Research article Special Issues

Protein clearance mechanisms and their demise in age-related neurodegenerative diseases

  • The accumulation of damaged proteins contributes to the etiology of neurodegenerative diseases such as Alzheimer's, Parkinson's, Huntington's or amyotrophic lateral sclerosis. Protein aggregation and deposition are common features of these disorders that share emergence patterns and are more frequent late in life, even though different toxic proteins are involved in their onset. The ability to maintain a functional proteome, or proteostasis, declines during the ageing process. Damaged proteins are degraded by the ubiquitin proteasome system or through autophagy-lysosome, key components of the proteostasis network. Here we review the links between neurodegenerative disorders and loss of protein clearance mechanisms with age.

    Citation: Isabel Saez, David Vilchez. Protein clearance mechanisms and their demise in age-related neurodegenerative diseases[J]. AIMS Molecular Science, 2015, 1(1): 1-21. doi: 10.3934/molsci.2015.1.1

    Related Papers:

    [1] Gonca Durmaz Güngör, Ishak Altun . Fixed point results for almost ($ \zeta -\theta _{\rho } $)-contractions on quasi metric spaces and an application. AIMS Mathematics, 2024, 9(1): 763-774. doi: 10.3934/math.2024039
    [2] Hieu Doan . A new type of Kannan's fixed point theorem in strong $ b $- metric spaces. AIMS Mathematics, 2021, 6(7): 7895-7908. doi: 10.3934/math.2021458
    [3] Pragati Gautam, Vishnu Narayan Mishra, Rifaqat Ali, Swapnil Verma . Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$-metric space. AIMS Mathematics, 2021, 6(2): 1727-1742. doi: 10.3934/math.2021103
    [4] Shaoyuan Xu, Yan Han, Suzana Aleksić, Stojan Radenović . Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications. AIMS Mathematics, 2022, 7(8): 14895-14921. doi: 10.3934/math.2022817
    [5] Abdullah Shoaib, Poom Kumam, Shaif Saleh Alshoraify, Muhammad Arshad . Fixed point results in double controlled quasi metric type spaces. AIMS Mathematics, 2021, 6(2): 1851-1864. doi: 10.3934/math.2021112
    [6] Mi Zhou, Naeem Saleem, Xiao-lan Liu, Nihal Özgür . On two new contractions and discontinuity on fixed points. AIMS Mathematics, 2022, 7(2): 1628-1663. doi: 10.3934/math.2022095
    [7] Amjad Ali, Muhammad Arshad, Awais Asif, Ekrem Savas, Choonkil Park, Dong Yun Shin . On multivalued maps for $ \varphi $-contractions involving orbits with application. AIMS Mathematics, 2021, 6(7): 7532-7554. doi: 10.3934/math.2021440
    [8] Tahair Rasham, Abdullah Shoaib, Shaif Alshoraify, Choonkil Park, Jung Rye Lee . Study of multivalued fixed point problems for generalized contractions in double controlled dislocated quasi metric type spaces. AIMS Mathematics, 2022, 7(1): 1058-1073. doi: 10.3934/math.2022063
    [9] Xun Ge, Songlin Yang . Some fixed point results on generalized metric spaces. AIMS Mathematics, 2021, 6(2): 1769-1780. doi: 10.3934/math.2021106
    [10] Qing Yang, Chuanzhi Bai . Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on $O$-complete metric spaces. AIMS Mathematics, 2020, 5(6): 5734-5742. doi: 10.3934/math.2020368
  • The accumulation of damaged proteins contributes to the etiology of neurodegenerative diseases such as Alzheimer's, Parkinson's, Huntington's or amyotrophic lateral sclerosis. Protein aggregation and deposition are common features of these disorders that share emergence patterns and are more frequent late in life, even though different toxic proteins are involved in their onset. The ability to maintain a functional proteome, or proteostasis, declines during the ageing process. Damaged proteins are degraded by the ubiquitin proteasome system or through autophagy-lysosome, key components of the proteostasis network. Here we review the links between neurodegenerative disorders and loss of protein clearance mechanisms with age.


    In various fields of science, nonlinear evolution equations practically model many natural, biological and engineering processes. For example, PDEs are very popular and are used in physics to study traveling wave solutions. They have played a crucial role in illustrating the nature of nonlinear problems. PDEs are collected to control the diffusion of chemical reactions. In biology, they play a fundamental role in describing various phenomena, such as population growth. In addition, natural phenomena such as fluid dynamics, plasma physics, optics and optical fibers, electromagnetism, quantum mechanics, ocean waves, and others are studied using PDEs. The qualitative and quantitative characteristics of these phenomena can be identified from the behaviors and shapes of their solutions. Therefore, finding the analytic solutions to such phenomena is a fundamental topic in mathematics. Scientists have developed sparse fundamental approaches to find analytic solutions for nonlinear PDEs. Among these techniques, I present integration methods from [1] and [2], the modified F-expansion and Generalized Algebraic methods, respectively. Bekir and Unsal [3] proposed the first integral method to find the analytical solution of nonlinear equations. Kumar, Seadawy and Joardar [4] used the improved Kudryashov technique to extract fractional differential equations. Adomian [5] proposed the Adomian decomposition technique to find the solution of frontier problems of physics. [6] uses an exploratory method to find explicit solutions of non-linear PDEs. Many different methods of solving equations arising from natural phenomena and some of their analytic solutions, such as dark and light solitons, non-local rogue waves, an occasional wave and mixed soliton solutions, are exhibited and can be found in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].

    The Novikov-Veselov (NV) system [41,42] is given by

    $ Ψt+αΓxxy+βΦxyy+γΓyΦ+γ(Ψ22)y+λΓΦx+λ(Ψ22)x=0,Γy=Ψx,Ψy=Φx,
    $
    (1.1)

    where $ \alpha, \, \beta, \, \gamma $ and $ \lambda $ are constants. Barman [42] declared that Eq (1.1) is involved to represent tidal and tsunami waves, electro-magnetic waves in communication cables and magneto-sound and ion waves in plasma. In [42], the generalized Kudryashov method was utilized to have traveling wave solutions for Eq (1.1). According to Croke [43], the Novikov-Veselov system is generalized from the KdV equations which were examined by Novikov and Veselov. Croke [43] used several approaches, (the extended mapping, the Hirota and the extended tanh-function approaches) in the proposed system to achieve numerous soliton solutions, such as breathers, and constrained analytic solutions. Boiti, Leon, and Manna [44] applied the inverse dispersion technique to solve (1.1) for a particular type of initial value. Numerical solutions and a study of the stability of solutions for the proposed equation were presented by Kazeykina and Klein [45]. The Nizhnik-Novikov-Veselov system for two dimensions was also solved using the Kansa technique to find the numerical results [46]. To the best of my knowledge, the stability and error analysis of the numerical scheme presented here has not yet been discussed for system (1.1). Therefore, this has motivated me enormously to do so. The primary purpose is to obtain multiple analytic solutions to system (1.1) by using both the modified F-expansion and Generalized Algebraic methods. In connection with the numerical solution, the method of finite differences is utilized to achieve numerical results for the studied system. I graphically and analytically compare the traveling wave solutions and numerical results. Undoubtedly, the presented results strongly contribute to describing physical problems in practice.

    The outline of this article is provided in this paragraph. Section 2 summarizes the employed methods. All the analytic solutions are extracted in Section 3. The shooting and BVP results for the proposed system are presented in Section 4. In addition, I examine the numerical solution of the system (1.1) in Section 5. Sections 6 and 7 study the stability and error analysis of the numerical scheme, respectively. Section 8 presents the results and discussion.

    Considering the development equation with physical fields $ \Psi(x, y, t) $, $ \Phi(x, y, t) $ and $ \Gamma(x, y, t) $ in the variables $ x, $ $ y $ and $ t $ is given in the following form:

    $ Q1(Ψ,Ψt,Ψx,Ψy,Γ,Γy,Γxxy,Φ,Φxyy,Φx,)=0.
    $
    (2.1)

    Step 1. We extract the traveling-wave solutions of System (1.1) that are formed as follows:

    $ Φ(x,y,t)=ϕ(η),η=x+ywt,Ψ(x,y,t)=ψ(η),Γ(x,y,t)=Θ(η),
    $
    (2.2)

    where $ w $ is the wave speed.

    Step 2. The nonlinear evolution (2.1) is reduced to the following ODE:

    $ Q2(ψ,ψη,Θ,Θη,Θηηη,ϕ,ϕηηη,ϕη,)=0,
    $
    (2.3)

    where $ Q_2 $ is a polynomial in $ \psi(\eta), \; \phi(\eta), $ $ \Theta(\eta) $ and their total derivatives.

    According to the modified F-expansion method, the solutions of (2.3) are given by the form

    $ ψ(η)=ρ0+Nk=1(ρkF(η)k+qkF(η)k),
    $
    (2.4)

    and $ F(\eta) $ is a solution of the following differential equation:

    $ F(η)=μ0+μ1F(η)+μ2F(η)2,
    $
    (2.5)

    where $ \mu_0 $, $ \mu_1 $, $ \mu_2 $, are given in Table 1 [1], and $ \rho_k $, $ q_k $ are to be determined later.

