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A non-linear reverse-engineering method
for inferring genetic regulatory networks
Siyuan Wu
1
, Tiangang Cui
1
, Xinan Zhang
2
and Tianhai Tian
1
1School of Mathematics, Monash University, Clayton, VIC, Australia
2School of Mathematics and Statistics, Central China Normal University, Wuhan, PR China
ABSTRACT
Hematopoiesis is a highly complex developmental process that produces various
types of blood cells. This process is regulated by different genetic networks that
control the proliferation, differentiation, and maturation of hematopoietic stem cells
(HSCs). Although substantial progress has been made for understanding
hematopoiesis, the detailed regulatory mechanisms for the fate determination of
HSCs are still unraveled. In this study, we propose a novel approach to infer the
detailed regulatory mechanisms. This work is designed to develop a mathematical
framework that is able to realize nonlinear gene expression dynamics accurately.
In particular, we intended to investigate the effect of possible protein heterodimers
and/or synergistic effect in genetic regulation. This approach includes the Extended
Forward Search Algorithm to infer network structure (top-down approach) and a
non-linear mathematical model to infer dynamical property (bottom-up approach).
Based on the published experimental data, we study two regulatory networks of
11 genes for regulating the erythrocyte differentiation pathway and the neutrophil
differentiation pathway. The proposed algorithm is first applied to predict the
network topologies among 11 genes and 55 non-linear terms which may be for
heterodimers and/or synergistic effect. Then, the unknown model parameters are
estimated by fitting simulations to the expression data of two different differentiation
pathways. In addition, the edge deletion test is conducted to remove possible
insignificant regulations from the inferred networks. Furthermore, the robustness
property of the mathematical model is employed as an additional criterion to choose
better network reconstruction results. Our simulation results successfully realized
experimental data for two different differentiation pathways, which suggests that the
proposed approach is an effective method to infer the topological structure and
dynamic property of genetic regulations.
Subjects Bioinformatics, Computational Biology
Keywords Genetic regulatory network, Network inference, Hematopoiesis, Probabilistic graphic
model, Differential equation
INTRODUCTION
Hematopoiesis is a highly complex process that controls the proliferation, differentiation
and maturation of hematopoietic stem cells (HSCs) (Ng & Alexander, 2017). It has been
widely accepted that genetic regulatory networks control the developmental processes
of various types of blood cells (Cedar & Bergman, 2011). Although the regulatory
mechanisms have been studied over a century, there are still many challenging questions
How to cite this article Wu S, Cui T, Zhang X, Tian T. 2020. A non-linear reverse-enginee ring method for inferring genetic regulatory
networks. PeerJ 8:e9065 DOI 10.7717/peerj.9065
Submitted 6 January 2020
Accepted 5 April 2020
Published 29 April 2020
Corresponding author
Tianhai Tian,
tianhai.tian@monash.edu
Academic editor
Walter de Azevedo Jr
Additional Information and
Declarations can be found on
page 20
DOI 10.7717/peerj.9065
Copyright
2020 Wu et al.
Distributed under
Creative Commons CC-BY 4.0
regarding the cell fate determination in hematopoiesis (Ottersbach et al., 2010). Thus, it
is imperative to unravel the regulatory mechanisms for the study of hematopoiesis.
In adult mammals, hematopoiesis occurs mostly in the bone marrow (Birbrair &
Frenette, 2016). HSCs have the feature of self-renewal and multipotent as well as the ability
to differentiate into multipotent progenitors (MPPs). Then, MPPs will differentiate
into two main lineages of blood cells, namely the myeloid lineage which starts at
common myeloid progenitors (CMPs) and the lymphoid lineage which starts at common
lymphoid progenitors (CLPs). In addition, the myeloid lineage has two distinct
progenitors, namely megakaryocyte-erythroid progenitors (MEPs) and granulocyte-
macrophage progenitors (GMPs). MEPs can differentiate into megakaryocytes and
erythrocytes, and GMPs can give rise to mast cells, macrophages and granulocytes.
Lymphoid lineage cells include T lymphocytes (T-cells), B lymphocytes (B-cells) and
natural killer cells (NK-cells) (Orkin & Zon, 2008). In this work, we focus on the fate
determination of HSCs in the myeloid lineage for the choice between erythrocytes and
neutrophils.
During the developmental process, a number of transcriptional factors (TFs) act as
regulators to control the fate determination of HSCs (Aggarwal et al., 2012). Among
them, the genetic complex Gata1-Gata2-PU.1 is a very important module for the cell-fate
choice of CMPs between erythrocytes or neutrophils (Friedman, 2007;Liew et al., 2006;
Ling et al., 2004). In particular, the Gata1-PU.1 complex forms a double negative
feedback module, in which each gene inhibits the expression of the other (Friedman, 2007).
Recently it has been elucidated that the fate determination of HSCs was defined not only by
the ratio of Gata1 and PU.1 (Hoppe et al., 2016), but also by a third party during the
regulation. For example, FOG-1 is a significant third party to regulate the Gata1-PU.1
module (Chang et al., 2002;Mancini et al., 2012). Erythropoietin receptor (EpoR) signaling
also acts the essential role in regulating the Gata1-PU.1 Module (Zhao et al., 2006).
Although the regulatory mechanisms of the Gata1-Gata2-PU.1 complex in hematopoiesis
are relatively well studied, the connection of this triad complex with other significant genes
as well as the role of these genes in hematopoiesis are mostly unclear (Liew et al., 2006).
Mathematical modeling is an important method for inferring the detailed regulatory
mechanisms. In 1910, Archibald Hill proposed a classical non-linear ordinary differential
equation (ODE or Hill equation) model to describe the sigmoidal oxygen binding curve of
hemoglobin (Hill, 1910). Since then, the Hill equation has been applied to explore the
mechanisms in a wide range of genetic regulatory networks and biological systems.
For example, the genetic toggle switching was achieved by the models with Hill equations
(Gardner, Cantor & Collins, 2000). In addition, the Hill equation was employed to
formalize the mechanisms of cell fate determination (Xiong & Ferrell, 2003;Laslo et al.,
2006;Huang et al., 2007). Recently, the Hill equation was also used to discover a regulatory
network of 52 genes with the uniform activation and repression strengthes (Li &
Wang, 2013). Another widely used approach is the Shea–Ackers formalism for studying
the thermodynamics of regulatory networks (Shea & Ackers, 1985). We developed a
mathematical model based on the Shea–Ackers formalism to study the regulations of the
Gata1-Gata2-PU.1 complex (Tian & Smith-Miles, 2014). A stochastic model was also
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 2/25
proposed to explore the function of noise in regulating the fate determination of HSCs.
