- Research article
- Open Access
An integer optimization algorithm for robust identification of non-linear gene regulatory networks
© Chemmangattuvalappil et al.; licensee BioMed Central Ltd. 2012
Received: 8 March 2012
Accepted: 27 August 2012
Published: 2 September 2012
Reverse engineering gene networks and identifying regulatory interactions are integral to understanding cellular decision making processes. Advancement in high throughput experimental techniques has initiated innovative data driven analysis of gene regulatory networks. However, inherent noise associated with biological systems requires numerous experimental replicates for reliable conclusions. Furthermore, evidence of robust algorithms directly exploiting basic biological traits are few. Such algorithms are expected to be efficient in their performance and robust in their prediction.
We have developed a network identification algorithm to accurately infer both the topology and strength of regulatory interactions from time series gene expression data in the presence of significant experimental noise and non-linear behavior. In this novel formulism, we have addressed data variability in biological systems by integrating network identification with the bootstrap resampling technique, hence predicting robust interactions from limited experimental replicates subjected to noise. Furthermore, we have incorporated non-linearity in gene dynamics using the S-system formulation. The basic network identification formulation exploits the trait of sparsity of biological interactions. Towards that, the identification algorithm is formulated as an integer-programming problem by introducing binary variables for each network component. The objective function is targeted to minimize the network connections subjected to the constraint of maximal agreement between the experimental and predicted gene dynamics. The developed algorithm is validated using both in silico and experimental data-sets. These studies show that the algorithm can accurately predict the topology and connection strength of the in silico networks, as quantified by high precision and recall, and small discrepancy between the actual and predicted kinetic parameters. Furthermore, in both the in silico and experimental case studies, the predicted gene expression profiles are in very close agreement with the dynamics of the input data.
Our integer programming algorithm effectively utilizes bootstrapping to identify robust gene regulatory networks from noisy, non-linear time-series gene expression data. With significant noise and non-linearities being inherent to biological systems, the present formulism, with the incorporation of network sparsity, is extremely relevant to gene regulatory networks, and while the formulation has been validated against in silico and E. Coli data, it can be applied to any biological system.
The progress in the field of experimental techniques in systems biology in recent years has contributed significantly to the analysis and understanding of gene regulatory networks . The simultaneous measurement of the expression levels of thousands of genes has become possible with these techniques. The time series data of gene expression obtained from the high-throughput techniques typically contain comprehensive information about the structure of the system. However, reverse engineering that data for identification of interactions between genes and reconstruction of the regulatory network is still a challenging problem.
A variety of modeling approaches have been developed recently for inferring genetic networks from gene expression data. Identification algorithms are dependent on how the network is modeled , and include Boolean logic [3, 4], Bayesian [5–7], and information-theoretic approaches . Several approaches use steady state information, the data of which typically coming from “structural perturbations” (such as gene knockout studies) , which might be difficult to obtain for some systems. Alternate approaches using time series data include dynamic Bayesian networks [10, 11] and differential equation-based models [12, 13]. Of the latter, initial reports on reverse engineering gene networks assumed linear model approximations [13, 14]. While such approximations retain simplicity in the identification algorithm, it may be inadequate in predicting strongly non-linear systems. One way of representing non-linear gene dynamics is the S-system model, a power-law formulation which incorporates both production and degradation terms of the genes. Previous studies have looked into network identification of non-linear systems with the S-system [1, 12, 15–20], which presents a more challenging task than identification of linear systems. In addition to non-linearities, gene regulatory networks are highly noisy and stochastic  which can lead to difficulties during network inference. Therefore, a strong need exists for robust network identification of non-linear systems in the presence of high system variability, while also being able to incorporate relevant biological information.
The performance of the developed bi-level integer programming algorithm is demonstrated on three case studies. In the first case study, we consider in silico gene expression data generated from a benchmark artificial 5-gene network model. In the second case study, the applicability of the algorithm on a larger network is tested using an in silico 10-gene network. In the third case study, the algorithm is applied to an experimental data set of the SOS DNA repair system in E.coli.
I Case Study 1: Five gene network model
The purpose of this case study is to validate the algorithm on a small network with and without experimental noise. The chosen 5-gene network model  has been used as a benchmark problem by different research groups to test the validity of their algorithms [15, 19].
