Designing a parallel evolutionary algorithm for inferring gene networks on the cloud computing environment
- Wei-Po Lee^{1}Email author,
- Yu-Ting Hsiao^{1} and
- Wei-Che Hwang^{1}
https://doi.org/10.1186/1752-0509-8-5
© Lee et al.; licensee BioMed Central Ltd. 2014
Received: 4 June 2013
Accepted: 6 January 2014
Published: 16 January 2014
Abstract
Background
To improve the tedious task of reconstructing gene networks through testing experimentally the possible interactions between genes, it becomes a trend to adopt the automated reverse engineering procedure instead. Some evolutionary algorithms have been suggested for deriving network parameters. However, to infer large networks by the evolutionary algorithm, it is necessary to address two important issues: premature convergence and high computational cost. To tackle the former problem and to enhance the performance of traditional evolutionary algorithms, it is advisable to use parallel model evolutionary algorithms. To overcome the latter and to speed up the computation, it is advocated to adopt the mechanism of cloud computing as a promising solution: most popular is the method of MapReduce programming model, a fault-tolerant framework to implement parallel algorithms for inferring large gene networks.
Results
This work presents a practical framework to infer large gene networks, by developing and parallelizing a hybrid GA-PSO optimization method. Our parallel method is extended to work with the Hadoop MapReduce programming model and is executed in different cloud computing environments. To evaluate the proposed approach, we use a well-known open-source software GeneNetWeaver to create several yeast S. cerevisiae sub-networks and use them to produce gene profiles. Experiments have been conducted and the results have been analyzed. They show that our parallel approach can be successfully used to infer networks with desired behaviors and the computation time can be largely reduced.
Conclusions
Parallel population-based algorithms can effectively determine network parameters and they perform better than the widely-used sequential algorithms in gene network inference. These parallel algorithms can be distributed to the cloud computing environment to speed up the computation. By coupling the parallel model population-based optimization method and the parallel computational framework, high quality solutions can be obtained within relatively short time. This integrated approach is a promising way for inferring large networks.
Keywords
Gene network inference Systems biology Evolutionary algorithm Swarm intelligence Parallel model Cloud computing MapReduceBackground
Model
Gene regulatory networks (GRNs) play important roles in genetic systems and are involved in various biological processes during the development of living organisms. Through analyzing the interactions between genes, we can uncover some complex behavior patterns and study genetic systems in detail. Gene network construction has been considered to be one of the most important issues in systems biology research. It is a procedure used to manipulate experimentally measured time-series data for building a model that can describe the observed phenotypic behavior of a system to be studied. To save the effort of testing experimentally which interactions in the gene networks are possible and then deriving the network accordingly, an automated reverse engineering procedure has been advocated[1, 2]. This work means to establish a practical computational methodology for the inference of real gene networks.
In the above equation, x _{ i } is the expression level of gene i and N is the number of genes in a genetic network. The non-negative parameters α _{ i } and β _{ i } are rate constants that indicate the direction of mass flow. The real number exponents g _{ i,j } and h _{ i,j } are kinetic orders that reflect the strength of the interactions from gene j to i. The above set of parameters defines an S-system model. To infer a coupled S-system model is, therefore, to determine all of the 2N(N + 1) parameters simultaneously.
The above inference of an S-system model is a large-scale parameter optimization problem that is very time-consuming. After analyzing the structural characteristics of gene networks, Maki et al. proposed an efficient strategy to divide this inference problem into N separated small sub-problems, each of which corresponds to one gene[4]. In other words, in a decoupled S-system, the original tightly coupled system of non-linear differential equations is decomposed into several differential equations[5, 6], each of which describes a specific gene that can be separately inferred. Although addressing gene network inference in the above-described way can substantially reduce the computational complexity, it is notable that the computational cost will still grow linearly when inferring large networks.
where${x}_{i}^{d}\left(t\right)$ is the desired expression level of gene i at time t, ${x}_{i}^{a}\phantom{\rule{.3em}{0ex}}\left(t\right)$ is the value generated from the inferred model, T is the number of time points in measuring gene profiles, and N is the number of genes in the network.
In real-world situations, the number of data points available is often smaller than that of the parameters to be determined. It is thus possible to obtain many feasible solutions with various combinations of network parameters (meaning different network structures/topologies). To solve the structure problem, prior knowledge or assumptions are required. Some researchers proposed to incorporate the structural/topological properties of the biological networks (such as the degree distribution of the nodes in a scale-free network or the presence of network motifs) with the gene expression data in the evaluation process. For example, some inference methods intend to limit the number of GRN connections to be as small as possible because gene regulatory networks are typically known to be sparsely connected. In such a case, a small penalty term that measures the connection between the genes can be added to the fitness function to discourage the connections[7, 8]. There are also other researchers who suggest using expert knowledge to determine the models, such as taking the form of parameter constraints to account for the prior domain knowledge to restrict the search of network parameters[9].
It is worth noting that prior knowledge is not always available. Additionally, the topological properties within the relatively small size genome-scale networks are not sufficient enough to provide practical information, even if they have been studied in sophisticated network modeling to better understand the corresponding biological details. Therefore, without losing generality, in this work, we simply take the mean square error (as shown in equation ([2])) to define evaluation function. We focus on how to design a parallel computational model to infer decoupled S-system and how to distribute the relevant computation to a cloud computing environment.
Algorithm
Although the non-linear ODEs mentioned above can more accurately capture the system dynamics of gene networks, they are difficult to solve by traditional local optimization techniques[3, 10], such as the conjugate gradient method, Newton’s method, and the Nelder-Mead simplex method. These techniques rely on the derivative of the evaluation function or a comparison of the evaluation function at the vertices of a simplex in the parameter space. Thus, they are not suitable for the optimization problem here. Global optimization techniques for parameter estimation are better choices than the above-described local techniques for finding the global optimum and are more suitable for biological systems.
Among global parameter estimation methods, the methods that use deterministic strategies are superior for finding the global optimum, but these are computationally more expensive. In contrast, the methods with stochastic strategies can have solutions that are near the global optimum within a reasonable amount of time. Population-based approaches (such as evolutionary algorithms, EAs) are stochastic methods, and they have been used in many studies to infer the S-system model (e.g.,[5, 6, 11–13]).