Simulations suggested that fluctuations of protein numbers may lead the HSC to different
developmental pathways. In recent years substantial process has been made to design
various types of mathematical models for describing the regulatory mechanisms of
gene networks, including stochastic differential equations, stochastic kinetic systems,
qualitative differential equations, Michaelis–Menten formalism, S-system and power-law
formalism (de Jong, 2002;Liu & Wang, 2008;Wang & Tian, 2010;Maetschke et al., 2013;
Woods et al., 2016;Olariu & Peterson, 2018;Yang et al., 2018;Yang & Bao, 2019).
In particular, a number of mathematical models have been designed to realize the stable
states of gene expression levels in the differentiation of HSCs (Chang et al., 2006;Huang
et al., 2007;Chickarmane, Enver & Peterson, 2009;Olariu & Peterson, 2018). However,
the majority of these models only considered the functions of each gene independently,
namely variable x
i
for the expression level of gene iin the model is in the form of Piaixi.
Nonetheless, this type of models fails to represent the co-operation functions of genes
together. There is a lack of investigations for the effect of possible protein heterodimers
and/or synergistic effect in genetic regulations, namely variables x
i
in the model are also
in the form of Pi;jbijxixj. Most recently, single-cell studies have been conducted to
explore the hematopoietic system. Compared with the analysis of bulk cells, the advantage
of single-cell analysis is the ability to understand the heterogeneity within the cell
population (Guo et al., 2010;Ye, Huang & Guo, 2017). With the development of single-cell
analysis, researchers have raised more novel computational and statistical methods to
explore the regulatory mechanism of hematopoiesis. For example, the partial correlation
method, Boolean model and ODE model were employed to construct the genetic
regulatory networks from the single-cell expression profiles (Hamey et al., 2017;Wei et al.,
2017). In addition, a deep learning method was applied to unravel the fate decision in
hematopoiesis (Athanasiadis et al., 2017).
Recently, we proposed a general approach that combines both top-down and bottom-up
approaches to reconstruct the genetic regulatory networks of the fate choice between
erythrocytes and neutrophils (Wu, Cui & Tian, 2018). The key issue in this work includes a
large number of unknown parameters and a high computational cost to add potential
regulations. For the issue of parameter number, a linear ODE model may have the least
number of unknown parameters among the models for all possible regulations between
genes. However, since the linear model is limited to describe the linear relationship,
it is not appropriate to use the linear model to study systems with complex non-linear
dynamics. Although the non-linearity has been addressed by the reverse-engineering
methods with the cost of more unknown parameters (Chickarmane & Peterson, 2008;
Crombach et al., 2012;Meister et al., 2013;Li & Wang, 2013;Wang et al., 2016), the issue
of protein heterodimers and/or synergistic effect between genes has not been discussed
in the majority of literature at all. This work is designed to address these issues by
proposing a novel approach for reconstructing genetic regulatory networks. The first
innovation of this approach is the new non-linear ODE model as the bottom-up approach
to study the effect of protein heterodimers and/or synergistic effect explicitly. The second
innovation of this work is the proposed Extended Forward Search Algorithm as the
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 3/25
top-down approach to infer the structure of networks in our newly proposed non-linear
model. The proposed approach thus is able to not only reduce the complex structure
of genetic regulatory networks but also improve the inference efficiency substantially
because the number of parameters in the mathematical model is decreased. We examined
the capability of our proposed method by studying the genetic regulatory networks for the
fate determination of HSCs.
METHODS
Experimental data
In this work, we used the sub-series GSE49987 as the experimental data from the published
microarray dataset GSE49991 (May et al., 2013). This dataset contains the expression
profiles collected by experiments using the cell line FDCPmix. This dataset was generated
with the probe name version of Agilent Whole Mouse Genome Microarray 4 × 44 K
(May et al., 2013). It provides microarray gene expression profiles of hematopoietic stem
cells (HSCs) differentiating into erythrocytes and neutrophils. This microarray dataset is
available at https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE49991. To convert
all microarray probe IDs to gene names, we pre-processed this dataset based on the
Ensembl BioMart and GO Enrichment Analysis (The Gene Ontology Consortium, 2017).
From a previous study, the regulatory network of 18 core genes during the hematopoiesis
has been curated (Moignard et al., 2013). Moreover, the same research team studied
the regulatory interaction of 26 core genes during the hematopoiesis (Moignard et al.,
2015). The total number of distinct genes in these two studies is 30. Thus, in our work we
considered 30 genes whose names are listed in Table S1. There are three repeated
experiments for each developmental process, each of which contains the expression levels
of 30 genes from HSCs to differentiated cells at 30 time points spanning over 1 week.
The observation time points are those starting from the HSCs/progenitors stage (1 point),
then every 2 h over the first day (12 points), every 3 h over the second day (8 points),
every 4 h over the third day (6 points), every 24 h until the fifth day (2 points), and the
seventh day (1 point). In this study, we used the average data of these three repeated tests as
the experimental data for each time point. The time points and expression data of four
genes can be found in the following Figs. 1 and 2.
Selection of candidate genes
Based on our research experience (Wang et al., 2016), it is challenging to study a dynamic
network with 30 genes. Thus, we conducted an extensive literature review for selecting
a smaller number of important genes based on their relationship with the three genes
Gata1, Gata2 and PU.1. These candidate genes should be essential for the cell-fate choice
in hematopoiesis, or they significantly interact with these three genes. For example, gene
Scl/Tal1 interacts with Gata1, Eto2/Cbfa2t3 and Ldb1 (Goardon et al., 2006), and is a
regulator in the differentiation of hematopoietic stem cells (HSCs) (Shivdasani,
Mayer & Orkin, 1995;Zhang et al., 2005;Porcher et al., 1996;Real et al., 2012). In addition,
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 4/25
Eto2/Cbfa2t3 regulates the differentiation of HSCs by repressing the expression of target
gene Scl/Tal1 (Goardon et al., 2006). Moreover, Ldb1 is a significant transcriptional factor
(TF) for the differentiation of erythroid lineage (Soler et al., 2010). According to the
ChIPSeq analysis, Ldb1 is necessary for HSCs to control their maintenance since it
binds to the majority of enhancer elements in hematopoiesis (Li et al., 2011).
We also included a number of genes with potential regulatory relationship with the
three genes Gata1, Gata2 and PU.1. For example, it was indicated that there might
be unclear regulations between Gata2 and Gfi1(Moignard et al., 2013). Gfi1is an
important TF in the regulation of HSCs differentiation (van der Meer, Jansen & van der
Reijden, 2010;Lancrin et al., 2012). Gfi1is required for the differentiation of common
lymphoid progenitors (CLPs) and common myeloid progenitors (CMPs) from HSCs
and exists in the majority of HSCs, CLPs and CMPs. Similar to gene Gfi1, gene Runx1 is
also expressed in most HSCs and progenitor cells as well. Then, Gfi1and/or Runx1 are
expressed continually in most cells which differentiate into the granulocyte lineage
(North et al., 2004). Lmo2 is a master regulator of hematopoiesis (Inouea et al., 2013).
