The identifiability of gene regulatory networks: the role of observation data

Abstract

Identifying gene regulatory networks (GRN) from observation data is significant to understand biological systems. Conventional studies focus on improving the performance of identification algorithms. However, besides algorithm performance, the GRN identification is strongly depended on the observation data. In this work, for three GRN S-system models, three observation data collection schemes are used to perform the identifiability test procedure. A modified genetic algorithm-particle swarm optimization algorithm is proposed to implement this task, including the multi-level mutation operation and velocity limitation strategy. The results show that, in scheme 1 (starting from a special initial condition), the GRN systems are of identifiability using the sufficient transient observation data. In scheme 2, the observation data are short of sufficient system dynamic. The GRN systems are not of identifiability even though the state trajectories can be reproduced. As a special case of scheme 2, i.e., the steady-state observation data, the equilibrium point analysis is given to explain why it is infeasible for GRN identification. In schemes 1 and 2, the observation data are obtained from zero-input GRN systems, which will evolve to the steady state at last. The sufficient transient observation data in scheme 1 can be obtained by changing the experimental conditions. Additionally, the valid observation data can be also obtained by means of adding impulse excitation signal into GRN systems (scheme 3). Consequently, the GRN systems are identifiable using scheme 3. Owing to its universality and simplicity, these results provide a guide for biologists to collect valid observation data for identifying GRNs and to further understand GRN dynamics.

Introduction

Gene regulatory network (GRN) plays a key role to control most cellular processes. They are considered to be an abstract function of complicated biochemical networks [1–3]. With the increasing development of microarray technology, the cellular dynamics for thousands of genes can be obtained simultaneously. Despite these advances, it is still a challenging task to identify GRN parameters from observation data (time series) [4, 5]. One way to address this problem is mathematical modeling, which provides an effective way to understand and investigate complex GRNs [6–10]. The S-system model consists of a set of nonlinear and strongly coupled ordinary differential equations (ODEs), which is considered to be realistic for characterizing GRN dynamics [11–14].

For inferring GRNs, a number of optimization methods are used to address this issue, such as genetic algorithm (GA) [15, 16], particle swarm optimization (PSO) [17, 18] and genetic algorithm-particle swarm optimization (GA-PSO) algorithm [19]. Conventional studies focus mainly on improving optimization algorithms performance to infer GRNs accurately. While applying these optimization methods, a strong underlying assumption is that the GRN parameters are thought to be inferred accurately, if the relative mean square error (RMSE) between the inferred and true GRN states is small enough. However, even though the RMSE is small enough, the GRN parameters cannot be inferred accurately owing to the observation data lacking sufficient transient process. Therefore, besides optimization methods, the observation data play a key role in inferring GRN parameters. Missing any of them, the GRN systems cannot be identified accurately.

In this work, three GRN S-system models are used in the identifiability test procedure. We focus on how the observation data influences the GRN identifiability, i.e., concerning the uniqueness of GRN parameters determined by input–output observation data. Additionally, the modified GA-PSO algorithm is used to illustrate that the GRN identifiability is also affected by the optimization method.

Three observation data collection schemes are compared to discuss this question. In schemes 1 and 2, the observation data are full and short of GRN transient process, respectively. Using scheme 1, the results show that three GRN models are of identifiability based on the modified GA-PSO algorithm, but incapable to do that based on the basic GA-PSO algorithm. The modified GA-PSO algorithm is of better performance. Therefore, in the schemes 2 and 3, we use the modified GA-PSO algorithm to perform the identifiability test procedure. In scheme 2, three GRN models are not of identifiability even the RMSE is small enough. In schemes 1 and 2, the observation data are obtained from zero-input GRN systems, which will evolve to the steady state at last. By means of changing the experimental conditions, these sufficient transient observation data in scheme 1 can be obtained. In addition, the valid observation data can be also obtained by means of adding excitation signal into GRN systems. In scheme 3, we give a workable observation data collecting method by means of adding the impulse excitation signals into the GRN systems. In this way, three GRN models are of identifiability. These results indicate that, besides optimization methods, the GRN identifiability is also highly dependent on the observation data. Therefore, though it is meaningful to improve the optimization algorithm performance, the validity of observation data must also be guaranteed for accurate identification. This paper is organized as follows: Section 2 gives a brief description of GRN identification, i.e., the GRN identification framework, the data collection schemes and the indices of GRN identifiability. Section 3 gives the modified GA-PSO algorithm for identifying GRN parameters. Section 4 gives the simulation results and some interesting findings. Section 5 gives the discussions. Section 6 gives the conclusion.

Problem description

Formulation of the identification problem

Generally speaking, gene regulation is a dynamical process. It describes the increasing or decreasing rate of one ingredient concentration, depending on the other ingredients concentrations. The S-system model is commonly used to describe GRN system [11–14], which is given by

$$\begin{array}{c}\hfill \frac{d{X}_i(t)}{\mathit{dt}}={\alpha }_i\prod _{j=1}^N{X}_j^{g_{\mathit{ij}}}(t)-{\beta }_i\prod _{j=1}^N{X}_j^{h_{\mathit{ij}}}(t),i=1,...,N\end{array}$$ 1

where t is the time, $X_i$ is the concentration of gene i; N is the number of genes in the GRN; ${\alpha }_i$ and ${\beta }_i$ are positive constants; the real numbers $g_{\mathit{ij}}$ and $h_{\mathit{ij}}$ are kinetic orders reflecting the interactive intensities from gene j to gene i. Equation (1) is an N-dimensional nonlinear dynamical system. There exist two terms in the right-hand side of Eq. (1). The first term and the second term describe the factor that makes the gene i increase and decrease, respectively.

Assuming that GRN has the model form given by Eq. (1), the states can be observed and recorded (as observation data), i.e., $X_i(t)$, $t=t_1,t_2,...,t_L$, L being the observation data length. GRN identification uses the observation data to identify the parameters, including ${\alpha }_i$, ${\beta }_i$, $g_{\mathit{ij}}$ and $h_{\mathit{ij}}$. One intuitive criterion of accurate identification is that the inferred GRN states should be the same as the actual ones. Practically, we observe and record the observation data (time series) from GRN system as given by

$$\begin{array}{c}\hfill X=\left[\begin{array}{c}{X}_1(t_1),X_1(t_2),...,...,X_1(t_L)\hfill \\ ⋮⋮⋮\hfill \\ X_N(t_1),X_N(t_2),...,...,X_N(t_L)\hfill \end{array}\right]\end{array}$$ 2

where $t_1,t_2,...,t_L$ are the observation time instants. If the GRN has the form as that of Eq. (1), we infer (identify) the parameters as ${\widehat{a}}_i$, ${\widehat{\beta }}_i$, ${\widehat{g}}_{\mathit{ij}}$ and ${\widehat{h}}_{\mathit{ij}}$; then the inferred GRN is given by

$$\begin{array}{c}\hfill \frac{d{\widehat{X}}_i(t)}{\mathit{dt}}={\widehat{\alpha }}_i\prod _{j=1}^N{\widehat{X}}_j^{{\widehat{g}}_{\mathit{ij}}}(t)-{\widehat{\beta }}_i\prod _{j=1}^N{\widehat{X}}_j^{{\widehat{h}}_{\mathit{ij}}}(t),i=1,...,N.\end{array}$$ 3