    Table 1.  The relations among $ \mu_0 $, $ \mu_1 $, $ \mu_2 $ and the function $ F(\eta) $.
    $ \mu_0 $ $ \mu_{1} $ $ \mu_{2} $ $ F(\eta) $
    $ \mu_0=0, $ $ \mu_1=1, $ $ \mu_2=-1, $ $ F(\eta)=\dfrac{1}{2}+\dfrac{1}{2} \tanh \left(\dfrac{1}{2} \eta\right). $
    $ \mu_0=0, $ $ \mu_1=-1, $ $ \mu_2=1, $ $ F(\eta)=\dfrac{1}{2}-\dfrac{1}{2} \operatorname{coth}\left(\dfrac{1}{2} \eta\right). $
    $ \mu_0=\dfrac{1}{2}, $ $ \mu_1=0, $ $ \mu_2=\dfrac{-1}{2}, $ $ F(\eta)=\operatorname{coth}(\eta) \pm \operatorname{csch}(\eta), $ $ \tanh (\eta) \pm \operatorname{sech}(\eta). $
    $ \mu_0=1, $ $ \mu_1=0, $ $ \mu_2=-1, $ $ F(\eta)=\tanh (\eta), $ $ \operatorname{coth}(\eta). $
    $ \mu_0=\dfrac{1}{2}, $ $ \mu_1=0, $ $ \mu_2=\dfrac{1}{2}, $ $ F(\eta)=\sec (\eta)+\tan (\eta), $ $ \csc (\eta)-\cot (\eta). $
    $ \mu_0=\dfrac{1}{2}, $ $ \mu_1=0, $ $ \mu_2=\dfrac{1}{2}, $ $ F(\eta)=\sec (\eta)-\tan (\eta), $ $ \csc (\eta)+\cot (\eta). $
    $ \mu_0=\pm1, $ $ \mu_1=0, $ $ \mu_2=\pm1, $ $ F(\eta)=\tan (\eta), $ $ \cot (\eta). $

     | Show Table
    DownLoad: CSV

    According to the generalized direct algebraic method, the solutions of (2.3) are given by

    $ ψ(η)=ν0+Nk=1(νkG(η)k+rkG(η)k),
    $
    (2.6)

    and $ G(\eta) $ is a solution of the following differential equation:

    $ G(η)=ε4k=0δkGk(η),
    $
    (2.7)

    where $ \nu_k, $ and $ r_k $ are to be determined, and $ N $ is an integer number obtained by the highest degree of the nonlinear terms and the highest order of the derivatives. $ \varepsilon $ is user-specified, usually taken with $ \varepsilon = \pm1 $, and $ \delta_k $, $ k = 0, 1, 2, 3, 4, $ are given in Table 2 [2].

    Table 2.  The relations among $ \delta_k $, $ k = 0, 1, 2, 3, 4, $ and the function $ G(\eta) $.
    $ \delta_{0} $ $ \delta_{1} $ $ \delta_{2} $ $ \delta_{3} $ $ \delta_{4} $ $ G(\eta) $
    $ \delta_{0}=0 $, $ \delta_{1}=0 $, $ \delta_{2} > 0 $, $ \delta_{3}=0 $, $ \delta_{4} < 0 $, $ G(\eta)= \varepsilon $ $ \sqrt{-\dfrac{\delta_{2}}{\delta_{4}}} \operatorname{sech}\left(\sqrt{\delta_{2}} \eta\right) $.
    $ \delta_{0}=\dfrac{\delta_{2}^{2}}{4 c 4} $, $ \delta_{1}=0 $, $ \delta_{2} < 0 $, $ \delta_{3}=0 $, $ \delta_{4} > 0 $, $ G(\eta)=\varepsilon \sqrt{-\dfrac{\delta_{2}}{2 \delta_{4}}} \tanh \left(\sqrt{-\dfrac{\delta_{2}}{2}} \eta\right) $.
    $ \delta_{0}=0 $, $ \delta_{1}=0 $, $ \delta_{2} < 0 $, $ \delta_{3}=0 $, $ \delta_{4} > 0 $, $ G(\eta)=\varepsilon $ $ \sqrt{-\dfrac{\delta_{2}}{\delta_{4}}} $ $ \sec $ $ \left(\sqrt{-\delta_{2}} \eta\right) $.
    $ \delta_{0}=\dfrac{\delta_{2}^{2}}{4 \delta_{4}}, $ $ \delta_{1}=0 $, $ \delta_{2} > 0 $, $ \delta_{3}=0, $ $ \delta_{4} > 0, $ $ G(\eta)=\varepsilon $ $ \sqrt{\dfrac{\delta_{2}}{2 \delta_{4}}} \tan \left(\sqrt{\dfrac{\delta_{2}}{2}} \eta\right) $.
    $ \delta_{0}=0, $ $ \delta_{1}=0, $ $ \delta_{2}=0, $ $ \delta_{3}=0, $ $ \delta_{4} > 0, $ $ G(\eta)=-\dfrac{\varepsilon}{\sqrt{\delta_{4}} \eta} $.
    $ \delta_{0}=0, $ $ \delta_{1}=0, $ $ \delta_{2} > 0, $ $ \delta_{3}\neq 0, $ $ \delta_{4}=0, $ $ G(\eta)=-\dfrac{\delta_{2}}{\delta_{3}}. \operatorname{sech}^{2}\left(\dfrac{\sqrt{\delta_{2}}}{2} \eta\right) $.

     | Show Table
    DownLoad: CSV

    Consider the Novikov-Veselov (NV) system

    $ Ψt+αΓxxy+βΦxyy+γΓyΦ+γ(Ψ22)y+λΓΦx+λ(Ψ22)x=0,Γy=Ψx,Ψy=Φx,
    $
    (3.1)

    a system of PDEs in the unknown functions $ \Psi = \Psi(x, y, t), \; \Phi = \Phi(x, y, t) $, $ \Gamma = \Gamma(x, y, t) $ and their partial derivatives. I plug the transformations

    $ Φ(x,y,t)=ϕ(η),η=x+ywt,Ψ(x,y,t)=ψ(η),Γ(x,y,t)=Θ(η),
    $
    (3.2)

    into Eq (3.1) to reduce it to a system of ODEs given by

    $ wψη+αΘηηη+βϕηηη+γΘηϕ+γ(ψ22)η+λΘϕη+λ(ψ22)η=0,Θη=ψη,ψη=ϕη.
    $
    (3.3)

    Integrating $ \Theta_\eta = \psi_\eta $ and $ \phi_\eta = \psi_\eta $ yields

    $ Θ=ψ,and ϕ=ψ.
    $
    (3.4)

    Substituting (3.4) into the first equation of (3.3) and integrating once with respect to $ \eta $ yields

    $ wψ+(α+β)ψηη+(γ+λ)ψ2=0.
    $
    (3.5)

    Balancing $ \psi_{\eta\eta} $ with $ \psi^2 $ in (3.5) calculates the value of $ N = 2. $

    According to the modified F-expansion method with $ N = 2 $, the solutions of (3.5) are

    $ ψ(η)=ρ0+ρ1F(η)+q1F(η)+ρ2F(η)2+q2F(η)2,
    $
    (3.6)

    and $ F(\eta) $ is a solution of the following differential equation:

    $ F(η)=μ0+μ1F(η)+μ2F(η)2,
    $
    (3.7)

    where $ \mu_0 $, $ \mu_1 $, $ \mu_2 $ are given in Table 1. To explore the analytic solutions to (3.5), I ought to follow the subsequent steps.

    $ \textbf{Step 1.} $ Placing (3.6) along with (3.7) into Eq (3.5) and gathering the coefficients of $ F(\eta)^j $, $ j $ = $ -4, -3, -2, -1, 0, 1, 2, 3, 4, $ to zeros gives a system of equations for $ \rho_0, \; \rho_k, \; q_k $, $ k = 1, 2. $

    $ \textbf{Step 2.} $ Solve the resulting system using mathematical software: for example, Mathematica or Maple.

    $ \textbf{Step 3.} $ Choosing the values of $ \mu_0, \; \mu_1 $ and $ \mu_2 $ and the function $ F(\eta) $ from Table 1 and substituting them along with $ \rho_0, \; \rho_k, \; q_k $, $ k = 1, 2, $ in (3.6) produces a set of trigonometric function and rational solutions to (3.5).

    Applying the above steps, I determine the values of $ \rho_0, \; \rho_1, \; \rho_2, \; q_1, \; q_2 $ and $ w $ as follows:

    $ (1). $ When $ \mu_0 = 0, $ $ \mu_1 = 1 $ and $ \mu_2 = -1 $, I have two cases.

    $ \textbf{Case 1.} $

    $ ρ0=0,ρ1=6(α+β)γ+λ,ρ2=6(α+β)γ+λ,q1=q2=0,andw=α+β.
    $
    (3.8)

    The solution is given by

    $ Ψ1(x,y,t)=3(α+β)2(γ+λ)sech2(12(x+y(α+β)t+x0)).
    $
    (3.9)

    $ \textbf{Case 2.} $

    $ \rho_0 = -\dfrac{\alpha+\beta}{\gamma+\lambda}, \rho_1 = \dfrac{6 (\alpha +\beta)}{\gamma +\lambda}, \\ \rho_2 = -\dfrac{6 (\alpha +\beta)}{\gamma + \lambda}, q_1 = q_2 = 0, \quad \text{and}\quad w = -\left( \alpha+\beta\right) . $ (3.10)

    The solution is given by

    $ Ψ2(x,y,t)=(α+β)2(γ+λ)(3tanh2(12(x+y+t(α+β))+x0)1).
    $
    (3.11)

    Figure 1 presents the time evolution of the analytic solutions $ (a) $ $ \Psi_1 $ and $ (b) $ $ \Psi_2 $ with $ t = 0, 10, 20 $. The parameter values are $ x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5, $ and $ \lambda = 1. $ Figure 2 presents the wave behavior by changing a certain parameter value and fixing the values of the others. Figure 2$ (a, b) $ presents the behavior of $ \Psi_1 $ when I change the values of (a) $ \alpha $ or $ \beta $ and (b) $ \gamma $ or $ \lambda $. In Figure 2$ (a) $ it can also be seen that the value of $ \alpha $ or $ \beta $ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $ \alpha, \, \beta \to 0 $, and its amplitude increases when $ \alpha, \, \beta \to \infty. $ In Figure 2$ (b) $ the value of $ \gamma $ or $ \lambda $ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $ \gamma $ or $ \lambda $ increases. In Figure 2, $ (c) $ and $ (d) $ present the wave behavior of $ \Psi_2. $

    Figure 1.  Time evolution of the analytic solutions $ (a) $ $ \Psi_1 $ and $ (b) $ $ \Psi_2 $ with $ t = 0, 10, 20 $. The parameter are given by $ x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5, $ and $ \lambda = 1 $.
    Figure 2.  This figure present the wave behavior when changing a certain parameter value and fixing the values of the others. $ (a) $ presents the behavior when I change the value of $ \alpha $ or $ \beta, $ and $ (b) $ presents when I change the value of $ \gamma $ or $ \lambda $ for the solution $ \Psi_1. $ $ (c) $ and $ (d) $ are for $ \Psi_2 $.

