using a uniform random number. () Mutation is applied to the gene with a certain probability (mutation rate), using a normal distributed random number. (5) If the value of the gene is less than a given threshold, the value is reset to 0 (in every specified generation) (structure skeletalizing). () With the exception of the elite individual, the design code of each child is created based on the design codes of two parents. Two parents are selected from the P individu als according to the probability proportional to their order of fitness (ranking or roulette strategy) [ Davis, 99]. The individual having the larger fitness will be selected as the parent having the higher probability. For each individual, the possibility to be selected is set between 0.9 and. according to its fitness. (5) From the two selected parents, each matrix element of the child is alternatively chosen from the corresponding elements of either parent at the same probalibity. Since the number of matrix elements is 2n(n + ), where n is the number of state variables, this operation corresponds to the 2n(n + ) - point crossover.

5 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 5 of 0 (6) Furthermore, miss-writing (mutation) is supposed to occur at mutation rate m when the matrix elements of the parents are copied to the child. The mutated value is calculated using the normal distriuted random number whose average and distribution are defined by the original value and d, respectively. Set d to d 0 at the initial generation, and change the value according to the following conditions: (i) When the fitness of the elite individual does not change during G m generations, change d to d (< d 0 ). (ii)after the change, if the fitness of the elite individual does not increase during next G d generations, change d to qd 0 (q > ). (iii)if the fitness is not improved during next G d generations, the value of distribution d is returned to d 0. (iv)when the fitness of the elite individual increases due to the (i)(iii) strategies, the distribution d is reset to d 0. The mutation rate m is changed to k times (k > ) when the fitness of the elite individual does not go up during G m generations. (7) By repeating procedures()(6), (P - ) children are created for the next generation. One child is the elite individual whose design code is not submitted to the genetic oprations. (8) Structure skeletalizing (the details are presented in the following section). (9) Return to step (2). Structure skeletalizing In Ssystem formalism, the number of estimated parameters increases with the order of n 2, which leads to a considerably large computational cost and ambiguous capture of interactive mechanisms among state variables (X i ). In order to capture optimized essen tial interactions among X i, the following structure skeletalizing procedure is introduced: )The term g i j (h i j ), the absolute value of which is less than a given threshold value (skeletalizing value) is reset to zero. 2)This procedure is performed at every specified generation and at the termination of optimization. APPLICATIONS In order to examine the effectiveness of the proposed procedures, we have applied our method to determine a set of parameters of the Ssystem. Oscillatory system Figure 3(A) shows the calculated timecourses of X and X 2 of the Ssystem (N = 2 in Equation ), the parameter values of which are shown in Table. Given these timecourses of X and X 2 (Figure 3(A) is presented as experimental data (X i;exp;t, N = 2, T = 50 in Equation 2)), we examined whether the proposed optimization procedures could explore and exploit a set of parameter values which can provide the best fitted timecourse shown in Figure 3(A). The major optimization conditions are summarized in Table 2 and other parameters are tuned as follows: G n = G max /2, G d = 0, q = 5, k = :0. Table : Given Ssystem parameters which provide the dynamics shown in Figure 3(A).

6 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 6 of 0 Figure 3: Given timecourses and obtained timecourses of X and X 2. (A), Given timecourses (parameter values are listed in Table ); (B), obtained timecourses in trial 2 (a set of parameters is listed in Table 3). Initial values of X andx 2 are.0 and.5, respectively. In (A), the number of sampling points of experimental data is 50 (T = 50 in Equation (2)). Table 2: Optimization conditions and obtained results of optimization. The time for optimiza tion was measured using a TITAN300MP (KGT Inc., Japan, 8.0 SPECfp95).

7 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 7 of 0 Figure 3(B) shows the obtained timecourses. The obtained parameter values are listed in Table 3. The obtained timecourses shown in Figure 3(B) are quite similar to those shown as in Figure 3(A). As shown in Table 3, the average relative error between calculated and given time courses per sampling point is 5.7%. By changing randomly initial guess values for opti mization, we obtained the result E=(9:32 + 0:)% after 00 trials. Table 3: Obtained Ssystem parameters which provide the dynamics shown in Figure 3(B). Gene network Next, the proposed method was applied to determine a set of parameters of the S system which represents a typical model of a gene network. Table : Given Ssystem parameters in Figure 5. Pool size (constant value, pool to pool 5 in Figure 5) corresponds to the values of to 5.

8 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 8 of 0 As a case study, we created several sets of time series data, shown in Figure, which were numerically calculated using the scheme shown in Figure 5 ( Savageau, 998). The Ssystem parameters in Figure 5 are shown in Table. The time series data in Figure (A) were calculated under the condition = 5:0, = 8:0 in Table, which corresponds to the situation in which both gene producing X (mrna) and gene producing X (mrna) are active (wildtype). In contrast, the time series data in Figures (B) and (C) were obtained under the condition = 0, = 8:0 and = 5:0, = 0, respectively. Figure (B) shows the case of the disruption of gene (corresponds to = 0), and (C) shows the case of the disruption of gene (corresponds to = 0). The optimization task is as follows: Can the proposed algorithm explore and exploit the best Ssystem parameters matching the observed time courses shown in Figure (A)(C)? Figure : Time series data calculated from the Ssystem shown in Table. (A) wild type ( = 5:0, = 8:0); (B) disruption of gene ( = 0:0, = 8:0); (C) disruption of gene ( = 5:0, = 0:0). These time course data contain 5 (sampling points) * 5 (components) points. Figure 5: Typical gene network. We attemped to estimate part of the system parameters in Table. The targets of optimization were twelve parameters:, g ;... ; g 5,, g ;... ; g 5. These parameters represent the interaction coefficients that give increasing effects of X and X (expression and regulation of mrna) respectively. The major condition of the optimization is shown in Table 5, and the other parameters were set to as follows: G n = G max /2, G d = 0, q = 5, and k = :0. Table 5: Optimization conditions and obtained results of optimization in the gene network. The time for

