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Alzheimer’s Disease: A short introduction to the calmodulin hypothesis

  • Received: 21 August 2019 Accepted: 28 September 2019 Published: 12 October 2019
  • At the cellular level, Alzheimer’s disease (AD) is characterized by the presence of intracellular plaques containing amyloid beta (Aβ) protein and neurofibrillary tangles consisting of phospho-tau (p-tau). These biomarkers are considered to contribute, at least in part, to the neurodegenerative events of the disease. But the accumulation of plaques and tangles is widely considered to be a later event with other factors likely being the cause of the disease. Calcium dysregulation—the unregulated accumulation of calcium ions—in neurons is an early event that underlies neurodegeneration. In 2002, O’Day and Myre extended this “Calcium Hypothesis” to include calmodulin (CaM) the primary target of calcium, suggesting the “Calmodulin Hypothesis” as an updated alternative. Here we overview the central role of CaM in the formation of the classic hallmarks of AD: plaques and tangles. Then some insight into CaM’s binding to various risk factor proteins is given followed by a short summary of specific receptors and channels linked to the disease that are CaM binding proteins. Overall, this review emphasizes the diversity of Alzheimer’s-linked CaM-binding proteins validating the hypothesis that CaM operates critically at all stages of the disease and stands out as a potential primary target for future research.

    Citation: Danton H. O’Day. Alzheimer’s Disease: A short introduction to the calmodulin hypothesis[J]. AIMS Neuroscience, 2019, 6(4): 231-239. doi: 10.3934/Neuroscience.2019.4.231

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  • At the cellular level, Alzheimer’s disease (AD) is characterized by the presence of intracellular plaques containing amyloid beta (Aβ) protein and neurofibrillary tangles consisting of phospho-tau (p-tau). These biomarkers are considered to contribute, at least in part, to the neurodegenerative events of the disease. But the accumulation of plaques and tangles is widely considered to be a later event with other factors likely being the cause of the disease. Calcium dysregulation—the unregulated accumulation of calcium ions—in neurons is an early event that underlies neurodegeneration. In 2002, O’Day and Myre extended this “Calcium Hypothesis” to include calmodulin (CaM) the primary target of calcium, suggesting the “Calmodulin Hypothesis” as an updated alternative. Here we overview the central role of CaM in the formation of the classic hallmarks of AD: plaques and tangles. Then some insight into CaM’s binding to various risk factor proteins is given followed by a short summary of specific receptors and channels linked to the disease that are CaM binding proteins. Overall, this review emphasizes the diversity of Alzheimer’s-linked CaM-binding proteins validating the hypothesis that CaM operates critically at all stages of the disease and stands out as a potential primary target for future research.


    Nonlinear evolution equations (NLEEs) model many complex phenomena in physics including plasma, solid state, chemical and optical fibers, nonlinear optics, fluid mechanics, etc. Exploring exact traveling wave solutions plays a significant role in nonlinear physics. For this purpose, a number of techniques were developed including method of modified Khater [1,2], first integral [3,4], functional variable [5], expansions [6,7] of new generalized $ (G'/G) $ [8,9,10], new $ \Phi6 $-model [11], Jacobi elliptic function [12,13], sine-Gordon [14], bifurcation [15,16], exp-function [17,18], new auxiliary equation [19], exp(-$ \phi (\xi) $)-expansion [20,21], fan sub-equation [22,23], inverse scattering [24], generalized Kudryshov [25,26,27], Hirota's bilinear [28,29], extended direct algebraic [30], Lie group [31].

    Consider the (2+1)-dimensional KK equations [32]

    $ 9ut+u5x+15uuxxx+752uxuxx+45u2ux+5σuxxy5σ1xuyy+15σuuy+15σux1xuy=0.
    $
    (1.1)

    where $ \sigma^2 = 1, \, \, \partial_x^{-1} = \int dx. $ This equation has been widely applied in many branches of physics like plasma physics, fluid dynamics, nonlinear optics, and so forth. If we take $ u(x, y, t) = u(x, t), $ Eq (1.1) becomes the (1+1)-dimensional KK equation [32]

    $ 9ut+u5x+15uuxxx+752uxuxx+45u2ux=0,
    $
    (1.2)

    In [33], method of exp-function was applied to Eq (1.2). In [32], symmetric method was applied to the nonlinear (2+1)-KK equation.