From a given initial condition, the Runge–Kutta method can be used to solve ordinary differential equations as in Eq. (3). The solution is recorded as a time series, which is given by

$$\begin{array}{c}\hfill \widehat{X}=\left[\begin{array}{c}{\widehat{X}}_1(t_1),{\widehat{X}}_1(t_2),...,...,{\widehat{X}}_1(t_L)\hfill \\ ⋮⋮⋮\hfill \\ {\widehat{X}}_N(t_1),{\widehat{X}}_N(t_2),...,...,{\widehat{X}}_N(t_L)\hfill \end{array}\right]\end{array}$$ 4

The relative mean-square error (RMSE) between the actual and inferred observation data is defined as

$$\begin{array}{c}\hfill RMSE=\sum _{i=1}^N{\sum _{t=t_1}^{t_L}\left\{\frac{X_i(t)-{\widehat{X}}_i(t)}{X_i(t)}\right\}}^2,\end{array}$$ 5

where N is the number of genes, L is the observation data length, and $X_i(t)$ and ${\widehat{X}}_i(t)$ are the actual and inferred GRN concentration of gene i, respectively.

To show the GRN identification procedure, the identification framework is given in Fig. 1, where the GRN model and RMSE are defined in Eqs. (3), (4) and (5). Using an optimization method, the GRN identification is performed to infer the parameters in the parameter searching space, which requires the RMSE to be minimum. Intuitively, if the RMSE is zero, that is, the inferred observation data (GRN states) are the same as the actual ones, then the common sense is that the inferred and actual GRN parameters are equal, i.e., ${\widehat{a}}_i=a_i$, ${\widehat{\beta }}_i={\beta }_i$, ${\widehat{g}}_{\mathit{ij}}=g_{\mathit{ij}}$ and ${\widehat{h}}_{\mathit{ij}}=h_{\mathit{ij}}$. In this work, we show that this conclusion depends strongly on the observation data, which we refer to as observation data validity.

Fig. 1.

GRN identification framework

Simplified GRN models

For the N-node S-system GRN model in Eq. (1), conventionally, there should be 2N(N+1) parameters needed to be inferred. To argue the major points of this work, three artificial GRNs are used in the identifiability test procedure. The GRN models 1 and 2 are both five-node systems, and the GRN model 3 is a four-node system. Their structures are assumed to be known. Correspondingly, there are remaining 23, 23 and 17 parameters in GRN models 1, 2 and 3, respectively [18–20]. Their actual and inferred models are given in Eqs. (6) and (7) (GRN model 1), Eqs. (8) and (9) (GRN model 2) and Eqs. (10) and (11) (GRN model 3), respectively. In Eqs. (7), (9) and (11), the ${\widehat{a}}_i$, ..., ${\widehat{\beta }}_i$, etc., are the parameters that need to be inferred.

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{X}}_1=15{X_3}^1{X}_5^{-0.1}-10X_1^2\hfill \\ {\dot{X}}_2=10X_1^2-10X_2^2\hfill \\ {\dot{X}}_3=10X_2^{-0.1}-10X_2^{-0.1}{X}_3^2\hfill \\ {\dot{X}}_4=8X_1^2{X}_5^{-0.1}-10X_4^2\hfill \\ {\dot{X}}_5=10X_4^2-10X_5^2\hfill \end{array}\end{array}$$ 6

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{\widehat{X}}}_1={\widehat{\alpha }}_1{\widehat{X}}_3^{{\widehat{g}}_{13}}{X}_5^{{\widehat{g}}_{15}}-{\widehat{\beta }}_1{\widehat{X}}_1^{{\widehat{h}}_{11}}\hfill \\ {\dot{\widehat{X}}}_2={\widehat{\alpha }}_2{\widehat{X}}_1^{{\widehat{g}}_{21}}-{\widehat{\beta }}_2{\widehat{X}}_2^{{\widehat{h}}_{22}}\hfill \\ {\dot{\widehat{X}}}_3={\widehat{\alpha }}_3{\widehat{X}}_2^{{\widehat{g}}_{32}}-{\widehat{\beta }}_3{\widehat{X}}_2^{{\widehat{h}}_{32}}{X}_3^{{\widehat{h}}_{33}}\hfill \\ {\dot{\widehat{X}}}_4={\widehat{\alpha }}_4{\widehat{X}}_1^{{\widehat{g}}_{41}}{X}_5^{{\widehat{g}}_{45}}-{\widehat{\beta }}_4{\widehat{X}}_4^{{\widehat{h}}_{44}}\hfill \\ {\dot{\widehat{X}}}_5={\widehat{\alpha }}_5{\widehat{X}}_4^{{\widehat{g}}_{54}}-{\widehat{\beta }}_5{\widehat{X}}_5^{{\widehat{h}}_{55}}\hfill \end{array}\end{array}$$ 7

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{X}}_1=5{X_3}^1{X}_5^{-1}-10X_1^2\hfill \\ {\dot{X}}_2=10X_1^2-10X_2^2\hfill \\ {\dot{X}}_3=10X_2^{-1}-10X_2^{-1}{X}_3^2\hfill \\ {\dot{X}}_4=8X_3^2{X}_5^{-1}-10X_4^2\hfill \\ {\dot{X}}_5=10X_4^2-10X_5^2\hfill \end{array}\end{array}$$ 8

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{\widehat{X}}}_1={\widehat{\alpha }}_1{\widehat{X}}_3^{{\widehat{g}}_{13}}{X}_5^{{\widehat{g}}_{15}}-{\widehat{\beta }}_1{\widehat{X}}_1^{{\widehat{h}}_{11}}\hfill \\ {\dot{\widehat{X}}}_2={\widehat{\alpha }}_2{\widehat{X}}_1^{{\widehat{g}}_{21}}-{\widehat{\beta }}_2{\widehat{X}}_2^{{\widehat{h}}_{22}}\hfill \\ {\dot{\widehat{X}}}_3={\widehat{\alpha }}_3{\widehat{X}}_2^{{\widehat{g}}_{32}}-{\widehat{\beta }}_3{\widehat{X}}_2^{{\widehat{h}}_{32}}{X}_3^{{\widehat{h}}_{33}}\hfill \\ {\dot{\widehat{X}}}_4={\widehat{\alpha }}_4{\widehat{X}}_3^{{\widehat{g}}_{43}}{X}_5^{{\widehat{g}}_{45}}-{\widehat{\beta }}_4{\widehat{X}}_4^{{\widehat{h}}_{44}}\hfill \\ {\dot{\widehat{X}}}_5={\widehat{\alpha }}_5{\widehat{X}}_4^{{\widehat{g}}_{54}}-{\widehat{\beta }}_5{\widehat{X}}_5^{{\widehat{h}}_{55}}\hfill \end{array}\end{array}$$ 9

