Bull Math Biol DOI 10.1007/s11538-014-9967-1 ORIGINAL ARTICLE
The Scaling and Shift of Morphogen Gene Expression Boundary in a Nonlinear Reaction Diffusion System Wei-Shen Li · Yuan-Zhi Shao
Received: 23 September 2013 / Accepted: 16 April 2014 © Society for Mathematical Biology 2014
Abstract The scaling and shift of the gene expression boundary in a developing embryo are two key problems with regard to morphogen gradient formation in developmental biology. In this study, a bigradient model was applied to a nonlinear reaction diffusion system (NRDS) to investigate the location of morphogen gene expression boundary. In contrast to the traditional synthesis–diffusion–degradation model, the introduction of NRDS in this study contributes to the precise gene expression boundary at arbitrary location along the anterior-posterior axis other than simply midembryo even when the linear characteristic lengths of two morphogens are equal. The scaling location depends on the ratio of two morphogen influxes (w) and concentrations (r ) as well as the nonlinear reaction diffusion parameters (α, n). We also formulate a direct relationship between the shift in the gene expression boundary and the influx of morphogen and find that enhancing the morphogen influx is helpful to build up a robust gene expression boundary. By analyzing the robustness of the morphogen gene expression boundary and comparing with the relevant results in linear reaction diffusion system, we determine the precise range of the ratio of the two morphogen influxes with a lower shift in the morphogen gene expression boundary and increased system robustness. Keywords Developmental embryo · Pattern formation and scaling · Morphogenetic gradient · Gene expression boundary · Reaction diffusion system · Robustness analysis 1 Introduction Morphogen gradient is a fundamental factor for the pattern formation of gene expression in development biological matrix (Wolpert 1969). The key problems are the W.-S. Li · Y.-Z. Shao (B) School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China e-mail: [email protected]
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formation of the morphogen gradient. The model proposed by Crick, famously as synthesis–diffusion–degradation (SDD) model, assumes that the morphogen is produced and synthesized from a source, is diffused and degraded along the anteriorposterior axis of the embryo (Crick 1970). However, Bollenbach and coworkers argue that transcytosis rather than diffusion plays a key role in the formation of the morphogen gradient (Bollenbach et al. 2005, 2007). Another viewpoint maintains that the cell growth driven by cell proliferation also contributes to the build up of the morphogen gradient without a diffusion term (Chisholm et al. 2010). Although the transportation process of morphogen in an embryo is still in controversy, the SDD model, as one of the most popular models, has been explored extensively from different aspects (Saunders and Howard 2009; Berezhkovskii et al. 2010, 2011; Gordon et al. 2011). Unfortunately, the traditional SDD model cannot explain the scaling law mechanism underlying the gene expression boundary and embryo size. Numerous studies have attempted to gain insight into the scaling law mechanism from the viewpoint of either proliferation growth or embryo volume using a simple morphogen gradient model (Wartlick et al. 2011; Cheung et al. 2011). Another representative viewpoint advocates that a bigradient model rather than a single gradient model is sufficient to explain the scaled settings of the gene expression boundary. Howard and Wolde (2005) assumed that the activator morphogen gradient originating from the anterior pole is bound to the second corepressor (inhibitor) morphogen from posterior-localized source. Their results indicate that such an interaction and binding between activator and inhibitor will result in the scaled and robust patterns of the gene expression boundary. Houchmandzadeh et al. revealed that two types of morphogen gradients, even without a binding, can also lead to a precise gene expression boundary domain (Houchmandzadeh et al. 2005). Furthermore, Mchale et al found that the scaling of the gene expression boundary is dependent on the ratio of the concentrations and influxes of two morphogens (Mchale et al. 2006). Although the bigradient model has made certain progress in this field, it has at least two limitations as below (Ben-Zvi et al. 2011). 