    The method of the present paper, a candid, succinct and efficient technique, considered as a generalization of ($ G' $/$ G $)-expansion technique [34,35,36,37] was developed in [38,39,40,41,42,43,44,45]. Main purpose of this paper is to investigate the applicability of the method to (1+1)-dimensional KK equation which was not considered in the history of research so far.

    We shortly overview the method in such a fashion that maintains four remarks and five basic postulates:

    Remark I. If we set up

    $ ϕ=G/G,ψ=1/G,
    $
    (2.1)

    in

    $ G(ξ)+λG(ξ)=β,
    $
    (2.2)

    then we must have the relations

    $ ϕ=ϕ2+βψλ,φ=φψ,
    $
    (2.3)

    wherein $ \lambda $ and $ \beta $ are parameters.

    Remark II. If $ \lambda $ is negative, general solution of (2.2) is:

    $ G(ξ)=D1sinh(ξλ)+D2cosh(ξλ)+βλ.
    $
    (2.4)

    and we receive the following relation

    $ ψ2=λλ2α1+β2(φ22βψ+λ),
    $
    (2.5)

    wherein $ D_{1} $ and $ D_{2} $ are arbitrary constants and $ \alpha _{1} = D_{1} ^{2} -D_{2} ^{2} $.

    Remark III. If $ \lambda $ is positive, general solution of (2.2) is:

    $ G(ξ)=D1sin(ξλ)+D2cos(ξλ)+βλ,
    $
    (2.6)

    consequently, we obtain

    $ ψ2=λλ2α2β2(φ22βψ+λ),
    $
    (2.7)

    wherein $ \alpha _{2} = D_{1} ^{2}+D_{2} ^{2} $.

    Remark IV. If $ \lambda = 0 $, the general solution of (2.2),

    $ G(ξ)=β2ξ2+D1ξ+D2,
    $
    (2.8)

    and therefore we get,

    $ ψ2=φ22βψD212βD2.
    $
    (2.9)

    Now let us consider:

    $ R(u,ut,ux,uy,utt,uxx,uyy,uxt,)=0,
    $
    (2.10)

    wherein $ R $ is a polynomial function in $ u $ and $ u_{t} = \frac{\partial u}{\partial t} $, $ u_{x} = \frac{\partial u}{\partial x} $, $ u_{y} = \frac{\partial u}{\partial y} $, $ u_{xx} = \frac{\partial ^{2} u}{\partial x^{2} } $, $ u_{yy} = \frac{\partial ^{2} u}{\partial y^{2} } $, $ u_{xy} = \frac{\partial ^{2} u}{\partial x \partial y} $ and so on.

    Postulate 1. Consider:

    $ u(x,y,t)=u(ξ),andξ=ηx+ωy+ct,
    $
    (2.11)

    wherein $ \eta $, $ \omega $ and $ c $ are parameters. By traveling wave transformations (2.11), the Eq.(2.10) can be reduced to:

    $ T(u,cu,ηu,ωu,c2u,η2u,ω2u,ηωu,cηu,)=0,
    $
    (2.12)

    wherein $ T $ is a polynomial.

    Postulate 2. Let us assume that the following relation is the general solution expressed by a polynomial:

    $ u(ξ)=a0+Ni=1(aiφi(ξ)+biφi1(ξ)ψ(ξ)),
    $
    (2.13)

    wherein $ a_{0} $, $ a_{i} $ and $ b_{i} (i = 1, 2, 3, ..., N) $ are the constant coefficients such that $ a_{N}^{2} +b_{N}^{2} \ne 0 $.

    Postulate 3. By homogeneous balance, we determine $ N $ in Eq (2.13).

    Postulate 4. To convert the left-hand-side of Eq (2.12) into a polynomial function in $ \psi $ and $ \phi $, we write Eq (2.13) into Eq (2.12) with Eq (2.3) and Eq (2.5). By solving polynomial, we obtain the system: in $ a_{0} $, $ a_{i} $, $ b_{i} (i = 1, 2, 3, ..., N) $, $ \lambda (<0) $, $ \beta $, $ \eta $, $ \omega $, $ c $, $ D_{1} $ and $ D_{2} $. We solve this system with Mathematica. Setting values of above algebraic constants in Eq (2.13), solutions by hyperbolic functions in Eq (2.12) are obtained.

