Bioinformatics Advance Access originally published online on March 5, 2008
Bioinformatics 2008 24(8):1085-1092; doi:10.1093/bioinformatics/btn075
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Inference of biochemical network models in S-system using multiobjective optimization approach
Department of Chemical Engineering, National Chung Cheng University, Chiayi 621-02, Taiwan, ROC
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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Motivation: The inference of biochemical networks, such as gene regulatory networks, protein–protein interaction networks, and metabolic pathway networks, from time-course data is one of the main challenges in systems biology. The ultimate goal of inferred modeling is to obtain expressions that quantitatively understand every detail and principle of biological systems. To infer a realizable S-system structure, most articles have applied sums of magnitude of kinetic orders as a penalty term in the fitness evaluation. How to tune a penalty weight to yield a realizable model structure is the main issue for the inverse problem. No guideline has been published for tuning a suitable penalty weight to infer a suitable model structure of biochemical networks.
Results: We introduce an interactive inference algorithm to infer a realizable S-system structure for biochemical networks. The inference problem is formulated as a multiobjective optimization problem to minimize simultaneously the concentration error, slope error and interaction measure in order to find a suitable S-system model structure and its corresponding model parameters. The multiobjective optimization problem is solved by the 
-constraint method to minimize the interaction measure subject to the expectation constraints for the concentration and slope error criteria. The theorems serve to guarantee the minimum solution for the -constrained problem to achieve the minimum interaction network for the inference problem. The approach could avoid assigning a penalty weight for sums of magnitude of kinetic orders.
Contact: chmfsw{at}ccu.edu.tw
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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The rapid development of systems biology over the past few years has been driven by the advances in experimental methods that generate in vivo time-course data characterizing biochemical network interactions. In recent years, researchers intend to use such data for inferring a model structure and its parameters in order to examine intracellular dynamic behaviors on a systemic level. The ultimate goal of inferred modeling is to obtain expressions that quantitatively understand every detail and principle of biological systems (Chang et al., 2005). How to select a suitable model structure and to estimate the parameter values involved is the main issue for mathematical modeling (Maki et al., 2001; Mendes and Kell, 1998; Tsai and Wang, 2005).
Given a model structure, parameter estimation remains the limiting step in the modeling of biological systems. There exists, however, no unique method for estimating model parameters for nonlinear dynamic models. Most of the traditional nonlinear regression algorithms involving gradient methods have the possibility of getting trapped at local optima, depending upon the degree of system nonlinearity and the initial starting point (Mendes and Kell, 1998). Alternating regression (Chou et al., 2006) dissects the nonlinear inverse problem of estimating parameter values into iterative steps of linear regression. The branch and bound algorithm (Polisetty et al., 2006) is employed to convert the inverse problem of generalized mass action (GMA) or S-system into a convex optimization problem in order to obtain a global solution. Stochastic optimization methods, such as genetic algorithms, evolution strategy and simulated annealing (Edwards et al., 1998; Gonzalez et al., 2007; Moles et al., 2003), are applied for parameter estimation in order to find a global solution. Many techniques have been employed to alleviate numerical integration burden. Voit and Almeida (2004) utilized a decoupling scheme to estimate the slopes of the dynamic processes. Tsai and Wang (2005) used the modified collocation method to approximate dynamic profiles at sampling points. The decomposing method (Kimura et al., 2005; Maki et al., 2002) is employed to convert the large network inference problem into subproblems. Such approximation techniques can be easily incorporated into an optimization method to avoid the computationally expensive numerical integrations for fitness evaluations.
To infer a realizable S-system structure, most articles have applied sums of magnitude of kinetic orders as a penalty term in the fitness evaluation (Ho et al., 2005; Kikuchi et al., 2003; Kimura et al., 2004, 2005; Noman and Iba, 2005). A weighting factor in the penalty term needs to be carefully tuned in order to infer a realizable S-system model structure. The weighting factor in general depends on the problem of interest. An improper weighting factor should make to yield a wrong structure. According to our knowledge, no guideline has been published for tuning a suitable penalty weight to infer model structures of biochemical networks. In this study, we introduce the multiple-objective optimization approach to inferring a realizable S-system structure for biochemical networks. Such an approach can avoid assigning a weighting factor for sums of magnitude of kinetic orders. One dry-lab and one wet-lab case studies are made to illustrate the performance of the proposed approach.