1. The model establishes the gene expression boundary only at specific scaled location, especially at the midembryo, when the characteristic lengths of two morphogens are equal. Such a “partial scaling” is incompatible with some realistic biological phenomena (Surkova et al. 2008; Lott et al. 2007). 2. The shift in the gene expression boundary due to certain biological effects is only related to the ratio of the two morphogen influxes rather than the influx of morphogen itself, which may restrict the quantitative evaluation of the robustness of the morphogen gene expression boundary in a developmental embryo. Judging from the viewpoint of mathematics, we suggest that these problems cannot be solved in the framework of traditional linear reaction diffusion system (LRDS) due to the two factors below: (i) the exponential morphogen gradient profile; (ii): the linear relationship between the exponential morphogen gradient profile and the morphogen influx. In nonlinear reaction diffusion system (NRDS), however, the power–law rather than the exponential morphogen gradient profile emerges, and it has a nonlinear relationship with the morphogen influx (Yuste et al. 2010; Boon et al. 2012a,b). As a
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Morphogen Gene Expression in NRDS
result, the newly discovered relationship between the morphogen gradient profile and influx inspired us to apply the bigradient model to the NRDS to investigate the scaling and shift of the morphogen gene expression boundary systematically. In the NRDS, whether are there any other factors, besides the variation in characteristic lengths of two morphogens responsible for contributing to the scaled morphogen gene expression boundary to be situated at different locations other than at midembryo (L/2)? Compared with the LRDS, how robust is the morphogen gene expression boundary in the NRDS? We will address these two questions in this paper. The remainder of the paper is organized as follows: In Sect. 2, the formalism for two types of morphogen gradients in an embryo is presented. In Sect. 3, we establish a precise gene expression boundary and analyze the shift of the gene expression boundary due to the influx fluctuation of both two morphogens and the maternal gene copies. The concluding remarks and prospects for future work are included in Sect. 4.
2 The General Formalism of the Bigradient Model in the Nonlinear Reaction Diffusion System In this study, the generalized reaction–diffusion equation derived by Boon and coworkers is utilized extensively to investigate the morphogen distribution in an embryo (Boon et al. 2012a,b). The dimensionless nonlinear reaction-diffusion equation is ∂2 ∂c(x, t) = D 2 cα (x, t) − kcn (x, t) ∂t ∂x
(1)
The symbols c, x, t, D, and k denote that the non-dimensional concentration of the morphogen, the distance to the anterior pole, time, diffusion, and the degradation coefficient, respectively; n and α stand for the indices of nonlinear reaction and diffusion, respectively. Furthermore, one can see that the Eq. (1) can be reduced to the traditional SDD model in the LRDS as α = n = 1. For an embryo with a finite size L, the steady-state solution of the first morphogen sourced from the anterior pole satisfies Eq. (2) with the boundary condition Eq. (3), ∂2 α c (x) − kcn (x) = 0, ∂x 2 ∂cα (x) ∂cα (x) −Dn s = j0 , Dn s ∂x ∂x
(2)
D
x=0
= 0,
(3)
x=L
where n s represents the standard and stationary local number density of morphogen particle solvent for acquiring the concentration of morphogen c(x) (Boon et al. 2012a,b). The variables are transformed as follows: x → z=
k x, D
j0 →
j0∗ =
j0 √ , c → g = cα , ns k D
L→L =
k L, D
(4)
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W.-S. Li, Y.-Z. Shao
Assuming that the embryo size L is large enough, we can gain the solution to Eq. (2) as follows c(x) =
j0∗
α−n x0 = α j0∗ α+n
dg ∗ j0 = − dz z=0
−2 n − α x n−α 1+ , n > α, 2 x0 α α + n n+α D , λ, λ = 2α k 2α α+n g 2α (0), = α+n
α+n 2α
2 n+α
(5) (6) (7)
