Tissue interactions, cell signaling and transcriptional control in the cranial mesoderm during craniofacial development

Xiaochen Fan; David A F Loebel; Heidi Bildsoe; Emilie E Wilkie; Jing Qin; Junwen Wang; Patrick P L Tam; Xiaochen Fan; David A F Loebel; Heidi Bildsoe; Emilie E Wilkie; Jing Qin; Junwen Wang; Patrick P L Tam

doi:10.3934/genet.2016.1.74

AIMS Genetics

2016, Volume 3, Issue 1: 74-98. doi: 10.3934/genet.2016.1.74

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Review Topical Sections

Tissue interactions, cell signaling and transcriptional control in the cranial mesoderm during craniofacial development

1.
Embryology Unit, Children’s Medical Research Institute, Westmead NSW 2145, Australia
2.
Bioinformatics Group, Children’s Medical Research Institute, Westmead NSW 2145, Australia
3.
School of Life Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
4.
Department of Health Sciences Research, Center for Individualized Medicine, Mayo Clinic, and Department of Biomedical Informatics, Arizona State University, Scottsdale AZ 85259, USA
5.
School of Medical Sciences, Sydney Medical School, University of Sydney, NSW 2006, Australia

Received: 29 February 2016 Accepted: 05 April 2016 Published: 11 April 2016

The cranial neural crest and the cranial mesoderm are the source of tissues from which the bone and cartilage of the skull, face and jaws are constructed. The development of the cranial mesoderm is not well studied, which is inconsistent with its importance in craniofacial morphogenesis as a source of precursor tissue of the chondrocranium, muscles, vasculature and connective tissues, mechanical support for tissue morphogenesis, and the signaling activity that mediate interactions with the cranial neural crest. Phenotypic analysis of conditional knockout mouse mutants, complemented by the transcriptome analysis of differentially enriched genes in the cranial mesoderm and cranial neural crest, have identified signaling pathways that may mediate cross-talk between the two tissues. In the cranial mesenchyme, Bmp4 is expressed in the mesoderm cells while its signaling activity could impact on both the mesoderm and the neural crest cells. In contrast, Fgf8 is predominantly expressed in the cranial neural crest cells and it influences skeletal development and myogenesis in the cranial mesoderm. WNT signaling, which emanates from the cranial neural crest cells, interacts with BMP and FGF signaling in monitoring the switch between tissue progenitor expansion and differentiation. The transcription factor Twist1, a critical molecular regulator of many aspects of craniofacial development, coordinates the activity of the above pathways in cranial mesoderm and cranial neural crest tissue compartments.

Keywords:

Citation: Xiaochen Fan, David A F Loebel, Heidi Bildsoe, Emilie E Wilkie, Jing Qin, Junwen Wang, Patrick P L Tam. Tissue interactions, cell signaling and transcriptional control in the cranial mesoderm during craniofacial development[J]. AIMS Genetics, 2016, 3(1): 74-98. doi: 10.3934/genet.2016.1.74

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Abstract

1. Introduction

Plasma physics is one of the most attractive branches of science where many scientists have been focusing their attention on discovering more properties of this field ^[2]. Plasma or cytoplasm is a distinct state of matter that can be described as an ionized gas in which the electrons are free and are not bound to an atom or a molecule ^[3]. If the substance is present in nature in three states: solid, liquid, and gas, then plasma can be classified as the fourth state in which the substance can exist ^[4]. Recently, investigating the heavy Langmuir turbulence's characterization becomes a very important tool for providing a good opportunity to overcome the Langmuir condensation problem ^[5,6]. Moreover, this investigation aims to raise the amount of long-wave disturbances through the condensation paradox in Langmuir ^[7]. At that condensation, the radiation can not dampen the vibration where at severe periods, the coulomb relation is unable to dampen the variations in the pulses because of their frequency ^[8]. Recently, these radiations with its distinct variations and interactions have been mathematically formulated by some nonlinear evolution equations such as KGZ model ^[9,10,11,12].

The ability of nonlinear partial differential equations with integer or fractional order for formulating different complicated phenomena in various fields including genetics, engineering, quantum mechanics, electro chemistry, chemistry, mechanical engineering, biology, mechanics, etc, makes it the ideal and direct way for discovering the indiscoverable properties of these phenomena ^{[13,14,15,16,17,18,19]}. Thus, many mathematicians and physics pay complete attention to derive computational, semi-analytical, numerical techniques for solving these equations such as the Adomian decomposition method, Elkalla expansion method, B-spline schemes, extended simplest equation method, modified Khater method, generalized Khater method, exponential expansion method, auxiliary equation method, direct algebraic expansion method, and so no ^{[20,21,22,23,24,25,26,27,28]}. These methods have been employed on several models but until now, there is no unified method can be applied to all the nonlinear evolution equation ^{[29,30,31,32]}.

In this context, this paper investigate the numerical solutions of the nonlinear KGZ model. This model is formulates as follows ^[33,34,35]:

$\begin{eqnarray} \left\lbrace \begin{array}{c} \mathcal{G}_{t\,t}-\mathcal{G}_{x\,x}+\mathcal{G}+\upsilon_{0}\,\mathcal{Q}\,\mathcal{G} = 0, \\ \\ \mathcal{Q}_{t\,t}-\mathcal{Q}_{x\,x}-\upsilon_{1}\, \left( |\mathcal{G}|^{2} \right)_{x\,x},\; \; = 0, \end{array}\right. \end{eqnarray}$ $

(1.1)

where $ \upsilon_{0}, \, \upsilon_{1} $ are nonzero real parameters describing the consistency of the initial data of the KGZ system while $ \mathcal{Q} = \mathcal{Q}(x, t), \, \mathcal{G} = \mathcal{G}(x, t) $ are receptively real and complex functions which represent the fast time scale component of the electric field raised by electrons and the derivation of ion density from its equilibrium. Eq (1.1) describes the interaction and contact between the Langmuir wave and the acoustic wave of the ions in a high frequency plasma. ^[1] have employed the generalized Khater method to Eq (1.1) and converted it into the following ordinary differential equation with the following initial and boundary conditions

$\begin{eqnarray} \left\lbrace \begin{array}{c} \mathfrak{P}''+\mathcal{L}_{1}\,\mathfrak{P}+\mathcal{L}_{2}\,\mathfrak{P}^{3} = 0,\; \; \; \; , \\ \\ \mathfrak{P}(0) = \mathcal{F}(\mathfrak{Z}),\, \mathfrak{P}_{\mathfrak{Z}}(0) = \mathcal{E}(\mathfrak{Z}). \end{array} \right. \end{eqnarray}$ $

(1.2)

The generalized Khater method have been constructed the values of $ \mathcal{F}, \, \mathcal{E} $ under the following value of the above-mentioned parameters $ \mathcal{L}_{0} = \frac{1}{2}, \, \mathcal{L}_{1} = -\frac{225}{32} $ as follows ^[1]:

$\begin{eqnarray} \left\lbrace\begin{array}{c} \mathcal{F}(\mathfrak{Z}) = \frac{1}{15} (-4) \tanh \left(\frac{\mathfrak{Z}}{2}\right), \\ \\ \mathcal{E}(\mathfrak{Z}) = -\frac{2}{15}.\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; . \end{array} \right. \end{eqnarray}$ $

(1.3)

This model can be used to calculate from Euler's equations for electrons and ions, with Maxwell's electromagnetic field law for ions, by disregarding the influence of magnetic fields ^[36,37]. The nonlinear KGZ model has numerically studied through some recent approximate schemes such as a finite difference method ^[38] where Chunmei Su and Wenfan Yi have investigated the numerical solutions and established the error estimates of a conservative finite difference method for the considered model with a dimensionless parameter $ 0 < \varepsilon \ll 1, $ which is inversely proportional to the speed of sound. While ^[39] has compared the obtained numerical solutions that have been obtained through applying Finite difference time domain (FDTD) methods, Exponential wave integrator (EWI) and Time-splitting (TS) method, Uniformly and optimally accurate (UOA) methods and Uniformly accurate (UA) methods that have been applied in ^[40,41] of the same model that give a precision, computational sophistication, and other properties are also addressed. ^[15] has employed the well-known Chebyshev Cardinal Functions for investigating the numerical solutions of the nonlinear KGZ model where operational matrices of derivatives have been used to convert partial differential equations into nonlinear algebraic equations. ^[16] has used a new conservative finite difference scheme with a parameter $ \theta $ has been employed for obtaining the numerical solutions of the considered model. Moreover, Convergence of the numerical solutions has been investigated. For further information of the numerical solutions of the nonlinear KGZ model, you can see ^[17,18].

The rest sections in this manuscript is organized as follows; Section 2 applies the above-mentioned numerical schemes to the nonlinear KGZ equation for estimating the numerical solutions. Section 3 discusses the obtained numerical solutions. Section 4 gives the conclusion of the whole paper.

2. Numerical investigation

Here, we give the headline of the used methods they we give the obtained results along with these approximate schemes

2.1. Methodology

2.1.1. Trigonometric Quintic B-spline scheme

Using the trigonometric Quintic B-spline scheme supposes the solutions of Eq (1.2) is formulated as following

$\begin{eqnarray} \mathfrak{P}(\mathfrak{Z}) = \sum\limits_{j = -2}^{r+2} \mathcal{C}_{j} \, \mathcal{G}_{j}(\mathfrak{Z}), \; \; \; j = (0,1,\cdots,r), \end{eqnarray}$ $

(2.1)

where $ \mathcal{C}_{j} $ are be determined form the collocation points $ \mathfrak{Z}_{j} $ and $ \mathcal{G}_{j}(\mathfrak{Z}) $ satisfies the following values

$\begin{eqnarray} \mathcal{G}_{j}(\mathfrak{Z}) = \left\lbrace \begin{array}{cc} \psi^{5}(\mathfrak{Z}_{j-3}),& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-3},\mathfrak{Z}_{j-2}] \\ \\ \psi^{4}(\mathfrak{Z}_{j-3})\,\Psi(\mathfrak{Z}_{j-1})+\cdots+\Psi(\mathfrak{Z}_{j+3})\,\psi^{4}(\mathfrak{Z}_{j-2})& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-2},\mathfrak{Z}_{j-1}]\\ \\ \psi^{3}(\mathfrak{Z}_{j-3})\,\Psi^{2}(\mathfrak{Z}_{j})+\cdots+\Psi^{2}(\mathfrak{Z}_{j+3})\, \psi^{3}(\mathfrak{Z}_{j-1}) & \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j-1},\mathfrak{Z}_{j}] \\ \\ \psi^{2}(\mathfrak{Z}_{j-3})\,\Psi^{3}(\mathfrak{Z}_{j+1})+\cdots+\Psi^{3}(\mathfrak{Z}_{j+3})\,\psi^{2}(\mathfrak{Z}_{j}) & \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j},\mathfrak{Z}_{j+1}]\\ \\ \psi(\mathfrak{Z}_{j-1})\,\Psi^{2}(j+2)+\cdots+\Psi^{4}(\mathfrak{Z}_{j+3})\,\psi(\mathfrak{Z}_{j+1}), & \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j+1},\mathfrak{Z}_{j+2}]\\ \\ \Psi^{5}(\mathfrak{Z}_{j+3})& \; \; \; \; \; \; \mathfrak{Z}\in [\mathfrak{Z}_{j+2},\mathfrak{Z}_{j+3}]\\ \\ 0, & \; \; \; \text{otherwise} \end{array} \right. \end{eqnarray}$ $

(2.2)

where $ \psi(\mathfrak{Z}_{j}) = sin\left(\frac{\mathfrak{Z}-\mathfrak{Z}_{j}}{2}\right), \, \Psi(\mathfrak{Z}_{j}) = sin\left(\frac{\mathfrak{Z}_{j}-\mathfrak{Z}}{2}\right) $. Consequently, we can find the values of $ \mathcal{G}_{j}(\mathfrak{Z}) $ as shown in the next Table 1 where the values of $ \mathcal{P}_{\mathcal{L}}, \, \mathcal{L} = 1, \cdots, 13. $

$\begin{eqnarray} \mathcal{P}_{1} = \frac{\sin ^5\left(\frac{h}{2}\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$ $

(2.3)

$\begin{eqnarray} \mathcal{P}_{2} = \frac{2 \sin ^5\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) \left(16 \cos ^2\left(\frac{h}{2}\right)-3\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$ $

(2.4)

$\begin{eqnarray} \mathcal{P}_{3} = \frac{2 \sin ^5\left(\frac{h}{2}\right) \left(48 \cos ^4\left(\frac{h}{2}\right)-16 \cos ^2\left(\frac{h}{2}\right)+1\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$ $

(2.5)

$\begin{eqnarray} \mathcal{P}_{4} = \frac{5 \sin ^4\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.6)

$\begin{eqnarray} \mathcal{P}_{5} = \frac{5 \sin ^4\left(\frac{h}{2}\right) \cos ^2\left(\frac{h}{2}\right) \left(8 \cos ^2\left(\frac{h}{2}\right)-3\right)}{\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)}, \end{eqnarray}$ $

(2.7)

$\begin{eqnarray} \mathcal{P}_{6} = \frac{5 \sin ^3\left(\frac{h}{2}\right) \left(5 \cos ^2\left(\frac{h}{2}\right)-1\right)}{4 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.8)

$\begin{eqnarray} \mathcal{P}_{7} = \frac{5 \sin ^3\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) \left(16 \cos ^4\left(\frac{h}{2}\right)-15 \cos ^2\left(\frac{h}{2}\right)+3\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.9)

$\begin{eqnarray} \mathcal{P}_{8} = -\frac{5 \sin ^3\left(\frac{h}{2}\right) \left(16 \cos ^6\left(\frac{h}{2}\right)-5 \cos ^2\left(\frac{h}{2}\right)+1\right)}{2 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.10)

