
Parathyroid hormone (PTH) is one of the primary phosphaturic hormones in the body. The type IIa sodium-phosphate cotransporter (Npt2a) is expressed in the apical membrane of the renal proximal tubule and is responsible for the reabsorption of the majority of the filtered load of phosphate. PTH acutely induces phosphaturia through the rapid stimulation of endocytosis of Npt2a and its subsequent lysosomal degradation. This review focuses on the homeostatic mechanisms underlying serum phosphate, with particular focus on the regulation of the phosphate transporter Npt2a by PTH within the renal proximal tubule. Additionally, the proximal tubular PTH-stimulated signaling events as they relate to PTH-induced phosphaturia are also highlighted. Lastly, we discuss recent findings by our lab concerning novel regulatory mechanisms of PTH-mediated reductions in Npt2a expression.
Citation: Rebecca D. Murray, Eleanor D. Lederer, Syed J. Khundmiri. Role of PTH in the Renal Handling of Phosphate[J]. AIMS Medical Science, 2015, 2(3): 162-181. doi: 10.3934/medsci.2015.3.162
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Parathyroid hormone (PTH) is one of the primary phosphaturic hormones in the body. The type IIa sodium-phosphate cotransporter (Npt2a) is expressed in the apical membrane of the renal proximal tubule and is responsible for the reabsorption of the majority of the filtered load of phosphate. PTH acutely induces phosphaturia through the rapid stimulation of endocytosis of Npt2a and its subsequent lysosomal degradation. This review focuses on the homeostatic mechanisms underlying serum phosphate, with particular focus on the regulation of the phosphate transporter Npt2a by PTH within the renal proximal tubule. Additionally, the proximal tubular PTH-stimulated signaling events as they relate to PTH-induced phosphaturia are also highlighted. Lastly, we discuss recent findings by our lab concerning novel regulatory mechanisms of PTH-mediated reductions in Npt2a expression.
The Lyapunov exponent (LE), as introduced by Oseledets [1] in the context of his multiplicative ergodic theorem, serves as a quantifier of the divergence between two proximal trajectories over time within a dynamical system. In an $ n $-dimensional dynamical system, there are $ n $ LEs available, with their quantity corresponding to the dimensionality of the system's phase space. Each LE characterizes the rate of convergence or divergence of the system's attractor in a specific direction. The spectrum of LEs offers a metric for assessing the local sensitivity of a system to initial conditions, as well as for providing crucial insights into the system's global dynamics. This spectrum facilitates the effective description and classification of system attractors based on their LEs. For stable equilibrium points, all LEs are real and negative; in the case of stable limit cycles, one LE is zero while the remainders are real and negative. An attractor is identified as a $ k $-torus if the first $ k $ LEs are zero and the others are negative. The presence of positive values within the LEs spectrum signifies a chaotic attractor, which is further classified as hyperchaotic if two or more LEs are positive [2].
The calculation or estimation of LEs constitutes a core aspect of research into nonlinear dynamical systems. Notably, the largest LE (LLE) holds particular significance as it directly influences the predictability of the system in question. Over the past several decades, a diverse array of studies focusing on the calculation schemes for LEs, especially the LLE, have been published. These studies have predominantly categorized them into two main methodologies: the determination of LEs from governing equations [3,4,5,6,7] and the estimation of LEs from time series data [8,9,10,11]. Benettin et al. [3] first introduced the method for calculating all LEs of dynamical systems, grounded in Oseledets's theory [1]. This methodology was later refined by Wolf et al. [8]. Additionally, Briggs [12] explored the calculation of LEs by using experimental data, proposing that the optimal estimation of Jacobian matrices in the presence of noisy data is attained through least-squares polynomial fitting. The QR decomposition (short for "QR factorization" is a process that decomposes a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R)) and the singular value decomposition (SVD) methods for determining all LEs of dynamical systems were developed by Von Bremen et al. [13] and Dieci and Elia [14], respectively. Dabrowski [15] numerically derived the LLE by calculating the LE in the direction of a disturbance, as based on the perturbation vector and its derivative's dot product. Liao [16] investigated the sensitivity gradients of the LLE in dynamical systems to address the deficiency of previous approaches, which were prohibitively time-consuming and resource-intensive for most optimization problems of reasonable size. Certain methods described above embody the direct approach, which quantifies the divergence growth rate between two trajectories that have an infinitesimal discrepancy in their initial conditions. Conversely, other methods aim to estimate the Jacobian matrices of systems, addressing the limitations inherent to direct approaches, especially in the context of noise [17]. However, approaches that rely on linear approximation often fall short of capturing nonlinear growth and typically necessitate laborious computations. In some instances, they may even result in erroneous LEs due to ill-conditioned Jacobian matrices [18]. Additionally, if the number of iterations is insufficient, the outcomes are likely to be imprecise [19]. Zhou et al. [20] introduced a groundbreaking method to derive the LLE by using two nearby pseudo-orbits, eliminating the need for phase space reconstruction and Jacobian matrix computation. Utilizing machine learning to forecast LEs from data was an approach adopted by Pathak et al. [21] and McAllister et al. [22]. The perturbation vectors method [23], the cloned dynamics approach [24], and the synchronization technique [25,26,27] have been developed as strategies to circumvent the direct calculation of Jacobian matrices or the need to solve variational equations. Recent advancements and applications of LLEs are documented in references [28].
The analytical estimation of LEs within a dynamical system represents a compelling and significant topic. According to the definition of LEs, their analytical formulas are readily derivable only for steady-state solutions, including fixed points in nonlinear systems with a limited number of degrees of freedom and steady-state, spatially homogeneous solutions in spatially extended systems. Analytical expressions for LEs have been derived for neural oscillator models [29,30,31], as well as for certain simple models of nonlinear oscillators applied for synchronization problems triggered by common external noise [32,33,34]. Caponetto and Fazzino [7] introduced a semi-analytical approach, utilizing the differential transform method, to compute LEs in fractional order systems. Hramov et al. [35] first presented the analytical formula for the zero LE. A zero LE exists within the spectrum of LEs for flow systems, characterizing the perturbation evolution along the phase trajectory [36]. To derive the analytical expression for the zero LE, Hramov et al. analyzed a model system that describes the behavior of a driven periodic oscillator with noise near the synchronization onset. In chaotic systems, analytical approximations of LEs typically retain validity within a very narrow range of control parameter values, despite their derivability [37]. The analytical characterization of LEs for chaotic oscillators remains to be a formidable challenge.
In this paper, we aim to derive an analytical expression for the LLE of a Rössler chaotic system [38] by utilizing the synchronization method [25,26,27,39]. It is established that synchronization between two diffusively coupled identical chaotic systems is invariably achievable with a sufficiently large coupling parameter [40]. A linear relationship exists between the synchronization threshold of the coupling parameter in two identical systems and the value of the LLE of the coupled systems. Consequently, the LLE can be estimated based on the critical coupling required for synchronization [39]. This paper focuses on the analytical criteria for synchronization between two identical Rössler chaotic systems from the perspective of the linear coupling of state variables. Unlike previous studies [41,42,43], we initially transform the synchronization error system between two identical Rössler chaotic systems into a set of Volterra integral equations, utilizing the Laplace transform and the convolution theorem. The critical coupling required for synchronization can be derived by applying the successive approximation method [44] within the framework of integral equation theory to resolve the error system's solution. Numerical simulations have been conducted to confirm the efficacy of our analytical estimation of the LLE for the Rössler chaotic system. Furthermore, this analytical estimation remains valid across a broad range of parameter variations.
The remainder of the paper is structured as follows. Section 2 introduces the theoretical foundation of the estimation procedure for the LLE based on the synchronization method. Section 3 details the analytical estimation of the LLE for a Rössler system. Section 4 validates the analytical findings through numerical simulations. Finally, Section 5 provides the conclusions.