where λ represents the linear characteristic length of the morphogen gradient profile. In the case of α > n, Boon et al introduce the Heaviside function to eliminate the imaginary or minus morphogen concentration c(x) for larger value of x c(x) =
j0∗
α+n 2α
2 n+α
2 α − n x α−n 2 1− x0 − x , α > n, Θ 2 x0 α−n
(8)
where Θ is the Heaviside step function (Boon et al. 2012a,b). For the second morphogen produced from the posterior pole (designated by c p ), its steady-state distribution can be acquired readily by analogy to the results of Eqs. (5–7) c p (x) =
j0∗p
x0 p = a j0∗p j0∗p =
α+n 2α α−n α+n
2 α − n x − L α−n 1+ , n > α, (9) 2 x0 p α Dp α + n n+α λp, λp = , (10) 2α kp 2 α+n
dg p 2α α+n g 2α (L ). = dz z=L α+n
(11)
Similar to Eq. (8), the Heaviside function is employed in the case of α > n c p (x) =
j0∗p
α+n 2α
Θ x−L+
2 α+n
2 α−n
α−n x −L 1+ 2 x0 p x0 p , n < α,
2 α−n
· (12)
where all of the variable denotations in Eqs. (9–12) are similar to the Eqs. (5–8), and λ p is the linear characteristic length of the second morphogen gradient profile. After completing the derivations of the above formulae, we turn our attention to a specific situation in which n > α, for two reasons:
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Morphogen Gene Expression in NRDS
1. the experiment data are compared well with the morphogen profile of n > α (Boon et al. 2012a,b) and 2. the morphogen profile of n > α which means the infinite support (i.e., fast diffusion and low degradation) is also compatible with the size of a large enough embryo (Boon et al. 2012a,b). Note that the results of the current study, unless specified otherwise, focus on the situation where the degradation, diffusion coefficients, and linear characteristic length of the two morphogens are equal (k = k p , D = D p , λ = λ p ). 3 The Scaling and Shift of Gene Expression Boundary Resulted From Different Biological Effects 3.1 The Scaling of the Gene Expression Boundary The ratios of two morphogen influxes and concentrations are supposed w and r , respectively, as indicated by Eqs. (13) and (14), j0∗ = w, j0∗p
(13)
c(x) = r c p (x).
(14)
Substituting Eqs. (5), (9), and (13) into Eq. (14), one can obtain the location of the morphogen gene expression boundary (xb ) strictly
xb =
r
α−n 2
1+r
α−n 2
α−n α−n r 2 − w α+n x0 . L + α−n
α−n 2 1 + r w α+n n−α 2
(15)
The right side of Eq. (15) includes two parts. The first term r
α−n 2
1+r
α−n 2
L
is the scaling location of xb , and the second term
α−n α−n r 2 − w α+n x0
α−n α−n 2 1 + r w α+n n−α 2 is the shift of xb due to the inequality between the two morphogen influxes and concentrations. Note that when r and w satisfy the certain relation below r
α−n 2
α−n
= w α+n ,
(16)
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W.-S. Li, Y.-Z. Shao
the scaled xb can be established xb =
r
α−n 2
1+r
α−n 2
L.
(17)
In comparison with the corresponding results in the framework of the LRDS (shown in Eq. 18) (Mchale et al. 2006), xb =
λ L + (ln w − ln r ), when λ = λ p . 2 2
(18)
Our results have two aspects of uniquenesses and significances 1. Establishing the precise morphogen gene expression boundary with an arbitrary scaling location strictly. The scaled location of xb is no longer restricted simply at midembryo, but is adjustable with an arbitrary scaling location along the anteriorposterior axis even if the linear characteristic lengths of two morphogens are equal (λ = λ p ). Our approach overcomes the weakness of the traditional bigradient model, and extends the results of Houchmandzadeh and Mchale et al effectively (Houchmandzadeh et al. 2005; Mchale et al. 2006). 2. Revealing the key factors that cause the deviation of the scaling location xb from L/2. Besides the variation of the linear characteristic lengths, the ratios of two morphogen concentrations (r = 1) as well as the environments of a nonlinear reaction diffusion (α = n) are the key factors for deviating the precise gene expression boundary from L/2. In fact, these two factors are indispensible. According to Eq. (17), in the LRDS (n = α = 1), the scaling location of xb is fixed at 0.5L regardless of the value of r . It verifies our statement in Sect. 1 that the weakness of traditional bigradient model about “partial scaling” fails to be solved in the LRDS. For the case that α < n, the scaling location of xb is smaller than L/2 as r > 1. Meanwhile, for a fixed r , the scaling location xb will deviate from the midembryo gradually with the increment of (n − α). These results, to some extent, can provide a potential explanation for the experimental observation by Surkova et al. (2008) that the shifts of the various morphogen gene expression boundaries occur toward the anterior or posterior. 