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Research article

Differential expression and functional analysis of micro RNAs in Papio anubis induced with endometriosis for early detection of the disease

  • Received: 07 July 2020 Accepted: 02 September 2020 Published: 10 September 2020
  • Endometriosis is a common gynecological disorder affecting approximately 10% of women of reproductive age who often experience chronic pelvic pain and infertility. Laparoscopy, which is invasive and expensive, is the gold standard for diagnosis of endometriosis. A simple minimally-invasive test for endometriosis-specific biomarkers which is yet to be realized would offer a timely and accurate diagnosis for the disease thereby allowing early treatment intervention. Although aberrant microRNA expression has been implicated in endometriosis in several studies, conflicting results have been reported. This study hypothesized that the use of an appropriate animal model will provide a unique entry point for the discovery of biomarkers for early diagnosis of endometriosis. The study aimed at identifying miRNAs that are differentially expressed in eutopic endometrium of induced endometriosis in Papio anubis for early detection of endometriosis. Female adult baboons (n = 3) were induced with endometriosis by intraperitoneal inoculation of autologous menstrual endometrium. We sequenced small RNA samples obtained from normal (control) and diseased eutopic endometrium. Quality reads from the sequences were subjected to differential expression analysis to identify dysregulated microRNAs and genes from other non-coding small RNA in the samples using a bioinformatics approach. Through in-silico analysis, gene targets of the dysregulated miRNA and their functions were determined. Our findings show significant high expression of seven microRNAs namely miR-199a-3p, miR-145-5p, miR-214-3p, miR-143-3p, miR-125b-5p, miR-199a-5p and miR-10b-5p. The study also reveals five microRNAs that were significantly down regulated and they include miR-29b-3p, miR-16-5p, miR-342-3p, miR-378a-3p and let-7g-5p. Seventeen genes from non-coding small RNAs were significantly dysregulated. The dysregulated microRNAs and genes play important roles in pathogenesis of endometriosis. Our findings indicate that specific miRNA signatures are associated with endometriosis, and the dysregulated miRNAs could constitute new and informative biomarkers for early diagnosis of endometriosis.

    Citation: Irene Mwongeli Waita, Atunga Nyachieo, Daniel Chai, Samson Muuo, Naomi Maina, Daniel Kariuki, Cleophas M. Kyama. Differential expression and functional analysis of micro RNAs in Papio anubis induced with endometriosis for early detection of the disease[J]. AIMS Molecular Science, 2020, 7(4): 305-327. doi: 10.3934/molsci.2020015

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  • Endometriosis is a common gynecological disorder affecting approximately 10% of women of reproductive age who often experience chronic pelvic pain and infertility. Laparoscopy, which is invasive and expensive, is the gold standard for diagnosis of endometriosis. A simple minimally-invasive test for endometriosis-specific biomarkers which is yet to be realized would offer a timely and accurate diagnosis for the disease thereby allowing early treatment intervention. Although aberrant microRNA expression has been implicated in endometriosis in several studies, conflicting results have been reported. This study hypothesized that the use of an appropriate animal model will provide a unique entry point for the discovery of biomarkers for early diagnosis of endometriosis. The study aimed at identifying miRNAs that are differentially expressed in eutopic endometrium of induced endometriosis in Papio anubis for early detection of endometriosis. Female adult baboons (n = 3) were induced with endometriosis by intraperitoneal inoculation of autologous menstrual endometrium. We sequenced small RNA samples obtained from normal (control) and diseased eutopic endometrium. Quality reads from the sequences were subjected to differential expression analysis to identify dysregulated microRNAs and genes from other non-coding small RNA in the samples using a bioinformatics approach. Through in-silico analysis, gene targets of the dysregulated miRNA and their functions were determined. Our findings show significant high expression of seven microRNAs namely miR-199a-3p, miR-145-5p, miR-214-3p, miR-143-3p, miR-125b-5p, miR-199a-5p and miR-10b-5p. The study also reveals five microRNAs that were significantly down regulated and they include miR-29b-3p, miR-16-5p, miR-342-3p, miR-378a-3p and let-7g-5p. Seventeen genes from non-coding small RNAs were significantly dysregulated. The dysregulated microRNAs and genes play important roles in pathogenesis of endometriosis. Our findings indicate that specific miRNA signatures are associated with endometriosis, and the dysregulated miRNAs could constitute new and informative biomarkers for early diagnosis of endometriosis.


    Several years ago, Bensoussan, Sethi, Vickson and Derzko [1] have been considered the case of a factory producing one type of economic goods and observed that it is necessary to solve the simple partial differential equation

    {σ22Δzαs+14|zαs|2+αzαs=|x|2forxRN,zαs=as|x|, (1.1)

    where σ(0,) denotes the diffusion coefficient, α[0,) represents psychological rate of time discount, xRN is the product vector, z:=zαs(x) denotes the value function and |x|2 is the loss function.