    $ (2). $ When $ \mu_0 = 0, $ $ \mu_1 = -1, $ and $ \mu_2 = 1 $, I have two cases.

    $ \textbf{Case 3.} $ The solution is given by

    $ Ψ3(x,y,t)=3(α+β)2(γ+λ)csch2(12(x+y(α+β)t)).
    $
    (3.12)

    $ \textbf{Case 4.} $ The solution is given by

    $ Ψ4(x,y,t)=(α+β)2(γ+λ)(3coth2(12(x+y+t(α+β)))1).
    $
    (3.13)

    $ (3). $ When $ \mu_0 = 1, $ $ \mu_1 = 0, $ and $ \mu_2 = -1 $, I have

    $ \textbf{Case 5.} $ The solution is given by

    $ Ψ5(x,y,t)=8(α+β)γ+λ(cosh(4(16t(α+β)+x+y))+2)csch2(2(16t(α+β)+x+y)).
    $
    (3.14)

    $ (4). $ When $ \mu_0 = \pm 1, $ $ \mu_1 = 0, $ and $ \mu_2 = \pm 1 $, I have one case.

    $ \textbf{Case 6.} $ The solution is given by

    $ Ψ6(x,y,t)=24(α+β)γ+λcsc2(2(16t(α+β)+x+y)).
    $
    (3.15)

    According to the generalized algebraic method, the solutions of (3.5) are given by the form

    $ ψ(η)=ν0+ν1G(η)+ν2G(η)2+r1G(η)+r2G(η)2,
    $
    (3.16)

    where $ \nu_k $, $ r_k $ are to be determined later. $ G(\eta) $ is a solution of the following differential equation:

    $ G(η)=ε4k=0δkGk(η),
    $
    (3.17)

    where $ \delta_k $, $ k = 0, 1, 3, 4 $, are given in Table 2. In all the cases mentioned above and the subsequent solutions, I used the mathematical software Mathematica to find the values of the constants $ \nu_0, \, \nu_1, \, \nu_2, \, r_1, \, r_2 $ and $ w $. Thus, the analytic solutions to (3.5) using the generalized algebraic method will be presented here with different values of the constants $ \delta_k $, $ k = 0, 1, 3, 4 $.

    $ (5). $ When $ \delta_0 = \dfrac{\delta^2_2}{4 \delta_4}, $ $ \delta_1 = \delta_3 = 0, $ $ \delta_2 < 0 $, $ \delta_4 > 0 $, and $ \varepsilon = \pm1, $

    $ ν0=±4δ22ε4(α+β)2(γ+λ)23δ2ε4(α+β)2(γ+λ)22δ2ε2(α+β)(γ+λ)(γ+λ)2,ν2=6δ4ε2(α+β)(γ+λ),ν1=r1=r2=0,ε=±1.w=±2(4δ223δ2)ε4(α+β)2(γ+λ)2(γ+λ).
    $
    (3.18)

    $ \textbf{Case 7.} $ The solution is given by

    $ \Psi_7(x, y, t) = -\frac{1}{{(\gamma+\lambda)^2}}\\ \left( -3 \delta_2 \varepsilon^4 (\alpha+\beta) (\gamma+\lambda) \tanh ^2\left(\frac{\sqrt{-\delta_2} \left(\frac{2 t \sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}}{\gamma+\lambda}+x+y\right)}{\sqrt{2}}\right)\right.\\ \left.+2 \delta_2 \varepsilon^2 (\alpha+\beta) (\gamma+\lambda)+\sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}\right) . $ (3.19)

    Figure 3 shows the time evolution of the analytic solutions. Figure 3$ (a) $ shows $ \Psi_7 $ with $ t $ = $ 0:2:6 $. The parameter values are $ \delta_2 = -1 $, $ \delta_4 = 1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10. $ Figure 3$ (b) $ shows $ \Psi_8 $ with $ t = 0:2:8 $. The parameter values are $ \delta_2 = 1 $, $ \delta_4 = -1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10. $ Figures 46 present the $ 3D $ time evolution of the analytic solutions $ \Psi_2 $ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $ t = 5, 15, 25 $, $ M_x = 1600, $ $ N_y = 100, $ $ x = 0\to60 $ and $ y = 0\to1. $

    Figure 3.  Time evolution of the analytic solutions. $ (a) $ $ \Psi_7 $ with $ t = 0:2:6 $. The parameter values are $ \delta_2 = -1 $, $ \delta_4 = 1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10. $ $ (b) $ $ \Psi_8 $ with $ t = 0:2:8 $. The parameter values are $ \delta_2 = 1 $, $ \delta_4 = -1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10 $.
    Figure 4.  3D graphs presenting the analytic (left) and the numerical (right) solutions of $ \Psi_2(x, y, t) $ at $ t = 5. $ The figures present the strength of agreement between analytic and numerical solutions.
    Figure 5.  3D graphs presenting the analytic (left) and the numerical (right) solutions of $ \Psi_2(x, y, t) $ at $ t = 15. $ The figures present the strength of agreement between analytic and numerical solutions.
    Figure 6.  3D graphs presenting the analytic (left) and the numerical (right) solutions of $ \Psi_2(x, y, t) $ at $ t = 25. $ The figures present the strength of agreement between analytic and numerical solutions.

    $ (6). $ When $ \delta_0 = 0, $ $ \delta_1 = \delta_3 = 0, $ $ \delta_2 > 0 $, $ \delta_4 < 0 $, and $ \varepsilon = \pm1, $

    $ ν2=6δ4ε2(α+β)γ+λ,ν1=ν0=r1=r2=0,w=4δ2ε2(α+β).
    $
    (3.20)

    $ \textbf{Case 8.} $ The solution is given by

    $ Ψ8(x,y,t)=6δ2ϵ4(α+β)sech2(δ2(4δ2ϵ2t(α+β)+x+y))γ+λ
    $
    (3.21)

    $ (7). $ When $ \delta_0 = 0, $ $ \delta_1 = \delta_4 = 0, $ $ \delta_3\neq0, $ $ \delta_2 > 0 $, $ \varepsilon = \pm1 $

    $ Set 1.ν1=3δ3ε2(α+β)2(γ+λ),ν0=ν2=r1=r2=0,w=δ2ε2(α+β).Set 2.ν0=δ2ε2(α+β)(γ+λ),ν1=3δ3ε2(α+β)2(γ+λ),ν2=r1=r2=0,w=δ2ε2(α+β).
    $
    (3.22)

    $ \textbf{Case 9.} $The solution is given by

    $ Ψ9(x,y,t)=3δ2(α+β)sech2(12δ2(δ2t((α+β))+x+y))2(γ+λ).
    $
    (3.23)

    $ \textbf{Case 10.} $ The solution is given by

    $ Ψ10(x,y,t)=3δ2(α+β)sech2(12δ2(δ2t(α+β)+x+y))2(γ+λ)δ2(α+β)γ+λ.
    $
    (3.24)

    In this section I extract numerical solutions to the resulting ODE system (3.5) using several numerical methods. The purpose of this procedure is to guarantee the accuracy of the analytic solutions. I picked one of the analytic solutions above to be a sample, (3.11). The nonlinear shooting and BVP methods, at $ t = 0, $ are used by taking the value of $ \psi $ at the right endpoint of the domain $ \eta = 0 $ with guessing the initial value for $ \psi_\eta. $ The new target is to obtain the second boundary condition of $ \psi $ at the left endpoint of a particular domain. Once the numerical result is obtained, I compare it with the analytic solution (3.11). The MATLAB solver ODE15s and FSOLVE [47] are used to get the numerical solution. The resulting ODE (3.5) is discretized as

    $ f(ψ)=0,f(ψi)=wψi+α+βΔη(ψi+12ψi+ψi1)+γ+λ2Δη(ψ2i+1ψ2i1),
    $
    (4.1)

    for the BVP method and

    $ ψηη=1α+β(wψ(γ+λ)ψ2),
    $
    (4.2)

    for the shooting method. Figure 7 presents the comparison between the numerical solutions obtained using the above numerical methods and the analytic solution. Figure 7 shows that the solutions are identical to the analytic solution.

    Figure 7.  Comparing the numerical solutions resulting from the shooting and BVP methods with the analytic solution (3.11) at $ t = 0. $ The parameter values are taken as $ \alpha = 0.50 $, $ \beta = 0.6 $, $ \gamma = -1.5, \lambda = 1.8 $, with $ N = 600 $.

    Thus, it is possible to verify the correctness of the analytic solution. I also accept the obtained numerical solution as an initial condition for the numerical scheme in the next section.

    In this section, I use the finite-difference method to obtain the numerical results of system (1.1) over the domain $ [a, \, b]\times [c, \, d]. $ Here, $ a, \; b, \; c $ and $ d $ represent the endpoints of the rectangular domain in the $ x $ and $ y $ directions, respectively, and $ T_f $ is a certain time. The domain $ [a, \, b]\times [c, \, d] $ is split into $ (M_x+1)\times (N_y+1) $ mesh points:

    $ xm=a+mΔx,m=0,1,2,,Mx,yn=c+nΔy,n=0,1,2,,Ny,
    $

    where $ \Delta_x $ and $ \Delta_y $ are the step-sizes of the $ x $ and $ y $ domains, respectively. The system (1.1) is converted to an ODE system by discretizing the space derivatives while keeping the time derivative continuous. Completing this yields

    $ Ψt|km,n+α2ΔyΔ2xδ2x(Γk+1m,n+1Γk+1m,n1)+β2ΔxΔ2yδ2y(Φk+1m+1,nΦk+1m1,n)γ4Δy((Φk+1m,n+1+Φk+1m,n)Γk+1m,n+1(Φk+1m,n+Φk+1m,n1)Γk+1m,n1)λ4Δx((Γk+1m+1,n+Γk+1m,n)Φk+1m+1,n(Γk+1m,n+Γk+1m1,n)Φk+1m1,n)+γ4Δy((Ψk+1m,n+1)2(Ψk+1m,n1)2)+λ4Δx((Ψk+1m+1,n)2(Ψk+1m1,n)2)=0,12Δy(Γk+1m,n+1Γk+1m,n1)=12Δx(Ψk+1m+1,nΨk+1m1,n),12Δy(Ψk+1m,n+1Ψk+1m,n1)=12Δx(Φk+1m+1,nΦk+1m1,n),
    $
    (5.1)

    where

    $ δ2xΓk+1m,n=(Γk+1m+1,n2Γk+1m,n+Γk+1m1,n),δ2yΦk+1m,n=(Φk+1m,n+12Φk+1m,n+Φk+1m,n1),
    $

    subject to the boundary conditions:

    $ Ψx(a,y,t)=Ψx(b,y,t)=0,y[c,d],Ψy(x,c,t)=Ψy(x,d,t)=0,x[a,b].
    $
    (5.2)

    Equation (5.2) permits us to use fictitious points in estimating the space derivatives at the domain's endpoints. The initial conditions are generated by

    $ Ψ2(x,y,0)=(α+β)2(γ+λ)(3tanh2(12(x+y+x0)1),
    $
    (5.3)

    where $ \alpha, \, \beta, \, \gamma $ and $ \lambda $ are user-defined parameters. In all the numerical results shown in this section, the parameter values are fixed as $ \alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0 $, $ y = 0\to1, $ $ x = 0\to60 $ and $ t = 0\to 25. $ The above system is solved by using an ODE solver in FORTRAN called the DDASPK solver [48]. This solver used a backward differentiation formula. Since I do not have the initial conditions for the space derivatives, I approximate the Jacobian matrix of the linearized system by using LU-Factorization. The obtained numerical results are acceptable. This can be observed from the Figures 8 and 9.

    Figure 8.  Time change for the numerical results while holding $ y = 0.5 $ and $ M_x = 1600 $ at $ t = 0:5:25. $ The wave at $ t = 25 $ illustrates that the numerical and the analytic solutions are quite identical.
    Figure 9.  The convergence histories of the scheme with the fixation of both $ y = 0.5 $ and $ M_x = 1600 $ at $ t = 5 $.

    The von Neumann analysis is used to examine the stability of the scheme (5.1). The von Neumann analysis is occasionally called Fourier analysis and is utilized exclusively when the scheme is linear. Hence, I suppose that the linear version is given by

    $ Ψt+αΓxxy+βΦxyy+s0Γy+s1Ψy+s2Φx+s3Ψx=0,Γy=Ψx,Ψy=Φx,
    $
    (6.1)

    where $ s_0 = \gamma \Phi $, $ s_1 = \gamma \Psi $, $ s_2 = \lambda \Gamma $, $ s_3 = \lambda \Psi $ are constants. Since $ \Gamma_y = \Psi_x, $ and $ \Psi_y = \Phi_x, $ the first equation of (6.1) is given by

    $ Ψt+αΨxxx+βΨyyy+s0Ψy+s1Ψy+s2Ψx+s3Ψx=0,
    $
    (6.2)

    where $ \alpha, \, \beta, \, \gamma, \, \lambda, \, s_0, \, s_1, \, s_2, \, s_3, \, l_4 $ are constants. I set directly

    $ Ψkm,n=μkexp(ιπξ0nΔx)exp(ιπξ1mΔy),
    $
    (6.3)

    and also I can have

    $ Ψk+1m,n=μΨkm,n,Ψkm+1,n=exp(ιπξ0Δx)Ψkm,n,Ψkm,n+1=exp(ιπξ1Δy)Ψkm,n,Ψkm1,n=exp(ιπξ0Δx)Ψkm,n,Ψkm,n1=exp(ιπξ1Δy)Ψkm,n,m=1,2,,Nx1,n=1,2,,Ny1.
    $

    Substituting (6.3) into (6.2) and doing some operations, I have

    $ 1=μ(1ιΔt(sin(ξ0πΔx)Δx(4αΔ2xsin2(ξ0πΔx2)s2s3)+sin(ξ1πΔy)Δy(4βΔ2ysin2(ξ1πΔy2)s0s1))).
    $

    Hence,

    $ μ=11aι,
    $
    (6.4)

    where

    $ a = \Delta_t\left( \frac{\sin(\xi_0\pi \Delta_x)\, }{\Delta_x} \left(\frac{4\, \alpha}{\Delta_x^2}\, \sin^2(\tfrac{\xi_0\pi\, \Delta_x}{2})-s_2-s_3\right)+\frac{ \sin(\xi_1\pi \Delta_y)\, }{\Delta_y} \left(\frac{4\, \beta}{\Delta_y^2}\, \sin^2(\tfrac{\xi_1\pi\, \Delta_y}{2})-s_0-s_1\right)\right). $

    Thus,

    $ |μ|2=11+a21.
    $
    (6.5)

    The stability condition of the von Neumann analysis is fulfilled. Consequently, from Eq (6.5), the scheme is unconditionally stable.

    To examine the accuracy of the numerical scheme (5.1), I study the truncation error utilizing Taylor expansions. Suppose that the error is

    $ ek+1m,n=Ψk+1m,nΨ(xm,yn,tk+1),
    $
    (7.1)

    where $ \Psi(x_m, y_n, t_{k+1}) $ and $ \Psi_{m, n}^{k+1} $ are the analytic solution and an approximate solution, respectively. Substituting (7.1) into (5.1) gives

    $ ek+1j,kekj,kΔt=Tk+1m,n(α12Δ3xδ2x(ek+1m+1,nek+1m1,n)+β12Δ3yδ2y(ek+1m,n+1ek+1m,n1)+s2+s32Δx(ek+1m+1,nek+1m1,n)+s0+s12Δy(ek+1m,n+1ek+1m,n1)),
    $

    where

    $ T^{k+1}_{m, n} = \frac{\alpha}{2\Delta_x^3 } \delta_x^2\left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right) \\ +\frac{\beta }{2\Delta_y^3 } \delta_y^2\left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right)\\ + \frac{ s_2+s_3}{2\Delta_x} \left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right)+\\ \frac{ s_0+s_1}{2\Delta_y} \left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right). $

    Hence,

    $ Tk+1m,nΔt22Ψ(xm,yn,ξk+1)t2Δ2x25Ψ(ζm,yn,tk+1)x5Δ2y25Ψ(xm,ηn,tk+1)x5Δ2y63Ψ(xm,ηn,tk+1)x3Δ2x63Ψ(ζm,yn,tk+1)x3.
    $

    Accordingly, the truncation error of the numerical scheme is

    $ Tk+1m,n=O(Δt,Δ2x,Δ2y).
    $

    I have prosperously employed several analytical methods to extract the traveling wave solutions to the two-dimensional Novikov-Veselov system, confirming the solutions with numerical results obtained using the numerical scheme (5.1). The major highlights of the results are shown in Table 3 and Figures 810, which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $ \Delta_x, \Delta_y\to 0 $. The numerical schemes are unconditionally stable for fixing the parameter values $ \alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0 $, $ y = 0\to1, $ $ x = 0\to60 $ and $ t = 0\to 25. $

    Table 3.  The relative error with $ L_2 $ norm and CPU at $ t = 20. $.
    $ \Delta_x $ The Relative Error CPU
    $ 0.6000 \qquad\qquad $ $ 5.600\times 10^{-3}\qquad $ $ 0.063\times10^{3}\, \text{m} \qquad $
    $ 0.3000 $ $ 2.100\times 10^{-3} $ $ 0.1524\times10^{3} \, s $
    $ 0.1500 $ $ 6.700\times 10^{-4} $ $ 0.3564\times10^{3}\, s $
    $ 0.0750 $ $ 2.100\times 10^{-4} $ $ 0.8892\times10^{3}\, s $
    $ 0.0375 $ $ 6.610\times 10^{-5} $ $ 1.7424\times10^{3}\, s $
    $ 0.0187 $ $ 2.310\times 10^{-5} $ $ 4.0230\times10^{3}\, s $

     | Show Table
    DownLoad: CSV
    Figure 10.  The convergence histories measured utilizing the relative error with $ l_2 $ norm as a function of $ \Delta_x $ (see Table 3). Here, I picked a certain value of the variable $ y = 0.5 $ at $ t = 20 $ and $ x = 0\to60 $.

    Figure 1 presents the time evolution of the analytic solutions $ (a) $ $ \Psi_1 $ and $ (b) $ $ \Psi_2 $ with $ t = 0, 10, 20 $. The parameter values are $ x0 = -20, \ \alpha = 0.50, \ \beta = 0.6, \ \gamma = -1.5, $ and $ \lambda = 1. $ Figure 2 presents the wave behavior by changing a certain parameter value and fixing the values of the others. Figure 2$ (a, b) $ presents the behavior of $ \Psi_1 $ when I change the values of (a) $ \alpha $ or $ \beta $ and (b) $ \gamma $ or $ \lambda $. In Figure 2$ (a) $ it can also be seen that the value of $ \alpha $ or $ \beta $ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $ \alpha, \, \beta \to 0 $, and its amplitude increases when $ \alpha, \, \beta \to \infty. $ In Figure 2$ (b) $ the value of $ \gamma $ or $ \lambda $ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $ \gamma $ or $ \lambda $ increases. In Figure 2$ (c, d) $ present the wave behavior of $ \Psi_2. $ Figure 3 shows the time evolution of the analytic solutions. Figure 3$ (a) $ shows $ \Psi_7 $ with $ t = 0:2:6 $. The parameter values are $ \delta_2 = -1 $, $ \delta_4 = 1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10. $ Figure 3$ (b) $ shows $ \Psi_8 $ with $ t = 0:2:8 $. The parameter values are $ \delta_2 = 1 $, $ \delta_4 = -1, $ $ \epsilon = -1, $ $ \alpha = 0.50, $ $ \beta = 0.6, $ $ \gamma = -1.5, $ $ \lambda = 1.8 $ and $ x0 = -10. $ Figures 46 present the $ 3D $ time evolution of the analytic solutions $ \Psi_2 $ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $ t = 5, 15, 25 $, $ M_x = 1600, $ $ N_y = 100, $ $ x = 0\to60 $ and $ y = 0\to1. $ These figures provide us with an adequate answer that the numerical and analytic solutions are quite identical. Barman et al. [42] accepted several traveling wave solutions for (1.1) as hyperbolic functions. The authors employed other parameters to develop new forms for the accepted solution. They proposed that Eq (1.1) describes tidal and tsunami waves, electromagnetic waves in transmission cables and magneto-sound and ion waves in plasma. In comparison, I have found numerous solutions also as hyperbolic functions. Furthermore, I obtained the numerical solutions to enhance the assurance that the solutions presented here are correct and accurate.