Consider a set of ordinary differential equations
$ ˙x=dxdt=f(x), $
|
(2.1) |
where $ x\in R^n $ represents the state variables and $ f:R^n\rightarrow R^n $ is a smooth vector function. Assume that $ s_t(x_0) $ is the solution of Eq (2.1) with the initial condition $ x = x_0 $ which has the components ($ x_{10}, x_{20}, \cdots, x_{n0} $); one has
$ dst(x0)dt=f[st(x0)],s0(x0)=x0. $
|
(2.2) |
Taking the variation with respect to $ x_0 $ on both sides of Eq (2.2) yields
$ dJt(x0)dt=∂f[st(x0)]∂xJt(x0), $
|
(2.3) |
where $ \frac{\partial f[s_t(x_0)]}{\partial x} = \frac{\partial f(x)}{\partial x}\bigg|_{x = s_t(x_0)} $, $ J_t(x_0) = \frac{\partial s_t(x_0)}{\partial x_0} $. Clearly, $ J_t(x_0) $ can be obtained by solving Eq (2.3), which describes the influence of infinitesimal disturbance $ \Delta x_0 $ to the initial condition $ x_0 $ on the trajectory $ s_t(x_0) $, that is,
$ Δs(t)≡st(x0+Δx0)−st(x0)=Jt(x0)Δx0. $
|
(2.4) |
Thus, the length of vector $ \Delta s(t) $ can be given as
$ |Δs(t)|=√Δs(t)TΔs(t)=√ΔxT0Jt(x0)TJt(x0)Δx0, $
|
(2.5) |
where the notation $ T $ denotes the transpose of vectors. Since $ J_t(x_0) $ is a real matrix, $ J_t(x_0)^TJ_t(x_0) $ is real symmetric and positive semi-definite. Assume that $ \xi_i(t) $, $ i = 1, 2, \cdots, n $, denotes the eigenvalues of the matrix $ J_t(x_0)^TJ_t(x_0) $. Obviously, $ \xi_i(t)\geq0 $. Assume that $ v_i(t) $ is the corresponding eigenvector of $ \xi_i(t) $. If $ \Delta x_0 $ has the same direction as $ v_i(t) $, Eq (2.5) becomes
$ |Δs(t)|=√ξi(t)|Δx0|. $
|
(2.6) |
The definition of LEs denoted by $ \lambda_i $, $ i = 1, 2, \cdots, n $, in system (2.1) is given as
$ λi=limt→∞ln√ξi(t)t=limt→∞ln|ξi(t)|2t, i=1,2,⋯,n. $
|
(2.7) |
From Eq (2.7), after long enough one has
$ √ξi(t)≈eλit, i=1,2,⋯,n. $
|
(2.8) |
Substituting Eq (2.8) into Eq (2.6), leads to
$ |Δsi(t)|=√ξi(t)|Δxi0|=eλit|Δxi0|, i=1,2,⋯,n. $
|
(2.9) |
The LEs are related to the expanding or contracting nature of different directions in phase space.
Consider a chaotic system in the following form [25]
$ ˙x=f(x), $
|
(2.10) |
where $ x\in R^n $ and $ f:R^n\rightarrow R^n $ is a smooth functional vector. Two such identical oscillators couple to undergo unidirectional coupling, as follows:
$ ˙x=f(x),˙y=f(y)+k(x−y), $
|
(2.11) |
where $ x, y\in R^n $ and $ k\in R $ is the coupling parameter. If $ k = 0 $, two separate dynamical systems are obtained
$ ˙x=f(x),˙y=f(y). $
|
(2.12) |
Assume that each system in Eq (2.12) evolves on an asymptotically stable chaotic attractor $ X $. The solutions of Eq (2.12) starting from different initial conditions represent two independent trajectories on the atrractor $ X $. If the two initial conditions are the same the two subsystems will exhibit identical behaviors ($ x = y $). If the initial conditions for the two subsystems in Eq (2.12) have a small difference, then a state difference exists between the two subsystems during the time evolution, which is defined by the expression
$ z=x−y, $
|
(2.13) |
where $ z\in R^n $.
Theorem 1. Assume that $ k_{min} > 0 $ is the boundary value of the coupling parameter $ k $ that is required to cause synchronization in system (2.11), and that $ \lambda_{max} $ is the LLE of system (2.10) such that $ \lambda_{max}\approx k_{min} $ holds.
Proof. To make further considerations easier, the following notations are first introduced:
$ * $ $ \lambda_j $ denotes the LEs in system (2.10) excluding $ \lambda_{max} $, $ j = 1, 2, \cdots, n-1 $,
$ * $ $ \Delta\lambda_{j} = \lambda_{max}-\lambda_j $ denotes the difference between $ \lambda_{max} $ and other LEs $ \lambda_{j} $, $ j = 1, 2, \cdots, n-1 $,
$ * $ $ \delta_0 $ is initial distance in the $ \lambda_{max} $ direction,
$ * $ $ \delta_{j0} = m_j\delta_0 $ denotes the initial distances in the $ \lambda_j $ direction, where $ m_j $ denotes constant values, $ j = 1, 2, \cdots, n-1 $.
The norm of vector $ z $ is given by
$ ||z||=(n∑i=1z2i)1/2. $
|
(2.14) |
Assume that $ z_0 = x_0-y_0 $ is an initial distance between two trajectories of subsystems in system (2.12), where $ z_0 $ has the components ($ z_{10}, z_{20}, \cdots, z_{n0} $). Obviously, $ z_0 $ is the total $ \lambda- $distance vector, which is a sum of $ \delta_0 $ and $ \delta_{j0} $, $ j = 1, 2, \cdots, n-1 $. From Eq (2.9), $ ||z|| $ can be written as
$ ||z||=(δ20e2λmaxt+n−1∑j=1δ2j0e2λjt)1/2=[δ20e2λmaxt(1+n−1∑j=1m2je−2Δλjt)]1/2. $
|
(2.15) |
Since $ \Delta\lambda_{j} < 0 $ holds for $ j = 1, 2, \cdots, n-1 $, the sum in Eq (2.15) finally decreases to zero during the time evolution; the norm of state difference $ z $ between two subsystems in Eq (2.12) approaches the following:
$ ||z||=δ0eλmaxt. $
|
(2.16) |
This implies that the distance associated with the $ \lambda_{max} $ direction becomes dominant after enough time.
Next, we consider the case of $ k\neq0 $ in Eq (2.11). For clarity, redefine the norm of the state difference between two subsystems in Eq (2.11) by $ Q $. Clearly, $ Q\geq0 $ for any values of $ x, y $ and $ k $. From Eq (2.16), if $ k > 0 $ one has
$ ˙Q=||f(x)−f(y)−k(x−y)||≥||f(x)−f(y)||−k||x−y||=λmaxδ0eλmaxt−kQ=(λmax−k)Q. $
|
(2.17) |
For $ k < 0 $,
$ ˙Q=||f(x)−f(y)−k(x−y)||≤||f(x)−f(y)||−k||x−y||=λmaxδ0eλmaxt−kQ=(λmax−k)Q. $
|
(2.18) |
Solving Eqs (2.17) and (2.18) yields
$ Q≥Q0e(λmax−k)t,fork>0,Q≤Q0e(λmax−k)t,fork<0, $
|
(2.19) |
where $ Q_0 $ is a constant determined by the initial conditions given in Eq (2.11). If $ k > 0 $ and the synchronization between two subsystems in Eq (2.11) is achieved, then $ Q\rightarrow0 $. From the first relation in Eq (2.19), it must follow that $ \lambda_{max} < k $. Assume that $ k_{min} > 0 $ is the boundary value of the coupling parameter $ k $ that is required to cause synchronization in system (2.11); then, the following inequality should be held: $ \lambda_{max} < k_{min} $.
On the contrary, if $ \lambda_{max}\geq k_{min} $, from the first inequality in Eq (2.19), the synchronization cannot be achieved. Therefore, one can have the following approximation
$ λmax≈kmin. $
|
(2.20) |
From Eq (2.18) and the second inequality in Eq (2.19), $ Q\rightarrow0 $ is impossible when $ \lambda_{max} > 0 $ and $ k < 0 $. It implies that two identical chaotic systems in Eq (2.11) cannot synchronize with each other when $ k < 0 $.