Adopting a reverse-engineering approach to derive GRNs in their early work, Tominaga and his colleagues employed a conventional simple genetic algorithm (SGA) to infer genetic networks with the S-system model[14]. Their results showed that SGA can successfully evolve small scale GRNs (around 5 genes). Following Tominaga’s work, Kikuchi et al. proposed an incremental optimization strategy with a novel crossover technique[5] for network inference. The modified SGA has a better performance while inferring the S-system model; however, the modified approach is computationally expensive since it has to deal with too many parameters simultaneously. Ho et al. also proposed an intelligent two-stage evolutionary algorithm (iTEA) that used orthogonal matrices to decide better solutions[7]. iTEA has shown its great power in the reconstruction of gene networks, but the orthogonal matrices will become bigger for large scale networks (e.g., 30 genes) and the computational time thus increases dramatically. There are other works proposed for further performance improvement, for example,[5, 13, 15–18]. However, they share the common problem of scalability. To tackle this problem, we adopt this type of global optimization technique and develop a new parallel model to enhance the search performance and speed up the computational process.
Using EAs to infer large gene networks that involve a large number of genes, two inherent features of EAs must be considered seriously. The first is premature convergence, which has the negative effect of losing population diversity before the goal is reached. To overcome this problem, parallel-model EAs have been proposed to divide the original population into several subpopulations to maintain population diversity, and their efficiency has been confirmed (e.g.,[19, 20]). The other problem that must be addressed is the inordinate amount of time that is required to perform evaluations for all of the individuals. To enhance the search performance and speed up the computation at the same time, it is therefore critical to develop parallel models for EAs and execute the parallel models in high-performance computing environments.
Computational platform
As mentioned above, a decoupled S-system model with N genes decomposes the original non-linear differential of S-system into N differential equations. It means that the original task that contains 2N(N + 1) parameters altogether is now transferred into N sub-tasks; each sub-task has 2(N + 1) parameters (i.e., α _{ i }, g _{ i, 1..N }, β _{ i }, h _{ i, 1..N } for a specific gene i) and can be independently performed. The decoupled S-system model allows us to perform the sub-tasks in a parallel way and then to assemble the results to obtain a final complete model. This characteristic perfectly matches the pre-requirement of distributed computing. Moreover, the population-based inference algorithm can also be parallelized to speed up the computation. Combining the above ways of parallelism, large scale networks containing hundreds or thousands of genes can thus be inferred.
Conceptually, the central idea of the parallel EA is to divide a large population used in a sequential version of EA into multiple smaller subpopulations and to distribute them on separate computational nodes so as to perform the evaluations on separate processors concurrently. Although parallel computers can be used to speed up EAs, they are expensive and usually not available in a campus environment. One promising choice is to utilize cloud computing, and the MapReduce programming model provides an easy-to-implement framework with fault tolerance capabilities[21–23]. This framework has been used to successfully solve many large-scale scientific computing problems, including problems in the life sciences[24–26].
Apache Hadoop is an open source implementation of MapReduce written in Java. Apart from MapReduce, Hadoop also provides the Hadoop Distributed File System (HDFS) to reliably store data across hundreds of computers. The cluster is a single master cluster with a varying number of slave nodes. The slave nodes can act as both the computing nodes for MapReduce and as data nodes for the HDFS. Hadoop MapReduce has been used in several bioinformatics research studies[27, 28]. For example, it has been employed to develop algorithms for the analysis of next-generation sequencing data[29, 30], to implement systems for sequence alignment[31], and to develop proteomic search engines[32].
Although the Hadoop MapReduce framework can effectively reduce the runtime for large-scale data-intensive applications, it has some limitations[33]. One limitation is job latency, which means the time spent on running background tasks, such as the time spent to schedule, start, and finish a MapReduce job. The other limitation is input data processing, which means the need to read the input data from the HDFS every time a MapReduce job is executed. These limitations decrease the efficiency of the framework, especially in applications that involve iterative computations. An alternative MapReduce framework is Twister, which can store input data in memory between iterations[34]. However, storing data in this way has another disadvantage: it requires the data to fit in the collective memory of the cluster so that the framework can be effective; this approach is unfeasible for tasks that have a large amount of data. In addition, Twister has a weak fault tolerance capability, but this capability is very important for population-based computations that require a large number of iterations and must be protected from hardware or network failures.
To infer large gene networks from the expression profiles, in this work, we present a parallel evolution-based optimization algorithm and distribute the algorithm to the cloud computing environment to speed up the computation. Considering the critical factors and the current use popularity, we chose to use Hadoop MapReduce for our application. To evaluate the proposed approach, experiments have been conducted, and the results have been analyzed. They show that our approach can be successfully used to infer networks with desired behaviors from four real-world biological datasets. Also, the evolution algorithm with a parallel model can improve the performance of network inference. Most importantly, using the MapReduce framework, the computation time for the inference algorithm can be substantially reduced, such that large networks can be inferred.
Methods
Optimization algorithm
The sequential model
The first step of using a population-based search method to solve an optimization problem is to define an appropriate representation to characterize the problem’s solution. To infer a gene network, we take a direct encoding scheme to represent solutions for both the GA and PSO parts, in which the network parameters related to the decoupled S-system model are arranged as a linear string chromosome of floating-point numbers. That is, for each gene i in a network with N gene nodes, the solution is represented as (α _{ i }, g _{ i, 1}, g _{ i, 2}, …, g _{ i,N }, β _{ i }, h _{ i,1 }, h _{ i,2 }, …, h _{ i,N }). As the parameter range of kinetic orders g _{ i, 1~N }, h _{ i, 1~N } ∈[-3, 3] have been widely used as the default search space for evolutionary algorithms (e.g.[5, 7, 8, 17]), we thus choose to use the same range in this study. For the rate constants, several range settings have been taken, such as α _{ i }, β _{ i } ∈[0, 10], [0, 15], and [0, 20] in the literature (e.g.[5, 8, 17]). Based on a preliminary test, we found no significant difference for using these ranges. The rate constants represent the ratio between the synthesis and degradation process. For simplicity, the range from 0 to 10 is considered suitable to evaluate the constant rate between the two processes. Hence, in this study the ranges for α _{ i }, and β _{ i } are both set to [0, 10].
The next step is to define a fitness function for the evaluation of candidates and to use the results as a form of selective pressure to derive better solutions. In this study, we directly use the mean squared error function shown in the above section (i.e., equation ([2])) as the fitness function. With the above representation and evaluation function, we can then perform the hybrid approach to determine the network parameters.