3.2 The Robustness of the Morphogen Gene Expression Boundary Provided that r and w do not satisfy Eq. (16) strictly, it will lead to the some extent of xb shift from scaling gene expression boundary. For comparing with the result of Mchale et al. (2006) more directly, we set r = 1 for further analysis. Then, Eqs. (15) and (18) can be simplified, respectively, as below
α−n α+n − 1 x w 0 L L = + xshift1 , in the NRDS; (19) xb = + α−n 2 2 (α − n)w α+n xb =
123
L λ ln w + , in the LRDS. 2 2
(20)
Morphogen Gene Expression in NRDS
The second term of Eq. (19) (xshift1 ) represents the shift of the morphogen gene expression boundary at the scaling location of L/2 due to the inequality between the two morphogen influxes in the framework of the NRDS. Different from Eq. (20), Eq. (19) suggests that xshift1 depends not only on w but also on j0∗ (x0 ) itself. Therefore, the feasibility to exactly assess the robustness of xb can be anticipated in this circumstance. Calculating the full derivative of Eq. (19), one may work out the robustness of the morphogen gene expression boundary in the NRDS, as indicated by Eq. (21) α−n −2n −2α n −α −n D ∗ α+n x0 w α+n − 1 n+α · 2 n+α · (α + n) n+α · dxb = · a d j0∗ . (21) w n+α dw + · j0 α−n n+α k α+n w Figure 1 shows the variation of xshift1 with different w and j0∗ . According to the Fig. 1a, the value of xshift1 in the NRDS will be smaller than the corresponding results in the LRDS at the certain range of w for larger morphogen influx j0∗ . The range of w for smaller xshift1 can be interpreted as the transcend inequality
1 − w n−α n+α 1 D
The Scaling and Shift of Morphogen Gene Expression Boundary in a Nonlinear Reaction Diffusion System Wei-Shen Li · Yuan-Zhi Shao
Received: 23 September 2013 / Accepted: 16 April 2014 © Society for Mathematical Biology 2014
Abstract The scaling and shift of the gene expression boundary in a developing embryo are two key problems with regard to morphogen gradient formation in developmental biology. In this study, a bigradient model was applied to a nonlinear reaction diffusion system (NRDS) to investigate the location of morphogen gene expression boundary. In contrast to the traditional synthesis–diffusion–degradation model, the introduction of NRDS in this study contributes to the precise gene expression boundary at arbitrary location along the anterior-posterior axis other than simply midembryo even when the linear characteristic lengths of two morphogens are equal. The scaling location depends on the ratio of two morphogen influxes (w) and concentrations (r ) as well as the nonlinear reaction diffusion parameters (α, n). We also formulate a direct relationship between the shift in the gene expression boundary and the influx of morphogen and find that enhancing the morphogen influx is helpful to build up a robust gene expression boundary. By analyzing the robustness of the morphogen gene expression boundary and comparing with the relevant results in linear reaction diffusion system, we determine the precise range of the ratio of the two morphogen influxes with a lower shift in the morphogen gene expression boundary and increased system robustness. Keywords Developmental embryo · Pattern formation and scaling · Morphogenetic gradient · Gene expression boundary · Reaction diffusion system · Robustness analysis 1 Introduction Morphogen gradient is a fundamental factor for the pattern formation of gene expression in development biological matrix (Wolpert 1969). The key problems are the W.-S. Li · Y.-Z. Shao (B) School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China e-mail: [email protected]
123
W.-S. Li, Y.-Z. Shao