    I have successfully utilized the generalized algebraic and modified F-expansion methods to acquire the soliton solutions for the two-dimensional Novikov-Veselov system, verifying these solutions with numerical results obtained by employing the numerical scheme (5.1). The major highlights of the results shown in Figures 810 and Table 3, which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $ \Delta_x, \Delta_y\to 0 $. The numerical schemes are unconditionally stable for fixing the parameter values $ \alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, $ $ x0 = -45.0 $, $ y = 0\to1, $ $ x = 0\to60 $ and $ t = 0\to 25. $ The Jacobi elliptic functions have effectively deteriorated to hyperbolic functions. The applied numerical schemes have provided reliable numerical solutions when using a small value of $ \Delta_x, \Delta_y\to0 $.

    Ultimately, I can deduce that the methods used are valuable and applicable to extract soliton solutions for other nonlinear evolutionary systems found in chemistry, engineering, physics and other sciences.

    The author declares that he has no potential conflict of interest in this article.

    [1] Powers ET, Morimoto RI, Dillin A, et al. (2009) Biological and chemical approaches to diseases of proteostasis deficiency. Annu Rev Biochem 78: 959-991. doi: 10.1146/annurev.biochem.052308.114844
    [2] Hartl FU, Bracher A, Hayer-Hartl M (2011) Molecular chaperones in protein folding and proteostasis. Nature 475: 324-332. doi: 10.1038/nature10317
    [3] Balch WE, Morimoto RI, Dillin A, et al. (2008) Adapting proteostasis for disease intervention. Science 319: 916-919. doi: 10.1126/science.1141448
    [4] Tanaka K (2013) The proteasome: from basic mechanisms to emerging roles. Keio J Med 62: 1-12. doi: 10.2302/kjm.2012-0006-RE
    [5] Wong E, Cuervo AM (2010) Integration of clearance mechanisms: the proteasome and autophagy. Cold Spring Harb Perspect Biol 2: a006734.
    [6] Finley D (2009) Recognition and processing of ubiquitin-protein conjugates by the proteasome. Annu Rev Biochem 78: 477-513. doi: 10.1146/annurev.biochem.78.081507.101607
    [7] Lopez-Otin C, Blasco MA, Partridge L, et al. (2013) The hallmarks of aging. Cell 153: 1194-1217. doi: 10.1016/j.cell.2013.05.039
    [8] Selkoe DJ (2011) Alzheimer's disease. Cold Spring Harb Perspect Biol 3.
    [9] Bosco DA, LaVoie MJ, Petsko GA, et al. (2011) Proteostasis and movement disorders: Parkinson's disease and amyotrophic lateral sclerosis. Cold Spring Harb Perspect Biol 3: a007500.
    [10] Finkbeiner S (2011) Huntington's Disease. Cold Spring Harb Perspect Biol 3.
    [11] Schmidt M, Finley D (2014) Regulation of proteasome activity in health and disease. Biochim Biophys Acta 1843: 13-25. doi: 10.1016/j.bbamcr.2013.08.012
    [12] Tanaka K, Matsuda N (2014) Proteostasis and neurodegeneration: the roles of proteasomal degradation and autophagy. Biochim Biophys Acta 1843: 197-204. doi: 10.1016/j.bbamcr.2013.03.012
    [13] Vernace VA, Schmidt-Glenewinkel T, Figueiredo-Pereira ME (2007) Aging and regulated protein degradation: who has the UPPer hand? Aging Cell 6: 599-606. doi: 10.1111/j.1474-9726.2007.00329.x
    [14] Taylor RC, Dillin A (2011) Aging as an event of proteostasis collapse. Cold Spring Harb Perspect Biol 3.
    [15] Buckley SM, Aranda-Orgilles B, Strikoudis A, et al. (2012) Regulation of pluripotency and cellular reprogramming by the ubiquitin-proteasome system. Cell Stem Cell 11: 783-798. doi: 10.1016/j.stem.2012.09.011
    [16] Okita Y, Nakayama KI (2012) UPS delivers pluripotency. Cell Stem Cell 11: 728-730. doi: 10.1016/j.stem.2012.11.009
    [17] Vilchez D, Boyer L, Morantte I, et al. (2012) Increased proteasome activity in human embryonic stem cells is regulated by PSMD11. Nature 489: 304-308. doi: 10.1038/nature11468
    [18] Iwai K (2012) Diverse ubiquitin signaling in NF-kappaB activation. Trends Cell Biol 22: 355-364. doi: 10.1016/j.tcb.2012.04.001
    [19] Thrower JS, Hoffman L, Rechsteiner M, et al. (2000) Recognition of the polyubiquitin proteolytic signal. EMBO J 19: 94-102. doi: 10.1093/emboj/19.1.94
    [20] Kisselev AF, Goldberg AL (2005) Monitoring activity and inhibition of 26S proteasomes with fluorogenic peptide substrates. Methods Enzymol 398: 364-378. doi: 10.1016/S0076-6879(05)98030-0
    [21] Jung T, Höhn A, Grune T (2013) The proteasome and the degradation of oxidized proteins: PartII - proteinoxidationandproteasomaldegradation. Redox Biology 2: 99-104.
    [22] Demartino GN, Gillette TG (2007) Proteasomes: machines for all reasons. Cell 129: 659-662. doi: 10.1016/j.cell.2007.05.007
    [23] Forster A, Masters EI, Whitby FG, et al. (2005) The 1.9 A structure of a proteasome-11S activator complex and implications for proteasome-PAN/PA700 interactions. Mol Cell 18: 589-599.
    [24] Dubiel W, Pratt G, Ferrell K, et al. (1992) Purification of an 11 S regulator of the multicatalytic protease. J Biol Chem 267: 22369-22377.
    [25] Ma CP, Slaughter CA, DeMartino GN (1992) Identification, purification, and characterization of a protein activator (PA28) of the 20 S proteasome (macropain). J Biol Chem 267: 10515-10523.
    [26] Realini C, Jensen CC, Zhang Z, et al. (1997) Characterization of recombinant REGalpha, REGbeta, and REGgamma proteasome activators. J Biol Chem 272: 25483-25492. doi: 10.1074/jbc.272.41.25483
    [27] Sijts A, Sun Y, Janek K, et al. (2002) The role of the proteasome activator PA28 in MHC class I antigen processing. Mol Immunol 39: 165-169. doi: 10.1016/S0161-5890(02)00099-8
    [28] Li X, Amazit L, Long W, et al. (2007) Ubiquitin- and ATP-independent proteolytic turnover of p21 by the REGgamma-proteasome pathway. Mol Cell 26: 831-842. doi: 10.1016/j.molcel.2007.05.028
    [29] Blickwedehl J, Agarwal M, Seong C, et al. (2008) Role for proteasome activator PA200 and postglutamyl proteasome activity in genomic stability. Proc Natl Acad Sci U S A 105: 16165-16170. doi: 10.1073/pnas.0803145105
    [30] Schmidt M, Haas W, Crosas B, et al. (2005) The HEAT repeat protein Blm10 regulates the yeast proteasome by capping the core particle. Nat Struct Mol Biol 12: 294-303. doi: 10.1038/nsmb914
    [31] Egan D, Kim J, Shaw RJ, et al. (2011) The autophagy initiating kinase ULK1 is regulated via opposing phosphorylation by AMPK and mTOR. Autophagy 7: 643-644. doi: 10.4161/auto.7.6.15123
    [32] Rubinsztein DC, Marino G, Kroemer G (2011) Autophagy and aging. Cell 146: 682-695. doi: 10.1016/j.cell.2011.07.030
    [33] Ravikumar B, Sarkar S, Davies JE, et al. (2010) Regulation of mammalian autophagy in physiology and pathophysiology. Physiol Rev 90: 1383-1435. doi: 10.1152/physrev.00030.2009
    [34] Nixon RA (2013) The role of autophagy in neurodegenerative disease. Nat Med 19: 983-997. doi: 10.1038/nm.3232
    [35] Martinez-Vicente M, Cuervo AM (2007) Autophagy and neurodegeneration: when the cleaning crew goes on strike. Lancet Neurol 6: 352-361. doi: 10.1016/S1474-4422(07)70076-5
    [36] Cuervo AM (2004) Autophagy: in sickness and in health. Trends Cell Biol 14: 70-77. doi: 10.1016/j.tcb.2003.12.002
    [37] Mizushima N, Levine B, Cuervo AM, et al. (2008) Autophagy fights disease through cellular self-digestion. Nature 451: 1069-1075. doi: 10.1038/nature06639
    [38] Mizushima N, Komatsu M (2011) Autophagy: renovation of cells and tissues. Cell 147: 728-741. doi: 10.1016/j.cell.2011.10.026
    [39] Cuervo AM (2010) Chaperone-mediated autophagy: selectivity pays off. Trends Endocrinol Metab 21: 142-150. doi: 10.1016/j.tem.2009.10.003
    [40] He C, Klionsky DJ (2009) Regulation mechanisms and signaling pathways of autophagy. Annu Rev Genet 43: 67-93. doi: 10.1146/annurev-genet-102808-114910
    [41] Klionsky DJ, Cregg JM, Dunn WA, Jr., et al. (2003) A unified nomenclature for yeast autophagy-related genes. Dev Cell 5: 539-545. doi: 10.1016/S1534-5807(03)00296-X
    [42] Dubouloz F, Deloche O, Wanke V, et al. (2005) The TOR and EGO protein complexes orchestrate microautophagy in yeast. Mol Cell 19: 15-26. doi: 10.1016/j.molcel.2005.05.020
    [43] Marzella L, Ahlberg J, Glaumann H (1981) Autophagy, heterophagy, microautophagy and crinophagy as the means for intracellular degradation. Virchows Arch B Cell Pathol Incl Mol Pathol 36: 219-234. doi: 10.1007/BF02912068
    [44] Mortimore GE, Lardeux BR, Adams CE (1988) Regulation of microautophagy and basal protein turnover in rat liver. Effects of short-term starvation. J Biol Chem 263: 2506-2512.