However, its specific role in regulation is still unclear. Experimental studies suggested
that the knockdown of Lmo2 does not affect the expression of Gata1 and Scl/Tal1
(Inouea et al., 2013). However, the overexpression of Lmo2 gene also inhibited erythroid
0.90
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1.00
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1.15
Expression level of Tal1
.B.A
.D.C
Figure 1 Simulation results and experimental data of the regulatory network for erythrocyte
differentiation Red solid line: experimental microarray data; Blue star dash line: simulation of the
regulatory network. (A) Gene Gata1; (B) gene PU.1; (C) gene Ets1; (D) gene Tal1.
Full-size
DOI: 10.7717/peerj.9065/fig-1
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 5/25
differentiation (Visvader et al., 1997). In addition, gene Ets1 is a suppressor in the
erythrocyte differentiation. It is downregulated in erythrocyte differentiation by binding
to and activating the Gata2 promoter (Lulli et al., 2006). The last candidate gene is
Notch1 that inhibits the differentiation of granulocyte lineage by maintaining the
expression of gene Gata2. It also enhances the HSCs differentiate to CLPs (Kumano et al.,
2001;Stier et al., 2002). Therefore, in this study we considered the regulatory networks
with the following 11 genes: Gata1, Gata2, PU.1/Sfpi1, Runx1, Eto2/Cbfa2t3, Ets1, Notch1,
Scl/Tal1, Ldb1, Gfi1and Lmo2. The detailed information of the references for these 11
genes is also given in Table S2 in Supplemental Information.
Top-down approach: extended forward search algorithm
To reduce the number of unknown parameters in our proposed mathematical model,
we used the probabilistic graphical models as the top-down approach to infer the
topological structure of gene regulatory networks. Probabilistic graphical model is a useful
tool for inferring the network structure (Noor et al., 2013). One type of probabilistic
graphical models is the Gaussian graphical model (GGM), which provides a simple and
effective method to characterize the regulatory relationship between genes. The GGM is
based on the calculation of the conditional dependencies among genes using the gene
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Expression level of Gata1
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Expression level of PU.1
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0.95
Expression level of Ets1
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1.00
1.05
1.10
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Expression level of Tal1
0.80
0.85
0.90
.B.A
.D.C
Figure 2 Simulation results and experimental data of the regulatory network for neutrophil
differentiation Red solid line: experimental microarray data; Blue star dash line: simulation of the
regulatory network. (A) Gene Gata1; (B) gene PU.1; (C) gene Ets1; (D) gene Tal1.
Full-size
DOI: 10.7717/peerj.9065/fig-2
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 6/25
expression data. The edge connecting two genes in the model is neglected if they are
conditionally independent given all other genes (Krämer, Schäfer & Boulesteix, 2009).
In this work it is assumed that a system includes genes {G
1
,…,G
m
} with expression
levels x
ij
for gene G
i
at time point j. Compared with the existing methods that study
networks with genes only, this work will study gene networks that include not only genes
in the form of monomers {G
1
,…,G
m
}, which are represented by the linear terms in
the model, but also protein heterodimers and/or synergistic effect {G
k
–G
l
}(k,l=1,…,m),
which are represented by the non-linear terms (NLTs) in the model. There are two
reasons for using the NLTs {G
k
–G
l
}. Firstly, we can use the product of two variables to
represent the synergistic effect of these two genes. Secondly, if the NLT represents the
protein heterodimer, we assumed that the binding and disassociation reactions for the
heterodimer {G
k
–G
l
} reach an equilibrium state quickly. Thus the level of the heterodimer
{G
k
–G
l
} can be written as C
kl
×G
k
×G
l
, where C
kl
is the equilibrium constant. We can
consider this constant C
kl
as a coefficient in our mathematical model. In both cases,
we only need to consider the product of the expression levels of these two genes, namely
y
klj
=x
kj
x
lj
, as the level of NLT {G
k
–G
l
} at time t
j
for our algorithm computation. Since
the number of possible regulations from NLTs to genes is much larger than that of possible
regulations among genes (i.e. 726 vs 110), the regulations from NLTs to genes will
dominate the whole genetic regulatory system with high probability. However, the
regulations among genes should be the core mechanisms rather than the regulations from
NLTs to genes. To avoid the dominance of NLTs regulations, we assume that the number
of regulations from NLTs to genes does not exceed that between genes.
According to the GGM (Wang, Myklebost & Hovig, 2003;Wang et al., 2016), we
proposed a new algorithm, named Extended Forward Search Algorithm (EFSA), to
infer the topological structure of regulatory networks that includes both genes and NLTs.
Let X=(x
1
,x
2
,:::,x
N
) be a vector that consists of mgenes and nNLTs (N=m+n).
The following three matrices are constructed, namely a m×mcovariance matrix Aof
mgenes, a m×ncovariance matrix Bto measure the covariance between mgenes
and nNLTs, and a n×ncovariance matrix Cof nNLTs. The N-dimensional matrix Mis
defined by
M¼AB
B0C
;(1)
where B′is the transpose of B. An initial empty graph Gis built by the N-dimensional
identity matrix. This initial graph Gconsists of four matrices G
1
,G
2
,G
3
and G
4
which have
the same dimensions as A,B,B′and C, respectively, namely
G¼G1G2
G3G4
;(2)
where G
1
and G
4
are identity matrix with dimensions mand n, respectively, and G
2
and G
3
are m×nand n×mzero matrices, respectively.
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 7/25
The proposed algorithm is given below.
Algorithm 1: extended forward search algorithm
1. Let X=(x
1
,x
2
,…,x
N
) be a vector with Nelements, and Nbe the number of components
consist of mgenes and nNLTs. An initial empty graph Gis built by the N-dimensional
identity matrix, which is defined by Eq. (2).
2. Substitute all covariance values from the diagonal positions of sub-matrix Ainto the
corresponding positions of sub-matrix G
1
, and then based on the updated G
1
, use the
Iterative Maximum Likelihood Estimates Algorithm (IMLEA) to compute the new
covariance matrix (Dempster, Laird & Rubin, 1977).
3. Add an undirected edge E1
ij ((i,j)∈[1, m]
2
) into G
1
, namely add the symmetrical
covariance value between the ith gene and jth gene from the positions A(i,j) and A(j,i)
into the positions G
1
(i,j) and G
1
(j,i), respectively. Then compute a new covariance
matrix by the IMLEA. Based on the deviance difference between the new covariance
matrix and that before addition, test the significance of the added edge E1
ij by using
the Chi-square distribution with one degree of freedom. The p-value of the Chi-square
test is used in the next step as the edge selection criterion. Record the p-value of this
tested edge and remove it from G
1
.
4. Add a new undirected edge into G
1
. Then, repeat the computation in Step 3.
After all possible undirected edges have been tested, sort all tested edges in ascending
order by their p-values. If the smallest p-value is lower than the predefined
cut-off value, add the edge with the smallest p-value into the sub-graph G
1
permanently.
