Incorporating time-delays in S-System model for reverse engineering genetic networks
- Ahsan Raja Chowdhury^{1, 2}Email author,
- Madhu Chetty^{1, 2} and
- Nguyen Xuan Vinh^{1}
https://doi.org/10.1186/1471-2105-14-196
© Chowdhury et al.; licensee BioMed Central Ltd. 2013
Received: 11 January 2013
Accepted: 7 June 2013
Published: 18 June 2013
Abstract
Background
In any gene regulatory network (GRN), the complex interactions occurring amongst transcription factors and target genes can be either instantaneous or time-delayed. However, many existing modeling approaches currently applied for inferring GRNs are unable to represent both these interactions simultaneously. As a result, all these approaches cannot detect important interactions of the other type. S-System model, a differential equation based approach which has been increasingly applied for modeling GRNs, also suffers from this limitation. In fact, all S-System based existing modeling approaches have been designed to capture only instantaneous interactions, and are unable to infer time-delayed interactions.
Results
In this paper, we propose a novel Time-Delayed S-System (TDSS) model which uses a set of delay differential equations to represent the system dynamics. The ability to incorporate time-delay parameters in the proposed S-System model enables simultaneous modeling of both instantaneous and time-delayed interactions. Furthermore, the delay parameters are not limited to just positive integer values (corresponding to time stamps in the data), but can also take fractional values. Moreover, we also propose a new criterion for model evaluation exploiting the sparse and scale-free nature of GRNs to effectively narrow down the search space, which not only reduces the computation time significantly but also improves model accuracy. The evaluation criterion systematically adapts the max-min in-degrees and also systematically balances the effect of network accuracy and complexity during optimization.
Conclusion
The four well-known performance measures applied to the experimental studies on synthetic networks with various time-delayed regulations clearly demonstrate that the proposed method can capture both instantaneous and delayed interactions correctly with high precision. The experiments carried out on two well-known real-life networks, namely IRMA and SOS DNA repair network in Escherichia coli show a significant improvement compared with other state-of-the-art approaches for GRN modeling.
Introduction
The availability of genome wide expression data has significantly increased interest in systems biology, in particular, reverse-engineering gene regulatory networks (GRNs). While static expression data allows the learning of only the network structure, i.e., transcription factors (TF) and target genes interactions, time-course data allows the modeling of detailed system dynamics over time. In our view, amongst different ways for classification [1-5], methods for reverse-engineering GRNs can be broadly categorized into six major groups, namely (i) co-expression network, (ii) Bayesian network, (iii) differential equation based approach, (iv) regression based approach, (v) meta approaches combining two or more different methods, and (vi) approaches that are based on other principles. Co-expression networks [6, 7] are coarse-scale, simplistic models that employ pairwise association measures, such as the partial correlation or conditional mutual information, for inferring the interactions between genes. These methods have low computational complexity and thus can easily scale up to very large networks of thousands of genes [8], but lack a mechanism for modeling system dynamics. Bayesian networks (BN) are more sophisticated models based on the strong foundation of probability and statistics, in which the dependencies between nodes are represented using directed edges and conditional probability distributions. A temporal form of BN, i.e., dynamic Bayesian network (DBN), allows the modeling of system dynamics in discrete time.
In this paper, we focus on differential equation (DE) based approaches, which belong to a sophisticated and well established class of methods for modeling biochemical phenomena, including GRNs [9-13]. A salient feature of all DE-based approaches is their ability to accurately model system dynamics in continuous time. Of the several linear and non-linear types of DE models employed for reconstructing GRNs, the S-System model has gained popularity recently [14-19]. Originating from the pioneering work of Savageau [20], the S-System has been considered as an excellent balance between model complexity and mathematical tractability: it is complex enough to represent a wide range of dynamics, yet is simple enough to allow certain analytical studies.
Most existing approaches for modeling GRNs attempt to capture instantaneous (non-temporal) interactions only. This is the case for all co-expression based approaches and static Bayesian networks, which do not differentiate between static and time-course expression data. There have been previous attempts for modeling time-delayed genetic interactions with dynamic Bayesian network using time-course data, such as the method proposed by Zou and Conzen [21], and also our recently proposed method GlobalMIT [23] and [24] (which we call BITGRN2 throughout this paper, as [24] is the improved version of BITGRN). The Recursive Neural Network (RNN) based methods [25-29], capable of interpreting complex temporal behavior of gene expression data, have the ability to work with time-delays. However, this time-delay issue is either not well-addressed [28, 29] or the delays are fixed for most of the existing approaches [25-27]. Further, so far RNN based methods are incapable of presenting regulations in the degradation phase, which is an inherent feature of S-System model. The ordinary differential equation (ODE) based methods [9-11] are limited to work with instantaneous interactions and are incapable of inferring time-delayed regulations. On the other hand, a delay differential equations (DDE) based model was employed in [12], that works with delay, but the time delay parameters were set manually rather than via learning from data. Kim et al.[13] proposed a DDE based method that is capable of working with time-delays. However, the method is limited to working with fixed delays, which are either set to their a-priori known values, or otherwise initialized randomly then fixed during the optimization. To the best of our knowledge, there is no differential equation based approach available that can model time-delayed and instantaneous interactions simultaneously, with the flexibility to adapt the delay parameters through optimization.