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{X}}_1=12{X_3}^{-0.8}-10X_1^{0.5}\hfill \\ {\dot{X}}_2=8X_1^{0.5}-3X_2^{0.75}\hfill \\ {\dot{X}}_3=3X_2^{0.75}-5X_3^{0.5}{X}_4^{0.2}\hfill \\ {\dot{X}}_4=2X_1^{0.5}-6X_4^{0.8}\hfill \end{array}\end{array}$$ 10

$$\begin{array}{c}\hfill \begin{array}{c}{\dot{\widehat{X}}}_1={\widehat{\alpha }}_1{\widehat{X}}_3^{{\widehat{g}}_{13}}-{\widehat{\beta }}_1{\widehat{X}}_1^{{\widehat{h}}_{11}}\hfill \\ {\dot{\widehat{X}}}_2={\widehat{\alpha }}_2{\widehat{X}}_1^{{\widehat{g}}_{21}}-{\widehat{\beta }}_2{\widehat{X}}_2^{{\widehat{h}}_{22}}\hfill \\ {\dot{\widehat{X}}}_3={\widehat{\alpha }}_3{\widehat{X}}_2^{{\widehat{g}}_{32}}-{\widehat{\beta }}_3{\widehat{X}}_3^{{\widehat{h}}_{33}}{X}_4^{{\widehat{h}}_{34}}\hfill \\ {\dot{\widehat{X}}}_4={\widehat{\alpha }}_4{\widehat{X}}_1^{{\widehat{g}}_{41}}-{\widehat{\beta }}_4{\widehat{X}}_4^{{\widehat{h}}_{44}}\hfill \end{array}\end{array}$$ 11

Observation data collection schemes

In this work, three schemes are used to illustrate how the observation data effects the identifiability of GRN system. In scheme 1, for GRN models 1, 2 and 3, the special initial conditions in Eqs. (6), (8) and (10) are given as [0.04, 0.06, 0.03, 0.07, 0.05], [2.00, 2.10, 0.05, 2.50, 0.02] and [0.39, 0.85, 0.98, 1.90], respectively. It guarantees that the observation data contain sufficient transient process of GRN system, shown in Subsection 4.2 (Fig. 2 for GRN model 1).

Fig. 2.

Based on the modified GA-PSO algorithm, using the observation data obtained by scheme 1 (GRN model 1), the actual and inferred states waveforms by one time running (with $RMSE=1.4\ast 10^{-2}$). A: The actual states waveforms. B: The inferred states waveforms

In scheme 2, for GRN models 1, 2 and 3, the initial conditions in Eqs. (6), (8) and (10) are given as [0.86, 0.63, 0.94, 0.65, 0.35], [0.45, 0.58, 0.83, 1.21, 0.65] and [0.41, 2.25, 2.19, 0.19], respectively. It makes the observation data be short of sufficient GRN transient process, shown in subsection 4.3 (Fig. 3 for GRN model 1). These two types of observation data (schemes 1 and 2) are both from zero-input response, and the GRN systems will evolve to a steady state at last.

Fig. 3.

Based on the modified GA-PSO algorithm, using the observation data obtained by scheme 2 (GRN model 1), the actual and inferred states waveforms by one time running (with $RMSE=6.3\ast 10^{-4}$). A: The actual states waveforms. B: The inferred states waveforms

In scheme 3, the impulse signals are added into the GRN models 1, 2 and 3, as given in Eqs. (12)–(15), respectively.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{X}}_1=15{X_3}^1{X}_5^{-0.1}-10X_1^2+impuls{e}_1\hfill \\ {\dot{X}}_2=10X_1^2-10X_2^2+impuls{e}_2\hfill \\ {\dot{X}}_3=10X_2^{-0.1}-10X_2^{-0.1}{X}_3^2+impuls{e}_3\hfill \\ {\dot{X}}_4=8X_1^2{X}_5^{-0.1}-10X_4^2+impuls{e}_4\hfill \\ {\dot{X}}_5=10X_4^2-10X_5^2+impuls{e}_5,\hfill \end{array}\right)\end{array}$$ 12

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{X}}_1=5{X_3}^1{X}_5^{-1}-10X_1^2+impuls{e}_1\hfill \\ {\dot{X}}_2=10X_1^2-10X_2^2+impuls{e}_2\hfill \\ {\dot{X}}_3=10X_2^{-1}-10X_2^{-1}{X}_3^2+impuls{e}_3\hfill \\ {\dot{X}}_4=8X_3^2{X}_5^{-1}-10X_4^2+impuls{e}_4\hfill \\ {\dot{X}}_5=10X_4^2-10X_5^2+impuls{e}_5,\hfill \end{array}\right)\end{array}$$ 13

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{X}}_1=12{X_3}^{-0.8}-10X_1^{0.5}+impuls{e}_1\hfill \\ {\dot{X}}_2=8X_1^{0.5}-3X_2^{0.75}+impuls{e}_2\hfill \\ {\dot{X}}_3=3X_2^{0.75}-5X_3^{0.5}{X}_4^{0.2}+impuls{e}_3\hfill \\ {\dot{X}}_4=2X_1^{0.5}-6X_4^{0.8}+impuls{e}_4\hfill \end{array}\right)\end{array}$$ 14

in which

$$impuls{e}_i(t)=\left\{\begin{array}{cc}0& t<t_{\mathit{im}}\\ {\lambda }_i& t_{\mathit{im}}\le t<t_{\mathit{im}}+t_w\\ 0& t_{\mathit{im}}+t_w\le t\end{array}\right)$$ 15

where i is the gene number, $impuls{e}_i$ is the excitation signal added into the $i^{\mathit{th}}$ gene state equation, $t_{\mathit{im}}$ is the starting time to add the impulse excitation signal and the observation data are collected from time $t_{\mathit{im}}+t_w$, ${\lambda }_i$ and $t_w$ are the intensity and time duration of impulse signal, respectively. The GRN system will go through a transient process; then, the steady-state is established.