5. Go back to step 3, add the second edge in the updated sub-graph G
1
. Repeat the
computation in steps 3 and 4 until the smallest p-value of an added edge is larger than
the cutoff p-value.
6. Based on the last updated undirected graph G
1
, the graph orientation rules are applied to
transform the undirected graph into a directed acyclic graph (DAG) (Meek, 1995).
The inferred DAG with m
1
directed edges, denoted as A
s
, represents the predicted
regulatory network among mgenes.
7. Test the possible edges between mgenes and nNLTs. Based on the latest matrix G,
add an undirected edge E2
ij between the ith gene and the jth NLT. That is, add the
symmetrical covariance value between the ith gene and jth NLT from the positions
B(i,j) and B′(j,i) into the positions G
2
(i,j) and G
3
(j,i), respectively. Then, compute a
new covariance matrix by the IMLEA. Based on the deviance difference between the new
covariance matrix and that before addition, test the significance of the added edge E2
ij by
using the Chi-square distribution with one degree of freedom. The p-value of the
Chi-square test is used as the edge selection criterion. Record the p-value of this tested
edge E2
ij and remove it from G.
8. Repeat the computation in steps 7 for the regulation between genes and NLTs. The last
updated sub-graph G
3
with n
1
edges, denoted as B′
s
, is the predicted directed regulatory
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 8/25
network from nNLTs to mgenes. Since we only consider regulations among genes
and those from NLTs to genes, the result matrix is given as follows:
Gs¼As
B0
s
:(3)
The output network includes m
1
directed edges among mgene and n
1
directed edges
from nNLTs to mgenes.
Note that we have initially applied the GGM in our previous work to the whole matrix
Mdirectly (Wang et al., 2016). However, since the number of NLTs is much larger
than that of genes, numerical results showed that the majority of selected edges connect
NLTs, but few edges are selected to connect genes. This result is not appropriate because
the regulations between genes should be the primary mechanisms of the network.
Then we conducted another test, in which we did not consider the regulations between
NLTs by changing matrix Cinto an identity matrix I
m
. Matrix Mnow is
M1¼AsB
B0Im
:(4)
However, when we applied the GGM to M
1
directly, the singular problem arose during
the computation of IMLEA. To satisfy our intention and make the algorithm stable,
we proposed EFSA which is executed in two steps. The first step selects regulations
between genes and the second step finds regulations from NLTs to genes. The EFSA can be
used to predict the gene-gene interactions and the effect from NLTs to genes based on the
time-course experimental data.
Bottom-up approach: mathematical model
For a regulatory network with mgenes, the expression levels of the i-th gene at time tis
denoted as x
i
(t). We used the following ordinary differential equation (ODE) model to
describe the dynamics of the network (de Jong, 2002)
dx
dt ¼Fðt;xÞ;(5)
where x=(x
1
,…,x
m
) is a vector representing the expression levels of mgenes. A number
of mathematical formalisms have been proposed to describe the dynamical interactions
between different genes in the network, such as the models with linear functions
(de Jong, 2002)
Fiðt;xÞ¼ X
n
j¼1;j6¼i
aijxjkixi(6)
or the models with non-linear functions (Olariu & Peterson, 2018)
Fiðt;xÞ¼ Pn
j¼1aijxj
1þPn
j¼1bijxj
kixi(7)
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 9/25
The advantage of the model (Eq. 5) with the linear functions (Eq. 6) is that it has a
much smaller number of unknown parameters than the non-linear functions (Eq. 7).
However, the non-linear model is able to describe the non-linear dynamics more precisely.
Therefore, we proposed a method that combines the feature of additive terms in the
linear model and the advantages of non-linear model. We applied the second truncated
Taylor series approach to approximate the non-linear function (Eq. 7). Here the
Taylor series is a mathematical formula to approximate a function by using a polynomial
function (Stewart, 2018). Thus, we proposed an ODE model (Eq. 5) with the following
functions
Fiðt;xÞ¼ X
m
j¼1;j6¼i
aijxjþX
1j,kn
bijkxjxkkixi(8)
where k
i
is the degradation rate of x
i
. This proposed model (Eq. 5) with the non-linear
function (Eq. 8) is based on the following assumptions:
1. The regulations from different genes to a particular gene are additive. Similarly, the
regulations from non-linear terms (NLTs) to a particular gene are also additive.
2. The regulations from gene jto gene iis represented by a
ij
x
j
, where a
ij
is the coefficient of
regulation strength.
3. The regulation of NLT x
j
x
k
to gene iis represented by β
ijk
x
j
x
k
, where β
ijk
consists of the
regulation strength and equilibrium constant C
ij
, as we discussed in the sub-section
Top-down Approach.
4. The auto-regulation is not considered, namely a
ii
= 0, to avoid confusion between
auto-regulation term a
ii
x
i
and degradation term k
i
x
i
. Note that the issue of
auto-regulation may be addressed using a model with non-linear function (Eq. 7).
In addition, we just consider the effect of NLTs x
j
x
k
for j≠ksince the expression levels of
x
j
may be highly correlated to that of x2
j. Therefore, we assume that β
ijj
=0.
5. If the value of a
ij
is positive (negative or zero), it means that gene x
j
activates (represses
or has no regulation to) the expression of gene x
i
. Similar assumption is applied to the
value of β
ijk
.
We emphasize that the proposed method in this work is substantially different from our
previous work (Wang et al., 2016). The first difference is that the proposed non-linear
model (Eq. 8) is different from the non-linear model in Wang et al. (2016). This new
model not only can study the regulations from genes to genes, as we considered in our
previously proposed model (Wang et al., 2016), but also can investigate the effects of
heterodimers and/or synergistic effect in genetic regulation. This new model also leads
to the second difference compared with our previous top-down approach, namely the
proposed Extended Forward Search Algorithm (EFSA) not only includes the probabilistic
graphical model in our previous work (Wang et al., 2016) but also can predict the
possible regulations from NLTs to genes. In addition, in this work, we will infer a
medium-sized network first by using EFSA and then reduce the network size by removing
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 10/25
regulations from the network in the Results section, rather than inferring a core
network first and then adding regulations to the core network in our previous approach
(Wang et al., 2016).
Parameter inference
When considering the full connected graph among mgenes and nnon-linear terms
(NLTs), we have an ordinary differential equation (ODE) system with mdifferential
equations. The total number of all unknown coefficients is m(m+n). After applying the
Extended Forward Search Algorithm (EFSA), we have an inferred regulatory network
which contains only m
1
edges among genes and n
1
edges from NLTs to genes. Thus,
the numbers of coefficients a
ij
and β
ijk
are reduced from m(m−1) to m
1
and from mn to
n
1
, respectively. It is easier to estimate the parameters for the inferred network than for
the fully connected network.
In this work, we used a MATLAB toolbox of Genetic Algorithm to estimate the
parameters in the proposed mathematical model (Chipperfield, Fleming & Fonseca, 1994).