    Postulate 5. Similar to Postulate 4, substituting Eq (2.13) into Eq (2.12), using Eq (2.3) and Eq (2.5) (or Eq (2.3) and Eq (2.7)), we obtain the exact traveling wave solutions of Eq (2.12) demonstrated by trigonometric functions.

    Let us consider transformation:

    $ u(x,t)=u(ξ),ξ=x+ct,
    $
    (3.1)

    wherein $ c $ is a parameter, which reduces Eq (1.2) to:

    $ 9cu+u(5)+15uu+752uu+45u2u=0.
    $
    (3.2)

    According to postulate 2, the positive number $ N = 2 $ is obtained by balancing between $ u^{(5)} $ and $ u^{2}u' $, thus general solutions of Eq (3.2) is:

    $ u(ξ)=a0+a1φ(ξ)+a2φ2(ξ)+b1ψ(ξ)+b2φ(ξ)ψ(ξ),
    $
    (3.3)

    wherein$ a_{0} $, $ a_{i} $ and $ b_{i} (i = 1, 2) $ are constant coefficients such that$ a_{N} ^{2}+b_{N} ^{2} \ne 0 (N = 1, 2) $, $ \phi (\xi) $ and $ \psi (\xi) $ are satisfied by the Eq (2.3). Now, there are three categories of solutions of Eq (3.2):

    Category 1: When $ \lambda < 0 $ (solutions by hyperbolic functions):

    Writing Eq (3.3) with Eq (2.3) and Eq (2.5) into Eq (3.2), Eq (3.2) forms a polynomial in $ \psi (\xi) $ and $ \phi (\xi) $. Solving this polynomial, we obtain a system: $ a_{0} $, $ a_{1} $, $ a_{2} $, $ b_{1} $, $ b_{2} $, $ \lambda (<0) $, $ \beta $, $ c $ and $ \alpha _{1} $. Solving this system with Mathematica, we obtain the values of $ a_{0} $ $ a_{1} $, $ a_{2} $, $ b_{1} $, $ b_{2} $, $ \beta $ and $ c $ as:

    Result 1:

    $ a0=10λ3,a1=0,a2=4,b1=4β,b2=±4β2+λ2α1λ,c=11λ29,β=β.
    $
    (3.4)

    Writing these constants from Eq (3.4) into (3.3) and by Eq (2.1) and Eq (2.4), we obtain explicit solutions of Eq (1.2):

    $ u(ξ)=10λ3+4λ{D1cosh(ξλ)D2sinh(ξλ)}2{D1sinh(ξλ+D2cosh(ξλ)+βλ}2+4β{D1sinh(ξλ+D2cosh(ξλ)+βλ}±4β2+λ2α1λ{D1cosh(ξλ)D2sinh(ξλ)}2{D1sinh(ξλ+D2cosh(ξλ)+βλ}2
    $
    (3.5)

    wherein $ \xi = x-\frac{11\lambda ^{2} t}{9} $ and $ \alpha _{1} = D_{1} ^{2} -D_{2} ^{2} $.

    In particular, if we choose $ D_{1} \ne 0 $, $ D_{2} = 0 $ and $ \beta = 0 $ in Eq (3.5), we get:

    $ u(x,t)=10λ3+4λcoth(λ(x11λ2t9)){coth(λ(x11λ2t9))±csch(λ(x11λ2t9))}.
    $
    (3.6)

    Similarly, if we choose $ D_{2} \ne 0 $, $ D_{1} = 0 $ and $ \beta = 0 $ in Eq (3.5), we get:

    $ u(x,t)=10λ3+4λtanh(λ(x11λ2t9)){tanh(λ(x11λ2t9))±isech(λ(x11λ2t9))},
    $
    (3.7)
    Figure 1.  3D, contour and 2D surfaces of absolute Eq (3.7) when $ \lambda = -1 $.

    wherein $ i = \sqrt{-1} $.

    Result 2:

    $ a0=5λ12,a1=0,a2=12, b1=β2,b2=±β2+λ2α12λ,c=λ2144,β=β.
    $
    (3.8)

    Explicit solutions of Eq (1.2) are given by:

    $ u(ξ)=5λ12+λ{D1cosh(ξλ)+D2sinh(ξλ)}22{D1sinh(ξλ)+D2cosh(ξλ)+βλ}2+β2{D1sinh(ξλ)+D2cosh(ξλ)+βλ}±β2+λ2α1{D1cosh(ξλ)+D2sinh(ξλ)}2{D1sinh(ξλ)+D2cosh(ξλ)+βλ}2,
    $
    (3.9)

    wherein $ \xi = x-\frac{\lambda ^{2} t}{144} $ and $ \alpha _{1} = D_{1} ^{2} -D_{2} ^{2} $.

    In particular, if we choose $ D_{1} \ne 0 $, $ D_{2} = 0 $ and $ \beta = 0 $ in Eq (3.9), we get:

    $ u(x,t)=5λ12+λ2coth(λ(xλ2t144)){coth(λ(xλ2t144))±csch(λ(xλ2t144))}.
    $
    (3.10)

    Similarly, if we choose $ D_{2} \ne 0 $, $ D_{1} = 0 $ and $ \beta = 0 $ in Eq (3.9), we get:

    $ u(x,t)=5λ12+λ2tanh(λ(xλ2t144)){tanh(λ(xλ2t144))±isech(λ(xλ2t144))},
    $
    (3.11)

    wherein $ i = \sqrt{-1} $.

    Result3:

    $ a0=11λβ2+8λ3α112(β2+λ2α1),a1=0,a2=1,b1=β,b2=0,c=λ2(β428λ2β2α1+16λ4α21)144(β2+λ2α1)2,β=β.
    $
    (3.12)

    wherein $ \beta ^{2} +\lambda ^{2} \alpha _{1} \ne 0 $.

    We get explicit solutions of Eq (1.2) as:

    $ u(ξ)=11λβ2+8λ3α112(β2+λ2α1)+λ{D1cosh(ξλ)+D2sinh(ξλ)}2{D1sinh(ξλ)+D2cosh(ξλ)+βλ}2+β{D1sinh(ξλ)+D2cosh(ξλ)+βλ},
    $
    (3.13)

    wherein $ \xi = x-\frac{\lambda ^{2} t (\beta ^{4} -28 \lambda ^{2} \beta ^{2} \alpha _{1} +16 \lambda ^{4} \alpha _{1}^{2})}{144 (\beta ^{2} +\lambda ^{2} \alpha _{1})^{2} } $ and $ \alpha _{1} = D_{1} ^{2} -D_{2} ^{2} $.

    In particular, if we choose $ D_{1} \ne 0 $, $ D_{2} = 0 $ and $ \beta = 0 $ in Eq (3.13), we get:

    $ u(x,t)=2λ3+λcoth2(λ(xλ2t9)).
    $
    (3.14)
    Figure 2.  3D, contour and 2D surfaces of absolute Eq (3.14) when $ \lambda = -5. $.

    Similarly, if we choose $ D_{2} \ne 0 $, $ D_{1} = 0 $ and $ \beta = 0 $ in Eq (3.14), we get:

    $ u(x,t)=2λ3+λtanh2(λ(xλ2t9)).
    $
    (3.15)

    Category 2: For $ \lambda > 0, $ (i.e. trigonometric functions),

    According to Postulate 5, if we execute as the category 1, we attain the values of $ a_{0} $, $ a_{1} $, $ a_{2} $, $ b_{1} $, $ b_{2} $, $ \beta $ and $ c $ as the following results:

    Result 1:

    $ a0=10λ3,a1=0,a2=4,b1=4β,b2=±4β2+λ2α1λ,c=11λ29,β=β.
    $
    (3.16)

    Writing constants in Eq (3.16) into Eq (3.3) and by Eq (2.1) and Eq (2.6), we get explicit solutions of Eq (1.2):

    $ u(ξ)=10λ34λ{D1cos(ξλ)D2sin(ξλ)}2{D1sin(ξλ)+D2cos(ξλ)+βλ}2+4βD1sin(ξλ)+D2cos(ξλ)+βλ±4β2+λ2α2{D1cos(ξλ)D2sin(ξλ)}{D1sin(ξλ)+D2cos(ξλ)+βλ}2,
    $
    (3.17)

    wherein $ \xi = x-\frac{ 11 \lambda ^{2} t }{9} $ and $ \alpha _{2} = D_{1} ^{2} +D_{2} ^{2} $.

    Result 2:

    $ a0=5λ12,a1=0,a2=12,b1=β2,b2=±β2+λ2α12λ,c=λ2144,β=β.
    $
    (3.18)

    We get explicit solutions of Eq (1.2) as:

    $ u(ξ)=5λ12λ{D1cos(ξλ)D2sin(ξλ)}22{D1sin(ξλ)+D2cos(ξλ)+βλ}2+β2{D1sin(ξλ)+D2cos(ξλ)+βλ}±β2+λ2α2{D1cos(ξλ)D2sin(ξλ)}2{D1sin(ξλ)+D2cos(ξλ)+βλ}2,.
    $
    (3.19)

    wherein $ \xi = x-\frac{ \lambda ^{2} t }{144} $ and $ \alpha _{2} = D_{1} ^{2} +D_{2} ^{2} $.

    Result 3:

    $ a0=11λβ28λ3α212(β2+λ2α2),a1=0,a2=1,b1=β,b2=0,c=λ2(β4+28λ2β2α2+16λ4α22)144(β2+λ2α2)2,β=β.
    $
    (3.20)

    wherein $ -\beta ^{2} +\lambda ^{2} \alpha _{2} \ne 0 $.

    We get explicit solutions of Eq (1.2) as:

    $ u(ξ)=11λβ28λ3α212(β2+λ2α2)λ{D1cos(ξλ)D2sin(ξλ)}2{D1sin(ξλ)+D2cos(ξλ)+βλ}2+β{D1sin(ξλ)+D2cos(ξλ)+βλ},
    $
    (3.21)
    Figure 3.  3D, contour and 2D surfaces of Eq (3.21) when $ \lambda = 3, D_{1} = 0.8 , D_{2} = 0.5, \beta = 3 $.

    wherein $ \xi = x-\frac{\lambda ^{2} t (\beta ^{4} +28 \lambda ^{2} \beta ^{2} \alpha _{2} +16 \lambda ^{4} \alpha _{2}^{2})}{144 (-\beta ^{2} +\lambda ^{2} \alpha _{2})^{2} } $ and $ \alpha _{2} = D_{1} ^{2} +D_{2} ^{2} $.

    Category 3: For $ \lambda = 0, $ (i.e.rational functions),

    According to Postulate 5, if we execute as the category 1, we attain the values of $ a_{0} $, $ a_{1} $, $ a_{2} $, $ b_{1} $, $ b_{2} $, $ \beta $ and $ c $ as the following results:

    $ a0=β24(D21+2βD2),a1=0,a2=1,b1=β,b2=0,c=5β416(D21+2βD)2,β=β.
    $
    (3.22)

    We get explicit solutions of Eq (1.2) as:

    $ u(ξ)=β24(D21+2βD2)(βξ+D1)2(β2ξ2+D1ξ+D2)2+β(β2ξ2+D1ξ+D2),
    $
    (3.23)

    wherein $ \xi = x-\frac{5 \beta ^{4} t}{16 (-D_{1} ^{2} +2 \beta D)^{2} } $ and $ -D_{1} ^{2} +2 \beta D\ne 0 $.

    If we set up the particular values of the arbitrary constants if we choose $ D_{1} $, $ D_{2} $ and $ \beta $in the above Eq (3.17), Eq (3.19), Eq (3.21) and Eq (3.23), we attain abundant new explicit wave solutions of KK equation which are unexposed for minimalism of length of the paper.

    We obtained new explicit solutions for the (1+1)-dimensional KK equation. We achieved solitary wave solutions for analogous traveling wave solutions of Eq (1.2). These affluent solutions including bell and anti-bell solitons, kink and anti-kink solitons, periodic and rational functions of KK equation indicate that double $ (G'/G, 1/G) $-expansion technique is more powerful than the method of $ (G'/G, 1/G) $-expansion. Comparing the solutions with the ones in [33], we presume that all the solutions are renewed which are un-indicted elsewhere. Our mentioned method is more powerful and also an offering method to demonstrate many higher order nonlinear PDEs. We will investigate the applicability of the method to (2+1)-dimensional KK equation in a future extension of the present work.

    The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


    Acknowledgments



    Initial research on this topic was supported by the Natural Sciences and Research Council of Canada.

    Conflicts of interest



    The author has no conflicts of interest.

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