    [45] Cuervo AM, Dice JF (1996) A receptor for the selective uptake and degradation of proteins by lysosomes. Science 273: 501-503. doi: 10.1126/science.273.5274.501
    [46] Rubinsztein DC, DiFiglia M, Heintz N, et al. (2005) Autophagy and its possible roles in nervous system diseases, damage and repair. Autophagy 1: 11-22. doi: 10.4161/auto.1.1.1513
    [47] Lee CK, Klopp RG, Weindruch R, et al. (1999) Gene expression profile of aging and its retardation by caloric restriction. Science 285: 1390-1393. doi: 10.1126/science.285.5432.1390
    [48] Ly DH, Lockhart DJ, Lerner RA, et al. (2000) Mitotic misregulation and human aging. Science 287: 2486-2492. doi: 10.1126/science.287.5462.2486
    [49] Ferrington DA, Husom AD, Thompson LV (2005) Altered proteasome structure, function, and oxidation in aged muscle. FASEB J 19: 644-646.
    [50] Vernace VA, Arnaud L, Schmidt-Glenewinkel T, et al. (2007) Aging perturbs 26S proteasome assembly in Drosophila melanogaster. FASEB J 21: 2672-2682. doi: 10.1096/fj.06-6751com
    [51] Grune T, Jung T, Merker K, et al. (2004) Decreased proteolysis caused by protein aggregates, inclusion bodies, plaques, lipofuscin, ceroid, and 'aggresomes' during oxidative stress, aging, and disease. Int J Biochem Cell Biol 36: 2519-2530. doi: 10.1016/j.biocel.2004.04.020
    [52] Andersson V, Hanzen S, Liu B, et al. (2013) Enhancing protein disaggregation restores proteasome activity in aged cells. Aging (Albany NY) 11: 802-812.
    [53] Bulteau AL, Petropoulos I, Friguet B (2000) Age-related alterations of proteasome structure and function in aging epidermis. Exp Gerontol 35: 767-777. doi: 10.1016/S0531-5565(00)00136-4
    [54] Carrard G, Dieu M, Raes M, et al. (2003) Impact of ageing on proteasome structure and function in human lymphocytes. Int J Biochem Cell Biol 35: 728-739. doi: 10.1016/S1357-2725(02)00356-4
    [55] Chondrogianni N, Stratford FL, Trougakos IP, et al. (2003) Central role of the proteasome in senescence and survival of human fibroblasts: induction of a senescence-like phenotype upon its inhibition and resistance to stress upon its activation. J Biol Chem 278: 28026-28037. doi: 10.1074/jbc.M301048200
    [56] Petropoulos I, Conconi M, Wang X, et al. (2000) Increase of oxidatively modified protein is associated with a decrease of proteasome activity and content in aging epidermal cells. J Gerontol A Biol Sci Med Sci 55: B220-227. doi: 10.1093/gerona/55.5.B220
    [57] Wagner BJ, Margolis JW (1995) Age-dependent association of isolated bovine lens multicatalytic proteinase complex (proteasome) with heat-shock protein 90, an endogenous inhibitor. Arch Biochem Biophys 323: 455-462. doi: 10.1006/abbi.1995.0067
    [58] Bardag-Gorce F, Farout L, Veyrat-Durebex C, et al. (1999) Changes in 20S proteasome activity during ageing of the LOU rat. Mol Biol Rep 26: 89-93. doi: 10.1023/A:1006968208077
    [59] Bulteau AL, Szweda LI, Friguet B (2002) Age-dependent declines in proteasome activity in the heart. Arch Biochem Biophys 397: 298-304. doi: 10.1006/abbi.2001.2663
    [60] Conconi M, Szweda LI, Levine RL, et al. (1996) Age-related decline of rat liver multicatalytic proteinase activity and protection from oxidative inactivation by heat-shock protein 90. Arch Biochem Biophys 331: 232-240. doi: 10.1006/abbi.1996.0303
    [61] Husom AD, Peters EA, Kolling EA, et al. (2004) Altered proteasome function and subunit composition in aged muscle. Arch Biochem Biophys 421: 67-76. doi: 10.1016/j.abb.2003.10.010
    [62] Keller JN, Hanni KB, Markesbery WR (2000) Possible involvement of proteasome inhibition in aging: implications for oxidative stress. Mech Ageing Dev 113: 61-70. doi: 10.1016/S0047-6374(99)00101-3
    [63] Shibatani T, Nazir M, Ward WF (1996) Alteration of rat liver 20S proteasome activities by age and food restriction. J Gerontol A Biol Sci Med Sci 51: B316-322.
    [64] Tomaru U, Takahashi S, Ishizu A, et al. (2012) Decreased proteasomal activity causes age-related phenotypes and promotes the development of metabolic abnormalities. Am J Pathol 180: 963-972. doi: 10.1016/j.ajpath.2011.11.012
    [65] Bergamini E (2006) Autophagy: a cell repair mechanism that retards ageing and age-associated diseases and can be intensified pharmacologically. Mol Aspects Med 27: 403-410. doi: 10.1016/j.mam.2006.08.001
    [66] Cuervo AM (2008) Autophagy and aging: keeping that old broom working. Trends Genet 24: 604-612. doi: 10.1016/j.tig.2008.10.002
    [67] Cuervo AM, Dice JF (2000) Age-related decline in chaperone-mediated autophagy. J Biol Chem 275: 31505-31513. doi: 10.1074/jbc.M002102200
    [68] Terman A (1995) The effect of age on formation and elimination of autophagic vacuoles in mouse hepatocytes. Gerontology 41 Suppl 2: 319-326.
    [69] Vittorini S, Paradiso C, Donati A, et al. (1999) The age-related accumulation of protein carbonyl in rat liver correlates with the age-related decline in liver proteolytic activities. J Gerontol A Biol Sci Med Sci 54: B318-323. doi: 10.1093/gerona/54.8.B318
    [70] Terman A, Brunk UT (2006) Oxidative stress, accumulation of biological 'garbage', and aging. Antioxid Redox Signal 8: 197-204. doi: 10.1089/ars.2006.8.197
    [71] Yen WL, Klionsky DJ (2008) How to live long and prosper: autophagy, mitochondria, and aging. Physiology (Bethesda) 23: 248-262. doi: 10.1152/physiol.00013.2008
    [72] Lipinski MM, Zheng B, Lu T, et al. (2010) Genome-wide analysis reveals mechanisms modulating autophagy in normal brain aging and in Alzheimer's disease. Proc Natl Acad Sci U S A 107: 14164-14169. doi: 10.1073/pnas.1009485107
    [73] Carames B, Taniguchi N, Otsuki S, et al. (2010) Autophagy is a protective mechanism in normal cartilage, and its aging-related loss is linked with cell death and osteoarthritis. Arthritis Rheum 62: 791-801.
    [74] Donati A, Cavallini G, Paradiso C, et al. (2001) Age-related changes in the regulation of autophagic proteolysis in rat isolated hepatocytes. J Gerontol A Biol Sci Med Sci 56: B288-293. doi: 10.1093/gerona/56.7.B288
    [75] Matecic M, Smith DL, Pan X, et al. (2010) A microarray-based genetic screen for yeast chronological aging factors. PLoS Genet 6: e1000921. doi: 10.1371/journal.pgen.1000921
    [76] Hars ES, Qi H, Ryazanov AG, et al. (2007) Autophagy regulates ageing in C. elegans. Autophagy 3: 93-95. doi: 10.4161/auto.3636
    [77] Toth ML, Sigmond T, Borsos E, et al. (2008) Longevity pathways converge on autophagy genes to regulate life span in Caenorhabditis elegans. Autophagy 4: 330-338. doi: 10.4161/auto.5618
    [78] Simonsen A, Cumming RC, Brech A, et al. (2008) Promoting basal levels of autophagy in the nervous system enhances longevity and oxidant resistance in adult Drosophila. Autophagy 4: 176-184. doi: 10.4161/auto.5269
    [79] Kuma A, Hatano M, Matsui M, et al. (2004) The role of autophagy during the early neonatal starvation period. Nature 432: 1032-1036. doi: 10.1038/nature03029
    [80] Levine B, Kroemer G (2008) Autophagy in the pathogenesis of disease. Cell 132: 27-42. doi: 10.1016/j.cell.2007.12.018
    [81] Hara T, Nakamura K, Matsui M, et al. (2006) Suppression of basal autophagy in neural cells causes neurodegenerative disease in mice. Nature 441: 885-889. doi: 10.1038/nature04724
    [82] Komatsu M, Waguri S, Chiba T, et al. (2006) Loss of autophagy in the central nervous system causes neurodegeneration in mice. Nature 441: 880-884. doi: 10.1038/nature04723
    [83] Komatsu M, Wang QJ, Holstein GR, et al. (2007) Essential role for autophagy protein Atg7 in the maintenance of axonal homeostasis and the prevention of axonal degeneration. Proc Natl Acad Sci U S A 104: 14489-14494. doi: 10.1073/pnas.0701311104
    [84] Liang CC, Wang C, Peng X, et al. (2010) Neural-specific deletion of FIP200 leads to cerebellar degeneration caused by increased neuronal death and axon degeneration. J Biol Chem 285: 3499-3509. doi: 10.1074/jbc.M109.072389
    [85] Masiero E, Agatea L, Mammucari C, et al. (2009) Autophagy is required to maintain muscle mass. Cell Metab 10: 507-515. doi: 10.1016/j.cmet.2009.10.008
    [86] Singh R, Kaushik S, Wang Y, et al. (2009) Autophagy regulates lipid metabolism. Nature 458: 1131-1135. doi: 10.1038/nature07976
    [87] Yang L, Li P, Fu S, et al. (2010) Defective hepatic autophagy in obesity promotes ER stress and causes insulin resistance. Cell Metab 11: 467-478. doi: 10.1016/j.cmet.2010.04.005