formation of the morphogen gradient. The model proposed by Crick, famously as synthesis–diffusion–degradation (SDD) model, assumes that the morphogen is produced and synthesized from a source, is diffused and degraded along the anteriorposterior axis of the embryo (Crick 1970). However, Bollenbach and coworkers argue that transcytosis rather than diffusion plays a key role in the formation of the morphogen gradient (Bollenbach et al. 2005, 2007). Another viewpoint maintains that the cell growth driven by cell proliferation also contributes to the build up of the morphogen gradient without a diffusion term (Chisholm et al. 2010). Although the transportation process of morphogen in an embryo is still in controversy, the SDD model, as one of the most popular models, has been explored extensively from different aspects (Saunders and Howard 2009; Berezhkovskii et al. 2010, 2011; Gordon et al. 2011). Unfortunately, the traditional SDD model cannot explain the scaling law mechanism underlying the gene expression boundary and embryo size. Numerous studies have attempted to gain insight into the scaling law mechanism from the viewpoint of either proliferation growth or embryo volume using a simple morphogen gradient model (Wartlick et al. 2011; Cheung et al. 2011). Another representative viewpoint advocates that a bigradient model rather than a single gradient model is sufficient to explain the scaled settings of the gene expression boundary. Howard and Wolde (2005) assumed that the activator morphogen gradient originating from the anterior pole is bound to the second corepressor (inhibitor) morphogen from posterior-localized source. Their results indicate that such an interaction and binding between activator and inhibitor will result in the scaled and robust patterns of the gene expression boundary. Houchmandzadeh et al. revealed that two types of morphogen gradients, even without a binding, can also lead to a precise gene expression boundary domain (Houchmandzadeh et al. 2005). Furthermore, Mchale et al found that the scaling of the gene expression boundary is dependent on the ratio of the concentrations and influxes of two morphogens (Mchale et al. 2006). Although the bigradient model has made certain progress in this field, it has at least two limitations as below (Ben-Zvi et al. 2011). 1. The model establishes the gene expression boundary only at specific scaled location, especially at the midembryo, when the characteristic lengths of two morphogens are equal. Such a “partial scaling” is incompatible with some realistic biological phenomena (Surkova et al. 2008; Lott et al. 2007). 2. The shift in the gene expression boundary due to certain biological effects is only related to the ratio of the two morphogen influxes rather than the influx of morphogen itself, which may restrict the quantitative evaluation of the robustness of the morphogen gene expression boundary in a developmental embryo. Judging from the viewpoint of mathematics, we suggest that these problems cannot be solved in the framework of traditional linear reaction diffusion system (LRDS) due to the two factors below: (i) the exponential morphogen gradient profile; (ii): the linear relationship between the exponential morphogen gradient profile and the morphogen influx. In nonlinear reaction diffusion system (NRDS), however, the power–law rather than the exponential morphogen gradient profile emerges, and it has a nonlinear relationship with the morphogen influx (Yuste et al. 2010; Boon et al. 2012a,b). As a
123
Morphogen Gene Expression in NRDS
result, the newly discovered relationship between the morphogen gradient profile and influx inspired us to apply the bigradient model to the NRDS to investigate the scaling and shift of the morphogen gene expression boundary systematically. In the NRDS, whether are there any other factors, besides the variation in characteristic lengths of two morphogens responsible for contributing to the scaled morphogen gene expression boundary to be situated at different locations other than at midembryo (L/2)? Compared with the LRDS, how robust is the morphogen gene expression boundary in the NRDS? We will address these two questions in this paper. The remainder of the paper is organized as follows: In Sect. 2, the formalism for two types of morphogen gradients in an embryo is presented. In Sect. 3, we establish a precise gene expression boundary and analyze the shift of the gene expression boundary due to the influx fluctuation of both two morphogens and the maternal gene copies. The concluding remarks and prospects for future work are included in Sect. 4.