    Regime switching refers to the situation when the characteristics of the state process are affected by several regimes (e.g., in finance bull and bear market with higher volatility in the bear market).

    It is important to point out that, when dealing with regime switching, we can describe a wide variety of phenomena using partial differential equations. In [1], the authors Cadenillas, Lakner and Pinedo [2] adapted the model problem in [1] to study the optimal production management characterized by the two-state regime switching with limited/unlimited information and corresponding to the system

    {σ212Δus1+(a11+α1)us1a11us2ρσ212ij2us1xixj|x|2=14|us1|2,xRN,σ222Δus2+(a22+α2)us2a22us1ρσ222ij2us2xixj|x|2=14|us2|2,xRN,us1(x)=us2(x)=as|x|, (1.2)

    where σ1,σ2(0,) denote the diffusion coefficients, α1,α2[0,) represent the psychological rates of time discount from what place the exponential discounting, xRN is the product vector, usr:=usr(x) (r=1,2) denotes the value functions, |x|2 is the loss function, ρ[1,1] is the correlation coefficient and anm (n,m=1,2) are the elements of the Markov chain's rate matrix, denoted by G=[ϑnm]2×2 with

    ϑnn=ann0,ϑnm=anm0andϑ2nn+ϑ2nm0fornm,

    the diagonal elements ϑnn may be expressed as ϑnn=Σmnϑnm.

    Furthermore, in civil engineering, Dong, Malikopoulos, Djouadi and Kuruganti [3] applied the model described in [2] to the study of the optimal stochastic control problem for home energy systems with solar and energy storage devices; the two regimes switching are the peak and the peak energy demands.

    After that, there have been numerous applications of regime switching in many important problems in economics and other fields, see the works of: Capponi and Figueroa-López [4], Elliott and Hamada [5], Gharbi and Kenne [6], Yao, Zhang and Zhou [7] and Wang, Chang and Fang [8] for more details. Other different research studies that explain the importance of regime switching in the real world are [9,10].

    In this paper, we focus on the following parabolic partial differential equation and system, corresponding to (1.1)

    {zt(x,t)σ22Δz(x,t)+14|z(x,t)|2+αz(x,t)=|x|2,(x,t)RN×(0,),z(x,0)=c+zαs(x),forallxRNandfixedc(0,),z(x,t)=as|x|,forallt[0,), (1.3)

    and (1.2) respectively

    {u1tσ212Δu1+(a11+α1)u1a11u2ρσ212ij2u1xixj|x|2=14|u1|2,(x,t)RN×(0,),u2tσ222Δu2+(a22+α2)u2a22u1ρσ222ij2u2xixj|x|2=14|u2|2,(x,t)RN×(0,),(u1(x,0),u2(x,0))=(c1+us1(x),c2+us2(x))forallxRNandforfixedc1,c2(0,),u1t(x,t)=u2t(x,t)=as|x|forallt[0,), (1.4)

    where zαs is the solution of (1.1) and (us1(x),us2(x)) is the solution of (1.2). The existence and the uniqueness for the case of (1.1) is proved by [10] and the existence for the system case of (1.2) by [11].

    From the mathematical point of view the problem (1.3) has been extensively studied when the space RN is replaced by a bounded domain and when α=0. In particular, some great results can be found in the old papers of Barles, Porretta [12] and Tchamba [13]. More recently, but again for the case of a bounded domain, α=0 and in the absence of the gradient term, the problem (1.3) has been also discussed by Alves and Boudjeriou [14]. The interest of these authors [12,13,14] is to give an asymptotic stable solution at infinity for the considered equation, i.e., a solution which tends to the stationary Dirichlet problem associated with (1.3) when the time go to infinity.

    Next, we propose to find a similar result as of [12,13,14], for the case of equation (1.3) and system (1.4) that model some real phenomena. More that, our first interest is to provide a closed form solution for (1.3) and (1.4). Our second objective is inspired by the paper of [14,15], and it is to solve the parabolic partial differential equation

    {zt(x,t)σ22Δz(x,t)+14|z(x,t)|2=|x|2,inBR×[0,T),z(x,T)=0,for|x|=R, (1.5)

    where T< and BR is a ball of radius R>0 with origin at the center of RN.

    Let us finish our introduction and start with the main results.