IA Network identification without noise
Figure 2(b) illustrates the 5-gene network identified using the above formulation. The kinetic orders (g ij ) are depicted over the connection and the kinetic rate constants (αij, βij) are depicted in brackets. The precision and recall value were both a perfect 1.0, indicating the accuracy with which the proposed algorithm predicted the network structure from time profile gene expression data. In addition, the identified kinetic orders and rate constants are also in agreement with the actual network model presented in Additional file 1 equation (1). These results validate the performance of the algorithm for a small network under deterministic conditions.
1B Network identification under data uncertainty
Figure 3(b) further illustrates the identified robust network connections screened for 45% occurrence, with frequency of occurrence of network connections being depicted over the connection. Quite encouragingly, the algorithm correctly identified all the existing connections in the actual network. However because of noise, the algorithm also identifies two false interactions involving gene 2, hence resulting in a recall and precision of 1 and 0.78, respectively.
Comparison of the identified S-system reaction order values to actual values
1.2 ± 0.09
-1.0 ± 0.03
2.4 ± 0.04
-3.4 ± 0.06
3.9 ± 0.05
-1.1 ± 0.03
1.9 ± 0.02
-1.0 ± 0.01
2.0 ± 0.02
Comparison of the identified S-system rate constant values to actual values
3.8 ± 0.2
18.0 ± 0.8
13.8 ± 0.9
16.2 ± 0.2
13.8 ± 0.2
11.2 ± 0.23
8.1 ± 0.1
11.8 ± 0.1
10.3 ± 0.05
8.9 ± 0.03
Effect of added noise on the network identification results
1C Deterministic network identification under data uncertainty
II Case Study 2: Ten Gene Network Model
III Case Study 3: Experimental Data of E.Coli SOS DNA repair
The proposed algorithm is next applied to the SOS DNA repair system of E.Coli, based on the gene data measured by Ronen et al. which is available online . In this model system, the response to DNA damage is governed by a few key genes, which in turn regulate the expression of more than 30 genes which have specific roles in DNA repair. A proposed model is that the RecA protein binds to single stranded DNA, and this nucleoprotein is integral in LexA cleavage, a transcription factor which is a major regulator of the DNA repair genes . The work of Ronen et al. investigates the Michaelis-Menten kinetic parameters associated with promoter activity for eight of the major genes in this system. Experimental kinetics were measured by first incorporating a GFP reporter plasmid for each of the gene’s promoter. DNA damage was induced, and the resulting GFP intensities were measured. The number of GFP molecules is proportional to the promoter activity, and can be taken to be analogous to the rate of transcription . We therefore used this promoter activity data  to represent gene expression (with the experimental intensity data normalized by the mean column intensity) and used it in our algorithm. Among the four data sets provided by the authors, we chose the third and fourth for this case study because these are measured at the same conditions. Our objective was to identify regulatory interactions between six genes: uvrD, lexA, umuD, recA, uvrA and polB.
Estimated reaction order values of the E. Coli SOS DNA repair network (connection coding: 1-uvrD, 2-lexA, 3-umuD, 4-recA, 5-uvrA, 6-polB)
0.9 ± 0.04
1.4 ± 0.03
0.9 ± 0.04
0.9 ± 0.04
0.9 ± 0.07
0.8 ± 0.08
1.0 ± 0.03
0.9 ± 0.03
1.3 ± 0.07
-0.7 ± 0.05
1.0 ± 0.14
Estimated rate constant values of the E. Coli SOS DNA repair network (connection coding: 1-uvrD, 2-lexA, 3-umuD, 4-recA, 5-uvrA, 6-polB)
3.2 ± 0.28
8.4 ± 0.12
1.5 ± 0.09
1.6 ± 0.19
1.7 ± 0.21
1.5 ± 0.17
5.3 ± 0.16
1.6 ± 0.07
1.6 ± 0.16
2.0 ± 0.15
4.3 ± 0.22
3.8 ± 0.12
In this work, we present an algorithm to identify robust regulatory networks from time profiles of gene expression data. Our identification algorithm is primarily developed on the hypothesis of sparsity of biological network connections. In our earlier work we established the validity of the hypothesis of sparsity using a simplified linear ode representation of gene expression dynamics in a deterministic system. Herein we further advance the algorithm by incorporating more realistic non-linear representation using an S-system formulation of gene expression dynamics. The identification algorithm is formulated as a bi-level optimization problem in which the upper level solves an integer programming problem while the lower level is a continuous parameter identification problem. Furthermore, we propose a framework to incorporate noisy experimental data towards identification of a robust regulatory network. This is done by first generating artificial experimental repeats using the bootstrapping technique, followed by solving the identification formulation at each of the bootstrap data sets. From this library of identified prospective networks we isolate the most-repeated network connections which we hypothesize to be a robust connection, having low variability to experimental noise.
