Fuzzy Dominance Based Multi-objective GA-Simplex Hybrid Algorithms Applied to Gene Network Models

Hybrid algorithms that combine genetic algorithms with the Nelder-Mead simplex algorithm have been effective in solving certain optimization problems. In this article, we apply a similar technique to estimate the parameters of a gene regulatory network for flowering time control in rice. The algorithm minimizes the difference between the model behavior and real world data. Because of the nature of the data, a multiobjective approach is necessary. The concept of fuzzy dominance is introduced, and a multi-objective simplex algorithm based on this concept is proposed as a part of the hybrid approach. Results suggest that the proposed method performs well in estimating the model parameters. 1 Gene Regulatory Network Models Molecular geneticists are rapidly deciphering the genomes of an increasing number of organisms. As of November 2003, 166 organisms had completely sequenced genomes with another 775 in progress [1]. The current challenge is to understand how the genes in each organism interact with each other and the environment to determine the characteristics (i.e., the phenotype) of the organism. In the agricultural contexts familiar to the authors, this is called the “genotype to phenotype” or “GP” problem. In [2] it has been stated that this problem is the most significant issue confronting crop improvement efforts today. For over 40 years plant physiologists, systems engineers, and computer scientists have been employing “top-down” analysis methods to predict plant phenotypes based on varietal and environmental inputs [3, 4]. Recently, a “bottom” up approach has been applied [5, 6, 7] that models gene interactions directly at the expression level. Small groups of one to four interacting genes can synthesize a wide variety of signal processing functions including Boolean logic gates, linear arithmetic units, delays, differentiators, integrators, oscillators, coincidence detectors, and bi-stable devices [8]. This is consistent with the apparent small-scale modularity of gene networks [9]. Models of this type extrapolate phenotypes by explicitly tracking the status of key genetic developmental switches, accumulators, etc. Fuzzy Dominance Based Multi-objective GA-Simplex Hybrid Algorithms 357 Models of this type require efficient, multi-dimensional, multi-objective, derivative-free, global methods for parameter estimation. The problem is characterized by high dimensionality due to the large numbers of genes. Multiobjective optimization methods are appropriate because (1) multiple data types (continuous, discrete, and/or categorical) for both dependent and independent variables make the design of a single objective function problematic, (2) individual data sets come from different sources and may contain withinor between-set inconsistencies not apparent in the metadata, and (3) the models are incomplete and, therefore, may not be equally consistent with every data set. Because actual biophysical systems cannot harbor internal inconsistencies, the Pareto fronts associated with these problems are ideally single points. However, when data and/or model inconsistency exists, the size of the front is a useful measure of its magnitude. Finally, nonlinearities and data discontinuities can lead to exceptionally rough, multimodal response surfaces (e.g., [7]) that mandate global, derivative free methods. The following sections of this paper present a new algorithm that posses these features. The algorithm is described and then the algorithm is tested with the following single-gene model that demonstrates the features just described. In [25] the levels of messenger RNA was measured every 3h under short-days (SD, 9h) and longdays (LD, 15h) for HEADING DATE 1 (Hd1), an important flowering time control gene in rice (Oryza sativa). In [8] the authors modeled this data with the equation(s): L D NN L D Hd t C g R R Hd dt d λ λ { } ) 1 ( )) ( ( ) 1 ( − = (1, 2) where R’s and λ’s are constants and L and D denote light and dark periods. The clock input is C(t) = A*Sin(2π/p + θ) + μ, where A is amplitude, p is period, θ is phase angle, μ is a phase factor and ) exp( 1 1 c g NN − + = [5]. The state variable, Hd1, is dimensionless as expression levels are routinely normalized. The parameters have to be found such that model satisfies both SD and LD data with minimal MSE error with experimental data. So the approach of multi-objective optimization is used to find the possible solutions. Agronomic research on this point is underway. Thus, possible objective functions are the MSE between the model predicted SD and LD time series data with actual data obtained experimentally. 2 The Multi-objective Evolutionary Approach Evolutionary algorithms have emerged as one of the most popular approaches for the complex optimization problems [10]. They draw upon Darwinian paradigms of evolution to search through the solution space (the set of all possible solutions). Starting with a set (or population) of solutions, in each generation of the algorithm, new solutions are created from older ones by means of two operations, mutation and crossover. Mutation is accomplished by imparting a small, usually random perturbation to the solution. In a manner similar to the Darwinian paradigm of survival of the fittest, only the better solutions are allowed to remain in a population, the degree of optimality of the solution being assessed through a measure called fitness. 358 P. Koduru et al. When dealing with optimization problems with multiple objectives, the conventional concept of optimality does not hold good [11, 12, 13, 14]. Hence, the concepts of dominance and Pareto-optimality are applied. Without a loss of generality, if we assume that the optimization problem involves minimizing each objective (.), i e M i ... 1 = , a solution u is said to dominate over another solution v iff }, , , 2 , 1 { M i ... ∈ ∀ ) ( ) ( v e u e i i ≤ with at least one of the inequalities being strict, i.e. for each objective, u is better than or equal to v and better in at least one objective. This relationship is represented as v u ≺ . In a population of solution vectors, the set of all non-dominating solutions is called the Pareto front. In other words, if S is the population, the Pareto Front Γ is given by, { } ) ( , | u v S v S u ¬ ∈ ∀ ∈ = Γ (3) The simplistic approach of aggregating multiple objectives into a single one often fails to produce good results. It produces only a single solution. Multi-objective optimization on the other hand involves extracting the entire Pareto front from the solution space. In recent years, many evolutionary algorithms for multi-objective optimization have been proposed [14, 15, 16, 17]. We propose a hybrid algorithm that combines genetic algorithms (GAs), an evolutionary algorithm, with a well-known approach for function optimization known as the simplex algorithm [18]. While several GA-simplex algorithms have been proposed, our version is the only one that is equipped to carry out multi-objective optimization. This is accomplished by means of a concept, that we introduce, called fuzzy dominance.