Identifiability of GRN

In order to quantitatively analyze the accuracy of GRN identification, five indices are defined as follows:

1. ${\widehat{\theta }}_n$, the average value of the $n^{\mathit{th}}$ inferred parameter (30 independent runs), is defined as

   $$\begin{array}{c}\hfill {\widehat{\theta }}_n=\frac{\sum _{m=1}^{30}{\widehat{\theta }}_n^m}{30},\end{array}$$ 16

   where n is the index of GRN parameters (for GRN models 1 and 2, n = 1, 2,..., 23; for GRN model 3, n = 1, 2,..., 17); m is the simulation runs. 
2. ${\xi }_n$, the relative identification error of the $n^{\mathit{th}}$ parameter is defined as

   $$\begin{array}{c}\hfill {\xi }_n=\left|\frac{{\widehat{\theta }}_n-{\theta }_n}{{\theta }_n}\right|\ast 100\%,\end{array}$$ 17

   where ${\theta }_n$ is the actual value of the $n^{\mathit{th}}$ parameter. If the ${\xi }_n$ is less than 5% (${\xi }_n$ < 5%), the $n^{\mathit{th}}$ parameter is thought to be inferred accurately.
3. $\overline{\xi }$, the average relative identification error of all parameters, is defined as

   $$\begin{array}{c}\hfill \overline{\xi }=\frac{\sum _{n=1}^{\mathit{Num}}{\xi }_n}{\mathit{Num}},\end{array}$$ 18

   where Num is the number of parameters in GRN model. There are $Nu{m}_1$ = 23, $Nu{m}_2$ = 23 and $Nu{m}_3$ = 17 parameters in GRN models 1, 2 and 3, respectively.
4. $\overline{\mathit{RMSE}}$, the average of RMSE (30 independent runs) is defined as

   $$\begin{array}{c}\hfill \overline{\mathit{RMSE}}=\frac{\sum _{m=1}^{30}RMS{E}^m}{30}\end{array}$$ 19

   where $RMS{E}^m$ is the relative mean squared error for the $m^{\mathit{th}}$ simulation.
5. P, the identification accuracy of GRN system, is defined as

   $$\begin{array}{c}\hfill P=\frac{Q}{\mathit{Num}}\ast 100\%,\end{array}$$ 20

   where Q is the number of parameters that can be inferred accurately.

In this work, the identifiability of GRN system is defined as follows:

1. more than 90% of the parameters are inferred accurately, i.e., $P>$ 90%. If ${\xi }_n$ < 5%, the $n^{\mathit{th}}$ parameter is considered to be inferred accurately.

2. the average relative error of all parameters is less than 5%, i.e., $\overline{\xi }<5\%$.

3. the $\overline{\mathit{RMSE}}$ (30 independent runs) is less than ${(2\ast {10}^{-2})}^2\ast N\ast L$, where N is the number of genes and L = 50 is the observation data length. That is, for GRN models 1 and 2, the $\overline{\mathit{RMSE}}<{(2\ast {10}^{-2})}^2\ast 5\ast 50=1.0\ast {10}^{-1}$; for GRN model 3, the $\overline{\mathit{RMSE}}<{(2\ast {10}^{-2})}^2\ast 4\ast 50=8.0\ast {10}^{-2}$.

The condition (1) is defined to guarantee that most parameters (more than 90%) can be inferred accurately. The condition (2) is defined to guarantee that even though some parameters have an estimation error, this error is not too large. The condition (3) is defined to guarantee the fitting degree of the data, which means that the average relative error between the inferred and actual GRN states is less than $2\ast {10}^{-2}=2\%$ for each gene. In other words, if the above three conditions are satisfied, the GRN system is considered to be of identifiability

Methods

In this work, a modified GA-PSO algorithm is used as the optimization method in our identification framework. It employs a multi-level mutation operation to promote the diversity of population [21]. At the same time, the velocity limitation strategy is used to guarantee the global exploration ability [22].

The individual representation and fitness function

In the optimization algorithm procedure, how to encode an individual and choose the fitness function are important issues. In this work, an individual is encoded by real number. Taking GRN model 1 as an example, each individual x is a 23-dimensional real-value vector, representing a parameter set with 23 parameters. An individual x in GRN model 1 is given as:

$$\begin{array}{c}\hfill x=[x_1,x_2,x_3,\dots ,x_{22},x_{23}]=[{\widehat{a}}_1,{\widehat{g}}_{{}_{13}},{\widehat{g}}_{{}_{15}},\dots ,{\widehat{\beta }}_{{}_5},{\widehat{h}}_{{}_{55}}]\end{array}$$ 21

In the proposed GA-PSO algorithm, the fitness function is defined as:

$$\begin{array}{c}\hfill fitness=\frac{1}{RMSE+\eta }\end{array}$$ 22

where RMSE is defined in Eq. (5); $\eta$ is a very small positive number to avoid the fitness being infinite (if RMSE = 0), $\eta ={10}^{-10}$ in this paper.

Multi-level mutation operation

After selection and crossover operations in GA-PSO algorithm, individual x is selected for the mutation operation with mutation probability $p_m$ to get $x^{\ast }$. We proposed the multi-level mutation strategy to perform mutation operation [21], given as:

$$\begin{array}{c}\hfill x^{\ast }=\left\{\begin{array}{c}x-0.1\ast (\mathrm{\Delta })\ast \sigma \mathrm{if}\mathrm{random}(0,1)=0\hfill \\ x+0.1\ast (\mathrm{\Delta })\ast \sigma \mathrm{if}\mathrm{random}(0,1)=1\hfill \\ \mathrm{\Delta }={\theta }^{\mathit{up}}-{\theta }^{\mathit{low}}\hfill \\ \sigma =\sum _{w=0}^{\delta }{a}_w{2}^{-w}\hfill \end{array}\right)\end{array}$$ 23

where $x^{\ast }$ is the new individual created by mutation operation; ${\theta }^{\mathit{low}}({\theta }^{\mathit{up}})$ is the lower (upper) bound for the parameter searching range; initially $a_w$ = 0, and w = 0, 1,..., $\delta$, then each $a_w$ is mutated to 1 with probability $p_{\delta }=1/\left(\delta +1\right)$; so that there is one and only one w such that $a_w$ = 1, and the rest $a_w$ = 0, $\delta$ decides the mutation precision.

Velocity limitation strategy

In the part of PSO operations, the maximum velocity $V^{max}$ determines the searching resolution (the velocity step) and acts as a constraint to control the global exploration ability of the population. Too large or too small $V^{max}$ makes it more likely to miss good solutions or be trapped at local optima for individuals [22]. In this work, the maximum searching velocity $V^{max}$ is set as $5\%\ast ({\theta }^{\mathit{up}}-{\theta }^{\mathit{low}})$, where the coefficient 5% is decided by a trial and error procedure, given by Eq. (24).

$$\begin{array}{c}\hfill v=\left\{\begin{array}{c}-V^{max}ifv<-V^{max}\hfill \\ V^{max}ifv>V^{max}\hfill \\ V^{max}=5\%\ast ({\theta }^{\mathit{up}}-{\theta }^{\mathit{low}})\hfill \end{array}\right)\end{array}$$ 24

The above-mentioned two strategies are important points different from the basic GA-PSO algorithm [19]. We show the effect of this modification in the subsection 4.2.