2 The General Formalism of the Bigradient Model in the Nonlinear Reaction Diffusion System In this study, the generalized reaction–diffusion equation derived by Boon and coworkers is utilized extensively to investigate the morphogen distribution in an embryo (Boon et al. 2012a,b). The dimensionless nonlinear reaction-diffusion equation is ∂2 ∂c(x, t) = D 2 cα (x, t) − kcn (x, t) ∂t ∂x
(1)
The symbols c, x, t, D, and k denote that the non-dimensional concentration of the morphogen, the distance to the anterior pole, time, diffusion, and the degradation coefficient, respectively; n and α stand for the indices of nonlinear reaction and diffusion, respectively. Furthermore, one can see that the Eq. (1) can be reduced to the traditional SDD model in the LRDS as α = n = 1. For an embryo with a finite size L, the steady-state solution of the first morphogen sourced from the anterior pole satisfies Eq. (2) with the boundary condition Eq. (3), ∂2 α c (x) − kcn (x) = 0, ∂x 2 ∂cα (x) ∂cα (x) −Dn s = j0 , Dn s ∂x ∂x
(2)
D
x=0
= 0,
(3)
x=L
where n s represents the standard and stationary local number density of morphogen particle solvent for acquiring the concentration of morphogen c(x) (Boon et al. 2012a,b). The variables are transformed as follows: x → z=
k x, D
j0 →
j0∗ =
j0 √ , c → g = cα , ns k D
L→L =
k L, D
(4)
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W.-S. Li, Y.-Z. Shao
Assuming that the embryo size L is large enough, we can gain the solution to Eq. (2) as follows c(x) =
j0∗
α−n x0 = α j0∗ α+n
dg ∗ j0 = − dz z=0
−2 n − α x n−α 1+ , n > α, 2 x0 α α + n n+α D , λ, λ = 2α k 2α α+n g 2α (0), = α+n
α+n 2α
2 n+α
(5) (6) (7)
where λ represents the linear characteristic length of the morphogen gradient profile. In the case of α > n, Boon et al introduce the Heaviside function to eliminate the imaginary or minus morphogen concentration c(x) for larger value of x c(x) =
j0∗
α+n 2α
2 n+α
2 α − n x α−n 2 1− x0 − x , α > n, Θ 2 x0 α−n
(8)
where Θ is the Heaviside step function (Boon et al. 2012a,b). For the second morphogen produced from the posterior pole (designated by c p ), its steady-state distribution can be acquired readily by analogy to the results of Eqs. (5–7) c p (x) =
j0∗p
x0 p = a j0∗p j0∗p =
α+n 2α α−n α+n
2 α − n x − L α−n 1+ , n > α, (9) 2 x0 p α Dp α + n n+α λp, λp = , (10) 2α kp 2 α+n
dg p 2α α+n g 2α (L ). = dz z=L α+n
(11)
Similar to Eq. (8), the Heaviside function is employed in the case of α > n c p (x) =
j0∗p
α+n 2α
Θ x−L+
2 α+n
2 α−n
α−n x −L 1+ 2 x0 p x0 p , n < α,
2 α−n
· (12)
where all of the variable denotations in Eqs. (9–12) are similar to the Eqs. (5–8), and λ p is the linear characteristic length of the second morphogen gradient profile. After completing the derivations of the above formulae, we turn our attention to a specific situation in which n > α, for two reasons:
123
Morphogen Gene Expression in NRDS
1. the experiment data are compared well with the morphogen profile of n > α (Boon et al. 2012a,b) and 2. the morphogen profile of n > α which means the infinite support (i.e., fast diffusion and low degradation) is also compatible with the size of a large enough embryo (Boon et al. 2012a,b). Note that the results of the current study, unless specified otherwise, focus on the situation where the degradation, diffusion coefficients, and linear characteristic length of the two morphogens are equal (k = k p , D = D p , λ = λ p ). 3 The Scaling and Shift of Gene Expression Boundary Resulted From Different Biological Effects 3.1 The Scaling of the Gene Expression Boundary The ratios of two morphogen influxes and concentrations are supposed w and r , respectively, as indicated by Eqs. (13) and (14), j0∗ = w, j0∗p
(13)
c(x) = r c p (x).