    We use the change of variable

    u(x,t)=ez(x,t)2σ2, (2.1)

    in

    zt(x,t)σ22Δz(x,t)+14|z(x,t)|2+αz(x,t)=|x|2

    to rewrite (1.3) and (1.5) in an equivalent form

    {ut(x,t)σ22Δu(x,t)+αu(x,t)lnu(x,t)+12σ2|x|2u(x,t)=0,if(x,t)Ω×(0,T)u(x,T)=u1,0,onΩ,u(x,0)=ec+zαs(x)2σ2,forxΩ=RN,c(0,) (2.2)

    where

    u1,0={1ifΩ=BR,i.e.,|x|=R,T<,0ifΩ=RN,i.e.,|x|,T=.

    Our first result is the following.

    Theorem 2.1. Assume Ω=BR, N3, T< and α=0.There exists a unique radially symmetric positive solution

    u(x,t)C2(BR×[0,T))C(¯BR×[0,T]),

    of (2.2) increasing in the time variable and such that

    limtTu(x,t)=us(x), (2.3)

    where usC2(BR)C(¯BR) is the unique positive radially symmetric solution of theDirichlet problem

    {σ22Δus=(12σ2|x|2+1)us,inBR,us=1,onBR, (2.4)

    which will be proved. In addition,

    z(x,t)=2σ2(tT)2σ2lnus(|x|),(x,t)¯BR×[0,T],

    is the unique radially symmetric solution of the problem (1.5).

    Instead of the existence results discussed in the papers of [12,13,14], in our proof of the Theorem 2.1 we give the numerical approximation of solution u(x,t).

    The next results refer to the entire Euclidean space RN and present closed-form solutions.

    Theorem 2.2. Assume Ω=RN, N1, T=, α>0 and c(0,) is fixed. There exists aunique radially symmetric solution

    u(x,t)C2(RN×[0,)),

    of (2.2), increasing in the time variable and such that

    u(x,t)uαs(x)astforallxRN, (2.5)

    where uαsC2(RN) is the uniqueradially symmetric solution of the stationary Dirichlet problem associatedwith (2.2)

    {σ22Δuαs=αuαslnuαs+12σ2|x|2uαs,inRN,uαs(x)0,as|x|. (2.6)

    Moreover, the closed-form radially symmetric solution of the problem (1.3) is

    z(x,t)=ceαt+B|x|2+D,(x,t)RN×[0,),c(0,), (2.7)

    where

    B=1Nσ2(12Nσ2α2+412Nασ2),D=12α(Nσ2α2+4Nασ2). (2.8)

    The following theorem is our main result regarding the system (1.4).

    Theorem 2.3. Suppose that N1, α1,α2(0,) and\ a11,a22[0,) with a211+a2220. Then, the system (1.4) has a uniqueradially symmetric convex solution

    (u1(x,t),u2(x,t))C2(RN×[0,))×C2(RN×[0,)),

    of quadratic form in the x variable and such that

    (u1(x,t),u2(x,t))(us1(x),us2(x))astuniformlyforallxRN, (2.9)

    where

    (us1(x),us2(x))C2(RN)×C2(RN)

    is the radially symmetric convex solution of quadratic form in the xvariable of the stationary system (1.2) which exists from the resultof [11].

    Our results complete the following four main works: Bensoussan, Sethi, Vickson and Derzko [1], Cadenillas, Lakner and Pinedo [2], Canepa, Covei and Pirvu [15] and Covei [10], which deal with a stochastic control model problem with the corresponding impact for the parabolic case (see [13,16] for details).

    To prove our Theorem 2.1, we use a lower and upper solution method and the comparison principle that can be found in [17].

    Lemma 2.1. If, there exist ¯u(x), u_(x)C2(BR)C(¯BR) two positive functions satisfying

    {σ22Δ¯u(x)+(12σ2|x|2+1)¯u(x)0σ22Δu_(x)+(12σ2|x|2+1)u_(x)inBR,¯u(x)=1=u_(x)onBR,

    then

    ¯u(x)u_(x)0forallx¯BR,

    and there exists

    u(x)C2(BR)C(¯BR),

    a solution of (2.4) such that

    u_(x)u(x)¯u(x),x¯BR,

    where u_(x) and ¯u(x) arerespectively, called a lower solution and an upper solution of (2.4).

    The corresponding result of Lemma 2.1 for the parabolic equations can be found in the work of Pao [18] and Amann [19]. To achieve our goal, complementary to the works [12,13,14,15] it can be used the well known books of Gilbarg and Trudinger [20], Sattinger [17], Pao [18] and a paper of Amann [19]. Further on, we can proceed to prove Theorem 2.1.

    By a direct calculation, if there exists and is unique, usC2(BR)C(¯BR), a positive solution of the stationary Dirichlet problem (2.4) then

    u(x,t)=etTus(x),(x,t)¯BR×[0,T],

    is the solution of the problem (2.2) and

    z(x,t)=2σ2(tT)2σ2lnus(x),(x,t)¯BR×[0,T],

    is the solution of the problem (1.5) belonging to

    C2(BR×[0,T))C(¯BR×[0,T]).