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2 METHODS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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A biochemical network system can be modeled as a set of S-system canonical forms (Voit, 2000):
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| (1) |
j and βj, and kinetic orders, gij and hij. fi is the net rate equation, which consists of both influx and efflux. The m-dimensional independent variables in the S-system equations are expressed as Xn+j, j = 1, ... , m. The parameter estimation is to determine model parameters, rate constants and kinetic orders, so that the dynamic profiles fit satisfactorily the measured observation.
2.1 Estimation criteria
The canonical biochemical network inference problem is formulated as a function optimization problem to minimize an objective function that measures the goodness-of-fit of the model with respect to a given experimental time-course dataset. The least-squared error criterion is a commonly used objective function and is expressed as
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| (2) |
denotes the measured data for the ith component at the sampling time tj, Xi(tj) is the computed concentration for the ith component at the sampling time tj, and 
is the maximum measured concentration of the ith component. Here, Ns denotes the number of sampled data points. The dynamic profiles Xi(tj) are in general obtained by applying a numerical integration method to solve the differential Equation (1). Numerical integration for parameter estimation is time consuming and may cause a run-time error during the computation progress. Wang (2000) has used the modified collocation method to convert ordinary differential equations into algebraic equations. The piecewise linear Lagrange polynomial is the simplest shape polynomial for obtaining the approximate dynamic profiles. The approximate equations are expressed as:
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| (3) |
Tsai and Wang (2005) have employed the algebraic equations to generate approximate profiles for parameter estimation in order to avoid solving the equations recursively. Such an approximation not only reduces computation time, but also converts the coupled algebraic equations into a set of uncoupled equations so that parallel computation can be straightforwardly applied for the parameter estimation.
The least-squared error criterion (2) is to directly employ concentration profiles of the system for evaluating fitness of the estimation. This error criterion refers to the concentration error in this study. An alternative error criterion is to use the slope information for evaluating fitness of the estimation (Voit and Almeida, 2004). The slope error criterion is therefore expressed as:
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| (4) |
is the approximate experimental slope for the ith component at the sampling time tj, 
is the computed slope for the ith component at tj, and 
is the maximum slope of the ith component. Using the model slope to compute the error criterion can avoid the numerical integration of differential equations so it alleviates the computational burden. However, a smoothing filter, such as artificial neural network or spline smoothing, has to be utilized to smooth the measured data in order to generate the approximate experimental slopes for each variable (Almeida and Voit, 2003).