The upper level integer programming problem is solved using GA. There are several advantages of using GA to solve the above problem, the most important being that it does not require gradient evaluation. This is a significant advantage for the above problem with non-linear ode as constraint function. In addition, GA starts its search not from a single point in the feasible parameter space, but from multiple locations specified in the starting population. Hence, it holds the chance of converging at global minima, although such convergence cannot be guaranteed with GA. However, it also suffers from the disadvantage of increased computational cost. All the computations reported here have been carried out on 2.66 Ghz processer and 16 GB RAM server. The computational time for the five gene network without noise was 1 hour and the same network with noise was 2.5 hours. The computational time for the experimental data was 3 hours. For the 10 gene network, the genetic algorithm needed more generations to converge, resulting in computational time of 11 hours. Hence, extension of the current solution procedure to a much larger data set will be expensive. While the same formulation will still be applicable in a larger system, alternate solution procedures are currently being investigated for its extension to larger networks.
Effect of error constraint on 5-gene network identification, 5% noise
Number of connections
The performance of the developed robust identification formulation is illustrated using three different systems. The first two case studies are based on in silico data which allows for detailed analysis of the performance of the algorithm. Overall the algorithm was found to demonstrate excellent predictive capability both in the small 5-gene network along with larger 10-gene network. The proposed bootstrapping scheme was found to adequately capture the precise network from the noisy data as well. Encouraged by the in silico results, we applied our algorithm to dynamic experimental data of a 6-gene network responsible for DNA damage repair in E. Coli. While verification of the identified network will be difficult for this system, the time profile of gene expression data predicted by the identified network is in good agreement with the experimental data set. A thorough literature search for existing knowledge of network interactions revealed that quite a few of the predicted connections have been reported in parallel studies. Our algorithm inferred the regulation of recA umuD and uvrA by lexA, which is consistent with the findings reported earlier . Another interesting finding is that our results suggest that polB does not influence any of the other genes in the system (pol B does not up- or down-regulate any other gene), a finding which was also reported by Kumura et al. Furthermore, our identified network shows the self-regulation of recA. This protein is the main factor responsible for sensing DNA damage, and has been reported to promote the transcription of itself, thereby promoting damage recognition, and other repair genes [27, 28].
The current approach offers an improvement on existing algorithms. Numerous studies have used the 5-gene network (the current case study I) to test the accuracy and efficiency of their network identification methods. A comparison between the methods is presented by Kimura et a l.  for the five gene network without noise. While most studies do not report the metrics of precision and recall, the accuracy of the results is still commented on. Most methods have a shorter computational time than the proposed method. However, our algorithm is able to predict a perfect network (recall and precision of 1), while the other algorithms deviate from this. Therefore, there is a trade-off between computational time and accuracy, and selection of the most appropriate method for the system of interest should be chosen judiciously. Nevertheless, this comparison shows that recall and precision are an improvement over many existing algorithms when analyzing the 5-gene network. Additional improvements could be made on the current approach to decrease computation time, such as parallel programming, or by altering the formulism (e.g. avoiding direct integration of the system of ode).
These results show that our bi-level integer optimization algorithm is able to effectively identify the topology and connection strength of gene regulatory networks, even when the gene dynamics are non-linear and noisy in nature. By using the biological trait of sparsity, the algorithm optimizes the number of connections in the network while maintaining agreement in gene temporal profiles with the experimental input data. Even with uncertainty and noise in the data, something which is unavoidable on an experimental level, our bootstrapping/identification combination was able to identify a robust network. While we have demonstrated the effectiveness of our algorithm on in silico and E. coli data, its formulation, biological relevancy, and results are applicable to any gene regulatory network, as long as time-series data is available.