$\begin{eqnarray} \mathcal{P}_{9} = \frac{5 \sin ^2\left(\frac{h}{2}\right) \cos \left(\frac{h}{2}\right) (25 \cos (h)-1)}{16 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.11)

$\begin{eqnarray} \mathcal{P}_{10} = -\frac{5 \sin ^2(h) (-27 \cos (h)+2 \cos (2 h)+1)}{32 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.12)

$\begin{eqnarray} \mathcal{P}_{11} = \frac{5 \sin \left(\frac{h}{2}\right) (44 \cos (h)+125 \cos (2 h)+23)}{128 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.13)

$\begin{eqnarray} \mathcal{P}_{12} = -\frac{5 \sin (h) (88 \cos (h)+127 \cos (2 h)+44 \cos (3 h)+125)}{128 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.14)

$\begin{eqnarray} \mathcal{P}_{13} = \frac{5 \sin \left(\frac{h}{2}\right) (2 \cos (h)+1) (125 \cos (h)+21 \cos (2 h)+23 \cos (3 h)+23)}{64 \left(\sin \left(\frac{5 h}{2}\right) \sin (2 h) \sin \left(\frac{3 h}{2}\right) \sin (h) \sin \left(\frac{h}{2}\right)\right)}, \end{eqnarray}$ $

(2.15)

Table 1. Value of $ \mathcal{G}_{j}(\mathfrak{Z}) $ and its principle two derivatives at the knot points.

$ \mathfrak{Z} $	$ \mathfrak{Z}_{j-3} $	$ \mathfrak{Z}_{j-2} $	$ \mathfrak{Z}_{j-1} $	$ \mathfrak{Z}_{j} $	$ \mathfrak{Z}_{j+1} $	$ \mathfrak{Z}_{j+2} $	$ \mathfrak{Z}_{j+3} $

$ \mathcal{G}_{j}(\mathfrak{Z}) $	0	$ \mathcal{P}_{1} $	$ \mathcal{P}_{2} $	$ \mathcal{P}_{3} $	$ \mathcal{P}_{2} $	$ \mathcal{P}_{1} $	0
$ \mathcal{G}'_{j}(\mathfrak{Z}) $	0	$ -\mathcal{P}_{4} $	$ -\mathcal{P}_{5} $	0	$ \mathcal{P}_{5} $	$ \mathcal{P}_{4} $	0
$ \mathcal{G}''_{j}(\mathfrak{Z}) $	0	$ \mathcal{P}_{6} $	$ \mathcal{P}_{7} $	$ \mathcal{P}_{8} $	$ \mathcal{P}_{7} $	$ \mathcal{P}_{6} $	0
$ \mathcal{G}'''_{j}(\mathfrak{Z}) $	0	$ \mathcal{P}_{9} $	$ \mathcal{P}_{10} $	0	$ \mathcal{P}_{10} $	$ \mathcal{P}_{9} $	0
$ \mathcal{G}''''_{j}(\mathfrak{Z}) $	0	$ \mathcal{P}_{11} $	$ \mathcal{P}_{12} $	$ \mathcal{P}_{13} $	$ \mathcal{P}_{12} $	$ \mathcal{P}_{11} $	0

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where $ h = \frac{q-p}{r}, \, q > p $ such that $ [p, q] $ is the problem's domain.

2.1.2. Exponential cubic B-spline schemes

Employing the exponential cubic spline technique to considered model with the above conditions, yields elicit its numerical solutions as following

$\begin{eqnarray} \mathfrak{P}(\mathfrak{Z}) = \sum\limits_{\mathfrak{T} = -1} ^{\mathfrak{M}+1} \mathfrak{C}_{\mathfrak{T}}\,\mathcal{E}_{\mathfrak{T}}, \end{eqnarray}$ $

(2.16)

where $ \mathfrak{C}_{\mathfrak{T}}, \, \mathcal{E}_{\mathfrak{T}} $ follow the next conditions, respectively:

$\begin{eqnarray*} \label{c2} \mathfrak{L}\,\mathfrak{B}(\mathfrak{Z}) = \mathcal{F}(\mathfrak{Z}_{\mathfrak{T}},\mathfrak{B}(\mathfrak{Z}_{\mathfrak{T}}))\,\,\text{where}\,\, (\mathfrak{T} = 0,1,...,n) \end{eqnarray*}$ $

and

$\begin{eqnarray} \mathcal{E}_{\mathfrak{T}}(\mathfrak{Z}) = \frac{1}{6\,\mathfrak{H}^{3}}\left\{ \begin{array}{cc} (\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-2})^{3},& \mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}-2},\mathfrak{Z}_{\mathfrak{T}-1}],\\ -3\,(\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})^{3}+3\,\mathfrak{H}\, (\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})^{2}+3\,\mathfrak{H}^{2}\,(\mathfrak{Z}-\mathfrak{Z}_{\mathfrak{T}-1})+\mathfrak{H}^{3},&\mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}-1},\mathfrak{Z}_{i}],\\ -3\,(\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})^{3}+3\,\mathfrak{H}\, (\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})^{2}+3\,\mathfrak{H}^{2}\, (\mathfrak{Z}_{\mathfrak{T}+1}-\mathfrak{Z})+\mathfrak{H}^{3},&\mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}},\mathfrak{Z}_{\mathfrak{T}+1}],\\ (\mathfrak{Z}_{\mathfrak{T}+2}-\mathfrak{Z})^{3},& \mathfrak{Z}\in [\mathfrak{Z}_{\mathfrak{T}+1},\mathfrak{Z}_{\mathfrak{T}+2}],\\ 0,&\text{otherwise}. \end{array}\right. \end{eqnarray}$ $

(2.17)

For $ \mathfrak{T} \in [-2, \mathfrak{M}+2] $, we obtain

$\begin{eqnarray} \mathfrak{B}_{\mathfrak{T}}(\mathfrak{Z}) = \mathfrak{C}_{\mathfrak{T}-1}+4\,\mathfrak{C}_{\mathfrak{T}}+\mathfrak{C}_{\mathfrak{T}+1}. \end{eqnarray}$ $

(2.18)

2.2. Method's results

Here, we apply the TQBS and ECBS schemes to Eq (1.1) with the evaluated initial and boundary conditions (1.3) as following.

2.2.1. Numerical solution via TQBS scheme

Applying the TQBS scheme to Eq (1.2) with above-conditions (1.3), gets the following numerical values in Tables 2, 3, and Figure 1.