Inference of regulatory interactions in a biochemical system provides fundamental biological knowledge and significant efforts. Several network inference algorithms estimate all of the S-system parameters from time-course data. The estimation for a large-scale S-system often causes bottlenecks, and fitting the model to experimentally observed data is not simple. The decoupling approach, such as modified collocation method (Tsai and Wang, 2005), slope approximation (Voit and Almeida, 2004) and decomposition method (Kimura et al., 2004), enables us to infer S-system models of genetic networks of a larger scale. To detect a suitable model structure for a large-scale S-system, the sum of magnitude of kinetic orders can be employed as a criterion to pruning a skeletal structure, and is expressed as (Kikuchi et al., 2003; Kimura et al., 2004; Voit and Almeida, 2003):
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| (5) |
2.2 Multiobjective optimization approach
The aim of this study is to minimize simultaneously the concentration error, slope error and interaction measure in order to find a suitable S-system model structure and its corresponding model parameters. The multiobjective parameter estimation problem is therefore expressed as:
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| (6) |
is a set of all admissible model parameters p that satisfy the corresponding S-system model Equation (1). Multiobjective optimization is a natural extension of the traditional optimization of a single-objective function. Typically, the objectives are incommensurable and often (partially or wholly) in conflict (Handl et al., 2007). The incommensurability between multiobjective functions gives rise to the distinguishing difference between multiobjective optimization and traditional single-objective optimization. This fact leads to the lack of a complete order for multiobjective optimization problems. Concept of Pareto optimality or noninferiority is therefore employed to characterize a solution to multiobjective optimization problems. The definition of Pareto optimal solution is introduced as follows:
DEFINITION. A vector of the model parameters, p*, is the Pareto optimal point if and only if there does not exist p 
such that
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The literature on multiobjective optimization is abundant (Sakawa, 1993), and we cannot hope to mention all the techniques that have been employed to generate a Pareto solution; however, one method is pervasive in multiobjective optimization literature. This technique is the weighted sum method for converting a multiobjective optimization problem such as (6) into a single-objective function problem. Such an approach is equivalent to introducing a penalty term to join with the concentration error criterion or slope error criterion as discussed in the literature (Kikuchi et al., 2003; Kimura et al., 2004; Tsai and Wang, 2005; Voit and Almeida, 2004). The penalty problem is therefore expressed as:
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| (7) |
. The weighting factor needs to be carefully tuned in order to infer a realizable S-system model structure. No guideline has been published for tuning a suitable penalty weight to infer model structures of regulatory networks. In this study, we introduce the -constraint method to overcome such a drawback.
The -constraint method for characterizing the Pareto optimal estimates is to solve the following constraint problem formulated by taking one criterion as the objective function and letting all other criteria be inequality constraints (Sakawa, 1993). The first goal of this study is to find a suitable S-system structure so the constraint problem is formulated as:
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| (8) |
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| (9) |
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| (10) |
are the expected values for the concentration criterion, slope criterion and interaction measure, respectively. In the following section, we will introduce an interactive computational algorithm to rationally provide these expected values. The relationships between the optimal solution p* to the constraint problem and the Pareto optimal concept of the primal multiobjective parameter estimation problem (6) can be characterized by the following theorems.
THEOREM 1. If p* is a unique optimal solution of the constraint problem for some 
and 
, then p* is the Pareto optimal solution to the multiobjective parameter estimation problem (6).
THEOREM 2. If p* is a Pareto optimal solution of the multiobjective parameter estimation problem (6), then p* is the optimal solution of the constraint problem for some 
and 
.
Both theorems can be immediately proved from the definition of the Pareto optimality by making use of contradictory arguments following the similar procedures discussed in the textbook (Sakawa, 1993), and are expressed in the Supplementary 1. This fact indicates that a Pareto optimal estimate for the multiobjective parameter estimation problem (6) can be obtained by solving the converted constraint problems (8)–(10) using a global optimization method. Several constrained optimization methods can be employed to solve the converted constraint problem. In this study, the popular penalty-function method is introduced to solve the constraint problem. The inference problem is therefore expressed as:
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| (11) |
. The second term in (11) indicates that a penalty is desired only if the point p is not feasible. If any or both C1 and C2 are positive, the worst value is employed to compute the penalty. If both inequality constraints are feasible, i.e. 
, then the penalty is zero, i.e. 
. This situation indicates that no penalty is incurred. From this result, it is clear that we can get arbitrarily close to the optimal interaction measure value of the constraint problem (8)–(10) by computing the inference problem (11) for a sufficiently large 
. To search easily a feasible point, the penalty parameter can be provided to be greater than the inverse for the minimum of the expected values, i.e. 
. Theorem 3 that serves to guarantee the minimum solution for the inference problem is also the optimal estimate for the constraint problems (8)–(10).
THEOREM 3. If 
and 
for some 
then p is a minimum solution to the constraint problems (8)–(10).