The algorithm begins by generating a population of initial parameter values, for example,
100 values. Each initial value is called an individual and the whole population is called
one generation. Then it calculates the fitness value for each individual of current
generation. Based on the fitness values, the algorithm next creates new values for each
individual and thus forms a population of the next generation. This process is repeated
until a pre-defined number of generations have been calculated. In this work, we used
the following functions, namely function crtbp to generate initially binary populations,
function reins to effect fitness-based reinsertion, function select to give a convenient
interface to the selection routines, function recombine to conduct crossover operators,
and function mut to conduct binary and integer mutations. The detailed information
of these functions and their alternatives can be found in the relevant reference
(Chipperfield, Fleming & Fonseca, 1994).
To ensure the accuracy of estimates, we set the number of generations as 1,000 and
the number of individuals for each generation as 300. For the parameter vector (a
ij
,β
ijk
,k
i
),
we used the uniform distribution over the interval (W
min
,W
max
) to generate the
initial estimates. Here W
min
and W
max
are the minimal value and maximal value,
respectively, for choosing the samples of the parameters. The values of W
min
and W
max
are adjusted by computation. For example, if the majority of estimated parameters all
are close to W
min
, then we will further decrease the value of W
min
. However, if the majority
of estimated values are well above W
min
, then we need to increase the value of W
min
accordingly. The similar consideration is applied to W
max
. In this study, for the erythroid
lineage pathway, numerical results suggest that the values of W
min
and W
max
for (a
ij
,
β
ijk
,k
i
) are ( −3, −3, 0) and (3, 3, 1), respectively. In addition, for the neutrophil lineage
pathway, numerical results suggest that the values of W
min
and W
max
for (a
ij
,β
ijk
,k
i
)
are (−2.5, −2.5, 0) and (2.5, 2.5, 1), respectively. We run the algorithm using an initial
random number to generate an initial set of model parameters, which leads to a set of
estimated parameters. For each model, we used 200 different initial random numbers,
which lead to 200 different sets of estimated model parameters. Denote x
i
(t
j
) and x
i
(t
j
)as
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 11/25
the observation data and numerical simulations at time point t
j
for j= {1,2,…,M},
respectively. The simulation error is calculated by
E¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
m
i¼1X
M
j¼1
ðxiðtjÞx
iðtjÞÞ2
v
u
u
t:(9)
We selected the top ten sets with the minimal estimated errors out of 200 estimates for
further analysis and comparison.
Robustness analysis
We noted that, if a model with the estimated parameters is not robust, a perturbation to
the parameters might lead to substantial variations of the model output. Thus, we next
used the robustness property of the model to select the inferred model parameter sets from
the Genetic Algorithm. This property was designed to examine the robustness of the
inferred model to the perturbations of model parameters (Kitano, 2004). Robustness
property is also an important method for understanding the variations in genetic
regulatory networks mathematically (Masel & Siegal, 2009). Note that our perturbation
test is a mathematical technique. It is different from the perturbation of biological
experiments, which may be conducted by the over-expression/knock-down tests. Although
we will conduct removal tests by removing edges from the developed model, these tests
are designed to remove the unnecessary (or unimportant) regulations in the network.
In this perturbation test, based on the inferred parameter k
i
that is assumed to be the
unperturbed one, the perturbed parameter is generated by
ki¼kið1þmεÞ(10)
where εis a sample generated from either the normal distribution or the uniform
distribution. In this work, we used the standard Gaussian random variable N(0,1) to
generate samples. In addition, mis a parameter to determine the values of perturbation
(Wang et al., 2016). The value of parameter mdetermines the variations of simulations.
Numerical results suggest that when the value of mis small, perturbation has small effect on
the system dynamics, and it is difficult to distinguish the robustness properties of the
model with different parameter sets. However, if the value of mis large, perturbation
will make the model output substantially different, and it will be difficult to measure the
robustness property. To make the variations of simulations appropriately for robustness
analysis, m= 0.4 was employed in this study.
For each of the top ten sets of parameters determined in the previous sub-section,
we firstly obtained N( = 5,000) sets of perturbed model parameters by using (Eq. 10) and
then used these parameter sets to obtain Ncorresponding simulations. We used xðkÞ
ij ðpÞ
and xðkÞ
ij to denote the simulation of variable x
i
at time point t
j
obtained by the k-th
perturbed and unperturbed model parameters, respectively. Then, we defined
EðkÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
m
i¼1X
M
j¼1
ðxðkÞ
ij ðpÞxðkÞ
ij Þ2
v
u
u
t(11)
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 12/25
as the measure for the robustness property of the model with the k-th perturbed parameter
set. Afterwards, we defined the robust average for the given parameter set as
RA ¼1
NX
N
k¼1
EðkÞ
;(12)
and robust standard deviation as
RSTD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
N1X
N
k¼1
ðEðkÞRAÞ2
v
u
u
t(13)
over Nperturbation tests. Smaller values of RA and RSTD mean that the model with the
given parameter set is more robust.
RESULTS
Inference of regulatory network
To reduce the complexity of regulatory networks, we first used the Extended Forward
Search Algorithm (EFSA) to predict the topological structure of genetic networks.
The algorithm controls the number of edges by adjusting a pre-defined cut-off value.
This value is equivalent to the significant value in statistics. If the threshold is too low,
we may miss some significant regulations. However, if the threshold is relatively high, it is
quite possible to select insignificant regulations. This work considers the networks
including 11 genes and 55 non-linear terms (NLTs). For the sub-network of 11 genes only
(i.e. matrix A
s
in Eq. 3), to ensure the statistically significant, we set a specific threshold
as 0.1 for both the erythroid regulatory network and neutrophil regulatory network.
The selection of this threshold value (i.e. 0.1) is based on the balance between neither
selecting much insignificant regulations nor choosing a small number of candidate
regulations. Then we had 46 and 40 directed edges for the erythroid regulatory network
and neutrophil network, respectively.
For the regulations from NLTs to genes (i.e. matrix B′
s
in Eq. 3), the size of matrix B′
s
is much larger than that of A
s
. To avoid the dominance of the regulations from NLTs
to genes, we also set the cut-off value as 0.1 for the two networks, or take the first 46 and
40 directed edges from NLTs to genes for the erythrocyte and neutrophil differentiation,
respectively, if more edges are selected when using the cut-off value 0.1. The reason we
still applied threshold 0.1 here is that the number of selected edges that satisfy this value is
much larger than the required number (i.e. 46 for the erythroid regulatory network
and 40 for the neutrophil regulatory network). Since the edges are selected and ranked by
their significance, we can simply select the top 46 edges and 40 edges for the erythroid and
neutrophil pathway, respectively, without conducting any further numerical tests.