The Rössler oscillator [38] is described as follows:
$ ˙u=−v−w,˙v=u+av,˙w=b+w(u−c), $
|
(3.1) |
where $ u $, $ v $, $ w $ are state variables, and $ a $, $ b $, $ c $ are parameters. For convenience, by moving the equilibrium ($ u_0, v_0, w_0 $) of system (3.1) to the origin, system (3.1) can be rewritten as follows:
$ ˙x=−y−z−v0−w0,˙y=x+ay+u0+av0,˙z=(x−c)(z+w0)+u0(z+w0)+b, $
|
(3.2) |
where $ x = u-u_0 $, $ y = v-v_0 $, $ z = w-w_0 $. Consider two unidirectionally coupled Rössler systems as follows:
$ ˙x1=−y1−z1−v0−w0,˙y1=x1+ay1+u0+av0,˙z1=(x1−c)(z1+w0)+u0(z1+w0)+b,˙x2=−y2−z2−v0−w0+k(x1−x2),˙y2=x2+ay2+u0+av0+k(y1−y2),˙z2=(x2−c)(z2+w0)+u0(z2+w0)+b+k(z1−z2), $
|
(3.3) |
where $ k $ is a coupling parameter. Synchronization is said to occur in system (3.3) if
$ ||x1−x2||→0, ||y1−y2||→0, ||z1−z2||→0for t→∞. $
|
(3.4) |
By introducing
$ e_1 = x_1-x_2, \quad e_2 = y_1-y_2, \quad e_3 = z_1-z_2, $ |
$ e_4 = x_1+x_2, \quad e_5 = y_1+y_2, \quad e_6 = z_1+z_2, $ |
to system (3.3), the dynamical behavior of errors denoted by $ e_i $, $ i = 1, 2, 3 $, can be described as follows:
$ ˙e1=−ke1−e2−e3,˙e2=e1−(k−a)e2,˙e3=w0e1−(k+c)e3+e1e6+e3e4. $
|
(3.5) |
Then the synchronization condition given by Eq (3.4) becomes $ \lim_{t\rightarrow \infty}{||e_i|| = 0}, \; i = 1, 2, 3. $ Consider the Laplace transform defined as follows:
$ Ei(s)=L[ei]=∫+∞0ei(t)e−stdt,ei(t)=L−1[Ei]=12πj∫σ+j∞σ−j∞Ei(s)estds,i=1,2,3. $
|
(3.6) |
By taking the Laplace transform on both sides of system (3.5), we obtain
$ (sI3−M)[E1E2E3]=[e10e20e30+W], $
|
(3.7) |
where $ e_{0i} $, $ i = 1, 2, 3 $, denotes the given initial values of system (3.5), $ I_{3} $ is the $ 3\times3 $ real identity matrix, and
$ M=[−k−1−11a−k0w00u0−k−c], $
|
(3.8) |
$ W $ is the Laplace transform of the nonlinear parts in the third equation in system (3.5)
$ W=∫+∞0[e1e6+e3e4]e−stdt. $
|
Solving Eq (3.7) by using the Cramer's rule, one has
$ E1=e10(s+k−a)(s+k+c+u0)D(s)−e20(s+k+c+u0)D(s)−e30(s+k−a)D(s)−(s+k−a)WD(s),E2=e10(s+k+c+u0)D(s)+e20(s+k)(s+k+c+u0)D(s)+e20w0−e30D(s)−WD(s),E3=e10w0(s+k−a)D(s)−e20w0−e30D(s)+e30(s+k)(s+k−a)D(s)+(s+k)(s+k−a)WD(s)+WD(s), $
|
(3.9) |
where $ D(s) = s^3+\beta_1s^2+\beta_2s+\beta_3 $ is the characteristic polynomial of the matrix of Eq (3.8), and
$ β1=3k+c−a−u0,β2=3k2+2(c−a−u0)k+1+w0−ac+au0,β3=k3+(c−a−u0)k2+(1+w0−ac+au0)k+c−aw0−u0. $
|
To investigate whether $ ||e_i||\rightarrow0 $, $ i = 1, 2, 3 $, when $ t\rightarrow \infty $, we take the inverse Laplace transform on both sides of three equations in system (3.9) and consider the convolution theorem in the Laplace transform, which yields
$ e1=e10γ5(t)−e20γ2(t)−e30γ1(t)−∫t0γ1(t−τ)[e1e6+e3e4]dτ,e2=e10γ2(t)+e20γ4(t)+(e20w0−e30)γ6(t)−∫t0γ6(t−τ)[e1e6+e3e4]dτ,e3=e10w0γ1(t)−(e20w0−e30)γ6(t)+e30γ3(t)+∫t0(γ3(t−τ)+γ6(t−τ))[e1e6+e3e4]dτ, $
|
(3.10) |
where $ \gamma_1 = L^{-1}[\frac{s+k-a}{D(s)}] $, $ \gamma_2 = L^{-1}[\frac{s+k+c+u_0}{D(s)}] $, $ \gamma_3 = L^{-1}[\frac{(s+k)(s+k-a)}{D(s)}] $, $ \gamma_4 = L^{-1}[\frac{(s+k)(s+k+c+u_0)}{D(s)}] $, $ \gamma_5 = L^{-1}[\frac{(s+k-a)(s+k+c+u_0)}{D(s)}] $, $ \gamma_6 = L^{-1}[\frac{1}{D(s)}] $.
Theorem 2. The necessary condition for $ e_{1, 2, 3}\rightarrow0 $ with $ t\rightarrow \infty $ in Eq (3.10) is that all eigenvalues of the matrix of Eq (3.8) have negative real parts.
Proof. Without loss of generality, consider the inverse Laplace transform of the following true fraction
$ A1s2+A2s+A3D(s), $
|
where $ D(s) $ is the characteristic polynomial of the matrix of Eq (3.8) and $ A_i $, $ i = 1, 2, 3 $, denotes constants. There exist the following four cases:
● $ D(s) $ has $ 3 $ single real roots: $ s_1, s_2, s_3 $
$ \frac{A_1s^2+A_2s+A_3}{D(s)} = \frac{B_1}{s-s_1}+\frac{B_2}{s-s_2}+\frac{B_3}{s-s_3} $,
where $ B_i = \frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_i)\bigg|_{s = s_i} $, $\quad i = 1, 2, 3 $.
$ L^{-1}[\frac{A_1s^2+A_2s+A_3}{D(s)}] = B_1e^{s_1t}+B_2e^{s_2t}+B_{3}e^{s_{3}t} $
● $ D(s) $ has a pair of conjugate complex roots $ s_{1, 2} = \omega_1\pm j\omega_2 $ and a real root $ s_3 = \omega_3 $
$ \frac{A_1s^2+A_2s+A_3}{D(s)} = \frac{A_1s^2+A_2s+A_3}{(s-\omega_1-j\omega_2)(s-\omega_1+j\omega_2)(s-\omega_3)} = \frac{B_1}{s-\omega_1-j\omega_2}+\frac{B_2}{s-\omega_1+j\omega_2}+\frac{B_3}{s-\omega_3} $,
where $ B_{1, 2} = \frac{A_1s^2+A_2s+A_3}{D'(s)}\bigg|_{s = \omega_1\pm j\omega_2}, \quad B_3 = \frac{A_1s^2+A_2s+A_3}{D(s)}(s-\omega_3)\bigg|_{s = \omega_3} $,
$ L^{-1}[\frac{A_1s^2+A_2s+A_3}{D(s)}] = B_1e^{(\omega_1+j\omega_2)t}+B_2e^{(\omega_1-j\omega_2)t}+B_3e^{\omega_3t} $
● $ D(s) $ has $ 2 $ repeated real roots $ s = s_0 $ and a single real root $ s = s_k $
$ \frac{A_1s^2+A_2s+A_3}{D(s)} = \frac{B_1}{s-s_0}+\frac{B_2}{(s-s_0)^2}+\frac{B_3}{s-s_k} $,
where $ B_{1} = \frac{1}{2}\frac{d^2}{ds^2}[\frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_0)^2]\bigg|_{s = s_0} $, $ B_{2} = \frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_0)^{2}\bigg|_{s = s_0} $,
$ B_3 = \frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_k)\bigg|_{s = s_k} $,
$ L^{-1}[\frac{A_1s^2+A_2s+A_3}{D(s)}] = (B_1+B_2t)e^{s_{0}t}+B_3e^{s_kt} $
● $ D(s) $ has $ 3 $ repeated real roots: $ s = s_0 $
$ \frac{A_1s^2+A_2s+A_3}{D(s)} = \frac{B_1}{s-s_0}+\frac{B_2}{(s-s_0)^2}+\frac{B_3}{(s-s_0)^{3}} $,
where $ B_{(3-i)} = \frac{1}{i!}\frac{d^i}{ds^i}[\frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_0)^{3}]\bigg|_{s = s_0} $, $\quad i = 1, 2 $,
$ B_{3} = [\frac{A_1s^2+A_2s+A_3}{D(s)}(s-s_0)^{3}]\bigg|_{s = s_0} $,
$ L^{-1}[\frac{A_1s^2+A_2s+A_3}{D(s)}] = (B_1+B_2t+B_{3}t^{2})e^{s_{0}t} $
It is evident that for $ ||e_i|| $ to approach zero ($ i = 1, 2, 3 $) in system (3.10), a necessary condition is that $ \gamma_{j} $ tends toward zero as time approaches infinity ($ j = 1, 2, 3, 4, 5, 6 $). This condition is satisfied if all roots of the equation $ D(s) = 0 $ possess negative real parts, implying that all eigenvalues of the matrix of Eq (3.8) also have negative real parts.