Results

In this paper, we analyzed the GRN identifiability using different observation data (schemes 1, 2 and 3). To explore the generality of our work, three S-system GRN models are used in this paper. Based on the Runge–Kutta method, the numerically solved time series in Fig. 1 is obtained by solving Eqs. (7), (9) and (11), respectively. All results reported in this section are obtained based on 30 independent runs to avoid issues associated with randomness and to evaluate the algorithms statistically [23].

Simulation parameters

In this work, the population size is Pop = 500, the largest iteration is iter = 500; for PSO operation, the learning factors are $c_1$ = $c_2$ = 2.05; for GA operation, the crossover probability is $p_c$ = 0.85 and the mutation probability is $p_m$ = 0.1; the elite probability is $p_{\mathit{elite}}$ = 0.7; the probability q is linearly increasing from 0.1 to 0.5 with the increase of iteration, the observation data length is L = 50. We use the same parameter searching ranges as that in [19]. The searching ranges for ${\widehat{a}}_i({\widehat{\beta }}_i)$ are $[0,2\ast {\theta }_n]$; the searching ranges for ${\widehat{g}}_{\mathit{ij}}({\widehat{h}}_{\mathit{ij}})$ are $[{\theta }_n-1,{\theta }_n+1]$; ${\theta }_n$ is the actual value of the $n^{\mathit{th}}$ parameter, as given in Eqs. (6), (8) and (10), respectively.

Identification results using the observation data of scheme 1

Identification results of GRN model 1 using scheme 1

In scheme 1, a special initial condition in Eq. (6) is given as [0.04, 0.06, 0.03, 0.07, 0.05] for GRN model 1. We use the modified GA-PSO algorithm to perform identifiability test procedure, including the multi-level mutation operation and velocity limitation strategy. The identification results are shown in Table 1 (30 independent runs). The actual and inferred states of one time running are shown in Fig. 2; subplots (A) and (B) are the actual and inferred states, respectively. From Fig. 2, we learn that the inferred GRN states are close to the actual GRN states, $RMSE=1.4\ast 10^{-2}$ being small.

Table 1.

The identification results of scheme 1 based on the modified GA-PSO algorithm (GRN model 1)

n	parameter	${\theta }_n$	${\widehat{\theta }}_n$	${\xi }_n$
1	${\widehat{a}}_1$	15	14.997	0.02%
2	${\widehat{g}}_{13}$	1	1.007	0.73%
3	${\widehat{g}}_{15}$	−0.1	−0.104	4.12%
4	${\widehat{\beta }}_1$	10	10.010	0.09%
5	${\widehat{h}}_{11}$	2	2.006	0.27%
6	${\widehat{a}}_2$	10	10.011	0.11%
7	${\widehat{g}}_{21}$	2	1.986	0.72%
8	${\widehat{\beta }}_2$	10	9.979	0.21%
9	${\widehat{h}}_{22}$	2	2.006	0.30%
10	${\widehat{a}}_3$	10	10.234	2.34%
11	${\widehat{g}}_{32}$	−0.1	−0.088	12.22%
12	${\widehat{\beta }}_3$	10	10.223	2.23%
13	${\widehat{h}}_{32}$	−0.1	−0.098	2.41%
14	${\widehat{h}}_{33}$	2	2.069	3.44%
15	${\widehat{a}}_4$	8	7.800	2.54%
16	${\widehat{g}}_{41}$	2	2.000	0.02%
17	${\widehat{g}}_{45}$	−0.1	−0.112	11.89%
18	${\widehat{\beta }}_4$	10	9.758	2.42%
19	${\widehat{h}}_{44}$	2	2.002	0.07%
20	${\widehat{a}}_5$	10	9.853	1.47%
21	${\widehat{g}}_{54}$	2	1.990	0.51%
22	${\widehat{\beta }}_5$	10	9.801	1.99%
23	${\widehat{h}}_{55}$	2	2.032	1.59%
$\overline{\xi }=2.25\%\overline{\mathit{RMSE}}=1.25\ast {10}^{-2}P=21/23=91.30\%$

As shown in Table 1 (GRN model 1), 21 out of 23 (P = $21/23=91.30\%$) parameters have the relative error $\xi$ being less than 5%, which are considered to be inferred accurately. Only ${\xi }_{11}$ and ${\xi }_{17}$ are larger than 5%, corresponding to parameters $g_{32}$ and $g_{45}$. The largest relative error is ${\xi }_{11}$ = 12.22%, the average relative error is $\overline{\xi }$ = 2.25%. In addition, the average of RMSE (30 independent runs) satisfies $\overline{\mathit{RMSE}}=1.25\ast 10^{-2}<{(2\ast {10}^{-2})}^2\ast N\ast L=1.0\ast {10}^{-1}$. Using the observation data of scheme 1, the GRN model 1 is of identifiability based on the modified GA-PSO algorithm.

Identification results of GRN models 2 and 3 using scheme 1

In scheme 1, for GRN models 2 and 3, a special initial condition is given as [2.00, 2.10, 0.05, 2.50, 0.02] and [0.39, 0.85, 0.98, 1.90], respectively. Based on the modified GA-PSO algorithm, for GRN models 2 and 3, there are 21 (P = 21/23 = $91.30\%$) and 16 (P = 16/17 = $94.12\%$) parameters can be inferred accurately, respectively. For GRN model 2, Only ${\xi }_{12}$ = 6.22% and ${\xi }_{13}$ = 7.06% are larger than 5%, corresponding to parameters ${\beta }_3$ and $h_{32}$. For GRN model 3, Only ${\xi }_{12}$ = 17.82% is larger than 5%, corresponding to parameter $h_{33}$. The average relative error is $\overline{\xi }$ = 2.38% and $\overline{\xi }$ = 2.60%, respectively. In addition, the average of RMSE is $\overline{\mathit{RMSE}}=1.5\ast 10^{-3}$ (less than $1.0\ast {10}^{-1}$), and $\overline{\mathit{RMSE}}=2.2\ast 10^{-3}$ (less than $8.0\ast {10}^{-2}$), respectively. Therefore, using the observation data of scheme 1, the GRN models 2 and 3 are of identifiability based on the modified GA-PSO algorithm.

Identification results based on the basic GA-PSO algorithm using scheme 1

Additionally, to show the influence of the multi-level mutation operation and velocity limitation strategy in the proposed GA-PSO algorithm, we use scheme 1 to identify three GRN models based on the basic GA-PSO algorithm. For GRN models 1, 2 and 3, 17 (P = 17/23 = 73.91%), 16 (P = 16/23 = 69.57%) and 10 (P = 10/17 = 58.82%) parameters can be inferred accurately, respectively. The average relative error is $\overline{\xi }$ = 4.57%, $\overline{\xi }$ = 4.13% and $\overline{\xi }$=7.25%, respectively. The average of RMSE is $\overline{\mathit{RMSE}}=7.52\ast {10}^{-2}$(less than $1.0\ast {10}^{-1}$), $\overline{\mathit{RMSE}}=1.39\ast {10}^{-2}$ (less than $1.0\ast {10}^{-1}$) and $\overline{\mathit{RMSE}}=1.66\ast {10}^{-2}$ (less than $8.0\ast {10}^{-2}$), respectively. In this condition, the GRN models 1, 2 and 3 are all not of identifiability.