Table 2. Analytical, numerical, and absolute error of Eq (1.2) through the TQBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $ \mathfrak{Z} $	Analytica	Numerical	Absolute Error	Value of $ \mathfrak{Z} $	Analytical	Numerical	Absolute Error
0	0	-2.71051E-20	2.71051E-20	0.2578125	-0.034185856	-0.034185856	4.37705E-14
0.0078125	-0.001041661	-0.001041661	1.00159E-15	0.265625	-0.035209885	-0.035209885	4.49224E-14
0.015625	-0.002083291	-0.002083291	2.52879E-15	0.2734375	-0.036232859	-0.036232859	4.60743E-14
0.0234375	-0.003124857	-0.003124857	3.91484E-15	0.28125	-0.037254747	-0.037254747	4.72261E-14
0.03125	-0.004166328	-0.004166328	5.33774E-15	0.2890625	-0.038275521	-0.038275521	4.83363E-14
0.0390625	-0.005207671	-0.005207671	6.74634E-15	0.296875	-0.039295151	-0.039295151	4.94466E-14
0.046875	-0.006248856	-0.006248856	8.1584E-15	0.3046875	-0.040313607	-0.040313607	5.05221E-14
0.0546875	-0.00728985	-0.00728985	9.56613E-15	0.3125	-0.041330861	-0.041330861	5.15907E-14
0.0625	-0.008330622	-0.008330622	1.09721E-14	0.3203125	-0.042346885	-0.042346885	5.26246E-14
0.0703125	-0.00937114	-0.00937114	1.23738E-14	0.328125	-0.043361648	-0.043361648	5.36515E-14
0.078125	-0.010411372	-0.010411372	1.37685E-14	0.3359375	-0.044375124	-0.044375124	5.46438E-14
0.0859375	-0.011451287	-0.011451287	1.51632E-14	0.34375	-0.045387282	-0.045387282	5.56291E-14
0.09375	-0.012490853	-0.012490853	1.65527E-14	0.3515625	-0.046398096	-0.046398096	5.65936E-14
0.1015625	-0.013530039	-0.013530039	1.7937E-14	0.359375	-0.047407536	-0.047407536	5.75304E-14
0.109375	-0.014568812	-0.014568812	1.93196E-14	0.3671875	-0.048415576	-0.048415576	5.84324E-14
0.1171875	-0.015607143	-0.015607143	2.06935E-14	0.375	-0.049422187	-0.049422187	5.93275E-14
0.125	-0.016644999	-0.016644999	2.20587E-14	0.3828125	-0.050427341	-0.050427341	6.02018E-14
0.1328125	-0.017682349	-0.017682349	2.34222E-14	0.390625	-0.051431011	-0.051431011	6.10553E-14
0.140625	-0.018719162	-0.018719162	2.47719E-14	0.3984375	-0.052433171	-0.052433171	6.1888E-14
0.1484375	-0.019755406	-0.019755406	2.6118E-14	0.40625	-0.053433792	-0.053433792	6.2686E-14
0.15625	-0.020791051	-0.020791051	2.74503E-14	0.4140625	-0.054432847	-0.054432847	6.34492E-14
0.1640625	-0.021826065	-0.021826065	2.87825E-14	0.421875	-0.055430311	-0.055430311	6.41917E-14
0.171875	-0.022860418	-0.022860418	3.01044E-14	0.4296875	-0.056426157	-0.056426157	6.48925E-14
0.1796875	-0.023894078	-0.023894078	3.14124E-14	0.4375	-0.057420357	-0.057420357	6.56003E-14
0.1875	-0.024927014	-0.024927014	3.26926E-14	0.4453125	-0.058412887	-0.058412887	6.62456E-14
0.1953125	-0.025959197	-0.025959197	3.39763E-14	0.453125	-0.059403719	-0.059403719	6.68632E-14
0.203125	-0.026990595	-0.026990595	3.52322E-14	0.4609375	-0.060392829	-0.060392829	6.74669E-14
0.2109375	-0.028021178	-0.028021178	3.64916E-14	0.46875	-0.06138019	-0.06138019	6.80359E-14
0.21875	-0.029050915	-0.029050915	3.77406E-14	0.4765625	-0.062365777	-0.062365777	6.8591E-14
0.2265625	-0.030079776	-0.030079776	3.89688E-14	0.484375	-0.063349565	-0.063349565	6.91253E-14
0.234375	-0.03110773	-0.03110773	4.01866E-14	0.4921875	-0.064331528	-0.064331528	6.95971E-14
0.2421875	-0.032134749	-0.032134749	4.13905E-14	0.5	-0.065311643	-0.065311643	7.00412E-14
0.25	-0.0331608	-0.0331608	4.25909E-14	0.5078125	-0.066289885	-0.066289885	7.04575E-14

| Show Table

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Table 3. Analytical, numerical, and absolute error of Eq (1.2) through the TQBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $ \mathfrak{Z} $	Analytical	Numerical	Absolute error	Value of $ \mathfrak{Z} $	Analytical	Numerical	Absolute error
0.515625	-0.067266228	-0.067266228	7.08461E-14	0.7578125	-0.096468595	-0.096468595	6.22696E-14
0.5234375	-0.068240649	-0.068240649	7.11931E-14	0.765625	-0.097372658	-0.097372658	6.1201E-14
e 0.53125	-0.069213124	-0.069213124	7.14984E-14	0.7734375	-0.098274146	-0.098274146	6.00908E-14
0.5390625	-0.070183629	-0.070183629	7.18037E-14	0.78125	-0.099173043	-0.099173043	5.88973E-14
0.546875	-0.071152141	-0.071152141	7.20674E-14	0.7890625	-0.100069331	-0.100069331	5.76483E-14
0.5546875	-0.072118635	-0.072118635	7.22755E-14	0.796875	-0.100962996	-0.100962996	5.63855E-14
0.5625	-0.07308309	-0.07308309	7.24559E-14	0.8046875	-0.101854021	-0.101854021	5.50393E-14
0.5703125	-0.074045483	-0.074045483	7.25808E-14	0.8125	-0.102742391	-0.102742391	5.36515E-14
0.578125	-0.075005789	-0.075005789	7.2678E-14	0.8203125	-0.103628091	-0.103628091	5.21666E-14
0.5859375	-0.075963988	-0.075963988	7.27196E-14	0.828125	-0.104511107	-0.104511107	5.06262E-14
0.59375	-0.076920057	-0.076920057	7.27474E-14	0.8359375	-0.105391423	-0.105391423	4.89747E-14
0.6015625	-0.077873973	-0.077873973	7.27057E-14	0.84375	-0.106269025	-0.106269025	4.72677E-14
0.609375	-0.078825716	-0.078825716	7.26502E-14	0.8515625	-0.107143899	-0.107143899	4.55053E-14
0.6171875	-0.079775264	-0.079775264	7.25253E-14	0.859375	-0.108016031	-0.108016031	4.36595E-14
0.625	-0.080722594	-0.080722594	7.23588E-14	0.8671875	-0.108885407	-0.108885407	4.17999E-14
0.6328125	-0.081667688	-0.081667688	7.21367E-14	0.875	-0.109752015	-0.109752015	3.98431E-14
0.640625	-0.082610522	-0.082610522	7.18869E-14	0.8828125	-0.110615841	-0.110615841	3.78308E-14
0.6484375	-0.083551078	-0.083551078	7.15816E-14	0.890625	-0.111476871	-0.111476871	3.57908E-14
0.65625	-0.084489334	-0.084489334	7.12486E-14	0.8984375	-0.112335095	-0.112335095	3.36675E-14
0.6640625	-0.08542527	-0.08542527	7.086E-14	0.90625	-0.113190498	-0.113190498	3.14748E-14
0.671875	-0.086358867	-0.086358867	7.04575E-14	0.9140625	-0.11404307	-0.11404307	2.92405E-14
0.6796875	-0.087290105	-0.087290105	6.99718E-14	0.921875	-0.114892798	-0.114892798	2.68674E-14
0.6875	-0.088218965	-0.088218965	6.94583E-14	0.9296875	-0.11573967	-0.11573967	2.44249E-14
0.6953125	-0.089145427	-0.089145427	6.88755E-14	0.9375	-0.116583676	-0.116583676	2.18991E-14
0.703125	-0.090069472	-0.090069472	6.82371E-14	0.9453125	-0.117424804	-0.117424804	1.92901E-14
0.7109375	-0.090991083	-0.090991083	6.75571E-14	0.953125	-0.118263043	-0.118263043	1.66117E-14
0.71875	-0.09191024	-0.09191024	6.68215E-14	0.9609375	-0.119098383	-0.119098383	1.38639E-14
0.7265625	-0.092826925	-0.092826925	6.60166E-14	0.96875	-0.119930814	-0.119930814	1.10745E-14
0.734375	-0.093741121	-0.093741121	6.51562E-14	0.9765625	-0.120760325	-0.120760325	8.18789E-15
0.7421875	-0.094652809	-0.094652809	6.42403E-14	0.984375	-0.121586906	-0.121586906	5.31519E-15
0.75	-0.095561973	-0.095561973	6.32688E-14	0.9921875	-0.122410548	-0.122410548	2.13718E-15