Figures S1 and S2 in Supplemental Information present the inferred regulatory
networks for the erythroid and neutrophil networks, respectively. Note that there are
11 and 17 isolated NLTs in the erythroid and neutrophil networks, respectively, since no
significant edges have been selected from these NLTs by our algorithm. All these isolated
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 13/25
NLTs are listed in the “Isolated NLTs Table”. Moreover, all arrows in these figures only
represent the direction of regulations, rather than the types of regulations (i.e. positive or
negative regulation). We will study the detailed regulatory mechanisms in the next
subsection. We found that the targeted gene of the protein heterodimer is a component
of that heterodimer in all situations. The possible explanation of this observation is that
the expression levels of a heterodimer are the product of the expression levels of the
two corresponding genes (namely x
i
x
j
for genes iand jwith expression levels x
i
and x
j
,
respectively). Thus, the expression data of the NLTs {x
i
x
j
} may be highly correlated to
those of the component genes, namely {x
i
}or{x
j
}.
Inference of mathematical model
After the success of constructing regulatory networks in the previous sub-section, we next
study the detailed dynamics of genetic networks in fate determination of hematopoietic
stem cells (HSCs) by using our proposed mathematical model. The major step is to
infer the values of unknown parameters in the model (Eq. 8). If we consider the fully
connected model, there should be 11 × (11 + 55) = 726 parameters. However, after the
application of EFSA, the number of unknown parameters is reduced to 103 (including
46 directed edges between genes, 46 directed edges from non-linear terms (NLTs) to genes
and 11 self-degradation rate constants) for the differentiation of erythrocytes and 91
(including 40 directed edges between genes, 40 directed edges from NLTs to genes and
11 self-degradation rate constants) for the differentiation of neutrophils. We next applied
the Genetic-Algorithm to estimate these unknown parameters for two networks.
We used 200 different random numbers to obtain different initial values of rate constants
(a
ij
,β
ijk
,k
i
) over the defined range (W
min
,W
max
), which was discussed in the Methods
section. This leads to 200 different sets of estimated parameters. Then, we chose the top
ten sets of estimated results for each differentiated lineage with the smallest estimation
errors for further robustness analysis. According to the definition of estimation error
(Eq. 9), the optimal inferred network for the erythrocyte differentiation in our tests has
estimation error 0.9902. In addition, the robust average (Eq. 12) and robust standard
deviation (Eq. 13) are 0.3977 and 0.1066, respectively. For the neutrophil differentiation,
the optimal inferred network has estimation error 0.8726, robust average 0.3983 and
robust standard deviation 0.1275.
Figures 1 and 2present the simulation results based on the optimal estimated
parameters for the expression levels of four genes, namely genes Gata1, PU.1, Ets1 and
Tal1, for the differentiation of erythrocyte and neutrophil, respectively. The expression
levels of Gata1 increase continuously in both simulated and experimental data during
the erythrocyte differentiation. However, during the neutrophil differentiation,
experimental data of Gata1 keep fluctuations and then turn to slightly decreasing at
the end of differentiation, which is matched by our simulation. For gene PU. 1, both
microarray and simulated data decline in the differentiation of erythrocyte but climb
during the differentiation of neutrophil. Similarly, the expression levels of Ets1 in
microarray data increase during erythrocyte differentiation but decrease during neutrophil
differentiation. Simulation results also fit the trends for both differentiation pathways.
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 14/25
The experimental data of Tal1 increase with fluctuations during the first 60 h of
erythrocyte differentiation, but then rises rapidly after the first 60 h. Our simulated results
are consistent with the expression levels of Tal1 with the same trend in expression levels.
Thus, our simulation results fit the trend of expression levels of these genes very well
during two developmental processes. Figures S3 and S4 in Supplemental Information give
the simulation results of the other six genes for the differentiation of erythrocyte and
neutrophil, respectively.
Reduction of network model—edge deletion
We have obtained two regulatory networks with 92 directed edges and 80 directed edges
for erythroid and neutrophil differentiation, respectively. Next we tested the possibility to
delete the potential insignificant edges from our predicted regulatory networks. In the
first step, we tested the deletion of regulations from non-linear terms (NLTs) to genes.
We removed one edge in each test to form a temporary system model, and then
examined the simulation error and robustness property of the new model. Afterwards, we
removed one specific edge permanently if the corresponding new system has the minimal
change in simulation error and robustness property, and then formed an updated
model. This test is repeated until both the simulation error and robustness property of the
updated model are much worse than the original network without any removal. In the
second step, we evaluated the regulatory interactions between 11 genes using the same
method in the first step.
For the erythrocyte differentiation, Table 1 suggests that after removing 3 regulations
from NLTs to genes, the estimation error (Eq. 9) is improved (shown in DEL1). Then,
we tested the regulation reduction from gene to gene. The final result suggests that, after we
deleted (Ldb1 →Lmo2), (Notch1 →Lmo2), (Cbfa2t3 →Lmo2) and (Runx1 →Lmo2)
edges, the estimation error (Eq. 9) is slightly increased. However, the robustness property
is better than that of the DEL1 model since the robust average (Eq. 12) is decreased.
Thus, for the erythroid differentiation, numerical tests recommended to remove total
Table 1 Edge deletion test for erythrocyte differentiation. RR, Removed regulation; SE, Simulation
error, defined by Eq. (9); RA, Robust average, defined by Eq. (12); RSTD, Robust standard deviation,
defined by Eq. (13).
Model RR SE RA RSTD
OES N/A 0.9902 0.3977 0.1066
DEL1 Gata2-Notch1 →Notch1 Tal1-Gfi1→
Gfi1 Cbfa2t3-Gfi1→Gfi1
0.9826 0.4594 0.1259
DEL2 Ldb1 →Lmo2 0.9955 0.3938 0.1124
DEL3 Notch1 →Lmo2 0.9861 0.4506 0.1263
DEL4 Cbfa2t3 →Lmo2 1.0451 0.3820 0.0962
DEL5 Runx1 →Lmo2 1.0298 0.3471 0.0904
Note:
Description of different models: OES, The original model without any deletion; DEL1, Model based on OES by removing
regulations from NLTs to genes; DEL2, Model based on DEL1 by removing a regulation among genes; DEL3, Model
based on DEL2 by removing a regulation among genes; DEL4, Model based on DEL3 by removing a regulation among
genes; DEL5, Model based on DEL4 by removing a regulation among genes.
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 15/25
seven edges from our predicted regulatory network. We stopped the deletion test after
obtaining the DEL5 model. If we proceed further deletion, both simulation error and
robustness property of the temporary network are much worse than the original network
without removal.
Table 2 shows that, for the neutrophil differentiation, there are no insignificant
regulations from NLTs to genes, because the removal of any edge from NLTs to genes will
increase the simulation error (Eq. 9) substantially and/or decrease the robustness property
by increasing the values of robust average (Eq. 12) and robust standard deviation
(Eq. 13). For the regulations between genes, we have removed the following four
regulations, namely (Gata2 →Ldb1), (Runx1 →Cbfa2t3), (Ldb1 →Lmo2) and (Tal1 →
Lmo2), and formed an updated system. Table 2 shows that the simulation error and
robustness property of the updated system are close to those of the original system
without any removal of edges. Thus, for the neutrophil differentiation, numerical tests
recommended to remove only four edges from our predicted regulatory network.