(14)
Substituting Eqs. (5), (9), and (13) into Eq. (14), one can obtain the location of the morphogen gene expression boundary (xb ) strictly
xb =
r
α−n 2
1+r
α−n 2
α−n α−n r 2 − w α+n x0 . L + α−n
α−n 2 1 + r w α+n n−α 2
(15)
The right side of Eq. (15) includes two parts. The first term r
α−n 2
1+r
α−n 2
L
is the scaling location of xb , and the second term
α−n α−n r 2 − w α+n x0
α−n α−n 2 1 + r w α+n n−α 2 is the shift of xb due to the inequality between the two morphogen influxes and concentrations. Note that when r and w satisfy the certain relation below r
α−n 2
α−n
= w α+n ,
(16)
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W.-S. Li, Y.-Z. Shao
the scaled xb can be established xb =
r
α−n 2
1+r
α−n 2
L.
(17)
In comparison with the corresponding results in the framework of the LRDS (shown in Eq. 18) (Mchale et al. 2006), xb =
λ L + (ln w − ln r ), when λ = λ p . 2 2
(18)
Our results have two aspects of uniquenesses and significances 1. Establishing the precise morphogen gene expression boundary with an arbitrary scaling location strictly. The scaled location of xb is no longer restricted simply at midembryo, but is adjustable with an arbitrary scaling location along the anteriorposterior axis even if the linear characteristic lengths of two morphogens are equal (λ = λ p ). Our approach overcomes the weakness of the traditional bigradient model, and extends the results of Houchmandzadeh and Mchale et al effectively (Houchmandzadeh et al. 2005; Mchale et al. 2006). 2. Revealing the key factors that cause the deviation of the scaling location xb from L/2. Besides the variation of the linear characteristic lengths, the ratios of two morphogen concentrations (r = 1) as well as the environments of a nonlinear reaction diffusion (α = n) are the key factors for deviating the precise gene expression boundary from L/2. In fact, these two factors are indispensible. According to Eq. (17), in the LRDS (n = α = 1), the scaling location of xb is fixed at 0.5L regardless of the value of r . It verifies our statement in Sect. 1 that the weakness of traditional bigradient model about “partial scaling” fails to be solved in the LRDS. For the case that α < n, the scaling location of xb is smaller than L/2 as r > 1. Meanwhile, for a fixed r , the scaling location xb will deviate from the midembryo gradually with the increment of (n − α). These results, to some extent, can provide a potential explanation for the experimental observation by Surkova et al. (2008) that the shifts of the various morphogen gene expression boundaries occur toward the anterior or posterior. 3.2 The Robustness of the Morphogen Gene Expression Boundary Provided that r and w do not satisfy Eq. (16) strictly, it will lead to the some extent of xb shift from scaling gene expression boundary. For comparing with the result of Mchale et al. (2006) more directly, we set r = 1 for further analysis. Then, Eqs. (15) and (18) can be simplified, respectively, as below
α−n α+n − 1 x w 0 L L = + xshift1 , in the NRDS; (19) xb = + α−n 2 2 (α − n)w α+n xb =
123
L λ ln w + , in the LRDS. 2 2
(20)
Morphogen Gene Expression in NRDS
The second term of Eq. (19) (xshift1 ) represents the shift of the morphogen gene expression boundary at the scaling location of L/2 due to the inequality between the two morphogen influxes in the framework of the NRDS. Different from Eq. (20), Eq. (19) suggests that xshift1 depends not only on w but also on j0∗ (x0 ) itself. Therefore, the feasibility to exactly assess the robustness of xb can be anticipated in this circumstance. Calculating the full derivative of Eq. (19), one may work out the robustness of the morphogen gene expression boundary in the NRDS, as indicated by Eq. (21) α−n −2n −2α n −α −n D ∗ α+n x0 w α+n − 1 n+α · 2 n+α · (α + n) n+α · dxb = · a d j0∗ . (21) w n+α dw + · j0 α−n n+α k α+n w Figure 1 shows the variation of xshift1 with different w and j0∗ . According to the Fig. 1a, the value of xshift1 in the NRDS will be smaller than the corresponding results in the LRDS at the certain range of w for larger morphogen influx j0∗ . The range of w for smaller xshift1 can be interpreted as the transcend inequality
1 − w n−α n+α 1 D