Theorem 3 can be immediately proved following the similar procedures discussed in the textbook (Bazaraa and Shetty, 1979), and are expressed in the Supplementary 1. The aim of the theorem is to determine a feasible point p to the inference problem (11), i.e. both concentration and slope error criteria are less than their expected values, such that the interaction measure is minimized. Using this theorem, we introduce an interactive algorithm as shown in Table 1 for inferring biochemical regulatory networks. In Steps 1 and 3 of Table 1, we used the hybrid differential evolution (HDE) to minimize each corresponding objective function toward obtaining a global optimal solution. HDE enables a smaller population to be used for finding a global solution (Chiou and Wang, 1999) and has succeeded in solving several biochemical optimization problems (Wang and Sheu, 2000).
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Equations (9) and (10) are the inequality constraints for the constraint problems (8)–(10). All parameters those satisfy both inequality constraints make up a feasible set, i.e.
. The expected values, 
and 
, are individually obtained from solving the corresponding single objective parameter estimation problems (2) and (4). Therefore, the aim of the constraint problem (8)–(10) is to determine the optimal parameters within the feasible set such that the interaction measure is minimized. In other words, given a super-structure, each single objective parameter estimation problem is first solved in order to yield its expected value. The multiobjective parameter estimation is then applied to find a compromised solution. The solution must be less than the expected value, and makes the state variables of the super-structure among the smallest impact. Many constrained optimization methods can be employed to solve the constraint problems (8)–(10). In this study, the problem is converted to the inference problem (11) through a penalty-function method. HDE is then applied to solve the inference problem. A large punishment should be added to the inference problem (11) in order to move the searching parameters into the feasible set, when the searching parameters violate the constraints (9) and (10) in the solution process. When the searching parameters satisfy the constraints (9) and (10), no penalty is incurred. The inference problem can continuously minimize the interaction measure toward obtaining a suitable structure. |
3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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In this article, we show two case studies, an artificial genetic network and a wet-lab system, for inferring a suitable interaction network. The detail of the computational results and three additional case studies, a 30 gene network (Kimura et al., 2004), a circadian oscillations of period protein in Drosophila (Ingalls, 2004) and an embryonic gene regulatory network in zebrafish (Huang et al., 2006), are also provided in the Supplementary 2 to illustrate the effectiveness of the proposed algorithm. All computations were carried out on a Pentium IV computer using Microsoft Windows XP. The interactive inference algorithm is implemented in Compaq Visual Fortran. HDE serves as a minimization solver in the interactive algorithm, and has to provide four setting factors by the user. The setting factors used for all runs in the case studies are listed as follows: The crossover factor is set to be 0.5. Two tolerances used in the migration are set to be 0.05. The population size of 5 is used in the computation.
3.1 Case I: small-scale gene network
The dry-lab case study is a two-gene regulatory network shown in Hlavacek and Savageau (1996). The ‘true’ system is described in the S-system equations as follows:
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| (12) |
We first consider noise-free time-course data for evaluating the penalty problem (7) and the constraint problems (8)–(10) for comparison. The eight sets of training data generated by Tsai and Wang (2005) were employed to infer an S-system model structure. In this example, we set the kinetic orders gii = 0, which precluded the direct effect of a variable on its own production and required the kinetic orders hii to be greater than zero, indicating that the degradation of compounds almost depends always on the concentration. The search ranges used for the regression were i and βi [0, 20], gij and hij [–4, 4], i 
j and hii [0, 4]. The HDE algorithm was employed to solve the penalty problem (7) with the weighting factor of 10–3, 10–4 and 10–6, respectively. The computational results were shown in Supplementary 2. For the cases using the weighting factor of 10–3 and 10–6, we cannot infer a convergent structure from the penalty problem (7) using several trials for HDE. For the weighting factor of 10–4, the inferred structure is identical to the ‘true system’ after four iterations. We next applied the proposed interactive inference algorithm, as shown in Table 1, except using a different penalty parameter in Step 3, to solve the inference problem (11). In this computation, we used, respectively, the penalty parameters of 1, 102 and 104 in Step 3 to solve each inference problem (11). The proposed algorithm enabled us to infer the identical S-system structure to the ‘true’ system although the assigned penalty parameters were widely different. The computation time for each case was about the same. 38.8 min and two iterations required on a single-CPU Pentium IV 3.0 GHz. The proposed algorithm requires one-fifth CPU times that of the result solved by Tsai and Wang (2005) because it uses decoupling computation to solve each subsystem. Table 2 summarizes the comparison between the proposed algorithm and the reported methods for this inference problem (Kikuchi et al., 2003; Kimura et al., 2004; Tsai and Wang, 2005).