Our results indicate that the multi-level mutation operation and velocity limitation strategy in the proposed GA-PSO algorithm are important to perform accurate identification. The performance of the optimization algorithm is important for GRN identification. Therefore, in the following scheme 2 (Sects. 4.3) and scheme 3 (Sects. 4.4), we use the modified GA-PSO algorithm to perform identifiability test procedure.

Identification results using the observation data of scheme 2

Identification results of scheme 2

In scheme 2, the initial condition in Eq. (6) is given as [0.86, 0.63, 0.94, 0.65, 0.35] for GRN model 1. Based on the modified GA-PSO algorithm, the identification results are shown in Table 2 (30 independent runs). The actual and inferred states of one time running are shown in Fig. 3; subplots (A) and (B) are the actual and inferred states, respectively. From Fig. 3, we learn that these observation data (50 data points) is short of sufficient GRN transient process. The inferred GRN states are very close to the actual GRN states, $RMSE=6.3\ast 10^{-4}$ being very small.

Table 2.

The identification results of scheme 2 based on the modified GA-PSO algorithm (GRN model 1)

n	parameter	${\theta }_n$	${\widehat{\theta }}_n$	${\xi }_n$
1	${\widehat{a}}_1$	15	16.086	7.24%
2	${\widehat{g}}_{13}$	1	1.072	7.20%
3	${\widehat{g}}_{15}$	−0.1	−0.071	29.52%
4	${\widehat{\beta }}_1$	10	10.612	6.12%
5	${\widehat{h}}_{11}$	2	2.052	2.58%
6	${\widehat{a}}_2$	10	10.353	3.53%
7	${\widehat{g}}_{21}$	2	1.971	1.46%
8	${\widehat{\beta }}_2$	10	10.389	3.89%
9	${\widehat{h}}_{22}$	2	1.961	1.95%
10	${\widehat{a}}_3$	10	10.705	7.05%
11	${\widehat{g}}_{32}$	−0.1	−0.109	8.52%
12	${\widehat{\beta }}_3$	10	10.733	7.33%
13	${\widehat{h}}_{32}$	−0.1	−0.115	15.29%
14	${\widehat{h}}_{33}$	2	2.076	3.82%
15	${\widehat{a}}_4$	8	8.248	3.10%
16	${\widehat{g}}_{41}$	2	2.013	0.62%
17	${\widehat{g}}_{45}$	−0.1	−0.088	11.75%
18	${\widehat{\beta }}_4$	10	10.335	3.35%
19	${\widehat{h}}_{44}$	2	2.017	0.87%
20	${\widehat{a}}_5$	10	10.092	0.92%
21	${\widehat{g}}_{54}$	2	2.005	0.24%
22	${\widehat{\beta }}_5$	10	10.114	1.14%
23	${\widehat{h}}_{55}$	2	1.989	0.53%
$\overline{\xi }=5.57\%\overline{\mathit{RMSE}}=6.8\ast {10}^{-4}P=14/23=60.87\%$

As shown in Table 2 (GRN model 1), based on the modified GA-PSO algorithm, 14 out of 23 (P = 14/23 = 60.87%) parameters can be inferred accurately, whose parameter indices are n = 5$\sim$9, 14$\sim$16, 18$\sim$23. The largest relative error is ${\xi }_3$ = 29.52% (corresponding to parameter $g_{15}$), the average relative error is $\overline{\xi }$ = 5.57%, and the average of RMSE is $\overline{\mathit{RMSE}}$= $6.8\ast {10}^{-4}$. Although the $\overline{\mathit{RMSE}}$ is small enough, about 40% parameters cannot be inferred accurately. Specially, when the system is in the steady state (GRN model 1), the observation data does not contain transient process, the identification results are less than satisfactory. Based on the modified GA-PSO algorithm, 12 out of 23 (P = 12/23 = 52.17%) parameters can be inferred accurately, the largest relative error is ${\xi }_{17}$ = 97.28% (corresponding to parameter $g_{45}$), the average relative error is $\overline{\xi }$ = 13.30%, and the average of RMSE is $\overline{\mathit{RMSE}}$ = $3.2\ast {10}^{-7}$.

Using scheme 2, the identifiability test procedure is performed based on the modified GA-PSO algorithm for GRN models 2 and 3. For GRN model 2, the initial condition in Eq. (8) is given as [0.45, 0.58, 0.83, 1.21, 0.65]. The identification results show that 15 out of 23 (P = $15/23=65.22\%$) parameters have the relative error $\xi$ being less than 5%, which are considered to be inferred accurately. The rest eight parameters cannot be inferred accurately, corresponding to parameters $g_{13}$, ${\alpha }_3$, ${\beta }_3$, $h_{33}$, $g_{43}$, ${\alpha }_5$, $g_{54}$ and ${\beta }_5$. The ${\xi }_2$ = 24.58%, ${\xi }_{10}$ = 5.25%, ${\xi }_{12}$ = 5.87%, ${\xi }_{14}$ = 6.73%, ${\xi }_{16}$ = 10.58%, ${\xi }_{20}$ = 5.16%, ${\xi }_{21}$ = 5.56% and ${\xi }_{22}$ = 5.89%, are larger than 5%. The average relative error is $\overline{\xi }$ = 4.46%. The average of RMSE (30 independent runs) is $\overline{\mathit{RMSE}}=1.0\ast 10^{-3}$ (less than $1.0\ast {10}^{-1}$).

For GRN model 3, the initial condition in Eq. (10) is given as [0.41, 2.25, 2.19, 0.19]. The identification results show that 6 out of 17 (P=$35.29\%$) parameters can be inferred accurately, corresponding to parameters ${\alpha }_2$, $g_{21}$, ${\beta }_2$, $h_{22}$, $g_{32}$ and ${\beta }_3$, whose relative identification error are ${\xi }_5$ = 0.32%, ${\xi }_6$ = 2.92%, ${\xi }_7$ = 2.19%, ${\xi }_8$ = 1.25%, ${\xi }_{10}$ = 4.38% and ${\xi }_{11}$ = 2.70%. The rest 11 parameters cannot be inferred accurately, whose relative identification error are larger than 5%. The average relative error is $\overline{\xi }$ = 9.70%. The average of RMSE (30 independent runs) is $\overline{\mathit{RMSE}}=7.83\ast 10^{-4}$ (less than $8.0\ast {10}^{-2}$).