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Figure 1. analytical, numerical, and absolute error for the nonlinear KGZ equations through the TQBS scheme.

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2.2.2. Numerical solution via ECBS scheme

Applying the ECBS scheme to Eq (1.2) with above-conditions (1.3), gets the following numerical values in Table 4, and Figure 2.

Table 4. Analytical, numerical, and absolute error of Eq (1.2) through the ECBS scheme under the shown values of boundary and initial conditions (1.3).

Value of $ \mathfrak{Z} $	Analytical	Numerical	Absolute error
0	0	0	0
0.001	-0.000133333	-0.000133333	2.75062E-16
0.002	-0.000266667	-0.000266667	5.33482E-16
0.003	-0.0004	-0.0004	7.58508E-16
0.004	-0.000533333	-0.000533333	9.33498E-16
0.005	-0.000666665	-0.000666665	1.04181E-15
0.006	-0.000799998	-0.000799998	1.06685E-15
0.007	-0.00093333	-0.00093333	9.9172E-16
0.008	-0.001066661	-0.001066661	7.99924E-16
0.009	-0.001199992	-0.001199992	4.75097E-16
0.01	-0.001333322	-0.001333322	0

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Figure 2. Exact, approximate, and absolute error of the nonlinear KGZ equations through the ECBS scheme.

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3. Results and discussion

Here, we explain the accuracy and novelty of the obtained numerical result in this research paper by comparing them with the previously calculated in ^[42] through four-different schemes (Adomian decomposition (AD), El-kalla (EK), cubic B-spline (CB), and extended cubic B-spline (ECB) schemes)) and one common scheme (exponential cubic B-spline (ExCB) scheme). Although, comparing our obtained result with each other, shows the accuracy of the ECBS scheme over the TQBS scheme where the absolute error is smaller than that have been obtained by the TQBS scheme which have been shown in Figures 1, 2 and Tables 2–4. Now, comparing the accuracy between our solutions and that have been evaluated in ^[42], shows our solution is more accurate than their solutions that have been explained in Table 5, and Figure 3.

Table 5. Numerical methods' absolute error.

Value of $ \mathfrak{Z} $	AD	EK	CBS	EtCBS	ECBS	ECBS	TQBS
0	0	0	0	0	5.45828E-18	0	2.71E-20
0.001	2.71051E-20	2.71051E-20	1.83257E-16	3.29993E-09	1.10002E-09	2.75E-16
0.002	5.42101E-20	5.42101E-20	3.55401E-16	6.39986E-09	2.13337E-09	5.33E-16
0.003	0	0	5.05455E-16	9.0998E-09	3.03339E-09	7.59E-16
0.004	0	0	6.22007E-16	1.11997E-08	3.7334E-09	9.33E-16
0.005	0	0	6.94215E-16	1.24997E-08	4.16674E-09	1.04E-15
0.006	1.0842E-19	1.0842E-19	7.10695E-16	1.27997E-08	4.26674E-09	1.07E-15
0.007	1.0842E-19	1.0842E-19	6.60821E-16	1.18997E-08	3.96674E-09	9.92E-16
0.008	0	0	5.32994E-16	9.59976E-09	3.20006E-09	8E-16
0.009	2.1684E-19	0	3.1637E-16	5.69985E-09	1.90003E-09	4.75E-16
0.01	0	2.1684E-19	0	0	2.1684E-19	0	2.53E-15

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Figure 3. Absolute value of error for AD, EK, CBS, ECBS, ExCBS, ECBS, and TQBS schemes.

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4. Conclusions

This manuscript has employed the TQBS and ECBS numerical schemes for evaluating the numerical solutions of the nonlinear KGZ model. The matching between analytical and numerical solutions has been explained through the shown tables and figures. The accuracy of the modified Khater method has been proved through six numerical schemes. The novelty and originality of our obtained solutions have been explained. the powerful and effectiveness of the used techniques are also explained and verified.

Acknowledgments

The authors would like to thank Taif University Researchers supporting Project number (TURSP-2020/159), Taif University-Saudi Arabia.

Conflict of interest

There is no conflict of interest.

