Department of Chemical Engineering, University of Pittsburgh, 1249 Benedum Hall, 3700 O’Hara Street, Pittsburgh, PA, 15261, USA

Abstract

Background

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.

Results

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

Conclusions

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

Background

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

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

In the current report, we model the dynamics of gene expression by S-system formulation. Upon doing so, we formulate the network identification algorithm as a bi-level optimization problem, governed by the hypothesis of network sparsity. Network sparsity has been experimentally observed in various biological systems such as the visual system of primates

Pseudo-code of the robust network identification algorithm implementation

**Pseudo-code of the robust network identification algorithm implementation.**

Results

The performance of the developed bi-level integer programming algorithm is demonstrated on three case studies. In the first case study, we consider

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

Using the S-system formulation, the 5-gene network model can be represented by the system of five coupled nonlinear ode, shown in Additional file ^{2} = 25 binary variables are introduced corresponding to each of the five connections. Genetic Algorithm (GA), used to solve the upper level integer programming problem, does not have a convergence criterion. Standard practice is to evolve the population for enough generations until no significant improvement is observed. Figure ^{-5}). Typically least square optimization routines are very sensitive to the user defined initial guess. To make sure that the algorithm can identify the underlying network structure even without any

**Additional Equations.** The two systems of ordinary differential equations shown in Additional file 1 were those used to create the

Click here for file

Identification of a 5-gene network without noise

**Identification of a 5-gene network without noise.** (**a**) Convergence study of the genetic algorithm. The number of connections identified in each of the solutions generated by GA is plotted. No feasible solution was found with less than 65 generations. (**b**) Identified network. Arrows represent positive regulation and the filled circles represent negative regulation of the genes. Kinetic orders of each connection are represented above the corresponding connecting lines and the rate constants for each gene are shown above the genes. All connections and parameters are consistent with the original differential equations used to generate the

Figure _{ij}) are depicted over the connection and the kinetic rate constants (α_{ij}, β_{ij}) 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

1B Network identification under data uncertainty

The performance of the algorithm is next analyzed in the presence of experimental noise, generated by adding 5% Gaussian noise to the time-course data generated from equation (1) shown in Additional file

Results from 5-gene network identified under data uncertainty with 5% noise

**Results from 5-gene network identified under data uncertainty with 5% noise. (a)** Number of bootstrap occurrences for each connection (1000 bootstrap samples total). **(b)** Identified network structure. Numbers above each connection represent percent occurrence, with the thick lines representing the number of connections appearing in more than 90% of the bootstrapped samples and the thin lines representing the connections appearing in more than 45% of bootstrapped samples. **(c)** frequency of specific connection values shown as heat map.

Figure

The expected values of the S-system parameters estimated at 90% confidence level are represented in Table _{ij}) and Table _{ij}, β_{ij}), which demonstrates the excellent performance of the algorithm in identifying network parameters even from noisy data. While the error of the rate constants (compared to the actual values) is relatively high, it should be noted that the results of the network identification would not be as sensitive to these parameters as to the connectivity values, and therefore the rate constant values could vary significantly and not affect the gene profiles or the recall/precision. Furthermore, the error on the reaction orders (g_{ij}) is very low, further demonstrating the accuracy of the network identification. The heat map in Figure

**Connection**

**g**_{actual}

**g**_{estimated}

G1G3

1

1.2 ± 0.09

G1G5

-1

-1.0 ± 0.03

G2G1

2

2.4 ± 0.04

G2G3

NA

-3.4 ± 0.06

G2G4

NA

3.9 ± 0.05

G3G2

-1

-1.1 ± 0.03

G4G3

2

1.9 ± 0.02

G4G5

-1

-1.0 ± 0.01

G5G4

2

2.0 ± 0.02

**Gene**

**α**_{i}

**β**_{i}

**actual**

**estimated**

**actual**

**estimated**

X1

5

3.8 ± 0.2

10

18.0 ± 0.8

X2

10

13.8 ± 0.9

10

16.2 ± 0.2

X3

10

13.8 ± 0.2

10

11.2 ± 0.23

X4

8

8.1 ± 0.1

10

11.8 ± 0.1

X5

10

10.3 ± 0.05

10

8.9 ± 0.03

To evaluate the accuracy of the formulism under increased uncertainty, the algorithm was tested under various amounts of added noise. As one would expect, the accuracy of the algorithm depends on the level of noise added to the

**Percent noise**

**Recall**

**Precision**

White Gaussian noise was added at different amounts to the five-gene network

5

1

0.78

7

0.57

1

10

0.57

1

Results from 5-gene network identified under data uncertainty with 10% noise

**Results from 5-gene network identified under data uncertainty with 10% noise. (a)** Identified network structure. Numbers above each connection represent percent occurrence. **(b)** Sensitivity of recall and precision to bootstrap occurrence threshold.