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Next, we test the performance of the proposed method in a real-world situation by conducting the experiment with the sets of noisy time-course data. To imitate real profiles, 10% random noises are added into the eight sets of ‘true’ time-course data. The rational method in the curve-fitting toolbox for MATLAB is then employed to smooth the measured data in order to yield the noise-free time-course profiles for evaluating concentration error criterion and slope error criterion. In this work, the proposed interactive inference algorithm not only can infer the S-system model structure for dependent variables, as discussed in the previous run, but can also infer interaction relations between dependent and independent variables. The super-structure for the S-system is therefore expressed as:
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| (13) |
In the first run, the HDE algorithm was employed to solve each single-objective parameter estimation problem. The expected values for the first subsystem were 
and 
, respectively. These expected values were then provided for the inference problem (11) in Step 2 of the interactive inference algorithm in Table 1. The optimal estimates for each subsystem, as shown in the Supplementary 2, were feasible. The optimal concentration error and slope error were 
and 
, respectively, so the inequality constraints in (9) and (10) were less than zero, i.e. 
. Many kinetic orders gij and hij were very small. We then deleted those smaller kinetic orders with scores <0.01. The interactive inference algorithm was then repeated to refit each pruned subsystem. For the second iteration, the inferred regulatory structure for independent variables approached to the ‘true’ system. One iteration is enough to achieve the minimum connective network for the first subsystem, three iterations for the second, fourth and fifth subsystem. For the third subsystem, after the fifth iteration, the concentration error criterion and slope error criterion were 5.326E–3 and 1.449E–2, respectively, both of which were almost equal to their expected values. Both inequality constraints for the third subsystem were less than zero. The score of the kinetic order g35 was smaller than 1.0E–2, after deleting this parameter; the inferred S-system structure was essentially identical to the ‘true’ system. The estimated parameter values were employed to evaluate an extra test-experiment in order to validate the model. The initial condition for the test-experiment is beyond the training dataset. The ‘true’ dynamic profiles are shown as the data-points in Figure 1. The model profiles shown as solid curves are capable of predicting the dynamic responses under the condition. In order to yield more accurate estimates, the solution obtained by the proposed algorithm is employed as the initial starting point for a gradient-based method, a subroutine BCONF in IMSL Math/Library, to solve the parameter estimation problem. The local search procedure employs Runge–Kutta pairs of various orders, a subroutine IVMRK in IMSL Math/Library, to solve differential equations towards obtaining time-course profiles of the system. The refined estimates are then employed to evaluate the extra test-experiment to validate the model. The model profiles shown as dashed curves in Figure 1 can satisfactorily fit the test-experiments.
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So far, the inference problem is to minimize simultaneously the concentration error, slope error and interaction measure in order to find the S-system model structure and its corresponding model parameters. The problem can be solved by minimizing two objective functions. That is to minimize simultaneously the concentration error and interaction measure or the slope error and interaction measure, respectively. The bi-objective minimization problem is therefore expressed as min{J1, J3} or min{J2, J3}. The interactive inference algorithm is then employed to solve both problems, respectively. For noise-free time-course data, both bi-objective minimization problems can achieve the exact model structure.
However, for noisy time-course data, we cannot obtain the exact structure for minimizing {J1, J3} after four iterations and {J2, J3} after five iterations, respectively. The optimal estimates for each subsystem are shown in the Supplementary 2. The optimal estimates from min{J1, J3} and min{J2, J3} are then employed to evaluate the extra test-experiment to validate the model, respectively. Although both structures are different from the ‘true’ system, the model profiles shown as dashed-dot curves (min{J1, J3}) and dashed-dot-dot curves (min{J2, J3}) in Figure 1 can suitably fit the test-experiments.