As shown in Figure 2, in this algorithm a random population is first generated and evaluated. Then, the individuals (particles) are ranked according to their fitness and are separated into two parts. The first part includes elites (i.e., the best p% individuals of the entire population); these individuals are preserved and enhanced by the PSO procedure and are sent directly to a candidate list that is being prepared for the next generation. The second part includes individuals with a lower performance compared with those in the first part (i.e., the worst (1 - p)% individuals); they are discarded. To replace the removed individuals, the same number of individuals is produced to form a parent pool, in which some individuals are randomly generated (i.e., r%), and the remainder (i.e., (1 - p - r)%) are randomly selected from the ones that were already improved by the PSO procedure. Then, this parent pool is used to create new individuals through the GA procedure, and the newly created individuals are sent to the candidate list. Once a new candidate list is formed, the individuals in this list are again ranked according to their fitness values, and the new population is used for the next generation. The above procedure is repeated until the termination criterion is met.
In our experiments, p has a fixed value (which is estimated from a preliminary test), while the randomness rate r is a variable whose value can be changed during the run to control the population diversity (i.e., to coordinate the progress of the PSO and GA parts). For example, the randomness rate can increase linearly in proportion to the generation number to maintain the overall population diversity; the PSO tends to perform a local search at the end of the run, which implies that a high rate of randomness is desirable.
In the above equations, v _{ id } and x _{ id } are the velocity and position of particle i in dimension d, p _{ id } is the previous best position of particle i, p _{ gd } is the best position in the swarm, and w is the inertia weight, which controls the momentum of the particle by weighting the contribution of the particle’s previous velocity. The coefficients c _{1} and c _{2} are two positive acceleration constants; they are often determined empirically. The variables r _{1} and r _{2} are random values within the range [0, 1]. The products c _{1} r _{1} and c _{2} r _{2} thus stochastically control the overall velocity of a particle. In addition, χ is the constriction factor that was originally introduced in[35] to constrict the velocities of particles and to achieve an exploration-exploitation balance during the search. Finally, φ is a parameter often used to control the convergence characteristics of the PSO. It is the summation of c _{1} and c _{2}, and has to be larger than 4.
In the above equations, LB and UB are the lower and upper bounds of the parameter p _{ k } and t is the iteration number. The function Δ(t, y) (the often-used function as described in[36]) returns a value between 0 and y such that the probability of Δ(t, y) will be close to 0 as t increases.
The parallel model
The GA-PSO approach described above has been extended with a parallel model to enhance the search performance and to reduce the computational time in the network inference. Conceptually, parallelizing the above procedure involves dividing a large population into multiple smaller subpopulations so that they can be addressed on separate computational nodes simultaneously. Depending on the subpopulation size, two types of parallel models are typically used: coarse-grained and fine-grained. The best choice depends mainly on the machine availability and the type of application task. Because of the hardware limitation, in this work we adopt a coarse-grained model and distribute the computation in the cluster computing environment to achieve parallelism.
A coarse-grained model divides the whole population into a small number of subpopulations; each subpopulation can be evaluated by the original GA-PSO independently on a separate processor. In this model, any change for a particle occurs only in a local subpopulation. A communication (migration) phase is introduced in the parallel model to periodically select some promising individuals from each subpopulation and send them to other subpopulations. A common strategy is to use the selected individuals to substitute for the worst individuals of the neighboring subpopulations. In this way, the algorithm has a higher probability of maintaining population diversity and protecting good solutions that are found locally.
Our coarse-grained GA-PSO performs island model parallelism (called iGA-PSO). Ideally, each subpopulation is distributed to one or more available computational node(s). In this way, the subpopulations can be independently executed at the same time to speed up the solution search process. However, because the MapReduce framework is adopted in this work to realize the distributed computing, the dispatch of islands to the computational nodes has to follow the corresponding design principles and to work in a non-traditional manner. For example, the particles in the same island may be arranged in different computational nodes in the map phase and later grouped together in the reduce phase. Here, we focus on the operational flow of iGA-PSO in the conceptual level. How the iGA-PSO works on the MapReduce framework is described in the following section.
The distributed iGA-PSO code continues for a certain number of iterations before migration occurs. Because of the variation in the machine specifications (e.g., the CPU and memory utilities) and the discrepancy of the evaluation time in the individuals, computers that run different subpopulations can take different amounts of time to complete the corresponding evolution. To perform migration as in the original island model, our work uses a synchronized strategy to organize the distributed computation. In other words, the communication phase can start only when all of the subpopulations have been executed for a predefined number of generations. After exchanging individuals in the communication phase, all of the subpopulations continue independently again. The above procedure operates iteratively until the termination criterion is met.
Parallelizing iGA-PSO on a cloud computing framework
The first user-defined object is the map method (designed for the Mapper). This object allows the algorithm to execute the operations independently without waiting for others, and thus carries out the parallel computing. In the decomposed S-system model, the set of parameters for each individual gene can be independently inferred, and this characteristic matches the purpose of using Mappers for problem solving. In our iGA-PSO algorithm, some operations for optimizing the parameters that correspond to a specific gene can be executed concurrently (including the computation of updating a particle’s position and velocity, the fitness evaluation for each particle, and the determination of the best-so-far result for each particle (i.e., p _{ best })). They are thus arranged to be executed in the mapper phase. Such an arrangement means creating some input files by following the format specified by the MapReduce framework (see the third object below). According to the files, this framework will distribute particles (when they are to perform the operations mentioned above) to different Mappers to achieve the parallelism automatically.
The second user-defined object is the reduce method (designed for the Reducer). In contrast to the above map method, this method deals with the operations that involve results generated by others. Some operations in our iGA-PSO algorithm need to wait for others and they are thus allocated in the reduce phase (including the execution of selection, crossover, and mutation in the GA procedure, the operations of migration among different islands in the parallel model, and the determination of the global best result, i.e., g _{ best }). In the MapReduce framework, the particles have to be re-distributed to the computational nodes (working as the Reducers in this phase) before they start to perform the above operations. Here, we group the particles that deal with the same genes together, and each group of particles is dispatched to a Reducer.