Under the condition given in Theorem 2, when $ t\rightarrow \infty $ system (3.10) becomes as follows:
$ e1=−∫t0γ1(t−τ)[e1e6+e3e4]dτ,e2=−∫t0γ6(t−τ)[e1e6+e3e4]dτ,e3=∫t0(γ3(t−τ)+γ6(t−τ))[e1e6+e3e4]dτ. $
|
(3.11) |
Theorem 3.2. $ e_{1, 2, 3} = 0 $ represents the unique continuous solutions to Eq (3.11).
Proof. System (3.11) is a set of Volterra integral equations that can be solved by using the successive approximation method [44]. Consider the integral equation of the following form
$ h(t)=Ψ(t)+∫t0g(t−τ)H(τ,h(τ))dτ, $
|
(3.12) |
where $ g $ is an $ n\times n $ matrix and $ \Psi(t) $ and $ H(t, h(t)) $ are vectors with $ n $ components. If the following conditions are satisfied
● $ |h| < \infty $;
● $ \Psi $ and $ h $ are continuous for $ 0 < t < t_0 $, where $ 0 < t_0 < +\infty $;
● $ |g|\in L[0, \epsilon] $ holds for any $ 0 < \epsilon < t_0 $;
● For any $ \eta > 0 $, if $ |h_1-h_2| < \eta $ there must exist a constant $ \kappa > 0 $ such that $ |H(t, h_1)-H(t, h_2)| < \kappa $,
from the successive approximation method [44], Eq (3.12) has a unique continuous solution. Moreover the successive approximations given by
$ ω0(t)=0,ωn+1(t)=Ψ(t)+∫t0g(t−τ)H(τ,ωn(τ))dτ,n=0,1,2,⋯ $
|
(3.13) |
will uniformly converge to the unique continuous solution of Eq (3.12).
Comparing Eq (3.11) with Eq (3.12), it is easy to verify that $ e_1 = e_2 = e_3 = 0 $ constitutes the unique continuous solution of Eq (3.11).
From Theorems 2 and 3, we have the following result:
Theorem 4. The necessary condition for $ e_{1, 2, 3}\to 0 $ in Eq (3.5) is that all eigenvalues of the matrix of Eq (3.8) have negative real parts.
From the Routh-Hurwitz stability criterion, the necessary condition in {Theorem 4} is equivalent to the following condition:
$ β1>0,β2>0,β3>0,β1β2−β3>0, $
|
(3.14) |
where $ \beta_{1, 2, 3} $ has been defined in Eq (3.9). Since $ \beta_i $, $ i = 1, 2, 3 $, denotes functions of $ k $, from Eq (3.14) one can determine the boundary value of $ k $.
Theorem 5. The boundary value of $ k $ for synchronization in system (3.3) can be given as
$ kc=max{m1,m2,m3}, $
|
(3.15) |
where $ \max\{\cdot\} $ represents taking the maximum value of elements in the set,
$ m1={−c−a3,a2+ac+c2−3(w0+1)≤0,−c−a3+√a2+ac+c2−3(w0+1)3,a2+ac+c2−3(w0+1)>0, $
|
(3.16) |
and $ m_{2, 3} $ are the maximum real roots of the equations
$ β3=k3+(c−a)k2+(1+w0−ac)k+c−aw0=0,β1β2−β3=k3+(c−a)k2+(c−a)2+1+w0−ac4k+a2c−(c2+1)a+cw08=0, $
|
(3.17) |
respectively.
Proof. $ \beta_{i} $, $ i = 1, 2, 3 $, denotes functions of $ k $; it is easy to check that
$ \frac{d\beta_1}{dk} = 3, \quad \frac{d\beta_2}{dk} = 2\beta_1, \quad \frac{d\beta_3}{dk} = \beta_2. $ |
Therefore, one has
$ d(β1β2−β3)dk=d(β1β2)dk−dβ3dk=dβ1dkβ2+β1dβ2dk−β2=2(β21+β2). $
|
If $ \beta_{1, 2} > 0 $, then $ \beta_{3} $ and $ \beta_1\beta_2-\beta_3 $ are always monotonically increasing functions of $ k $. $ \beta_1 > 0 $ leads to
$ k > -\frac{c-a-u_0}{3}. $ |
Solving $ \beta_2 > 0 $ yields the following:
$ Ifa2+ac+c2−3(w0+1)>0,k<−c−a3−√a2+ac+c2−3(w0+1)3,ork>−c−a3+√a2+ac+c2−3(w0+1)3Ifa2+ac+c2−3(w0+1)=0,k≠−c−a3Ifa2+ac+c2−3(w0+1)<0,k∈(−∞,+∞) $
|
Denote
$ m1={−c−a3,a2+ac+c2−3(w0+1)≤0,−c−a3+√a2+ac+c2−3(w0+1)3,a2+ac+c2−3(w0+1)>0. $
|
If $ k > m_1 $, $ \beta_{3} $ and $ \beta_1\beta_2-\beta_3 $ are always monotonically increasing with an increase of $ k $. Assume that $ k = m_{2, 3} $ denotes the maximum real roots of equations $ \beta_{3} $ and $ \beta_1\beta_2-\beta_3 $, respectively, and that Eq (3.14) holds if and only if $ k > \max\{m_1, m_2, m_3\} $.
Remark. Suppose that system (3.1) has more than one equilibrium point $ (u_{01}, v_{01}, w_{01}) $, $ (u_{02}, v_{02}, w_{02}) $, $ \cdots $, $ (u_{0n}, v_{0n}, w_{0n}) $. For each equilibrium point $ (u_{0i}, v_{0i}, w_{0i}) $, one from Eq (3.15) has one $ k_{c}^{i} $, $ i = 1, 2, \cdots, n $. Then the boundary value of $ k $ for synchronization in system (3.3) is given by
$ kmin=min{k1c,k2c,⋯,knc}, $
|
(3.18) |
where $ \min\{\cdot\} $ denotes the minimum value of elements in the set.
In this section, numerical simulations are presented to illustrate the correctness of the result given by Eq (3.18). If $ a = 0.15 $, $ b = 0.2 $, and $ c = 10.0 $, system (3.1) is chaotic [38]. Under such parameter conditions, the numerical result for the LLE of system (3.1) is $ \lambda_{max} = 0.092 $ [8,45], where the initial conditions are taken as $ u(0) = -1 $, $ v(0) = 1 $, and $ w(0) = 1 $.
Consider that the value of $ c $ is allowed to vary between $ 10.0 $ and $ 13.0 $. Using the numerical method proposed in [8], the LLEs of system (3.1) for different values of $ c $ can be obtained as shown in Figure 1, where the initial conditions are retained as $ u(0) = -1 $, $ v(0) = 1 $, and $ w(0) = 1 $.
From Theorem 5, under the certain limitation of the parameters, the LLE of system (3.1) is just the maximum real root of the following equation (obtained from the second equation in Eq (3.17))
$ H(k)=k3+h1k2+h2k+h3, $
|
(4.1) |
where
$ h1=c−(w0+1)a,h2=14{[(w0+1)2+w0]a2−(2w0+3)ac+c2+w0+1},h3=−18{[ac2−(w0+(1+2w0)a2)c+a3w0(w0+1)+a(w20+1)]},w0=12a(c−√−4ab+c2), $
|
$ a = 0.15 $, $ b = 0.2 $ and $ c $ varies in the range of $ 10 $ to $ 13 $.
From Appendix, the analytic expression of the LLE of system (3.1) can be given as
$ λmax=k=3√Y1+3√Y2−h13, $
|
(4.2) |
where
$ Y1=3B−2h1A+3√Δ2,Y2=3B−2h1A−3√Δ2A=h21−3h2,B=h1h2−9h3,C=h22−3h1h3,Δ=B2−4AC. $
|
Figure 2 illustrates the comparisons between the analytical results from Eq (4.2) and the numerical results obtained by using the method described in [8]. The numerical results exhibit minor differences relative to the analytical results as the value of $ c $ increases. Figure 3 displays the time series for $ x_{1, 2} $, $ y_{1, 2} $, and $ z_{1, 2} $ in system (3.3) with varying $ c $ values to identify the critical synchronization conditions for $ k $. The analytical estimation based on Eq (4.2) for the LLE of system (3.1) is confirmed to be valid and highly accurate, as evidenced by the data in Figures 2 and 3.
Consider that the value of $ a $ changes in the range of $ 0.15 $ to $ 0.2 $. Applying the numerical method in [8], one can obtain the LLEs of system (3.1) for different values of $ a $, as depicted in Figure 4. The initial conditions were applied as $ u(0) = -1 $, $ v(0) = 1 $, and $ w(0) = 1 $.