References

[1]	Brault V, Moore R, Kutsch S, et al. (2001) Inactivation of the beta-catenin gene by {Wnt1-Cre-mediated} deletion results in dramatic brain malformation and failure of craniofacial development. Development 128: 1253-1264.
[2]	Noden DM, Trainor PA (2005) Relations and interactions between cranial mesoderm and neural crest populations. J Anat 207: 575-601. doi: 10.1111/j.1469-7580.2005.00473.x
[3]	Trainor P, Krumlauf R (2000) Plasticity in mouse neural crest cells reveals a new patterning role for cranial mesoderm. Nat cell biol 2: 96-9102. doi: 10.1038/35000051
[4]	Trainor PA, Tam PP (1995) Cranial paraxial mesoderm and neural crest cells of the mouse embryo: co-distribution in the craniofacial mesenchyme but distinct segregation in branchial arches. Development 121: 2569-2582.
[5]	McBratney-Owen, Iseki S, Bamforth SD (2008) Development and tissue origins of the mammalian cranial base. Dev Biol 322: 121-132. doi: 10.1016/j.ydbio.2008.07.016
[6]	Lescroart F, Kelly RG, Le Garrec JF, et al. (2010) Clonal analysis reveals common lineage relationships between head muscles and second heart field derivatives in the mouse embryo. Development 137: 3269-3279. doi: 10.1242/dev.050674
[7]	Yoshida T, Vivatbutsiri P, Gillian MK, et al. (2008) Cell lineage in mammalian craniofacial mesenchyme. Mech dev 125: 797-808.
[8]	Lee Y-HH, Saint-Jeannet J-PP (2011) Sox9 function in craniofacial development and disease. Genesis 49: 200-208.
[9]	Bildsoe H, Loebel DAF, Jones VJ, et al. (2009) Requirement for Twist1 in frontonasal and skull vault development in the mouse embryo. Dev Biol 331: 176-188.
[10]	Bothe I, Tenin G, Oseni A, et al. (2011) Dynamic control of head mesoderm patterning. Development 138: 2807-2821.
[11]	Tam PPL, Meier S (1982) The establishment of a somitomeric pattern in the mesoderm of the gastrulating mouse embryo. Am J Anat 164: 209-225.
[12]	Meier S, Tam PPL (1982) Metameric pattern development in the embryonic axis of the mouse I. Differentiation 21: 95-108. doi: 10.1111/j.1432-0436.1982.tb01202.x
[13]	Tzahor E, Kempf H, Mootoosamy RC, et al. (2003) Antagonists of Wnt and {BMP} signaling promote the formation of vertebrate head muscle. Genes dev 17: 3087-3099. doi: 10.1101/gad.1154103
[14]	Michailovici I, Eigler T, Tzahor E (2015) Craniofacial Muscle Development. Curr top dev biol 115: 3-30.
[15]	Tzahor E (2015) Head muscle development. Results and problems in cell differentiation 56: 123-142. doi: 10.1007/978-3-662-44608-9_6
[16]	Tajbakhsh S, Cossu G (1997) Establishing myogenic identity during somitogenesis. Curr Opin Genet Dev 7: 634-641. doi: 10.1016/S0959-437X(97)80011-1
[17]	Harel I, Nathan E, Tirosh-Finkel L, et al. (2009) Distinct origins and genetic programs of head muscle satellite cells. Dev cell 16: 822-832. doi: 10.1016/j.devcel.2009.05.007
[18]	Kitamura K, Miura H, S M-T, et al. (1999) Mouse Pitx2 deficiency leads to anomalies of the ventral body wall, heart, extra- and periocular mesoderm and right pulmonary isomerism. Development 126: 5749-5758.
[19]	Dong F, Sun X, Liu W, et al. (2006) Pitx2 promotes development of splanchnic mesoderm-derived branchiomeric muscle. Development 133: 4891-4899. doi: 10.1242/dev.02693
[20]	Diehl AG, Zareparsi S, Qian M, et al. (2006) Extraocular muscle morphogenesis and gene expression are regulated by Pitx2 gene dose. Invest Ophthalmol Vis Sci 47: 1785-1793. doi: 10.1167/iovs.05-1424
[21]	Richardson L, Venkataraman S, Stevenson P, et al. (2014) EMAGE mouse embryo spatial gene expression database: 2014 update. Nucleic acids res 42: D835-844. doi: 10.1093/nar/gkt1155
[22]	Zhang Z, Huynh T, Baldini A (2006) Mesodermal expression of Tbx1 is necessary and sufficient for pharyngeal arch and cardiac outflow tract development. Development 133: 3587-3595. doi: 10.1242/dev.02539
[23]	Aggarwal VS, Carpenter C, Freyer L, et al. (2010) Mesodermal Tbx1 is required for patterning the proximal mandible in mice. Dev Biol 344: 669-681. doi: 10.1016/j.ydbio.2010.05.496
[24]	Franco HL, Casasnovas J, Rodríguez-Medina JR, et al. (2011) Redundant or separate entities? - Roles of Twist1 and Twist2 as molecular switches during gene transcription. Nucleic Acids Res 39: 1177-1186.
[25]	Hebrok M, Wertz K, Fuchtbauer EM (1994) M-twist is an inhibitor of muscle differentiation. Dev biol 165: 537-544.
[26]	Spicer DB, Rhee J, Cheung WL, et al. (1996) Inhibition of myogenic {bHLH} and {MEF2} transcription factors by the {bHLH} protein Twist. Science 272: 1476-1480.
[27]	Bildsoe H, Loebel DA, Jones VJ, et al. (2013) The mesenchymal architecture of the cranial mesoderm of mouse embryos is disrupted by the loss of Twist1 function. Dev Biol 374: 295-307.
[28]	Goodnough LH, Dinuoscio GJ, Atit RP (2016) Twist1 contributes to cranial bone initiation and dermal condensation by maintaining wnt signaling responsiveness. Dev dyns 245: 144-156. doi: 10.1002/dvdy.24367
[29]	Longabaugh WJR, Davidson EH, Bolouri H (2009) Visualization, documentation, analysis, and communication of large-scale gene regulatory networks. Biochim Biophysica Acta 1789: 363-374.
[30]	Katsuno Y, Lamouille S, Derynck R (2013) TGF-β signaling and epithelial-mesenchymal transition in cancer progression. Curr opin oncol 25: 76-84. doi: 10.1097/CCO.0b013e32835b6371
[31]	Lamouille S, Xu J, Derynck R (2014) Molecular mechanisms of epithelial-mesenchymal transition. Nat rev Mol cell biol 15: 178-196. doi: 10.1038/nrm3758
[32]	Diogo R, Kelly RG, Christiaen L, et al. (2015) A new heart for a new head in vertebrate cardiopharyngeal evolution. Nature 520: 466-473. doi: 10.1038/nature14435
[33]	Lewis AE, Vasudevan HN, O’Neill AK, et al. (2013) The widely used Wnt1-Cre transgene causes developmental phenotypes by ectopic activation of Wnt signaling. Dev biol 379: 229-234. doi: 10.1016/j.ydbio.2013.04.026
[34]	Wang X-HH, Wang Y, Liu AKK, et al. (2015) Genome-wide identification and analysis of basic helix-loop-helix domains in dog, Canis lupus familiaris. Mol genet genomics 290: 633-648.
[35]	Liem KF, Tremml G, Roelink H, et al. (1995) Dorsal differentiation of neural plate cells induced by BMP-mediated signals from epidermal ectoderm. Cell 82: 969-979. doi: 10.1016/0092-8674(95)90276-7