As shown above, the proposed approach can efficiently distribute the iGA-PSO algorithm to the map and reduce phases. In this way, our approach can fully exploit the computational power of the cloud computing system to perform parallel computation for network inference. The proposed framework is able to remedy the scalability problem and be used to infer large gene networks.
Results
To assess the performance of our parallel iGA-PSO in a cloud computing environment, we used the well-known open-source software GeneNetWeaver 3.1 (GNW 3.1,[37]) to create several datasets. GNW was designed for generating in silico benchmarks of gene expression profiles by extracting network modules from prior in vivo studies (such as S. cerevisiae[38, 39]) and connecting/expanding these modules to form test networks. In other words, each benchmark is composed of hundreds of genes (that form the real biological sub-networks), and the network can be inferred by computing algorithms without performing an in vivo experiment in advance. According to the GNW tutorial, the benchmarks used in the competition of gene network inference in DREAM challenges were created in this way[40].
In the experiments, four datasets were generated from the software GNW 3.1. They included a 25-node (dataset 1), a 50-node (dataset 2), a 100-node (dataset 3), and a 125-node (dataset 4) yeast S. cerevisiae sub-networks. Then we used this software to produce the time series data for each test network. In the generation of time series data, we left most of the settings at the default values suggested in GNW 3.1 as in DREAM4, for example, the perturbation of the time series, the model of noise in microarray, and the normalization after adding noise. To increase the complexity of the network modeling for rigorous verification, the duration of the time series and time points were approximately 1.5 times of the default settings (changing from 21 to 31 time points). The main reason for generating datasets by GNW 3.1 rather than directly taking the datasets of DREAM challenges lies in that our evaluation procedure requires a set of different sizes networks with certain scale differences to compare the corresponding speed-up rates. As far as is known, there has not been any work that infers S-systems to characterize the gene networks used in the DREAM challenges, so no results of the same type are yet available for performance comparison.
With the above datasets, two phases of experiments were arranged and conducted. The first phase was to evaluate the performance of our network inference method without (i.e., GA-PSO) and with the island model (i.e., iGA-PSO for virtual parallelism), by running the code on one sequential machine. In the second phase of the experiments, the iGA-PSO method was implemented on the Hadoop MapReduce framework to speed up the gene network reconstruction (i.e., real parallelism), and the amount of time required to complete a run of network modeling using different methods was compared.
Performance of the proposed algorithm on a sequential machine
The first experimental phase involves examining whether the coarse-grained parallel model can further improve the search quality of the original method. Therefore, for the above four datasets, the optimization algorithm with two different settings was used to infer the networks. The first setting was the original GA-PSO (without the island model), which can also be regarded as the iGA-PSO with only one island of the population. The second setting was the GA-PSO with the island model in which different numbers (2 and 4 in the experiments) of subpopulations were used. In the experiments, the migration procedure was activated every m iterations (10 or 20 in the experiments), and the number of migrants to be exchanged was 5% of the sub-population. For each subpopulation, the migrants were chosen randomly from the top 20% particles of a sub-population to replace the worst 5% of its neighbors. These values were chosen based on a small pilot study.
Results obtained by the proposed algorithms with different settings for dataset 1 to 4
GA-PSO | iGA-PSO (i = 2) | iGA-PSO (i = 4) | ||||
---|---|---|---|---|---|---|
Migration ( m) | - | m = 10 | m = 20 | m = 10 | m = 20 | |
25 genes (dataset 1) | Avg | 0.1437 | 0.1312 | 0.1246 | 0.1195 | 0.1240 |
Best | 0.1256 | 0.1117 | 0.1113 | 0.1026 | 0.1083 | |
Worst | 0.1678 | 0.1516 | 0.1429 | 0.1311 | 0.1460 | |
SD | 0.0149 | 0.0103 | 0.0085 | 0.0107 | 0.0093 | |
50 genes (dataset 2) | Avg | 0.2288 | 0.2031 | 0.2098 | 0.1944 | 0.2002 |
Best | 0.1813 | 0.1572 | 0.1606 | 0.1519 | 0.1515 | |
Worst | 0.2516 | 0.2305 | 0.2267 | 0.2158 | 0.2254 | |
SD | 0.0187 | 0.0174 | 0.0156 | 0.0147 | 0.0180 | |
100 genes (dataset 3) | Avg | 0.3947 | 0.3759 | 0.3765 | 0.3656 | 0.3711 |
Best | 0.3570 | 0.3304 | 0.3501 | 0.3337 | 0.3411 | |
Worst | 0.4571 | 0.4067 | 0.4140 | 0.3957 | 0.3961 | |
SD | 0.0272 | 0.0168 | 0.0176 | 0.0148 | 0.0160 | |
125 genes (dataset 4) | Avg | 0.2275 | 0.2197 | 0.2216 | 0.2177 | 0.2210 |
Best | 0.2111 | 0.2043 | 0.2053 | 0.2056 | 0.2087 | |
Worst | 0.2497 | 0.2383 | 0.2468 | 0.2299 | 0.2374 | |
SD | 0.0105 | 0.0096 | 0.0112 | 0.0068 | 0.0087 |
The results in Table 1 indicate that the parallel model can successfully enhance the performance of the original network inference method. This is consistent with other island-based EA works. It is because that using multiple subpopulations can effectively keep groups of particles evolving separately in different islands for some pre-defined generations, and this strategy has successfully maintained the diversity of each group of particles. Hence, each island would not be dominated by the elite particles in other islands. The algorithm can then prevent the so-called premature situation (meaning converging too fast) and can explore more regions of the solution space to find better solutions.
The number of successful runs and the computational cost of the different methods
Dataset 1 | Dataset 2 | Dataset 3 | Dataset 4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
num. of population | 1 | 2 | 4 | 1 | 2 | 4 | 1 | 2 | 4 | 1 | 2 | 4 |
population size | 8000 | 4000 | 2000 | 10000 | 5000 | 2500 | 10000 | 5000 | 2500 | 10000 | 5000 | 2500 |
num. of successful runs | 12 | 19 | 20 | 8 | 17 | 20 | 7 | 15 | 16 | 10 | 14 | 17 |
Expected Cost (C) | 1.69 | 1.10 | 1.05 | 2.44 | 1.22 | 1.05 | 2.75 | 1.38 | 1.29 | 2 | 1.47 | 1.22 |
With this estimation approach, if we assume that the cost of conducting a run using the method of one population of 8000 individuals is C, and for this method we have M = 12 and N = 8, then the expected computational cost of obtaining a successful run with this method is C × 22/13 (= 1.69C). Similarly, for the method with two sub-populations of 4000 individuals, M = 19, N = 1, the cost of a single run is C (the cost for exchanging individuals is relatively small and ignored); thus, the expected cost of this method is C × 22/20 (= 1.1C). In the same way, we can estimate that the expected cost for the method with four subpopulations of 2000 individuals is 1.05C. The estimated costs for all of the datasets are listed in Table 2 (assuming that the run with one population is C). These results indicate that using iGA-PSO can indeed yield successful solutions with relatively less computational effort.