    We prove that (2.4) has a unique radially symmetric solution. The existence of solution for (2.4) is obtained by a standard monotone iteration and the lower and the upper solution method, Lemma 2.1. Hence, starting from the initial iteration

    u0s(x)=eR2|x|22σ2,

    we construct a sequence {uks(x)}k1 successively by

    {σ22Δuks(x)=(12σ2|x|2+1)uk1s(x),inBR,uks(x)=1,onBR, (3.1)

    and this sequence will be pointwise convergent to a solution us(x) of (2.4).

    Indeed, since for each k the right-hand side of (3.1) is known, the existence theory for linear elliptic boundary-value problems implies that {uks(x)}k1 is well defined, see [20].

    Let us prove that {uks(x)}k1 is a pointwise convergent sequence to a solution of (2.4) in ¯BR. To do this, first we prove that {uks(x)}k1 is monotone nondecreasing of k. We apply the mathematical induction by verifying the first step, k=1.

    {σ22Δu1s(x)σ22Δu0s(x),inBR,u1s(x)=1=u0s(x),onBR.

    Now, by the standard comparison principle, Lemma 2.1, we have

    u0s(x)u1s(x)in¯BR.

    Moreover, the induction argument yields the following

    u0s(x)=eR2|x|22σ2...uks(x)uk+1s(x)...in¯BR, (3.2)

    i.e., {uks(x)}k1 is a monotone nondecreasing sequence.

    Next, using again Lemma 2.1, we find

    u_s(x):=u0s(x)=eR2|x|22σ2...uks(x)uk+1s(x)...¯us(x):=1in¯BR, (3.3)

    where we have used

    σ22Δu_s(x)=u_s(x)σ22(|x|2+σ2σ4+N1σ2)u_s(x)(12σ2|x|2+1)σ22Δ¯us(x)=σ22Δ1=0¯us(x)(12σ2|x|2+1)

    i.e., Lemma 2.1 confirm.Thus, in view of the monotone and bounded property in (3.3) the sequence {uks(x)}k1 converges. We may pass to the limit in (3.3) to get the existence of a solution

    us(x):=limkuks(x)in¯BR,

    associated to (2.4), which satisfies

    u_s(x)us(x)¯us(x)in¯BR.

    Furthermore, the convergence of {uks(x)} is uniformly to us(x) in ¯BR and us(x) has a radial symmetry, see [15] for arguments of the proof. The regularity of solution us(x) is a consequence of classical results from the theory of elliptic equations, see Gilbarg and Trudinger [20]. The uniqueness of us(x) follows from a standard argument with the use of Lemma 2.1 and we omit the details.

    Clearly, u(x,t) is increasing in the time variable. The regularity of u(x,t) follows from the regularity of us(x). Letting tT we see that (2.3) holds. The solution of the initial problem (1.5) is saved from (2.1).

    Finally, we prove the uniqueness for (2.2). Let

    u(x,t),v(x,t)C2(BR×[0,T))C(¯BR×[0,T]),

    be two solutions of the problem (2.2), i.e., its hold

    {ut(x,t)σ22Δu(x,t)+12σ2|x|2u(x,t)=0,if(x,t)BR×[0,T),u(x,T)=1,onBR,

    and

    {vt(x,t)σ22Δv(x,t)+12σ2|x|2v(x,t)=0,if(x,t)BR×[0,T),v(x,T)=1,onBR.

    Setting

    w(x,t)=u(x,t)v(x,t),inBR×[0,T],

    and subtracting the two equations corresponding to u and v we find

    {wt(x,t)=σ22Δw(x,t)12σ2|x|2w(x,t),if(x,t)BR×[0,T),w(x,T)=0,onBR.

    Let us prove that u(x,t)v(x,t)0 in ¯BR×[0,T]. If the conclusion were false, then the maximum of

    w(x,t),inBR×[0,T),

    is positive. Assume that the maximum of w in ¯BR×[0,T] is achieved at (x0,t0). Then, at the point (x0,t0)BR×[0,T), where the maximum is attained, we have

    wt(x0,t0)0,Δw(x0,t0)0,w(x0,t0)=0,

    and

    0wt(x0,t0)=σ22Δw(x0,t0)12σ2|x|2w(x0,t0)<0

    which is a contradiction. Reversing the role of u and v we obtain that u(x,t)v(x,t)0 in ¯BR×[0,T]. Hence u(x,t)=v(x,t) in ¯BR×[0,T]. The proof of Theorem 2.1 is completed.

    Finally, our main result, Theorem 2.2 will be obtained by a direct computation.

    In view of the arguments used in the proof of Theorem 2.1 and the real world phenomena, we use a purely intuitive strategy in order to prove Theorem 2.2.

    Indeed, for the verification result in the production planning problem, we need z(x,t) to be almost quadratic with respect to the variable x.

    More exactly, we observe that there exists and is unique

    u(x,t)=eh(t)+B|x|2+D2σ2,(x,t)RN×[0,),withB,D(0,),

    that solve (2.2), where

    h(0)=c, (4.1)

    and B, D are given in (2.8). The condition (4.1) is used to obtain the asymptotic behaviour of solution to the stationary Dirichlet problem associated with (2.2). Then our strategy is reduced to find B,D(0,) and the function h which depends of time and c(0,) such that

    12h(t)σ2σ22[Bσ4(σ2B|x|2)(N1)Bσ2]+α(h(t)+B|x|2+D2σ2)+12σ2|x|2=0,

    or, after rearranging the terms

    |x|2(1αBB2)+Nσ2BαDh(t)αh(t)=0,

    where (4.1) holds. Now, by a direct calculation we see that the system of equations

    {1αBB2=0Nσ2BαD=0h(t)αh(t)=0h(0)=c

    has a unique solution that satisfies our expectations, namely,

    u(x,t)=eceαt+B|x|2+D2σ2,(x,t)RN×[0,), (4.2)

    where B and D are given in (2.8), is a radially symmetric solution of the problem (2.2). The uniqueness of the solution is followed by the arguments in [10] combined with the uniqueness proof in Theorem 2.1. The justification of the asymptotic behavior and regularity of the solution can be proved directly, once we have a closed-form solution. Finally, the closed-form solution in (2.7) is due to (2.1)–(4.2) and the proof of Theorem 2.2 is completed.

    One way of solving this system of partial differential equation of parabolic type (1.4) is to show that the system (1.4) is solvable by

    (u1(x,t),u2(x,t))=(h1(t)+β1|x|2+η1,h2(t)+β2|x|2+η2), (5.1)

    for some unique β1,β2,η1,η2(0,) and h1(t), h2(t) are suitable chosen such that

    h1(0)=c1andh2(0)=c2. (5.2)

    The main task for the proof of existence of (5.1) is performed by proving that there exist

    β1,β2,η1,η2,h1,h2,

    such that

    {h1(t)2β1Nσ212+(a11+α1)[h1(t)+β1|x|2+η1]a11[h2(t)+β2|x|2+η2]|x|2=14(2β1|x|)2,h2(t)2β2Nσ222+(a22+α2)[h2(t)+β2|x|2+η2]a22[h1(t)+β1|x|2+η1]|x|2=14(2β2|x|)2,

    or equivalently, after grouping the terms

    {|x|2[a11β2+(a11+α1)β1+β211]β1Nσ21a11η2+(a11+α1)η1+h1(t)+(a11+α1)h1(t)a11h2(t)=0,|x|2[a22β1+(a22+α2)β2+β221]β2Nσ22a22η1+(a22+α2)η2+h2(t)+(a22+α2)h2(t)a22h1(t)=0,

    where h1(t), h2(t) must satisfy (5.2). Now, we consider the system of equations

    {a11β2+(a11+α1)β1+β211=0a22β1+(a22+α2)β2+β221=0β1Nσ21a11η2+(a11+α1)η1=0β2Nσ22a22η1+(a22+α2)η2=0h1(t)+(a11+α1)h1(t)a11h2(t)=0h2(t)+(a22+α2)h2(t)a22h1(t)=0. (5.3)

    To solve (5.3), we can rearrange those equations 1, 2 in the following way

    {a11β2+(a11+α1)β1+β211=0a22β1+(a22+α2)β2+β221=0. (5.4)

    We distinguish three cases:

    1.in the case a22=0 we have an exact solution for (5.4) of the form

    β1=12α112a11+12α21+a2114a11(12α212α22+4)+2α1a11+4β2=12α2+12α22+4

    2.in the case a11=0 we have an exact solution for (5.4) of the form

    β1=12α1+12α21+4β2=12α212a22+12α22+a2224a22(12α112α21+4)+2α2a22+4

    3.in the case a110 and a220, to prove the existence and uniqueness of solution for (5.4) we will proceed as follows. We retain from the first equation of (5.4)

    β1=12α21+2α1a11+a211+4β2a11+412a1112α1.

    and from the second equation

    β2=12α22+2α2a22+a222+4β1a22+412a2212α2.

    The existence of β1, β2(0,) for (5.4) can be easily proved by observing that the continuous functions f1,f2:[0,)R defined by

    f1(β1)=a11(12α22+2α2a22+a222+4β1a22+412a2212α2)+(a11+α1)β1+β211,f2(β2)=a22(12α21+2α1a11+a211+4β2a11+412a1112α1)+(a22+α2)β2+β221,

    have the following properties

    f1()=andf2()=, (5.5)

    respectively

    f1(0)=a11(12α22+2α2a22+a222+412a2212α2)1<0,f2(0)=a22(12α21+2α1a11+a211+412a1112α1)1<0. (5.6)

    The observations (5.5) and (5.6) imply

    {f1(β1)=0f2(β2)=0

    has at least one solution (β1,β2)(0,)×(0,) and furthermore it is unique (see also, the references [21,22] for the existence and the uniqueness of solutions).

    The discussion from cases 1–3 show that the system (5.4) has a unique positive solution. Next, letting

    (β1,β2)(0,)×(0,),

    be the unique positive solution of (5.4), we observe that the equations 3, 4 of (5.3) can be written equivalently as a system of linear equations that is solvable and with a unique solution

    (a11+α1a11a22a22+α2)(η1η2)=(β1Nσ21β2Nσ22). (5.7)

    By defining

    Ga,α:=(a11+α1a11a22a22+α2),

    we observe that

    G1a,α=(α2+a22α1α2+α2a11+α1a22a11α1α2+α2a11+α1a22a22α1α2+α2a11+α1a22α1+a11α1α2+α2a11+α1a22).

    Using the fact that G1a,α has all ellements positive and rewriting (5.7) in the following way

    (η1η2)=G1a,α(β1Nσ21β2Nσ22),

    we can see that there exist and are unique η1, η2(0,) that solve (5.7). Finally, the equations 5, 6, 7 of (5.3) with initial condition (5.2) can be written equivalently as a solvable Cauchy problem for a first order system of differential equations

    {(h1(t)h2(t))+Ga,α(h1(t)h2(t))=(00),h1(0)=c1andh2(0)=c2, (5.8)

    with a unique solution and then (5.1) solve (1.4). The rest of the conclusions are easily verified.

    Next, we present an application.

    Application 1. Suppose there is one machine producing two products (see [23,24], for details). We consider a continuous time Markov chain generator

    (12121212),

    and the time-dependent production planning problem with diffusion σ1=σ2=12 and let α1=α2=12 the discount factor. Under these assumptions, we can write the system (5.4) with our data

    {β21+β112β21=0β2212β1+β21=0

    which has a unique positive solution

    β1=14(171),β2=14(171).

    On the other hand, the system (5.7) becomes

    (112121)(η1η2)=(β1β2),

    which has a unique positive solution

    η1=43β1+23β2=12(171),η2=23β1+43β2=12(171).

    Finally, the system in (5.8) becomes

    {(h1(t)h2(t))+(112121)(h1(t)h2(t))=(00),h1(0)=c1andh2(0)=c2,

    which has the solution

    h1(t)=s1e12ts2e32t,h2(t)=s1e12t+s2e32t,withs1,s2R.

    Next, from

    h1(0)=c1andh2(0)=c2,

    we have

    {s1s2=c1s1+s2=c2s1=12c1+12c2,s2=12c212c1,

    and finally

    {h1(t)=12(c1+c2)e12t12(c2c1)e32t,h2(t)=12(c1+c2)e12t+12(c2c1)e32t,

    from where we can write the unique solution of the system (1.4) in the form (5.1).

    Let us point that in Theorem 2.3 we have proved the existence and the uniqueness of a solution of quadratic form in the x variable and then the existence of other different types of solutions remain an open problem.

    Some closed-form solutions for equations and systems of parabolic type are presented. The form of the solutions is unique and tends to the solutions of the corresponding elliptic type problems that were considered.

    The author is grateful to the anonymous referees for their useful suggestions which improved the contents of this article.

    The authors declare there is no conflict of interest.


    Abbreviation DE: Differential Expression/Differentially Expressed; DR miRNAs/MTGs: Down Regulated miRNAs/MTGs; EB: Endometrial Tissue Biopsy; GO: Gene Ontology; IPR: Institute of Primate Research; KEGG: Kyoto Encyclopedia of genes and genomes; MiRNA(s): Micro RNA(s); mRNA(s): messenger RNA(s); MTGs: MicroRNA Target Genes; : ; SRA: Sequence Reads Archive; sRNA(s): small RNA(s); UR miRNAs/MTGs: Up Regulated;
    Acknowledgments



    The study was supported by Pan African University-Institute of Basic sciences, Technology and Innovation (PAUISTI) and a grant from Research, Production and Extension Division (RPE)-JKUAT.

    Conflict of interest



    The authors declare that they have no competing interests.

    [1] Giudice LC, Kao LC (2004) Endometriosis. Lancet 364: 1789-1799. doi: 10.1016/S0140-6736(04)17403-5
    [2] Panir K, Schjenken JE, Robertson SA, et al. (2018) Non-coding RNAs in endometriosis: a narrative review.24: 497-515.
    [3] Tamaresis JS, Irwin JC, Goldfien GA, et al. (2014) Molecular classification of endometriosis and disease stage using high-dimensional genomic data. Endocrinology 155: 4986-4999. doi: 10.1210/en.2014-1490
    [4] Rogers PAW, Adamson GD, Al-Jefout M, et al. (2017) Research Priorities for Endometriosis: Recommendations from a Global Consortium of Investigators in Endometriosis. Reprod Sci 24: 202-226. doi: 10.1177/1933719116654991
    [5] Berkley KJ, Rapkin AJ, Papka RE (2005) The pains of endometriosis. Science 308: 1587-1589. doi: 10.1126/science.1111445
    [6] Nnoaham KE, Hummelshoj L, Webster P, et al. (2011) Impact of endometriosis on quality of life and work productivity: A multicenter study across ten countries. Fertil Steril 96: 366-373.e8. doi: 10.1016/j.fertnstert.2011.05.090
    [7] Simoens S, Dunselman G, Dirksen C, et al. (2012) The burden of endometriosis: Costs and quality of life of women with endometriosis and treated in referral centres. Hum Reprod 27: 1292-1299. doi: 10.1093/humrep/des073
    [8] Nisenblat V, Prentice L, Bossuyt PMM, et al. (2016) Combination of the non-invasive tests for the diagnosis of endometriosis. Cochrane Database Syst Rev 7: CD012281.
    [9] ESHRE (2013) Management of Women with Endometriosis. Guidel Eur Soc Hum Reprod Embryol 1-97.
    [10] Ahn SH, Singh V, Tayade C (2017) Biomarkers in endometriosis: challenges and opportunities. Fertil Steril 107: 523-532. doi: 10.1016/j.fertnstert.2017.01.009
    [11] Bokor A, Kyama CM, Vercruysse L, et al. (2009) Density of small diameter sensory nerve fibres in endometrium: A semi-invasive diagnostic test for minimal to mild endometriosis. Hum Reprod 24: 3025-3032. doi: 10.1093/humrep/dep283
    [12] D'Hooghe TM, Kyama CM, Chai D, et al. (2009) Nonhuman primate models for translational research in endometriosis. Reprod Sci 16: 152-161. doi: 10.1177/1933719108322430
    [13] Kyama CM, Mihalyi A, Chai D, et al. (2007) Baboon model for the study of endometriosis. Women's Heal 3: 637-646. doi: 10.2217/17455057.3.5.637
    [14] Fazleabas AT (2006) A baboon model for inducing endometriosis. Methods Mol Med 121: 95-99.
    [15] Fazleabas AT, Brudney A, Gurates B, et al. (2002) A modified baboon model for endometriosis. Ann N Y Acad Sci 955: 308-317. doi: 10.1111/j.1749-6632.2002.tb02791.x
    [16] Marí-Alexandre J, Sánchez-Izquierdo D, Gilabert-Estellés J, et al. (2016) miRNAs regulation and its role as biomarkers in endometriosis. Int J Mol Sci 17: 1-16. doi: 10.3390/ijms17010093
    [17] Saare M, Rekker K, Laisk-Podar T, et al. (2017) Challenges in endometriosis miRNA studies—From tissue heterogeneity to disease specific miRNAs. Biochim Biophys Acta Mol Basis Dis 1863: 2282-2292. doi: 10.1016/j.bbadis.2017.06.018
    [18] Yovich JL, Rowlands PK, Lingham S, et al. (2020) Pathogenesis of endometriosis: Look no further than John Sampson. Reprod Biomed Online 40: 7-11. doi: 10.1016/j.rbmo.2019.10.007
    [19] Yang Y, Wang Y, Yang J, et al. (2012) Adolescent Endometriosis in China: A Retrospective Analysis of 63 Cases. J Pediatr Adolesc Gynecol 25: 295-299. doi: 10.1016/j.jpag.2012.03.002
    [20] Louise Hull M, Nisenblat V (2013) Tissue and circulating microRNA influence reproductive function in endometrial disease. Reprod Biomed Online 27: 515-529. doi: 10.1016/j.rbmo.2013.07.012
    [21] Valencia-Sanchez MA, Liu J, Hannon GJ, et al. (2006) Control of translation and mRNA degradation by miRNAs and siRNAs. Genes Dev 20: 515-524. doi: 10.1101/gad.1399806
    [22] Ibrahim SA, Hassan H, Götte M (2014) MicroRNA-dependent targeting of the extracellular matrix as a mechanism of regulating cell behavior. Biochim Biophys Acta Gen Subj 1840: 2609-2620. doi: 10.1016/j.bbagen.2014.01.022