For the parameter values considered in this section, the analytical expression of the LLE of system (3.1) is still given by Eq (4.2). The comparisons between the analytical results obtained based on Eq (4.2) and the numerical results derived by using the method in [8] are depicted in Figure 5.
Figure 5 demonstrates that the analytical results linearly increase as the value of $ a $ rises. The numerical results, however, display slight fluctuations before reaching their peak. Figure 6 depicts the time series for $ x_{1, 2} $, $ y_{1, 2} $ and $ z_{1, 2} $ in system (3.3) for various values of $ a $ and $ k $. A comparison of Figures 5 and 6 reveals that the analytic approach outlined in Eq (4.2) is both effective and highly accurate as a tool to determine the LLE of system (3.1), especially when the value of $ a $ varies within the range of $ 0.15 $ to $ 0.2 $.
Considering that the values of $ a $ and $ c $ vary within the ranges of $ 0.15 $ to $ 0.2 $ and $ 10 $ to $ 13 $, respectively, Figure 7 was constructed based on the calculated results of Eq (4.2) to illustrate the variation in the LLE of system (3.1) as $ a $ and $ c $ change. Equation (4.2) facilitates the examination of how wide-ranging parameter values influence the LLE in system (3.1).
Although a linear relationship exists between the synchronization threshold of the coupling coefficient in two identical chaotic systems and their LLE, previous studies have primarily derived the boundary value of the coupling parameter by employing numerical methods to estimate the LLE, to the best of our knowledge. This paper has presented an approach to analytically estimate the LLE of the Rössler chaotic system by using the synchronization method. Unlike previous studies, this approach transforms the synchronization error system into a set of Volterra integral equations. The stability of these equations was then examined through the application of the successive approximation method in accordance with the theory of integral equations. Compared to the numerical results for the LLEs of Rössler chaotic systems, our analytical estimates demonstrate high accuracy. Moreover, these analytical estimates remain valid across a wide range of parameter variations.
Our findings reveal that the value of the LLE for the Rössler chaotic system corresponds to the maximum real root of a cubic algebraic equation. This insight simplifies the challenge associated with analytically determining the LLE to solve such an equation. Our research introduces a novel approach for the analysis and management of the impact of parameter variations on the LLE value in the Rössler chaotic system.
The authors declare that they have not used artificial intelligence tools in the creation of this article.
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China grant numbers 11672185 and 11972327.
The authors declare that there is no conflict of interest.
In the Appendix, expressions of real roots of a cubic equation are provided according to the Cardano formula. Consider the following cubic equation:
$ f(x)=ax3+bx2+cx+d=0, $
|
(A1) |
where $ a, b, c, d $ are real constants and $ a\neq0 $. Denote
$ A=b2−3ac,B=bc−9ad,C=c2−3bd,Δ=B2−4AC,Y1,2=3aB−2Ab±3a√Δ2. $
|
(A2) |
● If $ \Delta > 0 $, there is only one real root:
$ x = \frac{\sqrt[3]{Y_1}+\sqrt[3]{Y_2}-b}{3a}. $
● If $ \Delta = 0 $ and $ A = 0 $, there are three equal real roots:
$ x_1 = x_2 = x_3 = -\frac{b}{3a} $.
● If $ \Delta = 0 $ and $ A > 0 $, there are three real roots, where two roots are equal:
$ x_1 = \frac{B}{A}-\frac{b}{a} $, $\quad x_2 = x_3 = -\frac{B}{2A}. $
● If $ \Delta < 0 $ and $ A > 0 $, there are three different real roots:
$ x_i = \frac{2\sqrt{A}\cos(\frac{\phi+2(i-1)\pi}{3})-b}{3a} $, $\quad \phi = \arccos(\frac{3aB-2Ab}{2A\sqrt{A}}), \quad i = 1, 2, 3. $
[1] | Gates F, Grant J (1927) Experimental Observations on Irradiated, Normal, and Partially Parathyroidectomized Rabbits. J Exp Med 45:125-137. |
[2] |
Beck L, Karaplis A, Amizuka N, et al. (1998) Targeted inactivation of Npt2 in mice leads to severe renal phosphate wasting, hypercalciuria, and skeletal abnormalities. Proc Natl Acad Sci U S A 95:5372-5377. doi: 10.1073/pnas.95.9.5372
![]() |
[3] |
Hedbäck G, Odén A (1998) Increased risk of death from primary hyperparathyroidism-an update. Eur J Clin Invest 28:271-276. doi: 10.1046/j.1365-2362.1998.00289.x
![]() |
[4] |
Raue F (1998) Increased incidence of cardiovascular diseases in primary hyperparathyroidism-a cause for more aggressive treatment? Eur J Clin Invest 28:277-278. doi: 10.1046/j.1365-2362.1998.00290.x
![]() |
[5] | Conzo G, Perna A, Candela G, et al. (2012) Long-term outcomes following “presumed” total parathyroidectomy for secondary hyperparathyroidism of chronic kidney disease. G Chir 33:379-382. |
[6] | Bansal V (1990) Serum Inorganic Phosphorus. In: Walker HK, Hall WD, Hurst JW (eds) Clin. Methods Hist. Phys Lab Exam 895-899. |
[7] |
Menon M, Ix J (2013) Dietary phosphorus, serum phosphorus, and cardiovascular disease. Ann N Y Acad Sci 1301:21-26. doi: 10.1111/nyas.12283
![]() |
[8] | Baker S, Worthley L (2002) The essentials of calcium, magnesium and phosphate metabolism: part I. Physiology. Crit Care Resusc 4:301-306. |
[9] |
Penido M, Alon U (2012) Phosphate homeostasis and its role in bone health. Pediatr Nephrol 27:2039-2048. doi: 10.1007/s00467-012-2175-z
![]() |
[10] | Mason J (2011) Vitamins, trace minerals, and other micronutrients. In: Goldman L, Ausiello D, eds. Cecil Medicine. 24th ed. Philadelphia, Pa: Saunders Elsevier; chap 225. |
[11] |
Takeda E, Taketani Y, Sawada N, et al. (2004) The regulation and function of phosphate in the human body. Biofactors 21:345-355. doi: 10.1002/biof.552210167
![]() |
[12] |
Loghman-Adham M (1997) Adaptation to changes in dietary phosphorus intake in health and in renal failure. J Lab Clin Med 129:176-188. doi: 10.1016/S0022-2143(97)90137-2
![]() |
[13] | Katai K, Miyamoto K, Kishida S, et al. (1999) Regulation of intestinal Na+-dependent phosphate co-transporters by a low-phosphate diet and 1,25-dihydroxyvitamin D3. Biochem J 343 Pt 3:705-712. |
[14] | Hildmann B, Storelli C, Danisi G, et al. (1982) Regulation of Na+-Pi cotransport by 1,25-dihydroxyvitamin D3 in rabbit duodenal brush-border membrane. Am J Physiol 242:G533-G539. |
[15] | Danisi G, Caverzasio J, Trechsel U, et al. (1990) Phosphate transport adaptation in rat jejunum and plasma level of 1,25-dihydroxyvitamin D3. Scand J Gastroenterol 25:210-215. |
[16] |
Kido S, Kaneko I, Tatsumi S, et al. (2013) Vitamin D and type II sodium-dependent phosphate cotransporters. Contrib Nephrol 180:86-97. doi: 10.1159/000346786
![]() |
[17] |
Kaneko I, Segawa H, Furutani J, et al. (2011) Hypophosphatemia in vitamin D receptor null mice: Effect of rescue diet on the developmental changes in renal Na+-dependent phosphate cotransporters. Pflugers Arch Eur J Physiol 461:77-90. doi: 10.1007/s00424-010-0888-z
![]() |
[18] |
Ikeda K, Takeshita S (2014) Factors and mechanisms involved in the coupling from bone resorption to formation: how osteoclasts talk to osteoblasts. J bone Metab 21:163-167. doi: 10.11005/jbm.2014.21.3.163