Performance of the proposed method on parallel machines
After investigating the performance of the proposed algorithms (GA-PSO and iGA-PSO) running in a sequential manner for gene network modeling, in the second phase, we conducted two suites of experiments to examine how our iGA-PSO method can be performed in a parallel computing environment to speed up the computation in practice. The iGA-PSO was implemented on the Hadoop MapReduce framework and was executed on different computer clusters for verification. The first suite of experiments was conducted at one of the personal computer laboratories in our university computer center, and the second suite was conducted in a commercial cloud computing environment. In the campus computing laboratory, two sizes of computing clusters were arranged as slave machines, including 20 and 25 PCs (due to the hardware limitations), and one extra machine was set up as the master. These machines have the same hardware/software specification: 3.0 GHz Intel Core 2 Duo E8400 CPU, 4GB DDR2 667 RAM, and Linux CentOS 6.2 (64-bits) or Ubuntu 11.10 (64-bits) operation system with Hadoop 0.20.0 version. The transmission speed of the local area network is 1,000 Mbps. We also ran the sequential version of GA-PSO and iGA-PSO on a single node for a baseline comparison.
It is notable that the main purpose of this study is to provide a practical framework to speed up the computation so that large biological regulatory networks can be inferred, rather than to compare different algorithms or to perform optimal parameter settings. Therefore, in this experimental phase, we considered the island number to be 4 and migration interval to be 10 (the parameter combination with the best performance as shown in the previous section) in the experimental runs of real parallelism. Here, we focused on the data sets that had more gene nodes (i.e., 50, 100, 125 nodes). Because of the large amount of computational time needed, in these runs, the population size was reduced to 5,000 and the iteration number was 200 iterations.
Results for datasets 2, 3, and 4 by running the experiments in the computer center
Algorithm | Sequential | Sequential | Parallel | ||
---|---|---|---|---|---|
GA-PSO | iGA-PSO | iGA-PSO | |||
50 genes (dataset 2) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 25 |
Time cost (mins) | 923 | 908 | 218 | 176 | |
Speed-up | - | 1 | 4.1651 | 5.1591 | |
Fitness value per gene | 0.2876 | 0.2494 | 0.2593 | 0.2562 | |
100 genes (dataset 3) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 25 |
Time cost (mins) | 3,687 | 3,137 | 496 | 411 | |
Speed-up | - | 1 | 6.3246 | 7.6326 | |
Fitness value per gene | 0.4567 | 0.4225 | 0.4165 | 0.4114 | |
125 genes (dataset 4) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 25 |
Time cost (mins) | 5,788 | 4,927 | 702 | 595 | |
Speed-up | - | 1 | 7.0185 | 8.2807 | |
Fitness value per gene | 0.2822 | 0.2651 | 0.2670 | 0.2667 |
After running the proposed iGA-PSO method on the Hadoop MapReduce framework to prove its correctness and confirm its performance, we conducted the second suite of experiments to further evaluate this parallel method in the cloud. Among the suppliers of cloud computing services, the Amazon Elastic Compute Cloud (i.e., Amazon EC2) is one of the most famous and largest cloud computing and storage providers. It provides a basic but versatile computing environment at a low rental cost, including scalable CPUs, memory (RAM), disk storage, and several operating systems. In addition, the cloud services also allow users to expand or eliminate computing nodes and to pay only for the activated nodes. Fusaro et al. drew our attention to how projects in biomedical informatics were deployed on the Amazon cloud service and listed the pricing structure from Amazon EC2 for the CPUs and storage[42].
Similarly, one of the telecommunication service providers, Chunghwa Telecom, which is the largest cloud computing supplier in Taiwan, offers application developers ideal computing resources (called hicloud,http://hicloud.hinet.net/). Comparing their pricing strategies, hicloud is cheaper than Amazon EC with regard to the matching on-demand packages, which fulfilled our requirements. Because using a reverse engineering approach to construct GRNs is a CPU- and memory (RAM)-consuming task, especially in the modeling of large networks (e.g., with more than 100 genes), we need efficient CPUs, a large memory capacity, and high-speed intranet data transfer to achieve this task. Therefore, considering both the price (each node costs $0.128 per hour)^{a} and the performance of the on-demand package (each hicloud node is equipped with a 2.0 GHz 2007 Xeon processor (4 virtual cores), 8 GB of RAM, 100 GB of disk storage, and Linux Ubuntu 11.10 (64-bit)), we chose hicloud as the cloud environment for this suite of experiments.
Results for datasets 2, 3, and 4 by running experiments on hicloud
Algorithm | Sequential | Sequential | Parallel | ||
---|---|---|---|---|---|
GA-PSO | iGA-PSO | iGA-PSO | |||
50 genes (dataset 2) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 30 |
Time cost (mins) | 3019 | 2898 | 591 | 465 | |
Speed-up | - | 1 | 4.9036 | 6.2323 | |
Fitness value per gene | 0.2345 | 0.2115 | 0.2140 | 0.2076 | |
100 genes (dataset 3) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 30 |
Time cost (mins) | 11342 | 10284 | 1526 | 1316 | |
Speed-up | - | 1 | 6.7392 | 7.8146 | |
Fitness value per gene | 0.3843 | 0.3520 | 0.3642 | 0.3582 | |
125 genes (dataset 4) | Master / Slaves | 1 / 0 | 1 / 0 | 1 / 20 | 1 / 30 |
Time cost (mins) | 18019 | 16323 | 2152 | 1792 | |
Speed-up | - | 1 | 7.5850 | 9.1088 | |
Fitness value per gene | 0.2358 | 0.2056 | 0.2195 | 0.2195 |
Discussions
In the current era, scalable parallel computing allows application developers to exploit all of the degrees of freedom of in linear scaling to solve a time-consuming problem[43]. In this new epoch, it always comes to a critical issue that if a sequential computational problem can be divided into several parallel sub-problems (as shown in above section), then how to judge the speed-up performance obtained? Ideally, the computation in Hadoop MapReduce operates in parallel and different parts of the framework have no mutual influences with others. This way, the speed-up rate can increase linearly in proportion to the number of computational nodes. Taking our application as an example, a 20-node cluster should give a nearly 20 times speed-up rate in a completely independent MapReduce environment. Nonetheless, a particular portion of serial overhead and the parallel environment-enabling cost must be taken into account and which slows down the experiments in practice.