The main contributions of this paper are two-fold. First, it proposes a novel time-delayed S-System model based on a set of delay differential equations (DDE) which is capable of simultaneously capturing both - time-delayed and instantaneous interactions. Further, it incorporates time delay parameters which are not restricted to take only integer values (corresponding to time stamps in the data) as possible in other discrete-time approaches (e.g., dynamic BN), but they can take fractional values. This allows the model to capture the time delays in genetic interactions with higher accuracy, because in reality, the amount of delay takes continuous value. Second, to overcome the limitations of previous optimization approaches, our new search algorithm is designed systematically exploiting the sparse and scale-free nature of GRNs to effectively narrow down the search space. Compared to the existing two S-System based modeling approaches [16, 19], the proposed approach learns the parameters more accurately despite an increase in the number of model parameters to be learnt. Experimental studies on two synthetic and two real genetic networks show a significant improvement over recently proposed modeling techniques.
Background
Traditional S-System model
For solving Eqn. (2), only Y_{ i } = X_{ i } is computed by numerical integration, while ${Y}_{j}={\hat{X}}_{j},\forall j\ne i$, where ${\hat{X}}_{j}$ is obtained by a direct estimation based on the observed expression data of the j^{ t h } gene [15]. For direct estimation, the commonly used technique of linear spline interpolation [31] can be applied. Although this approximation may decrease the accuracy slightly, the significantly reduced computational burden allows the optimization process to converge to better solutions in much shorter time.
Model evaluation criteria
Here t denotes a specific time-stamp (TS) in the observed time series of T sample points. ${X}_{i}^{\mathit{\text{cal}}}\left(t\right)$ and ${X}_{i}^{\mathit{\text{exp}}}\left(t\right)$ denote the calculated and observed expression value of gene i at time-stamp t respectively. It is to be noted that if the data set consists of several separate time series, then the SRE criterion can simply be extended by summing over all the available time series. Due to decoupling, this SRE criterion for each gene can be minimized independently. The solution for this optimization problem is normally dense, i.e., it has many non-zero parameter values corresponding to many regulators for each gene. However, it is widely reported that GRNs are sparse in nature, and in fact follow a scale-free topology [32, 33]. Thus, a regularization term, similar to LASSO regression, is often added. Authors of [15] were the first to propose a penalty term for model complexity (Eqn. (1) of the supplementary document (Additional file 1)), which was subsequently improved by Noman and Iba [17, 34] as follows:
Here, r_{ i } is the number of transcription factors (total regulations) for gene-i. Details about this fitness function are available in [19] and a brief discussion is included in Section 1.2 of the supplementary document (Additional file 1). Although, the penalty graph generated by the model complexity part resembles the property of power-law formalism, the addition of another fractional term in the prefixes of the exponential term (J/r_{ i } and r_{ i }/I) makes the penalty term asymmetric. While a preliminary study [19] on this fitness criteria showed improvement, the applied penalty function being adhoc is not well justified.
Methods
The proposed time-delayed S-System model
Modeling time-delayed interactions
For both these matrices, {$0\le \{{\tau}_{i,j}^{g},{\tau}_{i,j}^{h}\}\le {\tau}_{\mathit{\text{max}}}$}, ∀_{i,j = 1 …N} and τ_{ max } is the maximum allowed delay of the network.
At any time, the production and degradation rate of the i^{ th } gene is affected by its own and other genes’ concentration level at their corresponding delays. If a delay τ_{ ij }, corresponding to an interaction (g_{ ij }/ h_{ ij }), is 0, we have an instantaneous interaction (provided that there is a regulation between genes i and j), whereas a non-zero value of τ_{ ij } gives a delayed interaction. Thus, the proposed Time-Delayed S-System (TDSS) model is capable of capturing both time delayed and instantaneous genetic interaction in GRNs.
Model parameters
For the traditional S-System model, in the i^{ th } sub-problem corresponding to the i^{ th } gene, the 2N + 2-parameter set Ω_{ i } = {α_{ i },β_{ i },{g_{ ij },h_{ ij }}_{j = 1…N}} needs to be estimated. In the Time-Delayed S-System model, apart from these parameters, we also have to estimate the 2N time-delay parameters ${\{{\tau}_{\mathit{\text{ij}}}^{g},{\tau}_{\mathit{\text{ij}}}^{h}\}}_{j=1\dots N}$. Thus, a 4N + 2-parameter set ${\Omega}_{i}=\{{\alpha}_{i},{\beta}_{i},{\{{g}_{\mathit{\text{ij}}},{h}_{\mathit{\text{ij}}},{\tau}_{\mathit{\text{ij}}}^{g},{\tau}_{\mathit{\text{ij}}}^{h}\}}_{j=1\dots N}\}$ needs to be learned. For learning the time-delay parameters, we follow a two-stage approach. First, we employ the Pearson correlation coefficient (PCC) technique to identify the most probable lag of the interaction between any pair of genes. For doing this, we use linear spline interpolation to intrapolate additional data points between any two actual measurements. For a given data set, the maximum time delay (τ_{ max }) permissible for the system is set by considering common regulation time scale (ranging within tens of minutes [22]) and the data sampling rate. Although the proposed TDSS is capable of dealing with any resolution of fractional delay, in this paper we have limited the minimum time-delay step size to 1/10 of the time between two time-stamps, provided that the data are regularly sampled. Else, the time-delay step size is set to 1/10 of the time between two closest time-stamp in non-regularly sampled data. While using PCC, we fix the expression profile of a regulator gene and shift the target gene’s expression profile forward incrementally one step at a time (minimum time-delay step). The time lag maximizing PCC is considered as the most probable time lag between these two genes. These most probable lag values are then used to initialize the time delay parameters for the evolutionary optimization phase.