![]() |
[19] |
Agus Z, Puscttrr J, Senesky D, et al. (1971) Mode of Action of Parathyroid Hormone and cyclic adenosine 3',5'-monophosphate on Renal Tubular Phosphate Reabsorption in the Dog. J Clin Invest 50:617-626. doi: 10.1172/JCI106532
![]() |
[20] |
Collins J, Bai L, Ghishan F (2004) The SLC20 family of proteins: dual functions as sodium-phosphate cotransporters and viral receptors. Pflugers Arch 447:647-652. doi: 10.1007/s00424-003-1088-x
![]() |
[21] |
Nishimura M, Naito S (2008) Tissue-specific mRNA Expression Profiles of Human Solute Carrier Transporter Superfamilies. Drug Metab Pharmacokinet 23:22-44. doi: 10.2133/dmpk.23.22
![]() |
[22] |
Villa-Bellosta R, Ravera S, Sorribas V, et al. (2009) The Na+-Pi cotransporter PiT-2 (SLC20A2) is expressed in the apical membrane of rat renal proximal tubules and regulated by dietary Pi. Am J Physiol Renal Physiol 296:F691-699. doi: 10.1152/ajprenal.90623.2008
![]() |
[23] |
Bacconi A, Virkki L V, Biber J, et al. (2005) Renouncing electroneutrality is not free of charge: switching on electrogenicity in a Na+-coupled phosphate cotransporter. Proc Natl Acad Sci U S A 102:12606-12611. doi: 10.1073/pnas.0505882102
![]() |
[24] |
Renkema K, Alexander R, Bindels R, et al. (2008) Calcium and phosphate homeostasis: concerted interplay of new regulators. Ann Med 40:82-91. doi: 10.1080/07853890701689645
![]() |
[25] |
Tenenhouse H (2007) Phosphate transport: molecular basis, regulation and pathophysiology. J Steroid Biochem Mol Biol 103:572-577. doi: 10.1016/j.jsbmb.2006.12.090
![]() |
[26] |
Forster IC, Hernando N, Biber J, et al. (2006) Proximal tubular handling of phosphate: A molecular perspective. Kidney Int 70:1548-1559. doi: 10.1038/sj.ki.5001813
![]() |
[27] | Custer M, Lötscher M, Biber J, et al. (1994) Expression of Na-P(i) cotransport in rat kidney: localization by RT-PCR and immunohistochemistry. Am J Physiol 266:F767-F774. |
[28] |
Picard N, Capuano P, Stange G, et al. (2010) Acute parathyroid hormone differentially regulates renal brush border membrane phosphate cotransporters. Pflugers Arch Eur J Physiol 460:677-687. doi: 10.1007/s00424-010-0841-1
![]() |
[29] |
Segawa H, Kaneko I, Takahashi A, et al. (2002) Growth-related renal type II Na/Pi cotransporter. J Biol Chem 277:19665-19672. doi: 10.1074/jbc.M200943200
![]() |
[30] |
Chau H, El-Maadawy S, McKee M, et al. (2003) Renal calcification in mice homozygous for the disrupted type IIa Na/Pi cotransporter gene Npt2. J Bone Miner Res 18:644-657. doi: 10.1359/jbmr.2003.18.4.644
![]() |
[31] | Levi M, Lötscher M, Sorribas V, et al. (1994) Cellular mechanisms of acute and chronic adaptation of rat renal P(i) transporter to alterations in dietary P(i). Am J Physiol 267:F900-908. |
[32] |
Pfister M, Hilfiker H, Forgo J, et al. (1998) Cellular mechanisms involved in the acute adaptation of OK cell Na/Pi-cotransport to high- or low-Pi medium. Pflugers Arch 435:713-719. doi: 10.1007/s004240050573
![]() |
[33] |
Tenenhouse H (2005) Regulation of phosphorus homeostasis by the type iia na/phosphate cotransporter. Annu Rev Nutr 25:197-214. doi: 10.1146/annurev.nutr.25.050304.092642
![]() |
[34] | Murer H, Hernando N, Forster I, et al. (2000) Proximal tubular phosphate reabsorption: molecular mechanisms. Physiol Rev 80:1373-1409. |
[35] |
Khan S, Canales B (2011) Ultrastructural investigation of crystal deposits in Npt2a knockout mice: are they similar to human Randall's plaques? J Urol 186:1107-1113. doi: 10.1016/j.juro.2011.04.109
![]() |
[36] |
Iwaki T, Sandoval-Cooper M, Tenenhouse H, et al. (2008) A missense mutation in the sodium phosphate co-transporter Slc34a1 impairs phosphate homeostasis. J Am Soc Nephrol 19:1753-1762. doi: 10.1681/ASN.2007121360
![]() |
[37] |
Myakala K, Motta S, Murer H, et al. (2014) Renal-specific and inducible depletion of NaPi-IIc/Slc34a3, the cotransporter mutated in HHRH, does not affect phosphate or calcium homeostasis in mice. Am J Physiol - Ren Physiol 306:F833-F843. doi: 10.1152/ajprenal.00133.2013
![]() |
[38] |
Segawa H, Onitsuka A, Kuwahata M, et al (2009) Type IIc sodium-dependent phosphate transporter regulates calcium metabolism. J Am Soc Nephrol 20:104-113. doi: 10.1681/ASN.2008020177
![]() |
[39] |
Haussler M, Whitfield G, Kaneko I, et al. (2012) The role of vitamin D in the FGF23, klotho, and phosphate bone-kidney endocrine axis. Rev Endocr Metab Disord 13:57-69. doi: 10.1007/s11154-011-9199-8
![]() |
[40] |
Stechman M, Loh N, Thakker R (2009) Genetic causes of hypercalciuric nephrolithiasis. Pediatr Nephrol 24:2321-32. doi: 10.1007/s00467-008-0807-0
![]() |
[41] |
Prié D, Beck L, Friedlander G, et al. (2004) Sodium-phosphate cotransporters, nephrolithiasis and bone demineralization. Curr Opin Nephrol Hypertens 13:675-681. doi: 10.1097/00041552-200411000-00015
![]() |
[42] |
Magen D, Berger L, Coady M, et al. (2010) A Loss-of-Function Mutation in NaPi-IIa and Renal Fanconi's Syndrome. N Engl J Med 362:1102-1109. doi: 10.1056/NEJMoa0905647
![]() |
[43] | Rajagopal A, Débora B, James T, et al. (2014) Exome sequencing identifies a novel homozygous mutation in the phosphate transporter SLC34A1 in hypophosphatemia and nephrocalcinosis. J Clin Endocrinol Metab jc20141517. |
[44] |
Kenny J, Lees M, Drury S, et al. (2011) Sotos syndrome, infantile hypercalcemia, and nephrocalcinosis: a contiguous gene syndrome. Pediatr Nephrol 26:1331-1334. doi: 10.1007/s00467-011-1884-z
![]() |
[45] | Schlingmann K, Ruminska J, Kaufmann M, et al. (2015) Autosomal-Recessive Mutations in SLC34A1 Encoding Sodium-Phosphate Cotransporter 2A Cause Idiopathic Infantile Hypercalcemia. J Am Soc Nephrol 1-11. |
[46] |
Kestenbaum B, Glazer N, Köttgen A, et al. (2010) Common genetic variants associate with serum phosphorus concentration. J Am Soc Nephrol 21:1223-1232. doi: 10.1681/ASN.2009111104
![]() |
[47] |
Silver J, Naveh-Many T (2009) Phosphate and the parathyroid. Kidney Int 75:898-905. doi: 10.1038/ki.2008.642
![]() |
[48] |
Bergwitz C, Jüppner H (2010) Regulation of phosphate homeostasis by PTH, vitamin D, and FGF23. Annu Rev Med 61:91-104. doi: 10.1146/annurev.med.051308.111339
![]() |
[49] |
Caniggia A, Lore F, di Cairano G, et al. (1987) Main endocrine modulators of vitamin D hydroxylases in human pathophysiology. J Steroid Biochem 27:815-824. doi: 10.1016/0022-4731(87)90154-3
![]() |
[50] |
Wang W, Li C, Kwon T, et al. (2004) Reduced expression of renal Na+ transporters in rats with PTH-induced hypercalcemia. Am J Physiol Ren Physiol 286:534-545. doi: 10.1152/ajprenal.00044.2003
![]() |
[51] | Haramati A, Knox F (1983) Tubular capacity of phosphate transport in phosphate-deprived rats: effects of nicotinamide and PTH. Am J Physiol 244:F178-F184. |
[52] |
Guo J, Song L, Liu M, et al. (2013) Activation of a non-cAMP/PKA signaling pathway downstream of the PTH/PTHrP receptor is essential for a sustained hypophosphatemic response to PTH infusion in male mice. Endocrinology 154:1680-1689. doi: 10.1210/en.2012-2240
![]() |
[53] |