9 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for Inver... Page 9 of 0 optimization was measured using a TITAN 300MP (KGT Inc., Japan, 8.0 SPECfp95). At the 266th generation (CPUtime is about hours), we found the parameter set (, g, g 2, g 3, g, g 5,, g, g, g, g, g 2 3 5,) to be completely identical to that shown in Table. In an early step of total optimization (at the 78th generation), a model which reflects the same structure as the given model was obtained, as shown in Table 6. The differences in parameter values between Table and Table 6 are not remarkable, indicating that the structure of the system was obtained by global searching in early stages of the optimization, and that local searching (fine tuning of the parameters) was performed after the 78th generation. Table 6: Obtained Ssystem parameters at the 78th generation in the gene network. The parameters, g, g 2, g 3, g, g 5,, g, g 2, g 3, g, g 5, were optimized. DISCUSSION In the applications, the proposed algorithm found the parameter set that was best matched to the given time series data. In Section 3., since the search range of g i j (i, j =, 2) is [-.0,.0] with an increment of 0.0, the number of possible searching points for g and h is [{.0 - ( -.0 )} * 00] 8 = 2 2 *0 6. Similarly, since the search range of i and ß i (i =, 2) is [0, 20] with an increment of 0.0, the number of possible searching points for and ß is (20*00) = 2 * 0 2. Therefore, this searching problem is to find the best fitted parameters among (2 2 * 6 6 )*(2 * 0 2 ) = 2 28 * 0 28 ~ 2.6 * 0 36 possibilities. As shown in Table 2, since the total generations for exploring and the number of individuals were 07 and 000, respectively, the proposed method found the best fitted parameters after evaluating only 07 * 000 ~. * 0 5 possibilities. Therefore, if the searching efficiency is defined by the ratio of the number of all possibilities to the number of possibilities searched, the searching efficiency can be estimated to be

10 Nonlinear Numerical Optimization Technique Based on a Genetic Algorithm for In... Page 0 of 0 which is considerable high. In the gene network, the final optimized structure was found at the 266th generation and its essential structure was found at the 78th generation. The fitness value of the best model (obtained at the 266th generation) is The relative error of the calculated time series data to the given data (Figure (A) (C)) per sampling point is about.0 * 0 5. The fitness and relative error at the 78th generation are 3.86 and 3.5 * 0 -, respectively. The fitness was remarkably improved during the 78th and the 226th generations; however, the relative eeroor per sampling point decreased slightly (from.0 * 0-5 to 3.5 * 0 - ). For practical use, the experimentally observed data generally include a + 0% measurement error, which indicates that the parameters obtained at the 78th generation are condidered to be the best fittest parameters. In Figure (A) to Figure (C), several experimental conditions of the gene network, such as wildtype and disruption of gene, were varied. Estimation of parameters in the Ssystem using experimentally observed timecourses is generally referred to as an inverse problem, and these timecourses correspond to the restricted conditions for an inverse problem. Since the proposed algorithm proposes candidates (parameter sets) matching the restricted conditions, the best candidate can most likely be found by preparing more timecourse data under the disruption and overexpression conditions in the gene network. ACKNOWLEDGEMENT This study was performed as a part of a research and development project of the Indus trial Science and Technology Program, which is supported by NEDO (New Energy and Industrial Technology Development Organization) (SW, YM, YE). REFERENCES Baker, J.E. (985). Adaptive selection methods for genetic algorithms, Proceedings of the International Conference on Genetic Algorithms, 0, Lawrene Erlbaum Associates. Davis, L. (99). Handbook of genetic algorithms, Van Nostrand Reinhold. Goldberg, D.E. (989). Genetic Algorithms in Search, Optimization and Machine Learn ing, Addison Wesley. Irvine, D.H, Savageau, M.A. (990). Efficient solution of nonlinear ordinary differential equations expressed in Ssystem canonical form, SIAM J. NUMER. ANAL. 27, Okamoto, M., Morita, Y., Tominaga, D., Tanaka, K., Kinoshita, N., Ueno, JI., Miura, Y. (997). Toward a VirtualLaboSystem for Metabolic Engineering: Development of Biochemical Engineering System Analyzing ToolKit (BESTKIT), Proceedings of Pacific Symposium on Biocomputing '97, Savageau, M.A. (998) Rules for the evolution of gene circuitry, Proceedings of the Pacific Symposium on Biocomputing '98, 565, World Scientific. Savageau, M.A. (976). Biochemical system analysis: a study of function and design in molecular biology, AddisonWesley. Tominaga, D., Okamoto, M. (998). Design of Canonical Model Describing Complex Nonlinear Dynamics, Proceedings of Computer Applications in Biotechnology 998, 8590, Elsevier Science. Tominaga, D., Okamoto, M. (999). Nonlinear Numerical Optimization Technique Based on Genetic Algorithm for Inverse Problem, Kagaku Kougaku Ronbunshu, 25(2), (in Japanese). Tominaga, D., Ueno, J., Miura, Y., Okamoto, M. (996). Discovery of a Skeletal Net work Describing Complex Nonlinear Dynamics: Optimized Essential Model for Temporal InputOutput Matching, In: Tutorials of the th Intl. Conf. on Soft Computing (IIZUKA 96), 2, 6580, Fuzzy Logic Systems Institute. Voit, E.O. (99). Canonical nonlinear modeling: Ssystem approach to understanding complexity, Van Nostrand Reinhold.