Coincidentally, we stopped the deletion test after obtaining the DEL5 model because of the
same reason for the erythrocyte differentiation.
Figures 3 and 4present the inferred regulatory networks after edge deletion test for
erythroid and neutrophil differentiation, respectively. Initially, we have 92 directed edges
for the erythrocyte pathway and 80 directed edges for the neutrophil pathway. After
the edges deletion, seven and four directed edges have been taken away from the
erythrocyte network and neutrophil network, respectively, since the removal of these edges
has not much negative influence on simulation error (Eq. 9), robust average (Eq. 12) and
robust standard deviation (Eq. 13). Thus, there are 85 and 76 directed edges left for
the erythrocyte and neutrophil pathways, respectively.
DISCUSSION
This work was designed to develop a mathematical framework that was able to realize
nonlinear gene expression dynamics accurately. In particular, we intended to investigate
the effect of possible protein heterodimers and/or synergistic effect in genetic regulation.
Table 2 Edge deletion test for neutrophil differentiation. RR, Removed regulation; SE, Simulation
error, defined by Eq. (9); RA, Robust average, defined by Eq. (12); RSTD, Robust standard deviation,
defined by Eq. (13).
Model RR SE RA RSTD
OES N/A 0.8726 0.3983 0.1275
DEL1 No Suggestion N/A N/A N/A
DEL2 Gata2 →Ldb1 0.8726 0.3943 0.1273
DEL3 Runx1 →Cbfa2t3 0.8726 0.3928 0.1265
DEL4 Ldb1 →Lmo2 0.8748 0.4183 0.1333
DEL5 Tal1 →Lmo2 0.8809 0.3925 0.1237
Note:
Description of different models: OES, The original model without any deletion; DEL1, Model based on OES by removing
regulations from NLTs to genes; DEL2, Model based on DEL1 by removing a regulation among genes; DEL3, Model
based on DEL2 by removing a regulation among genes; DEL4, Model based on DEL3 by removing a regulation among
genes; DEL5, Model based on DEL4 by removing a regulation among genes.
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 16/25
In this study, we designed the Extended Forward Search Algorithm (EFSA) to predict the
topology of regulatory networks connecting genes and heterodimers. We also proposed a
new mathematical model for inferring dynamic mechanisms of regulatory networks.
Using the EFSA, we derived two regulatory networks of 11 genes for erythrocyte and
neutrophil differentiation pathways. According to the predicted networks and
experimental data, we estimated parameters in our proposed mathematical model based
on the criteria of simulation error and robustness property. By removing regulations with
less importance based on simulation error and robustness property, we developed two
gene networks that regulate erythrocyte and neutrophil differentiation pathways.
Numerical results suggested that our proposed method is capable of reconstructing genetic
regulatory networks effectively and accurately.
To infer the regulatory mechanisms of heterodimers, we combined both the
top-down approach (i.e. probabilistic graphical model) and the bottom-up approach
(i.e. mathematical model). We used the top-down approach first to simplify the network
topology and reduced the number of unknown parameters in the mathematical
model. Then the Genetic-Algorithm was used to estimate the unknown parameters.
The combination of these two approaches reduced the errors in simulation and also
improved the robustness property of the mathematical model. In this work, we considered
Gfi1−Gata1
Gfi1−Runx1
Gfi1−Gata2
Gfi1−PU.1
Gfi1−Ldb1
Notch1−Cbfa2t3
Gfi1−Lmo2
Lmo2−Gata2
Gfi1−Notch1
Cbfa2t3
Tal1
PU.1−Lmo2
PU.1−Ldb1
PU.1−Tal1
Runx1−Cbfa2t3
PU.1−Gata2
Runx1
PU.1−Runx1
Ets1−Lmo2
Ets1−Tal1
Ets1−Notch1
Ets1−PU.1
Gata1
Ets1
Gata1−Cbfa2t3
Ets1−Runx1
Gata1−Ldb1
Gata1−Lmo2
Ldb1
Gata2
Gata1−Tal1
Gata1−Gata2
Gfi1
Gata1−Pu.1
Notch1−Runx1
Lmo2
Notch1−PU.1
Notch1
Notch1−Tal1
PU.1−Cbfa2t3
PU.1
Notch1−Lmo2
Notch1−Ldb1
Tal1−Lmo2
Runx1−Tal1
Ets1−Gata1
Tal1−Gata2
Ets1−Gata2
Ets1−Cbfa2t3
Ets1−Ldb1Runx1−Gata2
Cbfa2t3−Gata2
Regulation among genes
Regulation from NLTs to Ets1
Regulation from NLTs to Gata1
Regulation from NLTs to Tal1
Regulation from NLTs to Cbfa2t3
Regulation from NLTs to Runx1
Regulation from NLTs to Notch1
Regulation from NLTs to PU.1
Regulation from NLTs to Lmo2
Color Reference Table
Isolated NLTs Table
Gata1−Runx1
Gata1−Notch1
Gata2−Notch1
Gata2−Ldb1
Runx1−Lmo2
Runx1−Ldb1
Tal1−Ldb1
Tal1−Gfi1
Cbfa2t3−Lmo2
Cbfa2t3−Ldb1
Cbfa2t3−Gfi1
Tal1−Cbfa2t3
Lmo2−Ldb1
Gfi1−Ets1
Figure 3 Predicted genetic regulatory network of erythrocyte pathway. The genetic regulatory
network predicted by the Extended Forward Search Algorithm with 11 genes and 41 non-linear terms
(NLTs) (14 isolated NLTs excluded) after edges deletion test, which is related to the fate deter-
mination of erythrocyte pathway: regulatory network for hematopoietic stem cells differentiate to
megakaryocyte-erythroid progenitors. The network is visualized by Cytoscape software.
Full-size
DOI: 10.7717/peerj.9065/fig-3
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 17/25
the network with medium-sized complexity initially. We then reduced the network
complexity by removing edges from the network, rather than studying the core
network and then adding the edges to the network in our previous study (Wang et al.,
2016). The reason for changing the method from “adding edge”to “removing edge”in
this work is mainly due to the high computational cost in the “adding edge”tests
since the number of candidate edges in the “removing edge test”is much smaller than
that in the “adding edge test”. Thus, in this work, we used the EFSA to obtain more
candidate edges and then used the dynamic model to remove unimportant edges.
If the number of potential regulations derived from the probabilistic graphical model is
relatively large, the removal of one single regulation from the potential network may
not have any changes in simulation error. Numerical results suggested that a couple of
regulations should be removed simultaneously in order to achieve changes in simulation
error.