Yamamoto H, Tani Y, Kobayashi K, et al. (2005) Alternative promoters and renal cell-specific regulation of the mouse type IIa sodium-dependent phosphate cotransporter gene. Biochim Biophys Acta 1732:43-52. doi: 10.1016/j.bbaexp.2005.11.003
![]() |
[54] |
Silver J, Russell J, Sherwood L (1985) Regulation by vitamin D metabolites of messenger ribonucleic acid for preproparathyroid hormone in isolated bovine parathyroid cells. Proc Natl Acad Sci U S A 82:4270-4273. doi: 10.1073/pnas.82.12.4270
![]() |
[55] | Silver J, Yalcindag C, Sela-Brown A, et al. (1999) Regulation of the parathyroid hormone gene by vitamin D, calcium and phosphate. Kidney Int 73:S2-7. |
[56] |
Barthel T, Mathern D, Whitfield G, et al. (2007) 1,25-Dihydroxyvitamin D3/VDR-mediated induction of FGF23 as well as transcriptional control of other bone anabolic and catabolic genes that orchestrate the regulation of phosphate and calcium mineral metabolism. J Steroid Biochem Mol Biol 103:381-388. doi: 10.1016/j.jsbmb.2006.12.054
![]() |
[57] | Friedlaender M, Wald H, Dranitzki-Elhalel M, et al. (2001) Vitamin D reduces renal NaPi-2 in PTH-infused rats: complexity of vitamin D action on renal Pi handling. Am J Physiol Ren Physiol 281:428-433. |
[58] |
Weinman E, Steplock D, Wang Y, et al. (1995) Characterization of a Protein Cofactor that Mediates Protein Kinase A Regulation of the Renal Brush Border Membrane Na+-H+ Exchanger. J Clin Invest 95:2143-2149. doi: 10.1172/JCI117903
![]() |
[59] |
Weinman E, Lederer E (2012) NHERF-1 and the regulation of renal phosphate reabsoption: a tale of three hormones. Am J Physiol Renal Physiol 303:F321-327. doi: 10.1152/ajprenal.00093.2012
![]() |
[60] |
Weinman E, Biswas R, Peng G, et al. (2007) Parathyroid hormone inhibits renal phosphate transport by phosphorylation of serine 77 of sodium-hydrogen exchanger regulatory factor-1. J Clin Invest 117:3412-3420. doi: 10.1172/JCI32738
![]() |
[61] |
Mahon M, Segre G (2004) Stimulation by parathyroid hormone of a NHERF-1-assembled complex consisting of the parathyroid hormone I receptor, phospholipase Cbeta, and actin increases intracellular calcium in opossum kidney cells. J Biol Chem 279:23550-23558. doi: 10.1074/jbc.M313229200
![]() |
[62] |
Khundmiri S, Rane M, Lederer E (2003) Parathyroid hormone regulation of type II sodium-phosphate cotransporters is dependent on an A kinase anchoring protein. J Biol Chem 278:10134-10141. doi: 10.1074/jbc.M211775200
![]() |
[63] |
Lederer E, Khundmiri S, Weinman E (2003) Role of NHERF-1 in regulation of the activity of Na-K ATPase and sodium-phosphate co-transport in epithelial cells. J Am Soc Nephrol 14:1711-1719. doi: 10.1097/01.ASN.0000072744.67971.21
![]() |
[64] |
Hernando N, Deliot N, Gisler S, et al. (2002) PDZ-domain interactions and apical expression of type IIa Na/Pi cotransporters. Proc Natl Acad Sci U S A 99:11957-11962. doi: 10.1073/pnas.182412699
![]() |
[65] |
Mahon M, Cole J, Lederer E, et al. (2003) Na+/H+ exchanger-regulatory factor 1 mediates inhibition of phosphate transport by parathyroid hormone and second messengers by acting at multiple sites in opossum kidney cells. Mol Endocrinol 17:2355-2364. doi: 10.1210/me.2003-0043
![]() |
[66] |
Weinman E, Steplock D, Shenolikar S, et al. (2011) Dynamics of PTH-induced disassembly of Npt2a/NHERF-1 complexes in living OK cells. Am J Physiol Renal Physiol 300:F231-F235. doi: 10.1152/ajprenal.00532.2010
![]() |
[67] | Kempson S, Lötscher M, Kaissling B, et al. (1995) Parathyroid hormone action on phosphate transporter mRNA and protein in rat renal proximal tubules. Am J Physiol 268:F784-791. |
[68] |
Pfister M, Ruf I, Stange G, et al. (1998) Parathyroid hormone leads to the lysosomal degradation of the renal type II Na/Pi cotransporter. Proc Natl Acad Sci U S A 95:1909-1914. doi: 10.1073/pnas.95.4.1909
![]() |
[69] |
Pfister M, Lederer E, Forgo J, et al. (1997) Parathyroid Hormone-dependent Degradation of Type II Na+/Pi Cotransporters. J Biol Chem 272:20125-20130. doi: 10.1074/jbc.272.32.20125
![]() |
[70] |
Abou-Samra A, Jüppner H, Force T, et al. (1992) Expression cloning of a common receptor for parathyroid hormone and parathyroid hormone-related peptide from rat osteoblast-like cells: a single receptor stimulates intracellular accumulation of both cAMP and inositol trisphosphates and increases intracel. Proc Natl Acad Sci U S A 89:2732-2736. doi: 10.1073/pnas.89.7.2732
![]() |
[71] |
Watson P, Fraher L, Hendy G, et al. (2000) Nuclear localization of the type 1 PTH/PTHrP receptor in rat tissues. J Bone Miner Res 15:1033-1044. doi: 10.1359/jbmr.2000.15.6.1033
![]() |
[72] | Friedman P, Gesek F, Morley P, et al. (1999) Cell-specific signaling and structure-activity relations of parathyroid hormone analogs in mouse kidney cells. Endocrinology 140:301-309. |
[73] | Amizuka N, Lee H, Kwan M, et al. (1997) Cell-Specific Expression of the Parathyroid Hormone (PTH)/PTH-Related Peptide Receptor Gene in Kidney from Kidney-Specific and Ubiquitous Promoters. Endocrinology 138:469-481. |
[74] |
Taylor C, Tovey S (2012) From parathyroid hormone to cytosolic Ca2+ signals. Biochem Soc Trans 40:147-52. doi: 10.1042/BST20110615
![]() |
[75] |
Muff R, Fischer J, Biber J, et al. (1992) Parathyroid Hormone Receptors in Control of Proximal Tubule Function. Annu Rev Physiol 54:67-79. doi: 10.1146/annurev.ph.54.030192.000435
![]() |
[76] | Suarez F, Silve C (1992) Effect of Parathyroid Hormone on Arachidonic Acid Metabolism in Mouse Osteoblasts: Permissive Action of Dexamethasone. Endocrinology 130:592-598. |
[77] |
Cole J (1999) Parathyroid hormone activates mitogen-activated protein kinase in opossum kidney cells. Endocrinology 140:5771-5779. doi: 10.1210/endo.140.12.7173
![]() |
[78] |
Khundmiri S, Ameen M, Delamere N, et al. (2008) PTH-mediated regulation of Na(+)-K(+)-ATPase requires Src kinase-dependent ERK phosphorylation. Am J Physiol - Ren Physiol 295:F426-F437. doi: 10.1152/ajprenal.00516.2007
![]() |
[79] | Bacic D, Schulz N, Biber J, et al. (2003) Involvement of the MAPK-kinase pathway in the PTH-mediated regulation of the proximal tubule type IIa Na+/Pi cotransporter in mouse kidney. Pflugers Arch 446:52-60. |
[80] |
Yang S, Xiao L, Li J, et al. (2013) Role of guanine-nucleotide exchange factor Epac in renal physiology and pathophysiology. Am J Physiol Ren Physiol 304:F831-839. doi: 10.1152/ajprenal.00711.2012
![]() |
[81] |
Li Y, Konings I, Zhao J, et al. (2008) Renal expression of exchange protein directly activated by cAMP (Epac) 1 and 2. Am J Physiol Ren Physiol 295:F525-533. doi: 10.1152/ajprenal.00448.2007
![]() |
[82] |
Ostrom R, Bogard A, Gros R, et al. (2012) Choreographing the adenylyl cyclase signalosome: sorting out the partners and the steps. Naunyn Schmiedebergs Arch Pharmacol 385:5-12. doi: 10.1007/s00210-011-0696-9
![]() |
[83] |
Bek M, Zheng S, Xu J, et al. (2001) Differential expression of adenylyl cyclases in the rat nephron. Kidney Int 60:890-899. doi: 10.1046/j.1523-1755.2001.060003890.x
![]() |
[84] |
Murtazina R, Kovbasnjuk O, Zachos NC, et al. (2007) Tissue-specific regulation of sodium/proton exchanger isoform 3 activity in Na(+)/H(+) exchanger regulatory factor 1 (NHERF1) null mice. J Biol Chem 282:25141-25151. doi: 10.1074/jbc.M701910200