    [88] Jung HS, Chung KW, Won Kim J, et al. (2008) Loss of autophagy diminishes pancreatic beta cell mass and function with resultant hyperglycemia. Cell Metab 8: 318-324. doi: 10.1016/j.cmet.2008.08.013
    [89] Hartleben B, Godel M, Meyer-Schwesinger C, et al. (2010) Autophagy influences glomerular disease susceptibility and maintains podocyte homeostasis in aging mice. J Clin Invest 120: 1084-1096. doi: 10.1172/JCI39492
    [90] Dice JF (1982) Altered degradation of proteins microinjected into senescent human fibroblasts. J Biol Chem 257: 14624-14627.
    [91] Kiffin R, Kaushik S, Zeng M, et al. (2007) Altered dynamics of the lysosomal receptor for chaperone-mediated autophagy with age. J Cell Sci 120: 782-791. doi: 10.1242/jcs.001073
    [92] Zhang C, Cuervo AM (2008) Restoration of chaperone-mediated autophagy in aging liver improves cellular maintenance and hepatic function. Nat Med 14: 959-965. doi: 10.1038/nm.1851
    [93] Bandyopadhyay U, Kaushik S, Varticovski L, et al. (2008) The chaperone-mediated autophagy receptor organizes in dynamic protein complexes at the lysosomal membrane. Mol Cell Biol 28: 5747-5763. doi: 10.1128/MCB.02070-07
    [94] Soti C, Csermely P (2003) Aging and molecular chaperones. Exp Gerontol 38: 1037-1040. doi: 10.1016/S0531-5565(03)00185-2
    [95] Nardai G, Csermely P, Soti C (2002) Chaperone function and chaperone overload in the aged. A preliminary analysis. Exp Gerontol 37: 1257-1262. doi: 10.1016/S0531-5565(02)00134-1
    [96] Cohen E, Dillin A (2008) The insulin paradox: aging, proteotoxicity and neurodegeneration. Nat Rev Neurosci 9: 759-767. doi: 10.1038/nrn2474
    [97] Zabel C, Nguyen HP, Hin SC, et al. (2010) Proteasome and oxidative phoshorylation changes may explain why aging is a risk factor for neurodegenerative disorders. J Proteomics 73: 2230-2238. doi: 10.1016/j.jprot.2010.08.008
    [98] Matsuda N, Tanaka K (2010) Does impairment of the ubiquitin-proteasome system or the autophagy-lysosome pathway predispose individuals to neurodegenerative disorders such as Parkinson's disease? J Alzheimers Dis 19: 1-9.
    [99] Saez I, Vilchez D (2014) The Mechanistic Links Between Proteasome Activity, Aging and Age-related Diseases. Curr Genomics 15: 38-51. doi: 10.2174/138920291501140306113344
    [100] Massey AC, Kaushik S, Sovak G, et al. (2006) Consequences of the selective blockage of chaperone-mediated autophagy. Proc Natl Acad Sci U S A 103: 5805-5810. doi: 10.1073/pnas.0507436103
    [101] Ravikumar B, Vacher C, Berger Z, et al. (2004) Inhibition of mTOR induces autophagy and reduces toxicity of polyglutamine expansions in fly and mouse models of Huntington disease. Nat Genet 36: 585-595. doi: 10.1038/ng1362
    [102] Cuervo AM (2006) Autophagy in neurons: it is not all about food. Trends Mol Med 12: 461-464. doi: 10.1016/j.molmed.2006.08.003
    [103] Keck S, Nitsch R, Grune T, et al. (2003) Proteasome inhibition by paired helical filament-tau in brains of patients with Alzheimer's disease. J Neurochem 85: 115-122. doi: 10.1046/j.1471-4159.2003.01642.x
    [104] Grune T, Botzen D, Engels M, et al. (2010) Tau protein degradation is catalyzed by the ATP/ubiquitin-independent 20S proteasome under normal cell conditions. Arch Biochem Biophys 500: 181-188. doi: 10.1016/j.abb.2010.05.008
    [105] Dickey CA, Koren J, Zhang YJ, et al. (2008) Akt and CHIP coregulate tau degradation through coordinated interactions. Proc Natl Acad Sci U S A 105: 3622-3627. doi: 10.1073/pnas.0709180105
    [106] Petrucelli L, Dickson D, Kehoe K, et al. (2004) CHIP and Hsp70 regulate tau ubiquitination, degradation and aggregation. Hum Mol Genet 13: 703-714. doi: 10.1093/hmg/ddh083
    [107] Dange T, Smith D, Noy T, et al. (2011) Blm10 protein promotes proteasomal substrate turnover by an active gating mechanism. J Biol Chem 286: 42830-42839. doi: 10.1074/jbc.M111.300178
    [108] Jinwal UK, Koren J, 3rd, Borysov SI, et al. (2010) The Hsp90 cochaperone, FKBP51, increases Tau stability and polymerizes microtubules. J Neurosci 30: 591-599. doi: 10.1523/JNEUROSCI.4815-09.2010
    [109] Blair LJ, Nordhues BA, Hill SE, et al. (2013) Accelerated neurodegeneration through chaperone-mediated oligomerization of tau. J Clin Invest 123: 4158-4169. doi: 10.1172/JCI69003
    [110] Nixon RA, Wegiel J, Kumar A, et al. (2005) Extensive involvement of autophagy in Alzheimer disease: an immuno-electron microscopy study. J Neuropathol Exp Neurol 64: 113-122.
    [111] Berger Z, Ravikumar B, Menzies FM, et al. (2006) Rapamycin alleviates toxicity of different aggregate-prone proteins. Hum Mol Genet 15: 433-442.
    [112] Caccamo A, Majumder S, Richardson A, et al. (2010) Molecular interplay between mammalian target of rapamycin (mTOR), amyloid-beta, and Tau: effects on cognitive impairments. J Biol Chem 285: 13107-13120. doi: 10.1074/jbc.M110.100420
    [113] Tseng BP, Green KN, Chan JL, et al. (2008) Abeta inhibits the proteasome and enhances amyloid and tau accumulation. Neurobiol Aging 29: 1607-1618. doi: 10.1016/j.neurobiolaging.2007.04.014
    [114] Yu WH, Cuervo AM, Kumar A, et al. (2005) Macroautophagy--a novel Beta-amyloid peptide-generating pathway activated in Alzheimer's disease. J Cell Biol 171: 87-98. doi: 10.1083/jcb.200505082
    [115] Cataldo AM, Barnett JL, Berman SA, et al. (1995) Gene expression and cellular content of cathepsin D in Alzheimer's disease brain: evidence for early up-regulation of the endosomal-lysosomal system. Neuron 14: 671-680. doi: 10.1016/0896-6273(95)90324-0
    [116] Boland B, Kumar A, Lee S, et al. (2008) Autophagy induction and autophagosome clearance in neurons: relationship to autophagic pathology in Alzheimer's disease. J Neurosci 28: 6926-6937. doi: 10.1523/JNEUROSCI.0800-08.2008
    [117] Rozmahel R, Huang J, Chen F, et al. (2002) Normal brain development in PS1 hypomorphic mice with markedly reduced gamma-secretase cleavage of betaAPP. Neurobiol Aging 23: 187-194. doi: 10.1016/S0197-4580(01)00267-6
    [118] Pickford F, Masliah E, Britschgi M, et al. (2008) The autophagy-related protein beclin 1 shows reduced expression in early Alzheimer disease and regulates amyloid beta accumulation in mice. J Clin Invest 118: 2190-2199.
    [119] Jaeger PA, Pickford F, Sun CH, et al. (2010) Regulation of amyloid precursor protein processing by the Beclin 1 complex. PLoS One 5: e11102. doi: 10.1371/journal.pone.0011102
    [120] McNaught KS, Perl DP, Brownell AL, et al. (2004) Systemic exposure to proteasome inhibitors causes a progressive model of Parkinson's disease. Ann Neurol 56: 149-162. doi: 10.1002/ana.20186
    [121] Bedford L, Hay D, Devoy A, et al. (2008) Depletion of 26S proteasomes in mouse brain neurons causes neurodegeneration and Lewy-like inclusions resembling human pale bodies. J Neurosci 28: 8189-8198. doi: 10.1523/JNEUROSCI.2218-08.2008
    [122] Wahl C, Kautzmann S, Krebiehl G, et al. (2008) A comprehensive genetic study of the proteasomal subunit S6 ATPase in German Parkinson's disease patients. J Neural Transm 115: 1141-1148. doi: 10.1007/s00702-008-0054-3
    [123] Tofaris GK, Layfield R, Spillantini MG (2001) alpha-synuclein metabolism and aggregation is linked to ubiquitin-independent degradation by the proteasome. FEBS Lett 509: 22-26. doi: 10.1016/S0014-5793(01)03115-5
    [124] Anglade P, Vyas S, Javoy-Agid F, et al. (1997) Apoptosis and autophagy in nigral neurons of patients with Parkinson's disease. Histol Histopathol 12: 25-31.
    [125] Vogiatzi T, Xilouri M, Vekrellis K, et al. (2008) Wild type alpha-synuclein is degraded by chaperone-mediated autophagy and macroautophagy in neuronal cells. J Biol Chem 283: 23542-23556. doi: 10.1074/jbc.M801992200
    [126] Webb JL, Ravikumar B, Atkins J, et al. (2003) Alpha-Synuclein is degraded by both autophagy and the proteasome. J Biol Chem 278: 25009-25013. doi: 10.1074/jbc.M300227200
    [127] Cuervo AM, Stefanis L, Fredenburg R, et al. (2004) Impaired degradation of mutant alpha-synuclein by chaperone-mediated autophagy. Science 305: 1292-1295. doi: 10.1126/science.1101738
    [128] Spencer B, Potkar R, Trejo M, et al. (2009) Beclin 1 gene transfer activates autophagy and ameliorates the neurodegenerative pathology in alpha-synuclein models of Parkinson's and Lewy body diseases. J Neurosci 29: 13578-13588. doi: 10.1523/JNEUROSCI.4390-09.2009
    [129] Narendra D, Tanaka A, Suen DF, et al. (2008) Parkin is recruited selectively to impaired mitochondria and promotes their autophagy. J Cell Biol 183: 795-803. doi: 10.1083/jcb.200809125