[36]	Kanzler B, Foreman RK, Labosky PA, et al. (2000) BMP signaling is essential for development of skeletogenic and neurogenic cranial neural crest. Development 127: 1095-1104.
[37]	Dudas M, Sridurongrit S, Nagy A, et al. (2004) Craniofacial defects in mice lacking {BMP} type I receptor Alk2 in neural crest cells. Mech dev 121: 173-182. doi: 10.1016/j.mod.2003.12.003
[38]	Dudas M, Kim J, Li W-YY, et al. (2006) Epithelial and ectomesenchymal role of the type I TGF-beta receptor ALK5 during facial morphogenesis and palatal fusion. Dev biol 296: 298-314. doi: 10.1016/j.ydbio.2006.05.030
[39]	Liu W, Selever J, Murali D, et al. (2005) Threshold-specific requirements for Bmp4 in mandibular development. Dev biol 283: 282-293. doi: 10.1016/j.ydbio.2005.04.019
[40]	Ko SO, Chung IH, Xu X, et al. (2007) Smad4 is required to regulate the fate of cranial neural crest cells. Dev Biol 312: 435-447. doi: 10.1016/j.ydbio.2007.09.050
[41]	Zhao H, Oka K, Bringas P, et al. (2008) {TGF-β} type I receptor Alk5 regulates tooth initiation and mandible patterning in a type {II} receptor-independent manner. Dev Biol 320: 19-29. doi: 10.1016/j.ydbio.2008.03.045
[42]	Roybal PG, Wu NL, Sun J, et al. (2010) Inactivation of Msx1 and Msx2 in neural crest reveals an unexpected role in suppressing heterotopic bone formation in the head. Dev biol 343: 28-39.
[43]	Kim H, Rice D, Kettunen P, et al. (1998) FGF-, BMP- and Shh-mediated signalling pathways in the regulation of cranial suture morphogenesis and calvarial bone development. Development 125: 1241-1251.
[44]	Bonilla-Claudio M, Wang J, Bai Y, et al. (2012) Bmp signaling regulates a dose-dependent transcriptional program to control facial skeletal development. Development 139: 709-719.
[45]	Lu MF, Pressman C, Dyer R, et al. (1999) Function of Rieger syndrome gene in left-right asymmetry and craniofacial development. Nature 401: 276-278. doi: 10.1038/45797
[46]	Tapscott SJ, Weintraub H (1991) {MyoD} and the regulation of myogenesis by helix-loop-helix proteins. J clin invest 87: 1133-1138.
[47]	Langlands K, Yin X, Anand G, et al. (1997) Differential interactions of Id proteins with basic-helix-loop-helix transcription factors. J biol chem 272: 19785-19793.
[48]	Nie X, Luukko K, Kettunen P (2006) {FGF} signalling in craniofacial development and developmental disorders. Oral Dis 12: 102-111. doi: 10.1111/j.1601-0825.2005.01176.x
[49]	Suzuki A, Sangani DR, Ansari A (2015) Molecular mechanisms of midfacial developmental defects. Dev Dyn 245: 276-293.
[50]	Bachler M, Neubüser A (2001) Expression of members of the Fgf family and their receptors during midfacial development. Mech dev 100: 313-316.
[51]	Wilkie AOM (1997) Craniosynostosis: genes and mechanisms. Hum Mol Genet 6: 1647-1656. doi: 10.1093/hmg/6.10.1647
[52]	Abu-Issa R, Smyth G, Smoak I, et al. (2002) Fgf8 is required for pharyngeal arch and cardiovascular development in the mouse. Development 129: 4613-4625.
[53]	Griffin JN, Compagnucci C, Hu D, et al. (2013) Fgf8 dosage determines midfacial integration and polarity within the nasal and optic capsules. Dev biol 374: 185-197. doi: 10.1016/j.ydbio.2012.11.014
[54]	Wang C, Chang JY, Yang C, et al. (2013) Type 1 fibroblast growth factor receptor in cranial neural crest cell-derived mesenchyme is required for palatogenesis. J biol chem 288: 22174-22183. doi: 10.1074/jbc.M113.463620
[55]	Kasberg AD, Brunskill EW, Potter SS (2013) {SP8} regulates signaling centers during craniofacial development. Dev Biol 381: 312-323. doi: 10.1016/j.ydbio.2013.07.007
[56]	Yang X, Kilgallen S, Andreeva V, et al. (2010) Conditional expression of Spry1 in neural crest causes craniofacial and cardiac defects. Dev biol 10: 48.
[57]	Yu K, Xu J, Liu Z, et al. (2003) Conditional inactivation of {FGF} receptor 2 reveals an essential role for {FGF} signaling in the regulation of osteoblast function and bone growth. Development 130: 3063-3074.
[58]	Holmes G, Basilico C (2012) Mesodermal expression of Fgfr2S252W is necessary and sufficient to induce craniosynostosis in a mouse model of Apert syndrome. Dev Biol 368: 283-293. doi: 10.1016/j.ydbio.2012.05.026
[59]	Stottmann RW, Anderson RM, Klingensmith J (2001) The BMP antagonists Chordin and Noggin have essential but redundant roles in mouse mandibular outgrowth. Dev Biol 240: 457-473. doi: 10.1006/dbio.2001.0479
[60]	Tucker A, Sharpe P (2004) The cutting-edge of mammalian development; how the embryo makes teeth. Nat Rev Genet 5: 499-508.
[61]	Tirosh-Finkel L, Zeisel A, Brodt-Ivenshitz M, et al. (2010) BMP-mediated inhibition of FGF signaling promotes cardiomyocyte differentiation of anterior heart field progenitors. Development 137: 2989-3000.
[62]	Olwin BB, Rapraeger A (1992) Repression of myogenic differentiation by aFGF, bFGF, and K-FGF is dependent on cellular heparan sulfate. J cell biol 118: 631-639.
[63]	von Scheven G, Alvares LEE, Mootoosamy RC, et al. (2006) Neural tube derived signals and Fgf8 act antagonistically to specify eye versus mandibular arch muscles. Development 133: 2731-2745. doi: 10.1242/dev.02426
[64]	Michailovici I, Harrington HA, Azogui HH, et al. (2014) Nuclear to cytoplasmic shuttling of ERK promotes differentiation of muscle stem/progenitor cells. Development 141: 2611-2620.
[65]	Frank DU, Fotheringham LK, Brewer JA, et al. (2002) An Fgf8 mouse mutant phenocopies human 22q11 deletion syndrome. Development 129: 4591-4603.
[66]	Rinon A, Lazar S, Marshall H, et al. (2007) Cranial neural crest cells regulate head muscle patterning and differentiation during vertebrate embryogenesis. Development 134: 3065-3075.
[67]	Mani P, Jarrell A, Myers J, et al. (2010) Visualizing canonical Wnt signaling during mouse craniofacial development. Dev Dyn 239: 354-363.
[68]	Hari L, Brault V, Kléber M, et al. (2002) Lineage-specific requirements of beta-catenin in neural crest development. J cell biol 159: 867-880.
[69]	Goodnough LH, Chang AT, Treloar C, et al. (2012) Twist1 mediates repression of chondrogenesis by β-catenin to promote cranial bone progenitor specification. Development 139: 4428-4438. doi: 10.1242/dev.081679
[70]	Goodnough LH, Dinuoscio GJ, Ferguson JW, et al. (2014) Distinct requirements for cranial ectoderm and mesenchyme-derived wnts in specification and differentiation of osteoblast and dermal progenitors. Genetics 10: e1004152.
[71]	Wang Y, Song L, Zhou CJ (2011) The canonical Wnt/β-catenin signaling pathway regulates Fgf signaling for early facial development. Dev biol 349: 250-260. doi: 10.1016/j.ydbio.2010.11.004
[72]	Reid BS, Yang H, Melvin VS, et al. (2011) Ectodermal Wnt/β-catenin signaling shapes the mouse face. Dev biol 349: 261-269. doi: 10.1016/j.ydbio.2010.11.012
[73]	Klaus A, Saga Y, Taketo MM, et al. (2007) Distinct roles of Wnt/beta-catenin and Bmp signaling during early cardiogenesis. Proc Natl Acad Sci U S A 104: 18531-18536. doi: 10.1073/pnas.0703113104
[74]	Wang XP, O'Connell DJ, Lund JJ, et al. (2009) Apc inhibition of Wnt signaling regulates supernumerary tooth formation during embryogenesis and throughout adulthood. Development 136: 1939-1949.
[75]	Day TF, Guo X, Lisa G-B, et al. (2005) Wnt/beta-catenin signaling in mesenchymal progenitors controls osteoblast and chondrocyte differentiation during vertebrate skeletogenesis. Dev cell 8: 739-750. doi: 10.1016/j.devcel.2005.03.016
[76]	Reinhold MI, Kapadia RM, Liao Z, et al. (2006) The Wnt-inducible transcription factor Twist1 inhibits chondrogenesis. J biol chem 281: 1381-1388.
[77]	Hoang BH, Thomas JT, Abdul-Karim FW, et al. (1998) Expression pattern of two Frizzled-related genes, Frzb-1 and Sfrp-1, during mouse embryogenesis suggests a role for modulating action of Wnt family members. Dev Dyn 212: 364-372. doi: 10.1002/(SICI)1097-0177(199807)212:3<364::AID-AJA4>3.0.CO;2-F
[78]	Lescher B, Haenig B, Kispert A (1998) sFRP-2 is a target of the Wnt-4 signaling pathway in the developing metanephric kidney. Dev Dyn 213: 440-451. doi: 10.1002/(SICI)1097-0177(199812)213:4<440::AID-AJA9>3.0.CO;2-6
[79]	Yamashita S, Andoh M, Ueno-Kudoh H, et al. (2009) Sox9 directly promotes Bapx1 gene expression to repress Runx2 in chondrocytes. Exp Cell Res 315: 2231-2240. doi: 10.1016/j.yexcr.2009.03.008
[80]	Eames BF, Sharpe PT, Helms JA (2004) Hierarchy revealed in the specification of three skeletal fates by Sox9 and Runx2. Dev Biol 274: 188-200. doi: 10.1016/j.ydbio.2004.07.006
[81]	Zhou G, Zheng Q, Engin F, et al. (2006) Dominance of SOX9 function over RUNX2 during skeletogenesis. Proc Natl Acad Sci U S A 103: 19004-19009. doi: 10.1073/pnas.0605170103
[82]	Akiyama H, Lyons JP, Yuko M-A, et al. (2004) Interactions between Sox9 and beta-catenin control chondrocyte differentiation. Genes dev 18: 1072-1087.
[83]	Ting M-CC, Wu NL, Roybal PG, et al. (2009) EphA4 as an effector of Twist1 in the guidance of osteogenic precursor cells during calvarial bone growth and in craniosynostosis. Development 136: 855-864.
[84]	Chang AT, Liu Y, Ayyanathan K, et al. (2015) An evolutionarily conserved DNA architecture determines target specificity of the TWIST family bHLH transcription factors. Genes dev 29: 603-616.
[85]	Lee H-YY, Kléber M, Hari L, et al. (2004) Instructive role of Wnt/beta-catenin in sensory fate specification in neural crest stem cells. Science 303: 1020-1023.
[86]	Chen ZF, Behringer RR (1995) Twist Is Required in Head Mesenchyme for Cranial Neural Tube Morphogenesis. Genes Dev 9: 686-699.
[87]	Soo K, P OR, Meredith, Khoo P-LL, et al. (2002) Twist function is required for the morphogenesis of the cephalic neural tube and the differentiation of the cranial neural crest cells in the mouse embryo. Dev biol 247: 251-270. doi: 10.1006/dbio.2002.0699
[88]	Gitelman I (1997) Twist protein in mouse embryogenesis. Dev biol 189: 205-214. doi: 10.1006/dbio.1997.8614
[89]	Meng T, Huang Y, Wang S, et al. (2015) Twist1 Is Essential for Tooth Morphogenesis and Odontoblast Differentiation. J Biol Chem 290: 29593-29602. doi: 10.1074/jbc.M115.680546
[90]	Loebel DA, Hor AC, Bildsoe HK, et al. (2014) Timed deletion of Twist1 in the limb bud reveals age-specific impacts on autopod and zeugopod patterning. PLoS One 9: e98945. doi: 10.1371/journal.pone.0098945
[91]	Zhang Y, Blackwell EL, T M, Mitchell, et al. (2012) Specific inactivation of Twist1 in the mandibular arch neural crest cells affects the development of the ramus and reveals interactions with hand2. Dev dyn 241: 924-940. doi: 10.1002/dvdy.23776
[92]	Morris-Wiman J, Brinkley LL (1990) The role of the mesenchyme in mouse neural fold elevation. I. Patterns of mesenchymal cell distribution and proliferation in embryos developing in vitro. Am J anat 188: 121-132.
[93]	Teng Y, Li X (2014) The roles of HLH transcription factors in epithelial mesenchymal transition and multiple molecular mechanisms. Clin Exp Metastasis 31: 367-377. doi: 10.1007/s10585-013-9621-6
[94]	Lee K-WW, Lee NK, Ham S, et al. (2014) Twist1 is essential in maintaining mesenchymal state and tumor-initiating properties in synovial sarcoma. Cancer lett 343: 62-73. doi: 10.1016/j.canlet.2013.09.013
[95]	Tsai JH, Donaher JL, Murphy DA, et al. (2012) Spatiotemporal regulation of epithelial-mesenchymal transition is essential for squamous cell carcinoma metastasis. Cancer cell 22: 725-736.
[96]	Szklarczyk D, Franceschini A, Wyder S (2014) STRING v10: protein–protein interaction networks, integrated over the tree of life. Nucleic acids Res 43: D447-452.
[97]	Csardi G, Nepusz T (2006) The igraph software package for complex network research. Inter Journal.
[98]	Takada R, Satomi Y, Kurata T, et al. (2006) Monounsaturated fatty acid modification of Wnt protein: its role in Wnt secretion. Dev cell 11: 791-801. doi: 10.1016/j.devcel.2006.10.003
[99]	Grenier J, Teillet M-AA, Grifone R, et al. (2009) Relationship between neural crest cells and cranial mesoderm during head muscle development. PloS one 4: e4381. doi: 10.1371/journal.pone.0004381
[100]	Schneider RA (1999) Neural crest can form cartilages normally derived from mesoderm during development of the avian head skeleton. Dev biol 208: 441-455.

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