Time responses
In the traditional S-System model, numerical integration is normally performed with the well-known fourth order Runge-Kutta method (RK4). For the Time-Delayed S-System model, we adapt the traditional RK4 method for DDE which takes into account the time delay parameters as described in detail in [35]. For the adapted RK4, initial samples of the regulator gene’s expression profile of length τ_{ max } will be designated as history information, which reduces the available sample size for training. It should be noted that the step size for RK4 integration is set at a small value, allowing the numerical integration to capture the system dynamics accurately. Again, we use linear spline interpolation to generate a continuous history profile. A detailed description of the modified RK4 is presented in Sec. 2.3 in the supplementary document (Additional file 1).
Inference mechanism
Due to the intractable nature of optimization problem, S-System parameter learning is commonly carried out via evolutionary computation (EC), namely Genetic Algorithm (GA) or Differential Evolution (DE). Recently, DE and its variants, such as trigonometric differential evolution (TDE), have been used extensively because of their versatility [18, 19, 36-38]. As an optimization tool for learning model parameters, both DE and TDE perform better than the other conventional evolutionary computation approaches [18, 19]. In this paper, we employ a new TDE approach for learning TDSS parameters. We also employ the Multistage Refinement Algorithm (MRA) [19] as a pruning mechanism for eliminating the weak regulations from the resulting network. Details related to TDE, initial population generation, and MRA are presented in Sec. 1.3 and Sec. 2.4 of the supplementary document (Additional file 1).
Model evaluation criterion
with I and J being the maximum and minimum in-degree respectively. Note that in our formulation, r_{ i } and I are restricted to be smaller than or equal to N, since a transcription factor generally does not affect both its target gene’s production and degradation simultaneously. In our ASRE criterion, in contrast to a fixed weighting factor c as in Eqn. (4), the penalty factor C_{ i } takes the form of an inverse power-law function. This is motivated by the fact that biological networks often have a scale-free structure, in which the node connectivity degree x distributes according to a power-law distribution, P(x) ∝ x^{-γ}, with the scaling parameter γ ∈ [ 2,3] for various networks in nature, society and technology [33]. Gene regulatory networks generally have low in-degrees, with the number of genes having high in-degree diminishing according to a power-law form. Note that in our formulation, we also enforce a minimum in-degree J, thus genes with the number of in-degree falling in-between the min-max number of in-degree [J,I] are not penalized (C_{ i } = 1), while genes falling out of this region are penalized according to an inverse power law term (C_{ i } = 1 + d^{ γ }, where γ = 2 and d is the number of missing or violated regulations). Sec. 2.4.2 and 2.4.3 in the supplementary document (Additional file 1) explain how our algorithm adaptively adjusts the [J,I] region during the optimization process.
Salient features
We highlight the salient features of the proposed optimization framework as follows:
- (i)
Adaptive regulator set size: Our algorithm adaptively and continually adjusts the values of the min-max in-degree region [J,I]. Initially, we set J = 0 and I = a value less than or equal to N based on the size of the network. Then, for every l generations, we examine the smallest and largest in-degree within the population respectively and set these as new values for J and I.
- (ii)
Adaptive balancing factor B_{ i }: The balancing factor B_{ i } is included in Eqn. (12) to dynamically balance the terms corresponding to the network accuracy and the model complexity. For the first initial tens of generations, we set the value of B_{ i } to zero, i.e., we emphasize on network quality first. This allows the algorithm to quickly improve the network accuracy as there are no constraints on complexity. We allow the algorithm to proceed in this manner either until a fixed n_{ e } generations are executed or until the squared relative error is smaller than a specified threshold γ_{ i }. When the individuals in the population achieve stability and improved accuracy, the value of B_{ i } is updated as follows: from the top 50% individuals in the population, we calculate the average network accuracy ANA (first term of Eqn. (12)) and the average model complexity AMC (second term of Eqn. (12), i.e., 2N/(2N-r_{ i })), then set B_{ i } = ANA/AMC. With this, effect of the network accuracy is maintained in ‘balance’ with model complexity. Next, we replace the worst 50% individuals with randomly initialized individuals, and the optimization continues with the value of B_{ i } computed as above.
While our preliminary studies reported earlier [19] also used adaptation of I and J, the implementation was rather adhoc, and had static weight factor. The proposed model evaluation criteria represented by Eqn. (12) and Eqn. (13) are thus novel and perform systematic adaptation of I and J while also simultaneously carrying out adaptive balancing of network complexity and accuracy.
Results and discussions
The proposed TDSS model is evaluated experimentally using both synthetic and real-life networks. As the model parameters increase quadratically with the network size, large scale modeling with the S-System based models remains a long-standing challenge. For this reason, like previous research on the S-System [16, 19, 39-41], we mainly test our method on small and medium sized networks. We employ two synthetic network studies of different sizes, i.e., a small network with 5 genes and a 20-gene medium sized network. For real-life network studies, we present experiments on two small networks, namely IRMA that contains 5 genes, and SOS DNA Repair Network in Escherichia coli containing 8 genes.
With synthetic networks, we investigate network configurations having no delays (instantaneous interactions only) and also in the presence of delays (both instantaneous and time-delayed). For each of these configurations, along with noise free data, we have considered three different levels of Gaussian noise. The four well-known performance measures [24, 42] namely sensitivity (S_{ n }), specificity (S_{ p }), precision (P_{ r }) and F-score (F) have been applied for network evaluation. For the methods with executable code available, namely ALG [16, 34] and REGARD [19], we run the respective programs on our generated data. For other methods where no code is available, we extract the performance measure values from their respective original publications for comparison where possible.