Where f is the proportion of a problem that can be divided into parallel computing, and (1- f) means the proportion that cannot be parallelized (e.g., the non-parallelized task, or the other execution processes). Ideally, the maximum acceleration lies in S, meaning that f can be accelerated S times by using S parallel processors (i.e., the slaves on Hadoop). For example, suppose modeling a gene network took 200 minutes by running a sequential iGA-PSO, and there was 5% (i.e., 10 minutes) of computation belonging to the non-parallelized task. Under such circumstance, no matter how many slaves we applied and devoted to the parallel computing, the remaining 95% (190 minutes) led the maximum acceleration up to 20 times at most (the minimum time–cost cannot be less than 10 minutes). Similarly, if a task needed to take 10% of computation for running the mechanism of cloud computing and for executing the non-parallel processes, then the maximum speed-up rate was limited to 10 times.
In this study, the best acceleration rate we can obtain in the experiments is 9.1 (from the case of inferring gene network for dataset 4 with 30 computational nodes in the second suite of experiments, see Table 4). It means that we have parallelized 89% (1-(1792/16323)) of the whole modeling task, and only remained 11% of the original execution time for the non-parallelized overhead and for the cost of maintaining the Hadoop MapReduce framework.
In addition to the speed-up rates, other issues related to the performance of the network model are worth mentioning. Considering the use of a decoupled S-system model, some studies have pointed out the model’s characteristic, and the pros and cons of decomposing the original ODEs model (e.g.[4, 45]). In short, the decoupling strategy proposed by[4] enables us to infer one differential equation at a time to match the target expression data. Then, after the parametric solution for each single differential equation is provided, two strategies for deriving a final solution set are suggested.
The first strategy is to combine all single decoupled solutions into one and then to depict the entire network system directly. This strategy is suggested when every expression profile was precisely estimated. It means that the errors of the optimal solutions generated by the decoupled approach will not have a significant effect on the network behaviors when all the solution sets are assembled to form a coupled model[17]. However, when the solution sets obtained from the decoupled models cannot reproduce the expression profiles correctly, the second strategy, a recoupling procedure, has to be carried out. It arranges all the decoupled parametric sets to form a coupled S-system model, and then uses this model as the initial parameter population to continue the optimization process. This strategy, nevertheless, is computationally expensive as reported in[45]; an alternative approach has been suggested that is to turn to a cooperative evolutionary method to facilitate the recoupling process[17]. After evaluating the estimated expression profiles shown in Figure 7, in which all the estimated datasets have very high R-squared values (all above 0.98), we adopt the first strategy in this work.
To evaluate the performance of adopting the S-system model to infer both the gene expression profiles and the corresponding network structure, we propose another fitness function in the Additional file1. This fitness function concerns not only the network behaviors but also the sensitivity and specificity rates. The real positive and negative outcomes obtained from the real world resources are used to help the new fitness function to evolve more reasonable structural parameters while optimizing the expression profiles at the same time.
We have also selected one dataset of the DREAM challenges and conducted a set of experiments to compare our approach with other well-known reverse engineering methods. To be specific, in this set of experiments, the DREAM4 multifactorial network 1 with 100 nodes was used, and two well-known inference algorithms (GENIE3[46] and TIGRESS[47]) were chosen for comparison. Our approach was performed with two fitness functions: (a) the MSE function, and (b), the fitness function described in Part C of the Additional file1. The ensuing precision and the recall rates for fitness functions (a) and (b) are (91.57%, 5.08%) and (97.93%, 10.6%), respectively. These results reveal that if some useful structural information is integrated into the fitness function, the S-system model can be derived to optimize both the network’s behaviors (the profiles) and the structure. Note that the trade-off between optimizing the system dynamics and network structures always exists[48].
Before comparing our structure correctness rates to the results obtained by the state-of-the-art structural prediction algorithms, two concerns need to be mentioned. First of all, when using the MSE as the fitness function in inferring the S-system, we can only claim that the network profiles have been reconstructed properly if the fitness value is lower than an acceptable threshold. There is no guarantee that the structural correctness can be as acceptable as the numerical results. Therefore, it is helpful to apply some extra strategies to ensure the correctness of an inferred model. One promising way is to modify the MSE function: it means to include the structural information matrix to guide the search of genetic parameters. Through the constraints of the structural information, an outcome with correct network profiles and structure can be obtained.
The second concern is that the goal of our inference approach is different from that of the studies conducted for the DREAM challenges. They have focused on the prediction of the network structure. However, here, we rely on the structural information collected from the real world resources, so that we can use a classification matrix to guide the development of solution space. As a result, the correctness rate on the network structure thus depends on the available structural information. Most importantly, the inference method is presented to show how a network can be reconstructed to provide the required system dynamics and relative connection relationships among genes (but not for predicting a network structure of a general graph-based model). The computational model can be replaced by others.
After clarifying the primary differences between our approach and the predictive reverse-engineering methods, we are aware of the limitations of directly comparing our results to TIGRESS and GENIE3 methods. Therefore, by carefully examining and summarizing the results of the above two algorithms, we provide their precision and recall rates only for reference. Overall, if the precision is around 0.9, then its relative recall rate lies in the range of 0.1 to 0.15 for Dream4 networks[46, 47].
Conclusions
In this study, we have emphasized the importance of reverse engineering GRNs from gene expression profiles. Depending on the biological level to be studied, many models have been proposed to simulate GRNs, among which concrete models are more suitable for simulating biochemical processes realistically. To infer gene networks, we adopted a well-researched concrete model, the decomposed S-system model, to represent a network and to infer the relevant network parameters. Although this model has been simplified from its original form, the computational cost to infer such a model still grows linearly with the number of gene nodes that are involved. To overcome the scalability problem, we presented a practical framework that can efficiently determine network parameters and is scalable for inferring large-scale networks. In this framework, a hybrid GA-PSO optimization method was developed and parallelized for performance enhancement. To conduct the real parallelism to reduce the computation time, our parallel method was extended to work with the Hadoop MapReduce programming model and was executed in different cloud computing environments. Extensive sets of experiments and analyses have been conducted to evaluate the proposed framework. The results show that our approach can successfully infer networks with desired behaviors for real-world biological datasets. Most importantly, our approach can be used to infer large gene networks on the clouds, in which the Hadoop MapReduce framework has been shown to substantially reduce the computation time for the application here. Currently, we are investigating different ways to take both the fault tolerance ability and the computing performance into consideration and developing an even more efficient framework to infer networks with more nodes.
Endnote
^{a}The specifications of each node provided by Amazon EC2 are: (1) High-CPU Medium Instance with 2.5-3.0 GHz 2007 Xeon processor (2 virtual cores), 1.7 GB of memory, 350 GB of local instance storage. (2) High-CPU Extra Large Instance with 2.5-3.0 GHz 2007 Xeon processor (8 virtual cores), 7 GB of memory, 1690 GB of local instance storage. The High-CPU Medium package cost $0.185 per hour/node. The High-CPU Extra Large costs $0.740 per hour/node.
Declarations
Acknowledgement
This work was supported by National Science Council of Taiwan, under Grant No NSC-100-2221-E-110-086.
Authors’ Affiliations
References
- Ingolia NT, Weissman JS: Systems biology: reverse engineering the cell. Nature. 2008, 454: 1059-1062. 10.1038/4541059a.View ArticlePubMedGoogle Scholar
- Lee W-P, Tzou W-S: Computational methods for discovering gene networks from expression data. Brief Bioinform. 2009, 10 (4): 408-423.PubMedGoogle Scholar
- Ay A, Arnosti DN: Mathematical modeling of gene expression: a guide for the perplexed biologist. Crit Rev Biochem Mol Biol. 2011, 46 (2): 137-151. 10.3109/10409238.2011.556597.PubMed CentralView ArticlePubMedGoogle Scholar
- Maki Y, Ueda T, Okamoto M, Uematsu N, Inamura K, Uchida K, Takahashi 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
- Kikuchi S, Tominaga D, Arita M, Tomita M: Dynamic modeling of genetic networks using genetic algorithm and S-system. Bioinformatics. 2003, 19: 643-650. 10.1093/bioinformatics/btg027.View ArticlePubMedGoogle Scholar
- Noman N, Iba H: Inferring gene regulatory networks using differential evolution with local search. IEEE/ACM Trans Comput Biol Bioinform. 2007, 4: 634-647.View ArticlePubMedGoogle Scholar
- Ho S-Y, Ho S-Y, Hsieh C-H, Huang HL: "An intelligent two-stage evolutionary algorithm for dynamic pathway identification from gene expression profiles". IEEE/ACM Trans Comput Biol Bioinform. 2007, 4: 648-704.View ArticlePubMedGoogle Scholar
- Kabir M, Noman N, Iba H: Reversely engineering gene regulatory network from microarray data using linear time-variant model. BMC Bioinform. 2010, 11: S56-10.1186/1471-2105-11-S1-S56.View ArticleGoogle Scholar
- Lee W-P, Hsiao Y-T: Inferring gene regulatory networks using a hybrid GA-PSO approach with numerical constraints and network decomposition. Inform Sci. 2012, 188: 80-99.View ArticleGoogle Scholar
- Germany: it-weise.de (self-published). 2009, Available online:http://www.it-weise.de/,
- Bazil JN, Qi F, Beard DA: A parallel algorithm for reverse engineering of biological networks. Integr Biol. 2011, 3 (12): 1145-1145.View ArticleGoogle Scholar
- Sirbu A, Ruskin HJ, Crane M: Comparison of evolutionary algorithms in gene regulatory network model inference. BMC Bioinform. 2010, 11: 59-10.1186/1471-2105-11-59.View ArticleGoogle Scholar
- Jostins L, Jaeger J: Reverse engineering a gene network using an asynchronous parallel evolution strategy. BMC Syst Biol. 2010, 4: 17-10.1186/1752-0509-4-17.PubMed CentralView ArticlePubMedGoogle Scholar
- Tominaga D, Koga N, Okamoto M: Efficient numerical optimization algorithm based on genetic algorithm for inverse problem. Proc Genet Evol Comput Conf. 2000, 251-258.Google Scholar
- Moles CG, Mendes P, Banga J: Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res. 2003, 13 (11): 2467-2474. 10.1101/gr.1262503.PubMed CentralView ArticlePubMedGoogle Scholar
- Lee W-P, Hsiao Y-T: An adaptive GA-PSO approach with gene clustering to infer S-system models of gene regulatory networks. Comput J. 2011, 54 (9): 1449-1464. 10.1093/comjnl/bxr038.View ArticleGoogle 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 (7): 1154-1163. 10.1093/bioinformatics/bti071.View ArticlePubMedGoogle Scholar
- Spieth C, Streichert F, Speer N, Zell A: A memetic inference method for gene regulatory networks based on S-Systems. Proc Congress Evol Comput. 2004, 152-157.Google Scholar
- Alba E, Tomassini M: Parallelism and evolutionary algorithms. IEEE Trans Evol Comput. 2002, 6 (5): 443-462. 10.1109/TEVC.2002.800880.View ArticleGoogle Scholar
- Cantú-Paz E: Efficient and accurate parallel genetic algorithms. 2000, New York: SpringerGoogle Scholar
- Armbrust M, Fox A, Griffith R, Joseph AD, Katz R, Lee AG, Patterson D, Zaharia M: A view of cloud computing. Commun ACM. 2010, 53 (4): 50-58. 10.1145/1721654.1721672.View ArticleGoogle Scholar
- Dean J, Ghemawat S: MapReduce: simplified data processing on large clusters. Commun ACM. 2008, 51 (1): 107-113. 10.1145/1327452.1327492.View ArticleGoogle Scholar
- K–H L, Lee Y-J, Choi H, Chung YD, Moon B: Parallel data processing with MapReduce: A survey. ACM SIGMOD Record. 2011, 40 (4): 11-19.Google Scholar
- Qiu J, Ekanayake J, Gunarathne T, Choi JY, S–H B, Li H, Zhang B, Wu T-L, Ruan Y, Ekanayake S, Hughes A, Fox G: Hybrid cloud and cluster computing paradigms for life science applications. BMC Bioinform. 2010, 11 (Suppl 12): S3-10.1186/1471-2105-11-S12-S3.View ArticleGoogle Scholar
- Krampis K, Booth T, Chapman B, Tiwari B, Bicak M, Field D, Nelson K: Cloud BioLinux: pre-configured and on-demand bioinformatics computing for the genomics community. BMC Bioinform. 2012, 13: 42-10.1186/1471-2105-13-42.View ArticleGoogle Scholar
- Schatz MC, Labgmead B, Salzberg SL: Cloud computing and the DNA data race. Nat Biotechnol. 2010, 28 (7): 691-693. 10.1038/nbt0710-691.PubMed CentralView ArticlePubMedGoogle Scholar
- Pratt B, Howbert JJ, Tasman NI, Nilsson EJ: MR-Tandem: parallel X! Tandem using Hadoop MapReduce on Amazon Web Services. Bioinformatics. 2012, 28: 136-137. 10.1093/bioinformatics/btr615.PubMed CentralView ArticlePubMedGoogle Scholar
- Taylor RC: An overview of the Hadoop/MapReduce/HBase framework and its current applications in bioinformatics. BMC Bioinform. 2010, 11 (Suppl 12): S1-10.1186/1471-2105-11-S12-S1.View ArticleGoogle Scholar
- Schatz M: Cloudburst: highly sensitive read mapping with MapReduce. Bioinformatics. 2009, 25 (11): 1363-1369. 10.1093/bioinformatics/btp236.PubMed CentralView ArticlePubMedGoogle Scholar
- Langmead B, Hansen KD, Leek JT: Cloud-scale RNA-sequencing differential expression analysis with Myrna. Genome Biol. 2010, 11 (8): R83-10.1186/gb-2010-11-8-r83.PubMed CentralView ArticlePubMedGoogle Scholar
- Sadasivam GS, Baktavatchalam G: A novel approach to multiple sequence alignment using hadoop data grids. Proc Int Workshop Massive Data Anal Cloud. 2010, 1-7.Google Scholar
- Lewis S, Csordas A, Killcoyne S, Hermjakob H, Hoopmann MR, Moritz RL, Deutsch EW, Boyle J: Hydra: a scale proteomic search engine which utilizes the Hadoop distributed computing framework. BMC Bioinform. 2012, 13: 324-10.1186/1471-2105-13-324.View ArticleGoogle Scholar
- Srirama SN, Jakovits P, Vainikko E: Adapting scientific computing problems to clouds using MapReduce. Futur Gener Comput Syst. 2012, 28 (1): 184-192. 10.1016/j.future.2011.05.025.View ArticleGoogle Scholar
- Ekanayake J, Li H, Zhang B, Gunarathne T, S–H B, Qiu J, Fox G: Twister: A runtime for iterative MapReduce. Proc Nineteenth ACM Int Symp High Perform Distributed Comput. 2010, 810-818.Google Scholar
- Kennedy J, Eberhart R: Swarm intelligence. 2001, San Francisco, CA: Morgan Kaufman PublishersGoogle Scholar
- Michalewicz Z: Genetic algorithms + data structures = evolution programs. 1999, Berlin, Germeny: SpringerGoogle Scholar
- Schaffter T, Marbach D, Floreano D: GeneNetWeaver: in silico benchmark generation and performance profiling of network inference methods. Bioinformatics. 2011, 27 (16): 2263-2270. 10.1093/bioinformatics/btr373.View ArticlePubMedGoogle Scholar
- Kim SY, Imoto S, Miyano S: Inferring gene networks from time series microarray data using dynamic Bayesian networks. Brief Bioinform. 2003, 4 (3): 228-235. 10.1093/bib/4.3.228.View ArticlePubMedGoogle Scholar
- Balaji S, Babu MM, Iyer LM, Luscombe NM, Aravind L: Comprehensive analysis of combinatorial regulation using the transcriptional regulatory network of yeast. J Mol Biol. 2006, 360 (1): 213-227. 10.1016/j.jmb.2006.04.029.View ArticlePubMedGoogle Scholar
- Greenfield A, Madar A, Ostrer H, Bonneau R: DREAM4: Combining genetic and dynamic information to identify biological networks and dynamical models. PLoS One. 2010, 5 (10): e13397-10.1371/journal.pone.0013397.PubMed CentralView ArticlePubMedGoogle Scholar
- Gradshtenyn IS, Ryzhik IM: Table of integrals, series, and products. 1980, New York: Academic PressGoogle Scholar
- Fusaro VA, Patil P, Gafni E, Wall D, Tonellato P: Biomedical cloud computing with Amazon web servers. PLoS Comput Biol. 2011, 7 (8): e1002147-10.1371/journal.pcbi.1002147.PubMed CentralView ArticlePubMedGoogle Scholar
- Hill MD, Marty MR: Amdahl's law in the multicore era. IEEE Comput. 2008, 41 (7): 33-38.View ArticleGoogle Scholar
- Amdahl GM: Validity of the single processor approach to achieving large scale computing capabilities, Proceedings of Amer Federation of Information Processing Societies Conference. 1967, AFIPS PressGoogle Scholar
- Chou IC, Voit EO: Recent developments in parameter estimation and structure identification of biochemical and genomic systems. Math Biosci. 2009, 219 (2): 57-83. 10.1016/j.mbs.2009.03.002.PubMed CentralView ArticlePubMedGoogle Scholar
- Huynh-Thu VA, Irrthum A, Wehenkel L, Geurts P: Inferring regulatory networks from expression data using tree-based methods. PLoS One. 2010, 5 (9): e12776-10.1371/journal.pone.0012776.PubMed CentralView ArticlePubMedGoogle Scholar
- Haury AC, Mordelet F, Vera-Licona P, Vert J-P: TIGRESS: trustful inference of gene regulation using stability selection. BMC Syst Biol. 2012, 6 (1): 145-10.1186/1752-0509-6-145.PubMed CentralView ArticlePubMedGoogle Scholar
- Bornholdt S: Systems biology: less is more in modeling large genetic networks. Sci Signal. 2005, 310 (5747): 449-Google Scholar
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