3.2 Case studies II: kinetics model of ethanol fermentation
In this wet-lab case study, we analyze a batch fermentation process discussed by Wang et al. (2001). The fermentation process uses high-ethanol tolerance yeast, Saccharomyces diastaticus LORRE 316, to produce ethanol. The experimental materials and methods were illustrated in Wang et al. (2001). The yeast can utilize glucose to produce ethanol and glycerol so the super-structure of S-system is described as
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| (14) |
The time-course data with 1 h sampling were obtained from two batch fermentations using the initial glucose concentrations of 100 and 150 g/l. Each fermentation was carried out in two experiments giving data points shown in Figure 2a and b. These repeated time-course data were noisy so the curve-fitting toolbox in MATLAB software was first employed to smooth the observed data for evaluating concentration error criterion and slope error criterion. By following similar procedures as discussed in the previous dry-lab case study, the inference problem is decoupled into four subsystems. The search ranges for each parameter used in the interactive inference algorithm were i and βi [0, 5], and gij and hij [–3, 3].
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Table 3 shows the expected values for each iteration. The Pareto optimal error criteria for the concentration and slope are also listed in Table 3. Two iterations are enough to achieve the minimum connective network for the first, third and fourth subsystem, three iterations for the second subsystem. The rate constant of degradation for glycerol was almost zero so the degradation term was neglected. The inferred model and its parameters were then provided as the initial starting point for a gradient-based method, a subroutine BCONF in IMSL Math/Library, to yield the more accurate solution.
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The optimal model is obtained as follows:
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| (15) |
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4 CONCLUSION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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To infer a suitable interaction network for biological systems from time-course data poses many challenges. Numerical integration for differential equations and finding global parameter values are two major problems. Modified collocation and slope approximation can be employed to alleviate the computation burden. Hybrid differential evolution is utilized to obtain a global estimate. However, when inferring a minimum interaction network, sums of magnitude of kinetic orders serve as the penalty term to evaluate the fitness for the inference problem. How to tune a penalty weight to yield a realizable model structure is also a challenging problem. No guideline has been published for tuning a suitable penalty weight to infer a suitable model structure of biochemical networks. The multiobjective optimization approach could avoid assigning a penalty weight for sums of magnitude of kinetic orders. We have proved that the approach could guarantee the minimum solution for the constrained problem to achieve the minimum interaction network for the inference problem.
A multiobjective optimization can consider many goals at the same time. This study is to investigate the concentration error criterion, the slope error criterion and interaction measure. The concentration error criterion is employed to measure the goodness-of-fit of the model with respect to a given experimental time-course dataset. The slope error criterion is used to judge the accuracy of the dynamic function, i.e. the net rate equation in (1). Each kinetic order, gij or hij, is applied to quantify the effect of Xj variable on the production or degradation of Xi. A smaller parameter value means less interaction between state variables, Xj and Xi. The interaction measure sums up magnitude of kinetic orders which serves as an index to prune a skeletal structure of S-systems. Such a pruning strategy may be not suited for inferring whether genetic interactions are fragile or robust. Since the fragile interaction has higher sensitivity, a slight change in parameter value of this interaction, gij or hij, should cause a big difference of dynamic behaviors of gene expression. Under such circumstances, the interaction measure may be unable to infer such high-sensitive systems. An additional goal, such as dynamic sensitivities of state variables with respect to gij and hij, should be considered in the multiobjective parameter estimation problem toward inferring a suitable network structure for a high-sensitive gene-regulatory system.
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ACKNOWLEDGEMENT |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 CONCLUSIONACKNOWLEDGEMENTREFERENCES |
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The financial support from the National Science Council, Taiwan, ROC (Grant NSC96-2627-B-194-001), is highly appreciated.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: John Quackenbush
Received on November 20, 2007; revised on February 5, 2008; accepted on February 22, 2008
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