With real-life networks, for IRMA, the comparison is carried out with 7 other approaches, namely, ALG [16, 34], REGARD [19], BITGRN2 [24, 42], BANJO [43], TDARACNE [44], NIR and TSNI [45], ARACNE [7]. For the E. coli network, the performance has been evaluated with ALG [34], REGARD [19], S-Tree based approach [39], two approaches from Kimura et al.[40, 46], and several BN based approaches, e.g., [47], BANJO [43], BITGRN2 [24, 42] and GlobalMIT [23]. In addition, the time-responses of the inferred networks are compared with the actual time expression profiles to show the accuracy of the proposed model in capturing gene expression dynamics. All the inferred time-responses are shown for the best inference result (in terms of the objective function value, out of 5 separate runs) with error bars indicating the 95% confidence interval (CI).
The proposed algorithm is implemented in C++ using a 2.16 GHz Dual-core CPU PC with 3 GB of RAM. This code is made available upon request. The parameter values for the TDE algorithm were set as follows: Mutation Factor F_{ o } = 0.5, Trigonometric Mutation Factor F_{ t } =0.05, Crossover Factor CF = 0.8, population size Pop = 100. The number of generations when B_{ i } =0 is set to n_{ e } =50 and the specified threshold γ_{ i } to half the value of ASRE of the best individuals in initial population. Once B_{ i } is reset, the in-degrees are updated with ARGC algorithm (details in Sec. 2.4.3 of Additional file 1) in every l = 50 generations. The pruning factor ψ = 0.25 (details in Sec. 2.4.5 of Additional file 1) is used in both the stages of Multistage Refinement Algorithm (MRA). For synthetic network, M = 10 datasets are used for reverse-engineering, generated for each network from 10 different initial conditions. We have executed the proposed optimization method with TDSS for 1000 generations in the first phase while in the the second phase, MRA is executed for 250 generations. The maximum time delay value (τ_{ max }) was set to 3 time-stamps (TS) for all the synthetic networks, as the maximum delay among all delayed regulations was manually set to 3TS for synthetic data generation. For the IRMA networks, τ_{ max } was set to 100 minutes, equivalent to 10TS. For the E. coli network, we set τ_{ max } to 1h, which is also 10TS. The experiments are carried out with 5 separate runs of TDSS with different random initializations for each network. In the following, the best case result represents the best solution of these 5 separate runs, in terms of the objective function value, i.e., the adaptive squared relative error (ASRE) in Eqn. (12).
Synthetic networks
Small scale synthetic network
S-System parameters for the 5-gene synthetic network of [[14]]
Gene 1 | α_{1}=5.0,g_{1,3}=1.0,g_{1,5}=-1.0,β_{1}=10.0,h_{1,1}=2.0 |
Gene 2 | α_{2}=10.0,g_{2,1}=2.0,β_{2}=10.0,h_{2,2}=2.0 |
Gene 3 | α_{3}=10.0,g_{3,2}=-1.0,β_{3}=10.0,h_{3,2}=-1.0,h_{3,3}=2.0 |
Gene 4 | α_{4}=8.0,g_{4,3}=2.0,g_{4,5}=-1.0,β_{4}=10.0,h_{4,4}=2.0 |
Gene 5 | α_{5}=10.0,g_{5,4}=2.0,β_{5}=10.0,h_{5,5}=2.0 |
Remaining g_{i,j}=h_{i,j}=0,∀i,j = 1,2…,5 |
Three different delay configurations of the 5-gene synthetic network
Configuration 1 | ${\tau}_{i,j}^{g}={\tau}_{i,j}^{h}=0$ (Non-delayed network) |
(Conf-1) | ∀i,j = 1,2…,5 |
Configuration 2 | ${\tau}_{1,5}^{g}={\tau}_{2,1}^{g}={\tau}_{3,2}^{g}={\tau}_{3,2}^{h}={\tau}_{4,5}^{g}=1.0$ |
(Conf-2) | remaining ${\tau}_{i,j}^{g}={\tau}_{i,j}^{h}=0,\forall i,j=1,2\dots ,5$ |
Configuration 3 | ${\tau}_{1,3}^{g}=1.1,{\tau}_{2,1}^{g}=1.2,{\tau}_{3,2}^{g}=1.3,{\tau}_{5,4}^{g}=2.1$, ${\tau}_{4,5}^{g}={\tau}_{3,2}^{h}=1.0$, |
(Conf-3) | remaining ${\tau}_{i,j}^{g}={\tau}_{i,j}^{h}=0,\forall i,j=1,\dots ,5$ |
A. Network with no-delay (Conf-1)
Experimental results on Conf-1 (5-gene synthetic network)
Conf-1 (No-delay network) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0% Noise | 5% Noise | 10% Noise | 25% Noise | |||||||||||||
S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | |
TDSS (Best) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.95 | 0.87 | 0.93 | 1.00 | 0.87 | 0.72 | 0.84 |
TDSS | 1.00 ± | 1.00 ± | 1.00 ± | 1.00 ± | 1.00 ± | 0.98 ± | 0.95 ± | 0.97 ± | 1.00 ± | 0.93 ± | 0.84 ± | 0.91 ± | 1.00 ± | 0.84 ± | 0.68 ± | 0.81 ± |
(Average ±Std) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.05 | 0.03 | 0.00 | 0.02 | 0.05 | 0.03 | 0.00 | 0.01 | 0.02 | 0.01 |
ALG [34] | 1.00 | 0.35 | 0.35 | 0.52 | 1.00 | 0.68 | 0.52 | 0.68 | 0.92 | 0.65 | 0.48 | 0.63 | 0.91 | 0.64 | 0.46 | 0.60 |
REGARD [19] | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.97 | 0.93 | 0.96 | 1.00 | 0.92 | 0.80 | 0.86 | 1.00 | 0.84 | 0.68 | 0.81 |
Noman et al.[17] | 1.00 | 0.45 | 0.39 | 0.27 | 1.00 | 0.73 | 0.57 | 0.44 | 0.92 | 0.75 | 0.57 | 0.39 | 0.89 | 0.79 | 0.61 | 0.38 |
Kimura [48] | 1.00 | 0.84 | 0.68 | 0.58 | - | - | - | - | - | - | - | - | - | - | - | - |
S-Tree [39] | 1.00 | 1.00 | 1.00 | 1.00 | - | - | - | - | - | - | - | - | - | - | - | - |
Hasan et al.[49] | 1.00 | 0.45 | 0.39 | 0.27 | 1.00 | 0.73 | 0.57 | 0.44 | 1.00 | 0.68 | 0.52 | 0.40 | - | - | - | - |
DPSO -L1 [50] | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.81 | 0.65 | 0.54 | - | - | - | - | 0.89 | 0.75 | 0.55 | 0.32 |
LTV [51] | 1.00 | 0.73 | 0.80 | 0.72 | 1.00 | 0.70 | 0.75 | 0.66 | 0.90 | 0.60 | 0.69 | 0.51 | - | - | - | - |
BANJO [43] | 0.42 | 0.77 | 0.63 | 0.50 | 0.42 | 0.70 | 0.56 | 0.48 | 0.42 | 0.70 | 0.56 | 0.48 | 0.33 | 0.70 | 0.50 | 0.40 |
BITGRN2 [42] | 0.92 | 0.77 | 0.79 | 0.85 | - | - | - | - | - | - | - | - | - | - | - | - |
B. Networks with delay (Conf-2 and Conf-3)
Experimental results on Conf-2 and Conf-3 (5-gene synthetic network)
Conf-2 (Delayed network) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0% Noise | 5% Noise | 10% Noise | 25% Noise | |||||||||||||
S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | |
TDSS (Best) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.92 | 0.81 | 0.90 | 1.00 | 0.84 | 0.68 | 0.82 |
TDSS | 1.00 ± | 1.00 ± | 1.00 ± | 1.00 ± | 1.00 ± | 0.98 ± | 0.96 ± | 0.98 ± | 0.98 ± | 0.90 ± | 0.77 ± | 0.87 ± | 0.97 ± | 0.83 ± | 0.67 ± | 0.80 ± |
(Average ±Std) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.06 | 0.03 | 0.03 | 0.02 | 0.04 | 0.03 | 0.04 | 0.02 | 0.03 | 0.03 |
ALG [34] | 0.92 | 0.78 | 0.60 | 0.72 | 0.92 | 0.78 | 0.60 | 0.73 | 0.77 | 0.78 | 0.56 | 0.65 | 0.77 | 0.78 | 0.56 | 0.65 |
REGARD [19] | 0.92 | 0.95 | 0.86 | 0.89 | 0.92 | 0.95 | 0.86 | 0.89 | 0.85 | 0.87 | 0.69 | 0.76 | 0.77 | 084 | 0.63 | 0.70 |
BANJO [43] | 0.42 | 0.77 | 0.63 | 0.50 | 0.42 | 0.70 | 0.56 | 0.48 | 0.42 | 0.70 | 0.56 | 0.48 | 0.33 | 0.70 | 0.50 | 0.40 |
Conf-3 (Delayed network) | ||||||||||||||||
0% Noise | 5% Noise | 10% Noise | 25% Noise | |||||||||||||
S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | |
TDSS (Best) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.95 | 0.87 | 0.93 | 0.92 | 0.84 | 0.67 | 0.78 |
TDSS | 1.00 ± | 1.00 ± | 1.00 ± | 1.00 ± | 0.99 ± | 0.98 ± | 0.94 ± | 0.96 ± | 0.99 ± | 0.92 ± | 0.83 ± | 0.89 ± | 0.88 ± | 0.80 ± | 0.60 ± | 0.71 ± |
(Average ±Std) | 0.00 | 0.00 | 0.00 | 0.00 | 0.03 | 0.02 | 0.06 | 0.03 | 0.03 | 0.06 | 0.11 | 0.08 | 0.04 | 0.02 | 0.02 | 0.02 |
ALG [34] | 0.85 | 0.87 | 0.69 | 0.76 | 0.77 | 0.79 | 0.56 | 0.65 | 0.77 | 0.78 | 0.50 | 0.60 | 0.70 | 0.73 | 0.48 | 0.56 |
REGARD [19] | 0.85 | 0.95 | 0.85 | 0.85 | 0.77 | 0.92 | 0.77 | 0.77 | 0.77 | 0.81 | 0.67 | 0.71 | 0.77 | 0.73 | 0.50 | 0.60 |
BANJO [43] | 0.42 | 0.70 | 0.56 | 0.48 | 0.42 | 0.62 | 0.50 | 0.46 | 0.30 | 0.62 | 0.44 | 0.39 | 0.25 | 0.54 | 0.33 | 0.29 |
Medium scale synthetic network
S-System parameters for the 20-gene synthetic network [34]
α_{ i },β_{ i } | 10.0 |
---|---|
g _{i,j} | g_{ 3,15 } = -0.7,g_{ 5,1 } = 1.0,g_{ 6,1 } = 2.0,g_{ 7,2 } = 1.2,g_{ 7,3 } = -0.8,g_{ 7,10 } = 1.6,g_{ 8,3 } = -0.6,g_{ 9,4 } = 0.5,g_{ 9,5 } = 0.7,g_{ 10,6 } = -0.3,g_{ 10,14 } = 0.9,g_{ 11,7 } = 0.5,g_{ 12,1 } = 1.0,g_{ 13,10 } = -0.4,g_{ 13,17 } = 1.3,g_{ 14,11 } = -0.4,g_{ 15,8 } = 0.5,g_{ 15,11 } = -1.0,g_{ 15,18 } = -0.9,g_{ 16,12 } = 2.0,g_{ 17,13 } = -0.5,g_{ 18,14 } = 1.2,g_{ 19,12 } = 1.4,g_{ 19,17 } = 0.6,g_{ 20,14 } = 1.0,g_{ 20,17 } = 1.5, other g_{i,j} = 0 |
h _{i,j} | 1.0 if (i=j), 0.0 otherwise, ∀i,j = 1,2…,20 |
Two different delay configurations of the 20-gene synthetic network
Configuration 4 | ${\tau}_{i,j}^{g}={\tau}_{i,j}^{h}=0$ (Non-delayed network) |
(Conf-4) | ∀i,j = 1,…,20 |
Configuration 5 | ${\tau}_{3,15}^{g}=1.1,{\tau}_{5,1}^{g}=1.3,{\tau}_{7,10}^{g}=1.6,{\tau}_{10,6}^{g}=2.1,{\tau}_{14,11}^{g}=1.5,{\tau}_{18,14}^{g}=1.9,{\tau}_{19,17}^{g}=0.6$, |
(Conf-5) | ${\tau}_{20,14}^{g}=1.0$, and remaining ${\tau}_{i,j}^{g}={\tau}_{i,j}^{h}=0,\forall i,j=1,2\dots ,20$ |
A. Network with no-delay (Conf-4)
Experimental results on the 20-gene network
Conf-4 (No-delay network) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0% Noise | 5% Noise | 10% Noise | 25% Noise | |||||||||||||
S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | |
TDSS (Best) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.98 | 0.91 | 0.60 | 0.74 | 0.91 | 0.90 | 0.53 | 0.67 |
TDSS | 0.98 ± | 0.97 ± | 0.81 ± | 0.88 ± | 0.96 ± | 0.90 ± | 0.60 ± | 0.72 ± | 0.96 ± | 0.90 ± | 0.56 ± | 0.71 ± | 0.90 ± | 0.87 ± | 0.47 ± | 0.62 ± |
(Average ±Std) | 0.01 | 0.03 | 0.13 | 0.01 | 0.03 | 0.06 | 0.23 | 0.19 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.03 | 0.03 |
ALG [34] | 0.98 | 0.85 | 0.47 | 0.63 | 0.98 | 0.84 | 0.44 | 0.61 | 0.85 | 0.90 | 0.54 | 0.69 | 0.87 | 0.86 | 0.44 | 0.58 |
REGARD [19] | 0.98 | 0.90 | 0.56 | 0.71 | 0.98 | 0.87 | 0.49 | 0.65 | 0.96 | 0.86 | 0.56 | 0.70 | 0.89 | 0.87 | 0.47 | 0.61 |
DPSO-L1^{ * }[50] | 0.93 | 1.00 | 1.00 | 0.90 | - | - | - | - | 0.71 | 1.00 | 1.00 | 0.61 | - | - | - | - |
BANJO [43] | 0.67 | 0.85 | 0.35 | 0.46 | 0.62 | 0.79 | 0.27 | 0.38 | 0.56 | 0.75 | 0.22 | 0.31 | 0.44 | 0.70 | 0.16 | 0.24 |
BITGRN2 [42] | 0.70 | 0.85 | 0.40 | 0.50 | 0.60 | 0.85 | 0.38 | 0.45 | 0.55 | 0.84 | 0.30 | 0.40 | - | - | - | - |
Conf-5(Delayed network) | ||||||||||||||||
0% Noise | 5% Noise | 10% Noise | 25% Noise | |||||||||||||
S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F | |
TDSS (Best) | 1.00 | 0.96 | 0.73 | 0.84 | 0.98 | 0.96 | 0.73 | 0.89 | 0.96 | 0.90 | 0.56 | 0.67 | 0.93 | 0.89 | 0.51 | 0.66 |
TDSS | 0.96 ± | 0.95 ± | 0.72 ± | 0.82 ± | 0.96 ± | 0.92 ± | 0.61 ± | 0.75 ± | 0.95 ± | 0.88 ± | 0.51 ± | 0.66 ± | 0.92 ± | 0.89 ± | 0.50 ± | 0.65 ± |
(Average ±Std) | 0.02 | 0.01 | 0.07 | 0.04 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 |
ALG [34] | 0.89 | 0.85 | 0.43 | 0.58 | 0.87 | 0.87 | 0.46 | 0.60 | 0.83 | 0.81 | 0.45 | 0.60 | 0.78 | 0.79 | 0.37 | 0.51 |
REGARD [19] | 0.91 | 0.88 | 0.48 | 0.63 | 0.91 | 0.87 | 0.48 | 0.63 | 0.87 | 0.80 | 0.47 | 0.61 | 0.77 | 0.75 | 0.41 | 0.57 |
BANJO [43] | 0.60 | 0.79 | 0.26 | 0.36 | 0.56 | 0.73 | 0.21 | 0.30 | 0.49 | 0.70 | 0.17 | 0.26 | 0.44 | 0.69 | 0.15 | 0.23 |
B. Network with delay (Conf-5)
Real-life biological networks
The IRMA network
Experimental results for IRMA network, reconstructed from ON dataset
Original network | Simplified network | |||||||
---|---|---|---|---|---|---|---|---|
Methods | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F |
TDSS (Best) | 0.85 | 0.86 | 0.69 | 0.76 | 0.80 | 0.92 | 0.80 | 0.80 |
TDSS | 0.80 ± | 0.84 ± | 0.64 ± | 0.71 ± | 0.76 ± | 0.89 ± | 0.75 ± | 0.75 ± |
(Avg ±StDev) | 0.04 | 0.02 | 0.04 | 0.04 | 0.05 | 0.02 | 0.04 | 0.03 |
ALG [16] | 0.77 | 0.27 | 0.27 | 0.40 | 0.80 | 0.42 | 0.36 | 0.50 |
REGARD [19] | 0.69 | 0.83 | 0.60 | 0.64 | 0.70 | 0.75 | 0.54 | 0.61 |
0.63 | 1.00 | 1.00 | 0.77 | 0.67 | 1.00 | 1.00 | 0.80 | |
TDARACNE [44] | 0.63 | 0.88 | 0.71 | 0.67 | 0.67 | 0.90 | 0.80 | 0.73 |
ARACNE [7] | 0.60 | - | 0.50 | 0.54 | 0.33 | - | 0.25 | 0.28 |
NIR & TSNI [45] | 0.50 | 0.94 | 0.80 | 0.63 | 0.50 | - | 0.50 | 0.50 |
BANJO [43] | 0.24 | 0.76 | 0.33 | 0.29 | 0.50 | 0.70 | 0.50 | 0.50 |
Experimental results for IRMA network, reconstructed from OFF dataset
Original network | Simplified network | |||||||
---|---|---|---|---|---|---|---|---|
Methods | S _{ n } | S _{ p } | P _{ r } | F | S _{ n } | S _{ p } | P _{ r } | F |
TDSS (Best) | 0.85 | 0.81 | 0.65 | 0.73 | 1.00 | 0.92 | 0.83 | 0.91 |
TDSS | 0.80 ± | 0.83 ± | 0.63 ± | 0.70 ± | 0.90 ± | 0.87 ± | 0.75 ± | 0.81 ± |
(Avg ±StDev) | 0.04 | 0.02 | 0.03 | 0.02 | 0.07 | 0.03 | 0.06 | 0.06 |
ALG [16] | 0.76 | 0.56 | 0.38 | 0.57 | 0.80 | 0.75 | 0.57 | 0.67 |
REGARD [19] | 0.77 | 0.76 | 0.53 | 0.63 | 0.80 | 0.79 | 0.62 | 0.70 |
0.50 | 0.94 | 0.80 | 0.62 | 0.50 | 0.90 | 0.75 | 0.60 | |
TDARACNE [44] | 0.60 | - | 0.37 | 0.46 | 0.75 | - | 0.50 | 0.60 |
ARACNE [7] | 0.33 | - | 0.25 | 0.28 | 0.60 | - | 0.50 | 0.54 |
NIR & TSNI [45] | 0.38 | 0.88 | 0.60 | 0.47 | 0.50 | 0.90 | 0.75 | 0.60 |
BANJO [43] | 0.38 | 0.88 | 0.60 | 0.46 | 0.33 | 0.90 | 0.67 | 0.44 |
Regulations within the IRMA network inferred by TDSS with corresponding τ values
Inferred | IRMA-ON | IRMA-OFF | ||
---|---|---|---|---|
regulations | lag( τ) values | |||
by TDSS | Time-stamps | Minutes | Time-stamps | Minutes |
S W I5 → C B F1 | 9.4 | 94 | 9.2 | 92 |
S W I5 → G A L80 | 2.3 | 23 | 1.7 | 17 |
S W I5 → A S H1 | 1.8 | 18 | 1.0 | 10 |
G A L4 → S W I5 | 0.0 | 0 | 0.0 | 0 |
G A L4 → G A L80 | 0.0 | 0 | - | - |
G A L80 → G A L4 | 0.0 | 0 | - | - |
A S H1 ⊣ C B F1 | - | - | 0.4 | 4 |
C B F1 → G A L4 | - | - | 0.0 | 0 |
The SOS DNA repair network in Escherichia coli
Next, we consider the well-studied SOS DNA repair network within Escherichia coli (E. coli). While the entire DNA repair system of E.coli involves more than 100 genes [39, 47], only its 30 genes contribute towards key regulations at the transcription level. We make use of the expression data set collected by Ronen et al.[52], which contains information about 8 genes namely uvrD, lexA, umuD, recA, uvrA, uvrY, ruvA, and polB. The data sets are obtained from four different experiments under various UV light conditions, with the gene expression levels being measured at 50 instants evenly spaced at a 6-minute interval. Following [34, 40, 46], we normalize the input data by dividing the expression profile of each gene by its maximum value. Historically, there were two versions of this SOS network in the literature, one involving 6 genes (uvrD, lexA, umuD, recA, uvrA and polB) [19, 39, 46], and another involving all the 8 genes [23, 24, 43, 47], both inferred from Ronen et al.’s expression data [52]. Herein, we study both the networks.
“True”+“Novel” interactions of E. coli network inferred by TDSS and other state-of-the-art methods
Considering 6-gene subnetwork | Considering 8-gene subnetwork | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
True | Proposed | REGARD | S-Tree | NGnet | Kimura | Proposed | ALG | Perrin | BANJO | GlobalMIT | BITGRN2 |
positives | (TDSS) | [19] | [39] | [46] | et al[53] | (TDSS) | [34] | et al[47] | [43] | [23] | [24] |
lexA⊣recA | √ | √ | √ | √ | ⋆ | √ | √ | √ | √ | √ | |
lexA⊣lexA | √ | √ | √ | √ | ⋆ | √ | √ | √ | |||
lexA⊣umuD | √ | √ | √ | √ | ⋆ | √ | √ | √ | √ | ||
lexA⊣uvrD | √ | √ | √ | ⋆ | √ | √ | √ | ||||
lexA⊣uvrA | √ | √ | √ | ⋆ | √ | √ | √ | √ | √ | √ | |
lexA⊣polB | √ | √ | √ | ⋆ | √ | √ | |||||
recA→lexA | √ | ⋆ | ⋆ | √ | √ | √ | |||||
lexA⊣uvrY | √ | √ | |||||||||
lexA⊣ruvA | |||||||||||
Total TP inferred | 6 | 5 | 6 | 5 | 0 | 7 | 6 | 4 | 2 | 5 | 4 |
Novel Interactions | |||||||||||
umuD→lexA | √ | √ | ⋆ | √ | √ | √ | √ | ||||
uvrA⊣lexA | √ | √ | ⋆ | √ | √ | √ | |||||
uvrA⊣recA | √ | ⋆ | ⋆ | √ | √ | √ | |||||
Total Novel Interactions | 2 | 2 | 1 | 0 | 1 | 2 | 1 | 3 | 1 | 2 | 0 |
It should be noted that, other than the known regulations reported in Table 11, considered as true positives, the proposed TDSS also inferred some unknown regulations. These can be either novel regulatory interactions, or false positive findings. These interactions are shown as “Novel Interactions” in Table 11. We refer to the existing state-of-the-art methods where these unknown regulations were justified. For example, the regulation of lexA by umuD was previously discovered and discussed in [47] and [16],[34]. This regulation was also discovered by two of our previously proposed methods REGARD [19] and GlobalMIT [23]. This regulation is inferred by the proposed TDSS on both the 6-gene and 8-gene networks. Further, the regulation uvrA ⊣lexA was also inferred by TDSS for both networks. This interaction was also previously reported in [47] and [53]. Finally, the regulation uvrA ⊣recA was inferred by 4 existing methods [23],[39],[43],[47], while TDSS did not discover this connection. Historically, all these three novel regulations mentioned in Table 11 were first reported by Perrin et al.[47], and later re-discovered by other methods [16],[43]. However, for confirming the biological validity of these interactions, suitable wet-lab experiments are yet to be performed. It is noted that for TDSS and other S-System based methods, self-regulations in either or both the production or degradation phase is normally needed to balance the model. However we clarify that, self-regulations in DE based approaches reflect the self-dependency of a gene expression upon its own value at a previous time point, rather than a physical self-interaction.
Regulations of E. coli SOS network inferred by TDSS with corresponding τ values
Inferred | 6-gene network | 8-gene network | ||
---|---|---|---|---|
regulations | Lag( τ) values | Lag( τ) values | ||
by TDSS | Time-stamps | Minutes | Time-stamps | Minutes |
lexA⊣uvrD | 1.8 | 10.8 | 1.9 | 11.4 |
lexA⊣lexA | 0.7 | 4.2 | 0.6 | 3.6 |
lexA⊣umuD | 0.0 | 0.0 | 0.1 | 0.6 |
lexA⊣recA | 2.1 | 12.6 | 2.3 | 13.8 |
lexA⊣uvrA | 0.0 | 0.0 | 0.0 | 0.0 |
lexA⊣polB | 0.0 | 0.0 | 0.0 | 0.0 |
umuD→lexA | 0.0 | 0.0 | 0.0 | 0.0 |
uvrA⊣lexA | 2.1 | 12.6 | 1.9 | 11.4 |
lexA⊣uvrY | - | - | 0.0 | 0.0 |
Computational efficiency
Conclusion
Time-delayed regulations are an inherent characteristics of all biological networks. While there have been some recent efforts using Bayesian network (BN) approach to simultaneously model time-delayed and instantaneous interactions, the current state of the art S-System approaches cannot model time-delayed interactions. In this paper, we have proposed a novel method to incorporate time-delayed interactions in the existing S-System modeling approaches for reverse engineering genetic networks. The proposed Time-delayed S-System (TDSS) model is capable of simultaneously representing both instantaneous and time-delayed regulations. Apart from the kinetic order and rate constant parameters as in traditional S-System models, additional parameters for the time delays are necessary for TDSS full description. To make the optimization effective and efficient in the increased parameter space, we proposed a novel objective function based on the sparse and scale-free nature of genetic network. The inference method was also redesigned, based on adaptive systematic adaptation of the max and min in-degrees for gene cardinality, and systematic balancing between time response accuracy and network complexity during the optimization process. The RK4 numerical integration technique has also been suitably adapted for TDSS. Investigations carried on small and medium synthetic networks with various levels of noise, as well as on two real-life genetic networks show that our approach correctly captures the time-delayed interactions and outperforms other existing S-System based methods.
Declarations
Acknowledgements
This work is supported in part by NICTA (National Information and Communication Technology Australia) research in Systems Biology flagship program.
Authors’ Affiliations
References
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