    [130] Narendra DP, Jin SM, Tanaka A, et al. (2010) PINK1 is selectively stabilized on impaired mitochondria to activate Parkin. PLoS Biol 8: e1000298. doi: 10.1371/journal.pbio.1000298
    [131] Tashiro Y, Urushitani M, Inoue H, et al. (2012) Motor neuron-specific disruption of proteasomes, but not autophagy, replicates amyotrophic lateral sclerosis. J Biol Chem 287: 42984-42994. doi: 10.1074/jbc.M112.417600
    [132] Gusella JF, MacDonald ME, Ambrose CM, et al. (1993) Molecular genetics of Huntington's disease. Arch Neurol 50: 1157-1163. doi: 10.1001/archneur.1993.00540110037003
    [133] Vonsattel JP, DiFiglia M (1998) Huntington disease. J Neuropathol Exp Neurol 57: 369-384. doi: 10.1097/00005072-199805000-00001
    [134] (1993) A novel gene containing a trinucleotide repeat that is expanded and unstable on Huntington's disease chromosomes. The Huntington's Disease Collaborative Research Group. Cell 72: 971-983.
    [135] Sanchez I, Mahlke C, Yuan J (2003) Pivotal role of oligomerization in expanded polyglutamine neurodegenerative disorders. Nature 421: 373-379. doi: 10.1038/nature01301
    [136] Snell RG, MacMillan JC, Cheadle JP, et al. (1993) Relationship between trinucleotide repeat expansion and phenotypic variation in Huntington's disease. Nat Genet 4: 393-397. doi: 10.1038/ng0893-393
    [137] Nance MA, Myers RH (2001) Juvenile onset Huntington's disease--clinical and research perspectives. Ment Retard Dev Disabil Res Rev 7: 153-157. doi: 10.1002/mrdd.1022
    [138] Holmberg CI, Staniszewski KE, Mensah KN, et al. (2004) Inefficient degradation of truncated polyglutamine proteins by the proteasome. EMBO J 23: 4307-4318. doi: 10.1038/sj.emboj.7600426
    [139] Seo H, Sonntag KC, Kim W, et al. (2007) Proteasome activator enhances survival of Huntington's disease neuronal model cells. PLoS One 2: e238. doi: 10.1371/journal.pone.0000238
    [140] Vilchez D, Morantte I, Liu Z, et al. (2012) RPN-6 determines C. elegans longevity under proteotoxic stress conditions. Nature 489: 263-268.
    [141] Tonoki A, Kuranaga E, Tomioka T, et al. (2009) Genetic evidence linking age-dependent attenuation of the 26S proteasome with the aging process. Mol Cell Biol 29: 1095-1106. doi: 10.1128/MCB.01227-08
    [142] Hipp MS, Patel CN, Bersuker K, et al. (2012) Indirect inhibition of 26S proteasome activity in a cellular model of Huntington's disease. J Cell Biol 196: 573-587. doi: 10.1083/jcb.201110093
    [143] Kalchman MA, Graham RK, Xia G, et al. (1996) Huntingtin is ubiquitinated and interacts with a specific ubiquitin-conjugating enzyme. J Biol Chem 271: 19385-19394. doi: 10.1074/jbc.271.32.19385
    [144] Finkbeiner S, Mitra S (2008) The ubiquitin-proteasome pathway in Huntington's disease. ScientificWorldJournal 8: 421-433. doi: 10.1100/tsw.2008.60
    [145] Sieradzan KA, Mechan AO, Jones L, et al. (1999) Huntington's disease intranuclear inclusions contain truncated, ubiquitinated huntingtin protein. Exp Neurol 156: 92-99. doi: 10.1006/exnr.1998.7005
    [146] Waelter S, Boeddrich A, Lurz R, et al. (2001) Accumulation of mutant huntingtin fragments in aggresome-like inclusion bodies as a result of insufficient protein degradation. Mol Biol Cell 12: 1393-1407. doi: 10.1091/mbc.12.5.1393
    [147] Ardley HC, Hung CC, Robinson PA (2005) The aggravating role of the ubiquitin-proteasome system in neurodegeneration. FEBS Lett 579: 571-576. doi: 10.1016/j.febslet.2004.12.058
    [148] Ciechanover A, Brundin P (2003) The ubiquitin proteasome system in neurodegenerative diseases: sometimes the chicken, sometimes the egg. Neuron 40: 427-446. doi: 10.1016/S0896-6273(03)00606-8
    [149] Venkatraman P, Wetzel R, Tanaka M, et al. (2004) Eukaryotic proteasomes cannot digest polyglutamine sequences and release them during degradation of polyglutamine-containing proteins. Mol Cell 14: 95-104. doi: 10.1016/S1097-2765(04)00151-0
    [150] Bennett EJ, Bence NF, Jayakumar R, et al. (2005) Global impairment of the ubiquitin-proteasome system by nuclear or cytoplasmic protein aggregates precedes inclusion body formation. Mol Cell 17: 351-365. doi: 10.1016/j.molcel.2004.12.021
    [151] Pratt G, Rechsteiner M (2008) Proteasomes cleave at multiple sites within polyglutamine tracts: activation by PA28gamma(K188E). J Biol Chem 283: 12919-12925. doi: 10.1074/jbc.M709347200
    [152] Tellez-Nagel I, Johnson AB, Terry RD (1974) Studies on brain biopsies of patients with Huntington's chorea. J Neuropathol Exp Neurol 33: 308-332. doi: 10.1097/00005072-197404000-00008
    [153] Qin ZH, Wang Y, Kegel KB, et al. (2003) Autophagy regulates the processing of amino terminal huntingtin fragments. Hum Mol Genet 12: 3231-3244. doi: 10.1093/hmg/ddg346
    [154] Ravikumar B, Duden R, Rubinsztein DC (2002) Aggregate-prone proteins with polyglutamine and polyalanine expansions are degraded by autophagy. Hum Mol Genet 11: 1107-1117. doi: 10.1093/hmg/11.9.1107
    [155] Sarkar S, Perlstein EO, Imarisio S, et al. (2007) Small molecules enhance autophagy and reduce toxicity in Huntington's disease models. Nat Chem Biol 3: 331-338. doi: 10.1038/nchembio883
    [156] Martinez-Vicente M, Talloczy Z, Wong E, et al. (2010) Cargo recognition failure is responsible for inefficient autophagy in Huntington's disease. Nat Neurosci 13: 567-576. doi: 10.1038/nn.2528
    [157] Shibata M, Lu T, Furuya T, et al. (2006) Regulation of intracellular accumulation of mutant Huntingtin by Beclin 1. J Biol Chem 281: 14474-14485. doi: 10.1074/jbc.M600364200
    [158] Atwal RS, Truant R (2008) A stress sensitive ER membrane-association domain in Huntingtin protein defines a potential role for Huntingtin in the regulation of autophagy. Autophagy 4: 91-93. doi: 10.4161/auto.5201
    [159] Cattaneo E, Zuccato C, Tartari M (2005) Normal huntingtin function: an alternative approach to Huntington's disease. Nat Rev Neurosci 6: 919-930. doi: 10.1038/nrn1806
    [160] Ravikumar B, Acevedo-Arozena A, Imarisio S, et al. (2005) Dynein mutations impair autophagic clearance of aggregate-prone proteins. Nat Genet 37: 771-776. doi: 10.1038/ng1591
    [161] Hassan WM, Merin DA, Fonte V, et al. (2009) AIP-1 ameliorates beta-amyloid peptide toxicity in a Caenorhabditis elegans Alzheimer's disease model. Hum Mol Genet 18: 2739-2747. doi: 10.1093/hmg/ddp209
    [162] Yun C, Stanhill A, Yang Y, et al. (2008) Proteasomal adaptation to environmental stress links resistance to proteotoxicity with longevity in Caenorhabditis elegans. Proc Natl Acad Sci U S A 105: 7094-7099. doi: 10.1073/pnas.0707025105
    [163] Nakatogawa H, Suzuki K, Kamada Y, et al. (2009) Dynamics and diversity in autophagy mechanisms: lessons from yeast. Nat Rev Mol Cell Biol 10: 458-467. doi: 10.1038/nrm2708
    [164] Xie Z, Klionsky DJ (2007) Autophagosome formation: core machinery and adaptations. Nat Cell Biol 9: 1102-1109. doi: 10.1038/ncb1007-1102
    [165] Yang Z, Klionsky DJ (2009) An overview of the molecular mechanism of autophagy. Curr Top Microbiol Immunol 335: 1-32.
    [166] Hanada T, Noda NN, Satomi Y, et al. (2007) The Atg12-Atg5 conjugate has a novel E3-like activity for protein lipidation in autophagy. J Biol Chem 282: 37298-37302. doi: 10.1074/jbc.C700195200
    [167] Hanada T, Satomi Y, Takao T, et al. (2009) The amino-terminal region of Atg3 is essential for association with phosphatidylethanolamine in Atg8 lipidation. FEBS Lett 583: 1078-1083. doi: 10.1016/j.febslet.2009.03.009
    [168] Komatsu M, Waguri S, Ueno T, et al. (2005) Impairment of starvation-induced and constitutive autophagy in Atg7-deficient mice. J Cell Biol 169: 425-434. doi: 10.1083/jcb.200412022
  • This article has been cited by:

    1. Erdal Karapinar, Andreea Fulga, Seher Sultan Yeşilkaya, Santosh Kumar, Fixed Points of Proinov Type Multivalued Mappings on Quasimetric Spaces, 2022, 2022, 2314-8888, 1, 10.1155/2022/7197541
    2. Mi Zhou, Xiaolan Liu, Naeem Saleem, Andreea Fulga, Nihal Özgür, A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms, 2022, 14, 2073-8994, 93, 10.3390/sym14010093
    3. Salvador Romaguera, Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results, 2022, 10, 2227-7390, 3931, 10.3390/math10213931
    4. Salvador Romaguera, On Protected Quasi-Metrics, 2024, 13, 2075-1680, 158, 10.3390/axioms13030158
  • Reader Comments
  • © 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(8773) PDF downloads(1322) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog