Citation: Teresa M. Fonovich. Phospholipid synthetic and turnover pathways elicited upon exposure to different xenobiotics[J]. AIMS Molecular Science, 2020, 7(3): 211-228. doi: 10.3934/molsci.2020010
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Let p be a prime, f be a polynomial with k variable and Fp=Z/(p) be the finite field, where Z is the integer ring, and let
N(f;p)=#{(x1,x2,⋯,xk)∈Fkp|f(x1,x2,⋯,xk)=0}. |
Many scholars studied the exact formula (including upper bound and lower bound) for N(f;p) for many years, it is one of the main topics in the finite field theory, the most elementary upper bounds was given as follows (see [14])
N(f;p)≤pk−1degf. |
Let ordp denote the p-adic additive valuation normalized such that ordpp=1. The famous Chevalley-Warning theorem shows that ordpN(f;p)>0 if n>degf. Let [x] denote the least integer ≥x and e denote the extension degree of Fq/Fp. Ax (see [2]) showed that
ordpN(f;q)≥e[n−degfdegf]. |
In 1977, S. Chowla et al. (see [7]) investigated a problem about the number of solutions of a equation in finite field Fp as follow,
x31+x32+⋅⋅⋅+x3k≡0, |
where p is a prime with p≡1mod 3 and xi∈Fp, 1≤i≤k.
Let Mk denotes the number of solutions of the above equation. They proved that
M3=p2+d(p−1),M4=p2+6(p2−p),∞∑s=1Msxs=x1−px+x2(p−1)(2+dx)1−3px2−pdx3, |
where d is uniquely determined by 4p=d2+27y2 and d≡1 mod 3.
Myerson [12] extended the result in [2] to the field Fq and first studied the following equation over Fq,
x31+x32+⋅⋅⋅+x3k≡0. |
Recently J. Zhao et al. (see [17]) investigated the following equations over field Fp,
f1=x41+x42+x43,f2=x41+x42+x43+x44. |
And they give exact value of N(f1;p) and N(f2;p). For more general problem about this issue interested reader can see [6,9,10,11].
In this paper, let A(k,p) denotes the number of solutions of the following equation in Fp,
x61+x62+⋅⋅⋅+x6k≡0, |
where p is a prime with p≡1mod 3 and xi∈Fp, 1≤i≤k, and for simplicity, in the rest of this paper, we assume there exists an integer z such that z3≡2mod p, we use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid a lot of complicated calculations. We proved the following:
Theorem 1. For any positive integer k≥1, we have the recurrence formula
A(k+6,p)=5pA(k+4,p)+10dpA(k+3,p)+(46p2+5d2p+dp)A(k+2,p)+(2p2+120dp2+3d3p+d2p+dp)A(k+1,p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)A(k,p)+pk+5−pk+4−(10dp+2d2)pk+3−64pk+5/2−(429+121d+5d2)pk+2−2d2pk+3/2−(3d3+130d2+12d+6)pk+1−d4pk, |
with the initial condition
A(1,p)=1,A(2,p)=4(p−1)+p,A(3,p)=10d(p−1)+p2,A(4,p)=56p(p−1)+10d2(p−1)+p3,A(5,p)=188dp(p−1)+5d3(p−1)+16dC(p)(p−1)+p4,A(6,p)=p5+1400p2(p−1)+(388d2+8d−576)p(p−1)+d2p−d2, |
where d is uniquely determined by 4p=d2+27y2 and d≡1 mod 3, and C(p)=p∑a=1ep(a3).
Remark. Our method is suitable to calculus the number of solutions of the following equation in Fp,
xt1+xt2+⋅⋅⋅+xtk≡0, |
where p satisfied a certain congruence conditions, and t is any nature number.
Our Theorem 2 can be deduced from Theorem 1 and the theory of the Difference equations.
Theorem 2. Let ti (1≤i≤k) be the real root of the below equation with multiplicity si (1≤i≤k) respectively, and ρje±iwj (1≤j≤h) be the complex root of the below equation with multiplicity rj (1≤j≤h) respectively,
x6=5px4+10dpx3+(46p2+5d2p+dp)x2+(2p2+120dp2+3d3p+d2p+dp)x+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p). |
We have
A(n,p)=k∑i=1si∑a=1Ciansi−atni+h∑j=1rj∑b=1Djbnrj−bρnjcosnwj+h∑j=1rj∑b=1Ejbnrj−bρnjsinnwj, |
where Cia,Djb,Ejb, are determined by
A(6,p)=k∑i=1si∑a=1Cia6si−at6i+h∑j=1rj∑b=1Djb6rj−bρ6jcos6wj+h∑j=1rj∑b=1Ejb6rj−bρ6jsin6wj,A(5,p)=k∑i=1si∑a=1Cia5si−at5i+h∑j=1rj∑b=1Djb5rj−bρ5jcos5wj+h∑j=1rj∑b=1Ejb5rj−bρ5jsin5wj,A(4,p)=k∑i=1si∑a=1Cia4si−at4i+h∑j=1rj∑b=1Djb4rj−bρ4jcos4wj+h∑j=1rj∑b=1Ejb4rj−bρ4jsin4wj,A(3,p)=k∑i=1si∑a=1Cia3si−at3i+h∑j=1rj∑b=1Djb3rj−bρ3jcos3wj+h∑j=1rj∑b=1Ejb3rj−bρ3jsin3wj,A(2,p)=k∑i=1si∑a=1Cia2si−at2i+h∑j=1rj∑b=1Djb2rj−bρ2jcos2wj+h∑j=1rj∑b=1Ejb2rj−bρ2jsin2wj,A(1,p)=k∑i=1si∑a=1Ciati+h∑j=1rj∑b=1Djbρjcoswj+h∑j=1rj∑b=1Ejbρjsinwj. | (1.1) |
Before we prove these lemmas, we give some notations, χ2 denotes the second-order character of Fp, χ denotes the third-order character of Fp, ψ denotes the sixth order character of Fp.
ep(x)=e2πixp,τ(χ)=p∑m=1χ(m)ep(m),G(χ,m)=p∑a=1χ(a)ep(am). |
We call G(χ,m) the Gauss sum, and we have the following:
G(χ,m)=τ(χ)¯χ(m),(m,p)=1. | (2.1) |
And also we have
|τ(χ)|=√p, | (2.2) |
where χ is a primitive character of Fp. And let G(m,6;p)=p−1∑a=0ep(ma6). For the property of the exponential sum and the general Gauss sum, interested readers can see [1,4,5,8,13,15].
Lemma 1. Let p be a prime with p≡1mod 3. Then for any third-order character χ of Fp, we have the identity
τ3(χ)+τ3(¯χ)=dp, |
where d is uniquely determined by 4p=d2+27y2 and d≡1mod 3.
Proof. For the proof of this lemma see [3].
Lemma 2. Let χ be a third-order character of Fp with p≡1mod 3, and C(p)=τ(χ)+τ(¯χ), then C(p)=p∑a=1ep(a3).
Proof.
A=τ(χ)+τ(¯χ)=p∑a=1(1+χ(a)+¯χ(a))e(ap)=p∑a=1e(a3p). |
Lemma 3. Let p≡1mod 6, 2≡z3mod p for some z, and let χ be a third-order character of Fp, ψ be a sixth-order character of Fp, then we have the identity
τ(ψ)=τ2(χ)√p. |
Proof. This is Lemma 3 in [16].
Lemma 4. As the definition above, we have the identity
G(m,6;p)=√pχ2(m)+¯x2√pψ(m)+x2√p¯ψ(m)+¯xχ(m)+x¯χ(m), |
where (m,p)=1 and x=τ(χ).
Proof. Firstly we have the identity
1+χ2(m)+χ(m)+¯χ(m)+ψ(m)+¯ψ(m)={6, if m≡a6 mod p;0,otherwise. |
So we have
G(m,6;p)=p−1∑a=0(1+χ2(a)+χ(a)+¯χ(a)+ψ(a)+¯ψ(a))ep(ma)=G(χ2,m)+G(ψ,m)+G(¯ψ,m)+G(χ,m)+G(¯χ,m) |
By (2.1) and Lemma 3, we have
G(m,6;p)=τ(χ2)χ2(m)+τ(¯ψ)ψ(m)+τ(ψ)¯ψ(m)+τ(¯χ)χ(m)+τ(χ)¯χ(m)=√pχ2(m)+¯x2√pψ(m)+x2√p¯ψ(m)+¯xχ(m)+x¯χ(m). | (2.3) |
By (2.3), we complete the proof of our lemma.
Next we let,
Gn(m,6;p)=an+bnχ2(m)+cnψ(m)+dn¯ψ(m)+enχ(m)+fn¯χ(m). | (2.4) |
Then we have following Lemma 5.
Lemma 5. Let an,bn,cn,dn,en,fn are defined as above, then we have that an,bn,cn,dn,en,fn are uniquely determined by n, where n≥1.
Proof. By the orthogonality of characters of Fp, we have
p−1∑a=1χ(a)={p−1, if χ=χ0;0,otherwise. | (2.5) |
By (2.4) and (2.5) we have
p−1∑m=1Gn(m,6;p)=(p−1)an+bnp−1∑m=1χ2(m)+cnp−1∑m=1ψ(m)+dnp−1∑m=1¯ψ(m)+enp−1∑m=1χ(m)+fnp−1∑m=1¯χ(m)=(p−1)an. |
So we have
an=1p−1p−1∑m=1Gn(m,6;p). | (2.6) |
By the same method, we have
bn=1p−1p−1∑m=1χ2(m)Gn(m,6;p),cn=1p−1p−1∑m=1¯ψ(m)Gn(m,6;p),dn=1p−1p−1∑m=1ψ(m)Gn(m,6;p),en=1p−1p−1∑m=1¯χ(m)Gn(m,6;p),fn=1p−1p−1∑m=1χ(m)Gn(m,6;p). |
So now it is easy to see the conclusion of the lemma.
Lemma 6. The sequences {an}, {bn}, {cn}, {dn}, {en}, {fn} are defined above, then they satisfied the following recurrence formulae (n≥0):
an+1=√pbn+¯x2√pdn+x2√pcn+xen+¯xfn, | (2.7) |
bn+1=√pan+¯x2√pen+x2√pfn+xdn+¯xcn, | (2.8) |
cn+1=√pfn+¯x2√pan+x2√pen+xbn+¯xdn, | (2.9) |
dn+1=√pen+¯x2√pfn+x2√pan+xcn+¯xbn, | (2.10) |
en+1=√pdn+¯x2√pcn+x2√pbn+xfn+¯xan, | (2.11) |
fn+1=√pcn+¯x2√pbn+x2√pdn+xan+¯xen, | (2.12) |
with the initial condition
a0=1,b0=c0=d0=e0=f0=0. |
Proof. We only prove (2.7), the rest can be proved in the same way. By Lemma 5, we know an is unique determined by n. We can compare the coefficient of the equation
Gn+1(m,6;p)=Gn(m,6;p)G(m,6;p). |
We have
an+1=√pbn+¯x2√pdn+x2√pcn+xen+¯xfn. |
So we complete the proof of the lemma.
Lemma 7. Let an is defined as above, then we have
a0=1,a1=0,a2=5p,a3=10dp,a4=56p2+10d2p,a5=188dp2+5d3p+16dpC(p). |
Proof. By Lemma 4 and after some elementary calculations we have
G2(m,6;p)=5p+2dp1/2χ2(m)+4p1/2xψ(m)+4p1/2¯x¯ψ(m)+(p−1¯x4+3x2)χ(m)+(p−1x4+3¯x2)¯χ(m),G3(m,6;p)=10dp+(16p3/2+dp1/2)χ2(m)+(15p¯x+2dx2+p−1x5)χ(m)+(15px+2d¯x2+p−1¯x5)¯χ(m)+(4p−1/2x4+12p1/2¯x2+2dp1/2x)ψ(m)+(4p−1/2¯x4+12p1/2x2+2dp1/2¯x)¯ψ(m),G4(m,6;p)=60p2+9d2p+dp+48dp3/2χ2(m)+(p−2x8+17¯x4+46px2+16dp)χ(m)+(p−2¯x8+17x4+46p¯x2+16dp)¯χ(m)+(56p3/2x+4dp−1/2x4+12dp1/2¯x2+8p−1/2¯x5)ψ(m)+(56p3/2¯x+4dp−1/2¯x4+12dp1/2x2+8p−1/2x5)¯ψ(m),G5(m,6;p)=188dp2+5d3p+16dpC(p)+(52d2p3/2+208p5/2+16dp1/2(x2+¯x2))χ2(m)+(p−2/5x10+p−3/2¯x8+4dp−1/2¯x5+71p1/2x4+(46p3/2+16p1/2)x2+(129p3/2+10d2p1/2)¯x2+60dp3/2x+16dp3/2)ψ(m)+(p−2/5¯x10+p−3/2x8+4dp−1/2x5+71p1/2¯x4+(46p3/2+16p1/2)¯x2+(129p3/2+10d2p1/2)x2+60dp3/2¯x+16dp3/2)¯ψ(m)+(8p−1¯x7+p−1x7+25x5+52dpx2+(28dp+46p2)x+16d¯x4+112p2¯x)χ(m)+(8p−1x7+p−1¯x7+25¯x5+52dp¯x2+(28dp+46p2)¯x+16dx4+112p2x)¯χ(m), |
and comparing the above formulae with (2.6), we have
a0=1,a1=0,a2=5p,a3=10dp,a4=60p2+9d2p+dp,a5=188dp2+5d3p+16dpC(p). |
Lemma 8. Let an, bn, cn, dn, en, fn are defined as above, then we have
a6=5pa4+10dpa3+(46p2+5d2p+dp)a2+(2p2+120dp2+3d3p+d2p+dp)a1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)a0b6=5pb4+10dpb3+(46p2+5d2p+dp)b2+(2p2+120dp2+3d3p+d2p+dp)b1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)b0c6=5pc4+10dpc3+(46p2+5d2p+dp)c2+(2p2+120dp2+3d3p+d2p+dp)c1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)c0d6=5pd4+10dpd3+(46p2+5d2p+dp)d2+(2p2+120dp2+3d3p+d2p+dp)d1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)d0e6=5pe4+10dpe3+(46p2+5d2p+dp)e2+(2p2+120dp2+3d3p+d2p+dp)e1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)e0f6=5pf4+10dpf3+(46p2+5d2p+dp)f2+(2p2+120dp2+3d3p+d2p+dp)f1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)f0 |
Proof. We only proof the first formula, the rest can be proof in the same way. By Lemma 6, we have
a6=√pb5+¯x2√pd5+x2√pc5+xe5+¯xf5=5pa4+2dp1/2b4+4p1/2¯xc4+4p1/2xd4+(3¯x2+p−1x4)e4+(3x2+p−1¯x4)f4=5pa4+10dpa3+(d2p1/2+12p3/2)b3+(2dp1/2¯x+8p1/2x2+p−1/2¯x4)c3+(2dp1/2x+8p1/2¯x2+p−1/2x4)d3+(11px+¯x2+p−1¯x5)e3+(11p¯x+x2+p−1x5)f3=5pa4+10dpa3+(46p2+5d2p+dp)a2+(25dp3/2+2p3/2)b2+(p−3/2¯x7+2p−1/2x5+p−1/2¯x4+42p3/2¯x+2dp1/2x2+(d2+1)p1/2¯x)c2+(p−3/2x7+2p−1/2¯x5+p−1/2x4+42p3/2x+2dp1/2¯x2+(d2+1)p1/2x)d2+(10x4+(32p+d2)¯x2+(4dp+p)x)e2+(10¯x4+(32p+d2)x2+(4dp+p)¯x)f2=5pa4+10dpa3+(46p2+5d2p+dp)a2+(2p2+120dp2+3d3p+d2p+dp)a1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)a0. |
So we complete the proof of this lemma.
Lemma 9. Let an is defined as above, then for any integer n≥0, we have
an+6=5pan+4+10dpan+3+(46p2+5d2p+dp)an+2+(2p2+120dp2+3d3p+d2p+dp)an+1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)an. |
Proof. By (2.4) and Lemma 8, we have
G6(m,6;p)=5pG4(m,6;p)+10dpG3(m,6;p)+(46p2+5d2p+dp)G2(m,6;p)+(2p2+120dp2+3d3p+d2p+dp)G(m,6;p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p). |
We multiple Gn(m,6;p) to the both side of the above formula, we have
Gn+6(m,6;p)=5pGn+4(m,6;p)+10dpGn+3(m,6;p)+(46p2+5d2p+dp)Gn+2(m,6;p)+(2p2+120dp2+3d3p+d2p+dp)Gn+1(m,6;p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)Gn(m,6;p). |
By Lemma 5, we can compare the coefficient of the above equation, we have
an+6=5pan+4+10dpan+3+(46p2+5d2p+dp)an+2+(2p2+120dp2+3d3p+d2p+dp)an+1+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)an. |
In the formula below, we always let k≥1. By the following formula,
p−1∑a=0ep(ma)={p, if p∣m;0, otherwise, |
we have
A(k,p)=1pp−1∑m=0p−1∑x1=0,x2=0,⋅⋅⋅,xk=0ep(m(x61+x62+⋅⋅⋅+x6k))=1pp−1∑m=0Gk(m,6;p). | (3.1) |
By (8), we have
A(k,p)=1pp−1∑m=0Gk(m,6;p)=1p(p−1∑m=1Gk(m,6;p)+pk)=1p((p−1)ak+pk)=p−1pak+pk−1. | (3.2) |
So by Lemma 9, we have
A(k+6,p)−pk+5=5p(A(k+4,p)−pk+3)+10dp(A(k+3,p)−pk+2)+(46p2+5d2p+dp)(A(k+2,p)−pk+1)+(2p2+120dp2+3d3p+d2p+dp)(A(k+1,p)−pk)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)(A(k,p)−pk−1). |
So we have
A(k+6,p)=5pA(k+4,p)+10dpA(k+3,p)+(46p2+5d2p+dp)A(k+2,p)+(2p2+120dp2+3d3p+d2p+dp)A(k+1,p)+(−4p5+2d2p4+64p7/2+381p3+2d2p5/2+(129d2+11d+6)p2+d4p)A(k,p)+pk+5−pk+4−(10dp+2d2)pk+3−64pk+5/2−(429+121d+5d2)pk+2−2d2pk+3/2−(3d3+130d2+12d+6)pk+1−d4pk. |
And by Lemma 7 and (3.2), we have the initial conditions
A(1,p)=1,A(2,p)=4(p−1)+p,A(3,p)=10d(p−1)+p2,A(4,p)=56p(p−1)+10d2(p−1)+p3,A(5,p)=188dp(p−1)+5d3(p−1)+16dC(p)(p−1)+p4.A(6,p)=p5+1400p2(p−1)+(388d2+8d−576)p(p−1)+d2p−d2. |
So we complete the proof of the theorem.
The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. And we give an expression of the number of solutions of the above equation by the root of sixth degree polynomial. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.
The author thanks to referees for very important recommendations and warnings which improved the paper.
The author declares that there is no competing interest.
[1] |
Lou HY, Zhao W, Li X, et al. (2019) Membrane curvature underlies actin reorganization in response to nanoscale surface topography. Procced Natl Acad Sci 116: 23143-23151. doi: 10.1073/pnas.1910166116
![]() |
[2] |
Jensen MO, Mouritsen OG (2004) Lipids do influence protein function—the hydrophobic matching hypothesis revisited. Biochim Biophys Acta 1666: 205-226. doi: 10.1016/j.bbamem.2004.06.009
![]() |
[3] |
Dawaliby R, Trubbia C, Delporte C, et al. (2015) Phosphatidylethanolamine Is a Key Regulator of Membrane Fluidity in Eukaryotic Cells. J Biol Chem 291: 3658-3667. doi: 10.1074/jbc.M115.706523
![]() |
[4] |
Bieberich E (2018) Sphingolipids and lipid rafts: Novel concepts and methods of analysis. Chem Phys Lipids 216: 114-131. doi: 10.1016/j.chemphyslip.2018.08.003
![]() |
[5] |
Sezgin E, Levental I, Mayor S, et al. (2017) The mistery of membrane organization: composition, regulation and physiological relevance of lipid rafts. Nat Rev Mol Cell Biol 18: 361-374. doi: 10.1038/nrm.2017.16
![]() |
[6] |
Athenstaedt K, Daum G (1999) Phosphatidic acid, a key intermediate in lipid metabolism. Eur J Biochem 266: 1-16. doi: 10.1046/j.1432-1327.1999.00822.x
![]() |
[7] |
Bernat P, Gajewska E, Szewczyk R, et al. (2014) Tributyltin (TBT) induces oxidative stress and modifies lipid profile in the filamentous fungus Cunninghamella elegans. Environ Sci Pollut Res 21: 4228-4235. doi: 10.1007/s11356-013-2375-5
![]() |
[8] |
Voelker DR (2003) New perspectives on the regulation of intermembrane glycerophospholipid traffic. J Lipid Res 44: 441-449. doi: 10.1194/jlr.R200020-JLR200
![]() |
[9] |
Carman GM, Han GS (2018) Phosphatidate phosphatase regulates membrane phospholipid synthesis via phosphatidylserine synthase. Adv Biol Regul 67: 49-58. doi: 10.1016/j.jbior.2017.08.001
![]() |
[10] |
Zhang P, Csaki LS, Ronquillo E, et al. (2019) Lipin 2/3 phosphatidic acid phosphatases maintain phospholipid homeostasis to regulate chylomicron synthesis. J Clin Invest 129: 281-295. doi: 10.1172/JCI122595
![]() |
[11] |
Fonovich T, Magnarelli G (2013) Phosphoinositide and phospholipid phosphorylation and hydrolysis pathways – Organophosphate and organochlorine pesticides effects. Adv Biol Chem 3: 22-35. doi: 10.4236/abc.2013.33A004
![]() |
[12] | Fonovich de Schroeder TM, Pechén de D'Angelo AM (1991) Dieldrin effects on phospholipid metabolism in Buffo arenarum oocytes. Comp Biochem Physiol 98C: 287-292. |
[13] | Fonovich de Schroeder TM, Pechén de D'Angelo AM (1995) Dieldrin modifies the hydrolysis of PIP2 and decreases the fertilization rate in Buffo arenarum oocytes. Comp Biochem Physiol 112C: 61-67. |
[14] |
Nishio K, Sugimoto Y, Fujiwara Y, et al. (1992) Phospholipase C-mediated hydrolysis of phosphatidylcholine is activated by cis-diamminedichloroplatinum (II). J Clin Invest 89: 1622-1628. doi: 10.1172/JCI115758
![]() |
[15] |
Nakamura Y, Awai K, Masuda T, et al. (2005) A novel phosphatidylcholine-hydrolyzing phospholipase C induced by phosphate starvation in Arabidopsis. J Biol Chem 280: 7469-7476. doi: 10.1074/jbc.M408799200
![]() |
[16] |
Cruz-Ramírez A, Oropeza-Aburto A, Razo-Hernández F, et al. (2006) Phospholipase DZ2 plays an important role in extraplastidic galactolipid biosynthesis and phosphate recycling in Arabidopsis roots. Proc Natl Acad Sci USA 103: 6765-6770. doi: 10.1073/pnas.0600863103
![]() |
[17] |
Zavaleta-Pastor M, Sohlenkamp C, Gao JL, et al. (2010) Sinorhizobium meliloti phospholipase C required for lipid remodeling during phosphorus limitation. Proc Natl Acad Sci 107: 302-307. doi: 10.1073/pnas.0912930107
![]() |
[18] |
Billah MM, Anthes JM (1990) The regulation and cellular functions of phosphatidylcholine hydrolysis. Biochem J 269: 281-291. doi: 10.1042/bj2690281
![]() |
[19] |
Richmond GS, Smith TK (2011) Phospholipases A1. Int J Mol Sci 12: 588-612. doi: 10.3390/ijms12010588
![]() |
[20] |
Köhler GA, Brenot A, Haas-Stapleton E, et al. (2006) Phospholipase A2 and Phospholipase B Activities in Fungi. Biochim Biophys Acta 1761: 1391-1399. doi: 10.1016/j.bbalip.2006.09.011
![]() |
[21] |
Fonovich de Schroeder TM, Pechén de D'Angelo AM (2000) The turnover of phospholipid fatty acyl chains is activated by the insecticide Dieldrin in Buffo arenarum oocytes. J Biochem Molec Toxicol 14: 82-87. doi: 10.1002/(SICI)1099-0461(2000)14:2<82::AID-JBT3>3.0.CO;2-0
![]() |
[22] |
Wocławek-Potocka I, Rawińska P, Kowalczyk-Zieba I, et al. (2014) Lysophosphatidic Acid (LPA) Signaling in Human and Ruminant Reproductive Tract. Mediators Inflamm 2014: 1-14. doi: 10.1155/2014/649702
![]() |
[23] |
Ye X, Chun J (2010) Lysophosphatidic Acid (LPA) Signaling in Vertebrate Reproduction. Trends Endocrinol Metab 21: 1-17. doi: 10.1016/j.tem.2009.09.006
![]() |
[24] |
Kuriyama S, Theveneau E, Benedetto A, et al. (2014) In vivo collective cell migration requires an LPAR2-dependent increase in tissue fluidity. J Cell Biol 206: 113-127. doi: 10.1083/jcb.201402093
![]() |
[25] |
Jasieniecka-Gazarkiewicz K, Lager I, Carlsson AS, et al. (2017) Acyl-CoA: Lysophosphatidylethanolamine Acyltransferase Activity Regulates Growth of Arabidopsis. Plant Physiol 174: 986-998. doi: 10.1104/pp.17.00391
![]() |
[26] |
Cullis PR, De Kruijff B (1979) Lipid polymorphism and the functional roles of lipids in biological membranes. Biochim Biophys Acta 559: 399-420. doi: 10.1016/0304-4157(79)90012-1
![]() |
[27] | Dowhan W, Bogdanov M, Mileykovskaya E (2008) Functional roles of lipids in membranes. Biochemistry of lipids, lipoproteins and membranes Canada: Elsevier, 1-35. |
[28] |
Ball WB, Neff JK, Gohil VM (2018) The role of non-bilayer phospholipids in mitochondrial structure and function. FEBS Lett 592: 1273-1290. doi: 10.1002/1873-3468.12887
![]() |
[29] |
Baker CD, Ball WB, Pryce EN, et al. (2016) Specific requirements of nonbilayer phospholipids in mitochondrial respiratory chain function and formation. Mol Biol Cell 27: 2161-2171. doi: 10.1091/mbc.E15-12-0865
![]() |
[30] |
Gasanov SE, Kim AA, Yaguzhinsky LS, et al. (2018) Non-bilayer Structures in Mitochondrial Membranes Regulate ATP Synthase Activity. Biochim Biophys Acta 1860: 586-599. doi: 10.1016/j.bbamem.2017.11.014
![]() |
[31] | Fonovich TM, Perez-Coll CS, Fridman O, et al. (2016) Phospholipid changes in Rhinella arenarum embryos under different acclimation conditions to copper. Comp Biochem Physiol Part C 189: 10-16. |
[32] |
Garay LA, Boundy-Mills KL, Germa JB (2014) Accumulation of High-Value Lipids in Single-Cell Microorganisms: A Mechanistic Approach and Future Perspectives. J Agric Food Chem 67: 2709-2727. doi: 10.1021/jf4042134
![]() |
[33] |
Welte MA (2015) Expanding roles for lipid droplets. Curr Biol 25: R470-R481. doi: 10.1016/j.cub.2015.04.004
![]() |
[34] |
Meyers A, Weiskittel TM, Dalhaimer P (2017) Lipid Droplets: Formation to Breakdown. Lipids 52: 465-475. doi: 10.1007/s11745-017-4263-0
![]() |
[35] |
Olzmann JA, Carvalho P (2019) Dynamics and functions of lipid droplets. Nat Rev Mol Cell Biol 20: 137-155. doi: 10.1038/s41580-018-0085-z
![]() |
[36] |
Li Z, Thiel K, Thul PJ, et al. (2012) Lipid droplets control the maternal histone supply of Drosophila embryos. Curr Biol 22: 2104-2113. doi: 10.1016/j.cub.2012.09.018
![]() |
[37] |
Li Z, Johnson MR, Ke Z, et al. (2014) Drosophila lipid droplets buffer the H2Av supply to protect early embryonic development. Curr Biol 24: 1485-1491. doi: 10.1016/j.cub.2014.05.022
![]() |
[38] |
Huang X, Warren JT, Gilbert LI (2008) New players in the regulation of ecdysone biosynthesis. J Genet Genomics 35: 1-10. doi: 10.1016/S1673-8527(08)60001-6
![]() |
[39] |
Herms A, Bosch M, Ariotti N, et al. (2013) Cell-to-cell Heterogeneity in Lipid Droplets Suggests a Mechanism to Reduce Lipotoxicity. Curr Biol 23: 1489-1496. doi: 10.1016/j.cub.2013.06.032
![]() |
[40] |
Grygiel-Górniak B (2014) Peroxisome Proliferator-Activated Receptors and Their Ligands: Nutritional and Clinical Implications. A Review. Nutr J 13: 17-26. doi: 10.1186/1475-2891-13-17
![]() |
[41] |
Poursharifi P, Madiraju SRM, Prentki M (2017) Monoacylglycerol Signalling and ABHD6 in Health and Disease Diabetes. Obes Metab 19: 76-89. doi: 10.1111/dom.13008
![]() |
[42] |
Walker OLlS, Holloway AC, Raha S (2019) The role of the endocannabinoid system in female reproductive tissues. J Ovarian Res 12: 3-12. doi: 10.1186/s13048-018-0478-9
![]() |
[43] |
Fan C, Yan J, Qian Y, et al. (2006) Regulation of Lipoprotein Lipase Expression by Effect of Hawthorn Flavonoids on Peroxisome Proliferator Response Element Pathway. J Pharmacol Sci 100: 51-58. doi: 10.1254/jphs.FP0050748
![]() |
[44] |
Rotman N, Guex N, Gouranton E, et al. (2013) PPARβ interprets a chromatin signature of pluripotency to promote embryonic differentiation at gastrulation. PLoS One 8: e83300. doi: 10.1371/journal.pone.0083300
![]() |
[45] | Michalik L, Desvergne B, Dreyer C, et al. (2002) PPAR expression and function during vertebrate development. Int J Dev Biol 46: 105-114. |
[46] | Fonovich de Schroeder TM (1993) Efecto del Dieldrin sobre la transducción de señales en ovocitos de sapo Bufo arenarum, Hensel. PhD thesis. Pharmacy and Biochemistry Faculty. Buenos Aires University 1-181. |
[47] | Fonovich de Schroeder TM (1997) Pretreatment ofamphibian oocytes with the organochlorinated pesticide Dieldrin facilitates the formation of the fertilization membrane after insemination. Acta Toxicol Arg 5: 81-83. |
[48] |
Wozniak KL, Tembo M, Phelps WA, et al. (2018) PLC and IP 3-evoked Ca2+ Release Initiate the Fast Block to Polyspermy in Xenopus laevis Eggs. J Gen Physiol 150: 1239-1248. doi: 10.1085/jgp.201812069
![]() |
[49] |
Fonovich de Schroeder TM, Pechén de D'Angelo AM (1995) The effect of Dieldrin on Clostridium perfringens posphatidylcholine phospholipase C activity. Pest Biochem Physiol 51: 170-177. doi: 10.1006/pest.1995.1017
![]() |
[50] |
Carattino MD, Peralta S, Pérez-Coll C, et al. (2004) Effects of Long-Term Exposure to Cu2+ and Cd2+ on the Pentose Phosphate Pathway Dehydrogenase Activities in the Ovary of Adult Bufo Arenarum: Possible Role as Biomarker for Cu2+ Toxicity. Ecotoxicol Environ Saf 57: 311-318. doi: 10.1016/S0147-6513(03)00081-2
![]() |
[51] | Fonovich de Schroeder TM, Preller AF, Naab F, et al. (2000) Acumulación de Zn en ovocitos de sapo Bufo arenarum: efecto sobre el metabolismo de carbohidratos. Rev Bras Toxicol 13: 55-61. |
[52] |
Naab F, Volcomirsky M, Burlón A, et al. (2001) Metabolic Alterations Without Metal Accumulation in the Ovary of Adult Bufo Arenarum Females, Observed After Long-Term Exposure to Zn(2+), Followed by Toxicity to Embryos. Arch Environ Contam Toxicol 41: 201-207. doi: 10.1007/s002440010238
![]() |
[53] |
Fonovich de Schroeder TM (2005) The effect of Zn on glucose 6-phosphate dehydrogenase activity from Bufo arenarum toad ovary and alfalfa plants. Ecotoxicol Environ Saf 60: 123-131. doi: 10.1016/j.ecoenv.2004.07.008
![]() |
[54] |
Rokitskaya TI, Kotova EA, Agapov II, et al. (2014) Unsaturated lipids protect the integral membrane peptide gramicidin A from singlet oxygen. FEBS Lett 588: 1590-1595. doi: 10.1016/j.febslet.2014.02.046
![]() |
[55] | Kim SH, Kim BK, Park S, et al. (2019) Phosphatidylcholine extends lifespan via DAF-16 and reduces Amyloid-beta-Induced toxicity in Caenorhabditis elegans. Oxid Med Cell Longev 2019: 2860642. |