S-system representation of gene expression dynamics
Identification of the regulatory network from time series gene expression data first requires modeling the dynamic evolution of the individual genes constituting the network. Here we model gene dynamics as a set of coupled non-linear ode following the S-system formulation, which captures the non-linearity in gene expression profiles using a power-law kinetic representation.
Where X i is the concentration of the gene i, α and β represent the kinetic rate constants, g and h represent the kinetic orders for the production and degradation terms, respectively, and n is the total number of species in the system, in this case total number of genes in the network. In this work, we are using a modification of the above equation by assuming that species degradation follows a first order kinetics of the corresponding species and independent of other species (h ij = 1 for i = j; 0 otherwise). While being relevant to biological systems , this assumption also reduces the unknown parameters from 2n(n + 1) to n(n + 2).
Network Identification Algorithm
λ i j = binary variable
experimental and predicted gene expression levels, respectively
α i,, β i = kinetic rates constants of ith gene's production and degradation, respectively
g ij ,h ij = kinetic orders of production and degradation, respectively
nstep = number of time points
n = number of genes constituting the network
m = number of experimental time points
In the above formulation ∑λ represents the total number of network connections, minimizing which will promote sparsity in the network. The upper level integer programming is solved using combinatorial optimization techniques since combinatorial approach is known to handle L 0 minimization problems more efficiently than approximation algorithms . Of them, evolutionary algorithms are particularly efficient in finding a good approximate solution for combinatorial problems . In this work, we have used genetic algorithm (GA) for solving the integer programming problem, while the lower level non-linear programming problem is solved using a standard least square optimization routine.
Hence the constraint is imposed on the maximum number of binary variables assigned in the upper level, but does not constrain the total size of the analyzed network. Moreover, our primary objective being sparsity of network connections, the formulation essentially tries to minimize the number of connections assigned to 1. Hence, except for the very initial phase of GA evolution, the constraint defined in equation (2) typically does not become active, and never so in the final optimal solution.
Identification of Robust Networks
Real world data typically contains noise due to experimental uncertainty and system stochasticity. Biological data are particularly notorious for its inherent heterogeneity and stochasticity . Hence it is important to explicitly account for data variability in order to increase confidence in the predicted network. In the presence of large experimental repeats it may be possible to determine robustness of identified network by repeatedly solving the network identification problem at each of the experimental data sets and analyzing the connections which are heavily repeated. However, drawing statistically significant inference would necessitate a large data set which is impractical and infeasible.
An alternative to actual experimental repeats is to use bootstrapping. The purpose of this statistical technique is to estimate the distribution of the estimator around the unknown true value θ. However, instead of achieving this with a large number of individual replicates, bootstrapping utilizes resampling of the data. In this way, a large number artificial data sets can be generated from a limited number of experimental repeats. For each bootstrap run, data samples are randomly chosen, with replacement, from the empirical distribution, with the size of each artificial set being the same as the experimental set (e.g. if the experimental set has 20 data points, so would the bootstrap set). For each bootstrap, the estimators (e.g. mean, variance, or, as in the case of the current work, regression parameters) are calculated, and with sufficient number of resampled data sets, relevant statistical information, including confidence intervals, can be estimated [26, 33].
In our algorithm, we are dealing with limited experimental data. Hence, following the above methodology, we generate a large artificial data set by repeated resampling of the limited experimental repeats. Once the bootstrapped samples are obtained, the network identification algorithm previously described is applied to all bootstrap data sets to identify a network corresponding to each. The network sets thus obtained is further analyzed to determine the frequency of occurrence of each connection in the entire set of identified networks. We hypothesize that frequent occurrence of network connections in the bootstrap samples indicate the insensitivity of the corresponding network to experimental noise, and hence claim that connection to be robust.
Where: TP (True Positive) denotes the number of connections correctly captured; FN (False Negative) denotes existing connections which are not captured in the identified network; and FP (False Positive) denotes connections which are incorrectly captured in the identified network. Following the above equation: a low value of recall would indicate a more conservative estimate which is unable to capture many of the existing connections; a low value of precision will indicate prediction of incorrect connections not appearing in the actual network; and a value of 1 will indicate perfect network identification. The flow diagram of the overall network identification algorithm is shown in Figure 1.
This work was financed by NIH (DP2-16520).
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