Inference of Vohradský's Models of Genetic Networks by Solving Two-Dimensional Function Optimization Problems

The inference of a genetic network is a problem in which mutual interactions among genes are inferred from time-series of gene expression levels. While a number of models have been proposed to describe genetic networks, this study focuses on a mathematical model proposed by Vohradský. Because of its advantageous features, several researchers have proposed the inference methods based on Vohradský's model. When trying to analyze large-scale networks consisting of dozens of genes, however, these methods must solve high-dimensional non-linear function optimization problems. In order to resolve the difficulty of estimating the parameters of the Vohradský's model, this study proposes a new method that defines the problem as several two-dimensional function optimization problems. Through numerical experiments on artificial genetic network inference problems, we showed that, although the computation time of the proposed method is not the shortest, the method has the ability to estimate parameters of Vohradský's models more effectively with sufficiently short computation times. This study then applied the proposed method to an actual inference problem of the bacterial SOS DNA repair system, and succeeded in finding several reasonable regulations.


REX star /JGG
REX star /JGG [1] is a real-coded genetic algorithm, a sort of evolutionary algorithm, that uses JGG as a generation alternation model and REX star as a recombination operator. This section will describe each of the operators in greater detail.

JGG
JGG is a generation alternation model. The following is an algorithm of JGG. In the algorithm described below, a recombination operator uses m (≥ 2) parents to generate offsprings.

Initialization
As an initial population, create n p individuals. As REX star /JGG is a real-coded genetic algorithm, these individuals are represented as s-dimensional real number vectors, where s is the dimension of the search space. Set Generation = 0.

Selection for reproduction
Select m individuals without replacement from the population. The selected individuals, that are expressed here as p 1 , p 2 , · · · , p m , are used as the parents for the recombination operator in the next step.

Generation of offsprings
Generate n c children by applying the recombination operator to the parents selected in the previous step. This study uses REX star as the recombination operator.

Selection for survival
Select the best m individuals from the family containing the m parents (p 1 , p 2 , · · · , p m ) and their children.
Then, replace the m parents with the selected individuals. In the original JGG, the best m individauls are selected only from the children. As its optimization process seemed to be unstable, however, this study slightly modified its algorithm.

Termination
Stop if the halting criteria are satisfied. Otherwise, Generation ← Generation + 1, and then return to the step 2.

REX star
REX star is a real-coded crossover operator. REX star uses s + 1 parents, where s is the dimension of the search space, and generate n c (> s + 1) children according to the following algorithm. In the following algorithm, the parents are represented as p 1 , p 2 , · · · , p s+1 .
[Algorithm: REX star ] 1. Generate reflection points, p 1 , p 2 , · · · , p s+1 , of the parents, i.e., where 2. Compute the objective values of the s + 1 reflection points generated in the previous step. In REX star , these reflection points are treated as the children.
3. From the parents and their reflection points, select the best s + 1 individuals, and then compute the center of the gravity of the selected individuals. This study represents it as G b .
4. Generate n c − s − 1 children by applying the following equation n c − s − 1 times. Note that the s + 1 reflection points generated in the step 1 are treated as the children. The total number of the children generated is therefore n c .
where ξ t i 's and ξ i 's are random numbers drawn from uniform distributions [0, t] and , respectively, where t is a hyper-parameter named a step-size parameter.

Back-propagation through time
The discrete form of the Vohradský's model can be viewed as a recurrent neural network. The existing inference methods [2,5,6] have thus designed on the basis of the learning algorithm for the recurrent neural network, i.e., the back-propagation through time [4]. In the back-propagation through time, all of the parameters of the Vohradský's model are estimated simultaneously by minimizing where α = (α 1 , α 2 , · · · , α N ), β = (β 1 , β 2 , · · · , β N ), w = (w 1,1 , w 1,2 , · · · , w N,N ), and b = (b 1 , b 2 , · · · , b N ) are the model parameters, N is the number of genes contained in the network, and K is the number of measurements. X n | t k and X n | cal t k are the observed and the computed expression level of the n-th gene at time t k , respectively. In the back-propagation through time, X n | cal t k is computed from the discrete form of the Vohradský's model, i.e., where ∆t = t k − t k−1 . This study constructed two inference methods based on the back-propagation through time, i.e., BPTTLS and BPTTGA. BPTTLS and BPTTGA used a local search, i.e., the conjugate gradient method [3], and an evolutionary algorithm, i.e., REX star /JGG [1], respectively, as function optimization algorithms. The following recommended values were used for the parameters of REX star /JGG applied here; the population size n p is 20s, the number of children generated per selection n c is 3s, and the step-size parameter t is 2.5, where s is the dimension of the search space. Each run was continued until the best objective value did not improved over 2 × n p generations.