![]() |
[85] |
Courbebaisse M, Leroy C, Bakouh N, et al. (2012) A new human NHERF1 mutation decreases renal phosphate transporter NPT2a expression by a PTH-independent mechanism. PLoS One 7:e34764. doi: 10.1371/journal.pone.0034764
![]() |
[86] |
Ferrandon S, Feinstein T, Castro M, et al. (2009) Sustained cyclic AMP production by parathyroid hormone receptor endocytosis. Nat Chem Biol 5:734-742. doi: 10.1038/nchembio.206
![]() |
[87] |
Ahlström M, Lamberg‐Allardt C (1997) Rapid Protein Kinase A—Mediated Activation of Cyclic AMP-Phosphodiesterase by Parathyroid Hormone in UMR-106 Osteoblast-like Cells. J Bone Miner Res 12:172-178. doi: 10.1359/jbmr.1997.12.2.172
![]() |
[88] |
Whitfield J, Isaacs R, Chakravarthy B, et al. (2001) Stimulation of protein kinase C activity in cells expressing human parathyroid hormone receptors by C- and N-terminally truncated fragments of parathyroid hormone 1-34. J Bone Miner Res 16:441-447. doi: 10.1359/jbmr.2001.16.3.441
![]() |
[89] |
Mahon M, Donowitz M, Yun C, et al. (2002) Na+/H+ exchanger regulatory factor 2 directs parathyroid hormone 1 receptor signalling. Nature 417:858-861. doi: 10.1038/nature00816
![]() |
[90] | Jouishomme H, Whitfield J, Gagnon L, et al. (1994) Further definition of the protein kinase C activation domain of the parathyroid hormone. J Bone Miner Res 9:943-949. |
[91] |
Whitfield J, Isaacs R, Chakravarthy B, et al. (2001) Protein Kinase C Activity in Cells Expressing Human Parathyroid Hormone Receptors by C‐and N‐Terminally Truncated Fragments of Parathyroid Hormone 1. J Bone Miner Res 16:441-447. doi: 10.1359/jbmr.2001.16.3.441
![]() |
[92] |
Wang B, Yang Y, Abou-Samra A, et al. (2009) NHERF1 regulates parathyroid hormone receptor desensitization: interference with beta-arrestin binding. Mol Pharmacol 75:1189-1197. doi: 10.1124/mol.108.054486
![]() |
[93] |
Alonso V, Magyar C, Wang B, et al. (2011) Ubiquitination-deubiquitination balance dictates ligand-stimulated PTHR sorting. J bone Miner Res 26:2923-2934. doi: 10.1002/jbmr.494
![]() |
[94] |
Chauvin S, Bencsik M, Bambino T, et al. (2002) Parathyroid hormone receptor recycling: role of receptor dephosphorylation and beta-arrestin. Mol Endocrinol 16:2720-2732. doi: 10.1210/me.2002-0049
![]() |
[95] |
Wang B, Bisello A, Yang Y, et al. (2007) NHERF1 regulates parathyroid hormone receptor membrane retention without affecting recycling. J Biol Chem 282:36214-36222. doi: 10.1074/jbc.M707263200
![]() |
[96] |
Khundmiri S, Weinman E, Steplock D, et al. (2005) Parathyroid hormone regulation of NA+,K+-ATPase requires the PDZ 1 domain of sodium hydrogen exchanger regulatory factor-1 in opossum kidney cells. J Am Soc Nephrol 16:2598-2607. doi: 10.1681/ASN.2004121049
![]() |
[97] |
Salyer S, Lesousky N, Weinman E, et al. (2011) Dopamine regulation of Na+-K+-ATPase requires the PDZ-2 domain of sodium hydrogen regulatory factor-1 (NHERF-1) in opossum kidney cells. Am J Physiol Cell Physiol 300:C425-C434. doi: 10.1152/ajpcell.00357.2010
![]() |
[98] | Tawfeek H, Abou-Samra A (2004) Important role for the V-type H+-ATPase and the Golgi apparatus in the recycling of PTH/PTHrP receptor. Am J Physiol Endocrinol Metab 286:704-710. |
[99] |
Pickard B, Hodsman A, Fraher L, et al. (2007) Type 1 parathyroid hormone receptor (PTH1R) nuclear trafficking: regulation of PTH1R nuclear-cytoplasmic shuttling by importin-alpha/beta and chromosomal region maintenance 1/exportin 1. Endocrinology 148:2282-2289. doi: 10.1210/en.2007-0157
![]() |
[100] |
Silverstein D, Spitzer A, Barac-Nieto M (2005) Parathormone sensitivity and responses to protein kinases in subclones of opossum kidney cells. Pediatr Nephrol 20:721-724. doi: 10.1007/s00467-005-1832-x
![]() |
[101] |
Nagai S, Okazaki M, Segawa H, et al. (2011) Acute down-regulation of sodium-dependent phosphate transporter NPT2a involves predominantly the cAMP/PKA pathway as revealed by signaling-selective parathyroid hormone analogs. J Biol Chem 286:1618-1626. doi: 10.1074/jbc.M110.198416
![]() |
[102] | Cole J, Eber S, Poelling R, et al. (1987) A dual mechanism for regulation of kidney phosphate transport by parathyroid hormone. Am J Physiol 253:E221-227. |
[103] |
Cole J, Forte L, Eber S, et al. (1988) Regulation of sodium-dependent phosphate transport by parathyroid hormone in opossum kidney cells: Adenosine 3',5'-monophosphate-dependent and -independent mechanisms. Endocrinology 122:2981-2989. doi: 10.1210/endo-122-6-2981
![]() |
[104] | Fenton R, Murray F, Dominguez J, et al. (2014) Renal phosphate wasting in the absence of adenylyl cyclase 6. J Am Soc Nephrol 1-13. |
[105] | Weinstein L, Yu S, Warner D, et al. (2001) Endocrine manifestations of stimulatory G protein alpha-subunit mutations and the role of genomic imprinting. Endocr Rev 22:675-705. |
[106] |
Mantovani G (2011) Clinical review: Pseudohypoparathyroidism: diagnosis and treatment. J Clin Endocrinol Metab 96:3020-3030. doi: 10.1210/jc.2011-1048
![]() |
[107] | Carpenter T, McPhee M, Bort R, et al. (1992) Dissociation of phosphaturia and 25(OH)D-1a-hydroxylase trophism using a novel analogue of parathyroid hormone. Am J Physiol 25:483-487. |
[108] | Cunningham R, Biswas R, Brazie M, et al. (2009) Signaling pathways utilized by PTH and dopamine to inhibit phosphate transport in mouse renal proximal tubule cells. Am J Physiol Renal Physiol 296:F355-361. |
[109] |
Ranch D, Zhang M, Portale A, et al. (2011) Fibroblast growth factor 23 regulates renal 1,25-dihydroxyvitamin D and phosphate metabolism via the MAP kinase signaling pathway in Hyp mice. J Bone Miner Res 26:1883-1890. doi: 10.1002/jbmr.401
![]() |
[110] | Kilav R, Silver J, Biber J, et al. (1995) Coordinate regulation of rat renal parathyroid hormone receptor mRNA and Na-Pi cotransporter mRNA and protein. Am J Physiol 268:F1017-1022. |
[111] |
Moe S, Radcliffe J, White K, et al. (2011) The pathophysiology of early-stage chronic kidney disease-mineral bone disorder (CKD-MBD) and response to phosphate binders in the rat. J Bone Miner Res 26:2672-2681. doi: 10.1002/jbmr.485
![]() |
[112] | Hilfiker H, Hartmann C, Stange G, et al. (1998) Characterization of the 5 J-flanking region of OK cell type II Na-Pi cotransporter gene. 12:197-204. |
[113] |
Murray R, Holthouser K, Clark B, et al. (2013) Parathyroid hormone (PTH) decreases sodium-phosphate cotransporter type IIa (NpT2a) mRNA stability. Am J Physiol Renal Physiol 304:F1076-1085. doi: 10.1152/ajprenal.00632.2012
![]() |
[114] |
Moz Y, Silver J, Naveh-Many T. (1999) Protein-RNA Interactions Determine the Stability of the Renal NaPi-2 Cotransporter mRNA and Its Translation in Hypophosphatemic Rats. J Biol Chem 274:25266-25272. doi: 10.1074/jbc.274.36.25266
![]() |
[115] |
Moz Y, Silver J, Naveh-Many T. (2003) Characterization of cis-acting element in renal NaPi-2 cotransporter mRNA that determines mRNA stability. Am J Physiol Renal Physiol 284:F663-670. doi: 10.1152/ajprenal.00332.2002
![]() |
[116] | Noronha-Blob L, Sacktor B. (1986) Inhibition by glucocorticoids of phosphate transport in primary cultured renal cells. J Biol Chem 261:2164-2169. |