While the analysis is performed on 1000 bootstrap samples, it is computationally expensive to solve 1000 network identification problems. Hence, we investigated the sensitivity of the identified robust network on the number of bootstrap samples by considering a broad range of samples from 200 to 1000. Figure

Convergence study on network identification results using bootstrapping with 5% noise

**Convergence study on network identification results using bootstrapping with 5% noise.**

1C Deterministic network identification under data uncertainty

To assess the necessity of this bootstrapping technique, the aforementioned results were compared to a control group which did not utilize bootstrapping. To do this, a more deterministic approach was employed. Experimental replicates were generated as detailed: 10% white Gaussian noise was added to the 5-gene

Deterministic approach to network identification under noisy data

**Deterministic approach to network identification under noisy data.** Increasing number of replicates were generated using 10% noise from the

II Case Study 2: Ten Gene Network Model

In this example we investigate the performance of the developed algorithm in a larger network consisting of ten genes, as depicted in equation (2) shown in Additional file 1. For the deterministic case study the tolerance was specified at a low value of 10^{-5}. Because the 10-gene network increases the number of binary variables in the upper level to 100, more GA generations are needed to obtain a converged solution; therefore, the number of generations was increased to 1000. The identified connections and kinetic parameters are shown in Figure _{ij}) depicted over the connections and kinetic rate constants (α_{ij}, β_{ij}) in brackets over the genes. The comparison of actual and identified time series profiles is shown in Figure

Results from the 10-gene network

**Results from the 10-gene network.** (**a)** Identified network. Arrows represent positive regulation and the filled circles represent negative regulation of the genes. The kinetic orders of each connection are represented above the corresponding connecting lines and the rate constants for each gene are shown above the genes. (**b)** Time profile for the ten gene network. The triangles represent the profile generated from the

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

Identification of this 6 gene network will require 36 binary variables; hence the GA parameters were retained similar to our first case study presented earlier: 20 populations evolved through 200 generations. The error tolerance, however, had to be relaxed to a higher value of .7 because of noise inherent in the experimental data set. Figure

Results from the 5-gene experimental

**Results from the 5-gene experimental****data.** (**a)** Time profile for the gene network, based off of the mean experimental data. The triangles represent the experimental data and the lines represent the predicted profile. (**b**) Identified network structure from experimental data for six gene system. The percentage of connections in the bootstrapping samples are marked on the connections. (**c**) frequency of specific connection values shown as heat map (connection coding: 1-uvrD, 2-lexA, 3-umuD, 4-recA, 5-uvrA, 6-polB).

In the next step the robust connections of the identified network are further analyzed by bootstrapping the experimental data set. Since our previous analysis on the first case study demonstrated 200 bootstrap samples to be adequate, in this example we generated 300 artificial data sets from the original experimental repeats. The network identification algorithm was solved at each of the data sets to generate 300 alternate networks. The frequency of occurrence of each network connection is analyzed over the array of alternate network and connections appearing with over 45% frequency are considered to be robust. Figure _{ij}) and rate constants (α_{ij}, β_{ij}) with 90% confidence level are shown in Table

**Connection**

**g**_{estimated}

G1G2

0.9 ± 0.04

G1G3

1.4 ± 0.03

G1G4

0.9 ± 0.04

G2G3

0.9 ± 0.04

G2G4

0.9 ± 0.07

G2G5

0.8 ± 0.08

G3G4

1.0 ± 0.03

G3G5

0.9 ± 0.03

G4G4

1.3 ± 0.07

G4G5

-0.7 ± 0.05

G5G1

1.0 ± 0.14

**Gene**

**α**_{i}

**β**_{i}

X1

3.2 ± 0.28

8.4 ± 0.12

X2

1.5 ± 0.09

1.6 ± 0.19

X3

1.7 ± 0.21

1.5 ± 0.17

X4

5.3 ± 0.16

1.6 ± 0.07

X5

1.6 ± 0.16

2.0 ± 0.15

X6

4.3 ± 0.22

3.8 ± 0.12

Discussion

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.

In the formulation presented in equation (2), the only user defined parameter is the value of the tolerance which dictates how closely the model prediction must agree with experimental dynamics in order for the network to be considered in the overall algorithm. While for an ^{-3}) in our algorithm under noisy data failed to identify any network, as would be expected. Moreover, using a low tolerance is not advisable when using data sets with noisy replicates since we are not targeting a profile which exactly fits the noisy data; the target is to identify network profiles which describe all the noisy scenarios relatively well. On the other hand, a relaxed tolerance runs the risk of compromised prediction quality. In order to quantitatively evaluate the effect of specified tolerance on the identified network structure, the bootstrap/ bi-level optimization algorithm was repeated on the same 5-gene dataset with different tolerance values. Table

**Error**

**Precision**

**Recall**

**Number of connections**

0.13

0.78

1.0

9

0.20

0.88

0.88

7

0.25

0.78

0.78

7

0.30

0.88

0.7

5

0.35

0.88

0.7

5

The performance of the developed robust identification formulation is illustrated using three different systems. The first two case studies are based on

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

Conclusions

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

Methods

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.

For a system with N-genes, the S-system model can be represented using equation (1):

Where _{i} is the concentration of the gene _{ij} = 1 for

Network Identification Algorithm

Our network identification algorithm is primarily based on the hypothesis of sparsity of network connections governing biological systems. Hence our overall objective is to determine the sparsest network which can satisfactorily capture the observed network dynamics. Following this idea, the network identification problem is formulated as an optimization problem with the objective of promoting sparsity given the constraint of maximizing predictive capacity. Such problem definition results in a bi-level optimization problem, where the constraint itself is an unconstrained optimization problem. In the current formulation using S-system to model the gene expression level (equation (1)), the kinetic orders (_{ij}) are decomposed into two parts: binary part, λ_{ij}, which determines the existence of the connection; and continuous part, ρ_{ij}, representing the nature and strength of interaction for an existing connection. A value of 1 of the binary variable λ_{ij} would indicate the presence of the corresponding connection _{ij} are optimized to maximize network prediction and hence minimize deviation of the network predictions from the observables. The lower level essentially optimizes both strength (magnitude) and nature (sign) of the existing connections (ρ_{ij} , reactions orders) as well as the strengths of the production and degradation rate constants (_{i}_{i} respectively). Hence it results in a continuous non-linear programming problem where the objective is to minimize the deviation of the predicted profiles from experimental data in a least square sense. A constraint of tolerance (

_{i j} = binary variable

_{i,,}_{i} = kinetic rates constants of ith gene's production and degradation, respectively

_{ij}_{ij} = kinetic orders of production and degradation, respectively

In the above formulation _{0} minimization problems more efficiently than approximation algorithms

GA is typically designed to handle unconstrained optimization problems. One technique for constraint handling in GA is by penalty function, where the constraint is conditionally incorporated in the objective function. For conditions violating the constraint the objective function is penalized, and not so otherwise. In the current formulation the constraint is incorporated in the objective function using the following modification of the objective function:

where

A significant advantage of the bi-level formulation is that it allows optimum utilization of experimental data by sequentially reducing the number of unknown parameters in the lower level. In a conventional least-square parameter estimation problem, the connectivity is fixed and includes all possible network connections. Therefore, the size of the identifiable system is restricted, governed by the availability of experimental data points so that number of unknown parameters is less than the number of data points. For instance, a single level algorithm, using the above S-System formulation, would be restricted to less than m-3 genes. However, in the current bi-level formulation, this restriction is relaxed. Because the number of network connections are first reduced in the upper level, the number of genes to be analyzed is not so restricted, with the only constraint coming from the connectivity:

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

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

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.

In order to quantify the quality of prediction of the proposed algorithm the measures of

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

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

NC, KT and IB developed the algorithm. NC and KT analyzed the data and performed output analysis. NC, KT, and IB drafted the manuscript. IB conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Acknowledgements

This work was financed by NIH (DP2-16520).