These results show that the GRN models 1, 2 and 3 are all not of identifiability using scheme 2. The reason is that these observation data (50 data points) is short of sufficient GRN transient process. It is not easy to enhance the accuracy of GRN identification only by means of improving algorithm performance (reducing relative mean-square error). Even though the inferred GRN can reproduce all the actual state trajectories (the optimization algorithm has good performance, RMS E is small enough), the GRN system is not identifiable using the observation data short of sufficient transient process.

Why steady-state data are invalid for identification

Using the observation data of scheme 2, it can be seen that three GRN systems are not of identifiability. To elaborate on the reasons, we use the GRN model 1 to give explanations from the view of equilibrium points analysis. To simplify the description, in GRN model 1, only 8 parameters are considered to be independently unknown variables and the remaining 15 parameters are set as constants, as given in Eq. (25). Let ${\dot{X}}_1={\dot{X}}_2={\dot{X}}_3={\dot{X}}_4={\dot{X}}_5=0$, we derive the equilibrium points corresponding to steady-state, as given in Eq. (26), the analytical solutions of equilibrium with 8 unknown variables are given by Eq. (27).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{X}}_1={\alpha }_1{X}_3{X}_5^{-0.1}-{\beta }_1{X}_1^{2.0}\hfill \\ {\dot{X}}_2={\alpha }_2{X}_1^{2.0}-{\beta }_2{X}_2^{2.0}\hfill \\ {\dot{X}}_3=10X_2^{-0.1}-10X_2^{-0.1}{X}_3^{2.0}\hfill \\ {\dot{X}}_4={\alpha }_4{X}_1^{2.0}{X}_5^{-0.1}-{\beta }_4{X}_4^{2.0}\hfill \\ {\dot{X}}_5={\alpha }_5{X}_4^{2.0}-{\beta }_5{X}_5^{2.0}\hfill \end{array}\right)\end{array}$$ 25

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\alpha }_1{X}_3{X}_5^{-0.1}-{\beta }_1{X}_1^{2.0}=0\hfill \\ {\alpha }_2{X}_1^{2.0}-{\beta }_2{X}_2^{2.0}=0\hfill \\ 10X_2^{-0.1}-10X_2^{-0.1}{X}_3^{2.0}=0\hfill \\ {\alpha }_4{X}_1^{2.0}{X}_5^{-0.1}-{\beta }_4{X}_4^{2.0}=0\hfill \\ {\alpha }_5{X}_4^{2.0}-{\beta }_5{X}_5^{2.0}=0\hfill \end{array}\right)\end{array}$$ 26

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{X}_1^{\ast }={(({\beta }_5{\beta }_4{(({\alpha }_1{\alpha }_5{\alpha }_4)/({\beta }_1{\beta }_5{\beta }_4))}^{\frac{21}{22}})/({\alpha }_5{\alpha }_4))}^{\frac{1}{2}}\hfill \\ X_2^{\ast }={(({\alpha }_2{\beta }_5{\beta }_4{(({\alpha }_1{\alpha }_5{\alpha }_4)/({\beta }_1{\beta }_5{\beta }_4))}^{\frac{21}{22}})/({\beta }_2{\alpha }_5{\alpha }_4))}^{\frac{1}{2}}\hfill \\ X_3^{\ast }=1\hfill \\ X_4^{\ast }={({\beta }_5/{\alpha }_5)}^{\frac{1}{2}}{(({\alpha }_1{\alpha }_5{\alpha }_4)/({\beta }_1{\beta }_5{\beta }_4))}^{\frac{5}{11}}\hfill \\ X_5^{\ast }={(({\alpha }_1{\alpha }_5{\alpha }_4)/({\beta }_1{\beta }_5{\beta }_4))}^{\frac{5}{11}}\hfill \end{array}\right)\end{array}$$ 27

As shown in Eq. (27), for the analytical solutions of equilibrium point, there are 4 equations to determine 8 independent variables, including ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_4$, ${\alpha }_5$, ${\beta }_1$, ${\beta }_2$, ${\beta }_4$ and ${\beta }_5$. The steady-state observation corresponds to $X_1^{\ast }=1.22$, $X_2^{\ast }=1.22$, $X_3^{\ast }=1.00$, $X_4^{\ast }=1.086$ and $X_5^{\ast }=1.086$, as given in Figs. 2 and 3. In other words, there are uncountable number of solutions (${\alpha }_1$, ${\alpha }_2$, ${\alpha }_4$, ${\alpha }_5$, ${\beta }_1$, ${\beta }_2$, ${\beta }_4$ and ${\beta }_5$), which can yield the same steady-state observation data. Using only 4 equations to determine 8 parameters is an indetermined problem. If the unknown variable (parameter) number is 23, it gives more freedom to choose parameters to be the same steady-states but different parameter values. The conclusion is that we fail to infer GRN accurately when using the observation data short of transient process. That is why these observation data of scheme 2 cannot be used for GRN identification.

Identification results using valid observation data of scheme 3

The scheme 1 is a feasible way to obtain the valid observation data, i.e., the observation data with sufficient transient process. This asks for the zero-input GRN systems starting from a special initial condition, which can be achieved by means of changing the experimental conditions. In addition, adding excitation signal into GRN systems is another effective way. In scheme 3, the impulse excitation signals are designed and added into the GRN systems for obtaining the valid observation data.

In GRN model 1, as shown in Figs. 2 and 3, the GRN system will evolve to the steady-state at last. Using scheme 3, the impulsive excitation signals are added into the GRN system after the time $t>1.5$s. The intensity of impulse signal are ${\lambda }_1$ = −122.0, ${\lambda }_3$ = −100.0, ${\lambda }_2={\lambda }_4={\lambda }_5=0$, $t_w=0.01$, as given in Eqs. (12) and (15). Based on the modified GA-PSO algorithm, the inferred results of scheme 3 are shown in Table 3. The actual and inferred states of one time running (with $RMSE=1.66\ast 10^{-2}$) are shown in Fig. 4; subplots (A) and (B) are the actual and inferred states, respectively.

Table 3.