The inferred regulatory networks from our proposed models are partially supported
by experimental observations. For example, the regulation of Gata1-Gata2-PU.1 complex
in our inferred networks agrees with the experimental results (May et al., 2013).
The Gata1-PU.1 heterodimer plays an important role in regulating the hematopoiesis
(Zhang et al., 2000), which is also included in our inferred model. In addition, the Ldb1-
Lmo2 dimer is activated with significantly expression profiles during the erythroid
Notch1−PU.1
Lmo2−Tal1
Notch1−Gata2
Notch1−Ldb1
Notch1−Gfi1
Notch1−Cbfa2t3
Notch1−Lmo2
Notch1−Gata1
Notch1
PU.1
Lmo2−Cbfa2t3
Ldb1−Cbfa2t3
Lmo2
Runx1
Gfi1−Tal1
Gfi1−Runx1
Notch1−Tal1
Ets1
Cbfa2t3
Notch1−Runx1 Tal1
PU.1−Cbfa2t3
Gata1−Cbfa2t3
Gfi1−Gata1
Gata1−Runx1
Ets1−Tal1
Gfi1−Cbfa2t3
Gfi1−Gata2
Ets1−Notch1
Gata1−Tal1
Ets1−Runx1
Ets1−PU.1
Gata1−Pu.1
Ets1−Cbfa2t3
Ets1−Lmo2
PU.1−Runx1
PU.1−Gata2
Ets1−Ldb1PU.1−Ldb1
Ets1−Gata2
PU.1−Tal1 Ets1−Gata1
Gata2
Gfi1
Gfi1−Lmo2
Gfi1−Ldb1
Ldb1
Gata1
Gfi1−PU.1
Regulation among genes
Regulation from NLTs to Ets1
Regulation from NLTs to Gata1
Regulation from NLTs to PU.1
Regulation from NLTs to Lmo2
Regulation from NLTs to Notch1
Regulation from NLTs to Ldb1
Color Reference Table
Isolated NLTs Table
Gata1−Gata2
Gata1−Lmo2
Gata1−Ldb1
Gata2−Runx1
Gata2−Tal1
Gata2−Cbfa2t3
Runx1−Cbfa2t3
Runx1−Lmo2
Runx1−Ldb1
PU.1−Lmo2
Tal1−Cbfa2t3
Gata2−Lmo2
Tal1−Ldb1
Lmo2−Ldb1
Gata2−Ldb1
Runx1−Tal1
Gfi1−Ets1
Figure 4 Predicted genetic regulatory network of neutrophil pathway. The genetic regulatory
networks predicted by the Extended Forward Search Algorithm with 11 genes and 38 non-linear terms
(NLTs) (17 isolated NLTs excluded) after edges deletion test, which is related to the fate deter-
mination of neutrophil pathway: regulatory network for hematopoietic stem cells differentiate to
granulocyte-macrophage progenitors. The network is visualized by Cytoscape software.
Full-size
DOI: 10.7717/peerj.9065/fig-4
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 18/25
differentiation process (Xu et al., 2003), which is consistent with our prediction. Moreover,
there are evidences to show the existence of synergistic effect of Tal1, Lmo2 and Gata1
(Mead et al., 2001), which has been inferred in our regulatory networks as well.
However, not all of the predictions can be confirmed by the existing experimental
observations. The first explanation is that the non-linear terms in our mathematical
model are introduced by mathematical operation (i.e. the Taylor series). Some of these
non-linear terms may be needed for realizing the nonlinear dynamics accurately, but not
supported by biological mechanisms. Note that another inference method, called
semi-supervised method, can include the validated regulations first and then infer the
invalidated regulations (Maetschke et al., 2013). Secondly, our inferred regulatory network
may predict some potential possible regulations between genes and from non-linear terms
to genes, which may be confirmed by future experimental studies. Thus, the inferred
regulations in this work may provide testable prediction for further experimental studies to
explore the detailed mechanism of hematopoiesis.
This work also raised a number of important issues in the study of genetic regulations.
One question is that our non-linear model still cannot fit all the expression data very well
due to noise in the data. Figures 1 and 2show that the noise in expression data may
increase the simulation error of our proposed model. If the noise ratio in expression data is
large, it is a challenging issue in mathematical modeling. Large variations in the data may
lead to incorrect inference results. In that case, stochastic modeling may be a more
appropriate approach to describe the noise in gene expression data (Samad et al., 2005;
Tian, 2010;Chowdhury, Chetty & Evans, 2015). In addition, the Gaussian graphical model
is based on the covariance matrix. However, the correlation coefficient is suitable to
measure the linear correlation relationship. Currently, other approaches, such as mutual
information and conditional mutual information, have been used to measures both
linear and non-linear correlation relationships between the gene expression data
(Zhang et al., 2012,2015;Zhao et al., 2016). Finally, this research determines the regulatory
mechanisms based on numerical simulation and robustness property. More information
from experimental studies will be important to improve the accuracy of the model and
make more reasonable predictions. In addition, we may use other key criteria to select
mathematical models, such as Akaike’s Information Criterion (AIC), Bayesian
Information Criterion (BIC), and Bayesian factor (Kadane & Lazar, 2004). All these
issues will be the interesting topics of our future research.
CONCLUSION
In conclusion, this study proposes a new method to construct the network topology
from genes and heterodimers by a new top-down approach and then develops a
non-linear ordinary differential equation model to infer the dynamic mechanisms of
regulatory networks. The derived two networks may provide insights regarding the
genetic regulations in the cell fate determination of hematopoietic stem cells.
The proposed method can also be applied to model other regulatory pathways and
biological systems.
Wu et al. (2020), PeerJ, DOI 10.7717/peerj.9065 19/25
ADDITIONAL INFORMATION AND DECLARATIONS
Funding
This work was supported by the National Natural Science Foundation of China
(Nos.11871238 and 11931019). The funders had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript.
Grant Disclosures
The following grant information was disclosed by the authors:
National Natural Science Foundation of China: 11871238 and 11931019.
Competing Interests
The authors declare that they have no competing interests.
Author Contributions
Siyuan Wu performed the experiments, analyzed the data, prepared figures and/or
tables, authored or reviewed drafts of the paper, and approved the final draft.
Tiangang Cui analyzed the data, prepared figures and/or tables, and approved the final
draft.
Xinan Zhang analyzed the data, authored or reviewed drafts of the paper, and approved
the final draft.
Tianhai Tian conceived and designed the experiments, analyzed the data, prepared
figures and/or tables, authored or reviewed drafts of the paper, and approved the final
draft.
Data Availability
The following information was supplied regarding data availability:
The data is available at NCBI GEO: GSE49991. The code is available at GitHub:
https://github.com/ThaddeusWu/PeerJ_Submission.
Supplemental Information
Supplemental information for this article can be found online at http://dx.doi.org/10.7717/
peerj.9065#supplemental-information.
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