The identification results of scheme 3 based on the modified GA-PSO algorithm (GRN model 1)

n	parameter	${\theta }_n$	${\widehat{\theta }}_n$	${\xi }_n$
1	${\widehat{a}}_1$	15	15.561	3.74%
2	${\widehat{g}}_{13}$	1	1.036	3.64%
3	${\widehat{g}}_{15}$	−0.1	−0.102	1.97%
4	${\widehat{\beta }}_1$	10	10.349	3.49%
5	${\widehat{h}}_{11}$	2	2.008	0.40%
6	${\widehat{a}}_2$	10	9.932	0.68%
7	${\widehat{g}}_{21}$	2	2.004	0.19%
8	${\widehat{\beta }}_2$	10	9.978	0.22%
9	${\widehat{h}}_{22}$	2	1.986	0.72%
10	${\widehat{a}}_3$	10	10.169	1.69%
11	${\widehat{g}}_{32}$	−0.1	−0.096	3.78%
12	${\widehat{\beta }}_3$	10	10.180	1.80%
13	${\widehat{h}}_{32}$	−0.1	−0.095	4.60%
14	${\widehat{h}}_{33}$	2	1.922	3.89%
15	${\widehat{a}}_4$	8	7.884	1.45%
16	${\widehat{g}}_{41}$	2	2.008	0.42%
17	${\widehat{g}}_{45}$	−0.1	−0.128	27.86%
18	${\widehat{\beta }}_4$	10	9.911	0.89%
19	${\widehat{h}}_{44}$	2	1.973	1.34%
20	${\widehat{a}}_5$	10	9.868	1.32%
21	${\widehat{g}}_{54}$	2	2.031	1.57%
22	${\widehat{\beta }}_5$	10	9.848	1.52%
23	${\widehat{h}}_{55}$	2	2.012	0.58%
$\overline{\xi }=2.95\%\overline{\mathit{RMSE}}=1.77\ast {10}^{-2}P=22/23=95.65\%$

Fig. 4.

Based on the modified GA-PSO algorithm, using the observation data obtained by scheme 3 (GRN model 1), the actual and inferred states waveforms by one time running (with $RMSE=1.66\ast 10^{-2}$). A: The actual states waveforms. B: The inferred states waveforms

As shown in Table 3 (GRN model 1), 22 out of 23 (P = 22/23 = 95.65%) GRN parameters are with relative error $\xi <5\%$, which are considered to be inferred accurately. Only ${\xi }_{17}$ (corresponding to parameter $g_{45}$) is larger than 5%. The average relative error is $\overline{\xi }=2.95\%$. The average of RMSE (30 independent runs) is $\overline{\mathit{RMSE}}$ = $1.77\ast 10^{-2}<$ ${\left(2\ast 10^{-2}\right)}^2\ast N\ast L$ = $1.0\ast 10^{-1}$.

In GRN model 2, the intensity of impulse signal are ${\lambda }_1$ = −58.5, ${\lambda }_2$ = −61.5, ${\lambda }_3$ = −78.4, ${\lambda }_4$ = −53.6, ${\lambda }_5$ = −60.8, $t_w$ = 0.01, as given in Eqs. (13) and (15). Based on the modified GA-PSO algorithm, using scheme 3, 22 out of 23 (P = 95.65%) GRN parameters are with the relative error $\xi <5\%$, which are considered to be inferred accurately. Only ${\xi }_{14}=9.84\%$ (corresponding to parameter $h_{33}$) is larger than 5%. The average relative error is $\overline{\xi }=2.34\%$. The average of RMSE (30 independent runs) satisfies $\overline{\mathit{RMSE}}$ = $5.8\ast 10^{-3}<$ $1.0\ast 10^{-1}$.

In GRN model 3, the intensity of impulse signal are ${\lambda }_1$ = −23.40, ${\lambda }_2$ = −108.00, ${\lambda }_3$ = –128.76, ${\lambda }_4$ = −7.28, $t_w$ = 0.01, as given in Eqs. (14)–(15). Based on the modified GA-PSO algorithm, using scheme 3, 16 out of 17 (P = 94.12%) GRN parameters can be inferred accurately. Only ${\xi }_{15}=6.10\%$ (corresponding to parameter $g_{41}$) is larger than 5%. The average relative error is $\overline{\xi }=2.93\%$. The average of RMSE (30 independent runs) satisfies $\overline{\mathit{RMSE}}$ = $7.79\ast 10^{-2}<$ $8.0\ast 10^{-2}$.

For GRN models 1, 2 and 3, it can be seen that three GRN systems are of identifiability using the observation data of scheme 3. This strategy is helpful to obtain the valid observation data; then, address the problem associated with scheme 2.

Discussion

Discussion 1:

While comparing schemes 1, 2 and 3, three GRN models are of identifiability when using schemes 1 or 3, but not of identifiability when using scheme 2. The schemes 1 and 3 are feasible ways to obtain the valid observation data, which is helpful for accurate GRN identification. In scheme 1, the zero-input GRN systems start from a special initial condition, which can be achieved by means of changing the experimental conditions. In scheme 3, the impulsive excitation signals can be designed and added into the GRN systems.

Discussion 2:

In scheme 3, how to choose the parameters of impulse excitation signals? It is important for GRN identification. The impulse excitation signal adding into the gene state equation can be different, as given in Eqs. (12)–(15). The $impuls{e}_i$ larger or less than zero means that the positive or negative excitation is added into the $i^{\mathit{th}}$ gene state equation, and it equal to zero means that it is absent. The principle of choosing impulse excitation parameters is that the system can be sufficiently excited; then, the valid observation data can be obtained.

Discussion 3:

In the condition of reasonable parameter setting, other types of excitation signals are also possible to excite the GRN system sufficiently and then obtain the valid observation data. The step excitation is an alternative option to achieve the same objective. For the length of the paper, it is not presented here.

Discussion 4:

In this work, three GRN models used to perform identifiability test procedure are monostable systems. Our conclusions are valid for these three GRN models. For bistable, multistable, excitable and oscillatory networks, the relationship between GRN observation data and the identification accuracy is needed to be further studied and systematically investigated.

Conclusion

For identifying GRNs accurately, it is indeed important to enhance the performance of optimization algorithm, where the modified and basic GA-PSO algorithm are used to illustrate the influence. However, besides optimization algorithm performance, the observation data also play a key role in GRN identification. This point is often overlooked, but is very important in practice. In this work, three data collection schemes are used to discuss the identifiability of GRN system. In scheme 1, using the transient observation data, three GRN systems are of identifiability. In scheme 2, using the observation data short of sufficient dynamic process, three GRN systems are not of identifiability even though the inferred GRN can reproduce the actual state trajectories ($\overline{\mathit{RMSE}}$ is small enough), the reason for which is given by mean of equilibrium point analysis. Additionally, in scheme 3, we propose to add a special designed impulse excitation signal into GRN systems. The observation data with sufficient dynamic can be collected. Consequently, three GRN systems are of identifiability using scheme 3. Owing to its universality and simplicity, the proposed methodology can provide a guide to collect valid observation data, which is help to identify GRN accurately and to further understand GRN system.

Funding

Funding This study was funded by the Natural Science Foundation of Gansu Province (grant number 21JR1RG309), the Higher Education Innovation Foundation of Gansu Province (grant number 2021B-236), the Scientific Research Program Funded by Shaanxi Provincial Education Department (grant number 19JK0574), Natural Science Basic Research Plan in Shaanxi Province of China (grant number 2019JQ-317) and the Scientific Research Foundation of Hexi University.

Declarations

Conflicts of interest

The authors declare that they have no conflict of interest.