    [23] Weber JA, Baxter DH, Zhang S, et al. (2010) The microRNA spectrum in 12 body fluids. Clin Chem 56: 1733-1741. doi: 10.1373/clinchem.2010.147405
    [24] Nothnick WB (2017) MicroRNAs and Endometriosis: Distinguishing Drivers from Passengers in Disease Pathogenesis. Semin Reprod Med 35: 173-180. doi: 10.1055/s-0037-1599089
    [25] Teague EMCO, Van der Hoek KH, Van der Hoek MB, et al. (2009) MicroRNA-regulated pathways associated with endometriosis. Mol Endocrinol 23: 265-275. doi: 10.1210/me.2008-0387
    [26] Santamaria X, Taylor H (2014) MicroRNA and gynecological reproductive diseases. Fertil Steril 101: 1545-1551. doi: 10.1016/j.fertnstert.2014.04.044
    [27] Burney RO, Hamilton AE, Aghajanova L, et al. (2009) MicroRNA expression profiling of eutopic secretory endometrium in women with versus without endometriosis. Mol Hum Reprod 15: 625-631. doi: 10.1093/molehr/gap068
    [28] Datta A, Das P, Dey S, et al. (2019) Genome-wide small RNA sequencing identifies micrornas deregulated in non-small cell lung carcinoma harboring gain-of-function mutant P53. Genes (Basel) 10: 852. doi: 10.3390/genes10110852
    [29] Marí-Alexandre J, Barceló-Molina M, Belmonte-López E, et al. (2018) Micro-RNA profile and proteins in peritoneal fluid from women with endometriosis: their relationship with sterility. Fertil Steril 109: 675-684.e2. doi: 10.1016/j.fertnstert.2017.11.036
    [30] Haikalis ME, Wessels JM, Leyland NA, et al. (2018) MicroRNA expression pattern differs depending on endometriosis lesion type. Biol Reprod 98: 623-633. doi: 10.1093/biolre/ioy019
    [31] Panir K, Schjenken JE, Robertson SA, et al. (2018) Non-coding RNAs in endometriosis: A narrative review. Hum Reprod Update 24: 497-515. doi: 10.1093/humupd/dmy014
    [32] Rekker K, Tasa T, Saare M, et al. (2018) Differentially-expressed mirnas in ectopic stromal cells contribute to endometriosis development: The plausible role of miR-139-5p and miR-375. Int J Mol Sci 19: 1-11. doi: 10.3390/ijms19123789
    [33] Zhao L, Gu C, Ye M, et al. (2018) Integration analysis of microRNA and mRNA paired expression profiling identifies deregulated microRNA-transcription factor-gene regulatory networks in ovarian endometriosis. Reprod Biol Endocrinol 16: 18-22. doi: 10.1186/s12958-018-0335-0
    [34] Yang L, Liu HY (2014) Small RNA molecules in endometriosis: Pathogenesis and therapeutic aspects. Eur J Obstet Gynecol Reprod Biol 183: 83-88. doi: 10.1016/j.ejogrb.2014.10.043
    [35] Vodolazkaia A, Yesilurt BT, Kyama CM, et al. (2016) Vascular endothelial growth factor pathway in endometriosis: Genetic variants and plasma biomarkers. Fertil Steril 105: 988-996. doi: 10.1016/j.fertnstert.2015.12.016
    [36] D'Hooghe TM, Bambra CS, Raeymaekers BM, et al. (1999) Pelvic inflammation induced by diagnostic laparoscopy in baboons. Fertil Steril 72: 1134-1141. doi: 10.1016/S0015-0282(99)00406-9
    [37] D'Hooghe TM, Bambra CS, Raeymaekers BM, et al. (1995) Intrapelvic injection of menstrual endometrium causes endometriosis in baboons (Papio cynocephalus and Papio anubis). Am J Obstet Gynecol 173: 125-134. doi: 10.1016/0002-9378(95)90180-9
    [38] Clontech (2016)  Supplement: SMARTer® smRNA-Seq Kit for Illumina®. 1-15.
    [39]  Agilent Technologies, Agilent Technologies Agilent DNA 1000 Kit Guide, 2016. Available from: https://www.agilent.com/cs/library/usermanuals/public/G2938-90014_DNA1000Assay_KG.pdf.
    [40] Illumina Inc (2011) Sequencing Library qPCR Quantification Guide. Illumina Tech Manuals 1-27.
    [41] Illumina Inc (2017) Illumina sequencing introduction. Illumina Seq Introd 1-8. Available from: http://www.illumina.com/content/dam/illumina-marketing/documents/products/illumina_sequencing_introduction.pdf.
    [42] Saeidipour B, Bakhshi S (2013) The relationship between organizational culture and knowledge management,& their simultaneous effects on customer relation management. Adv Environ Biol 7: 2803-2809.
    [43] Barturen G, Rueda A, Hamberg M, et al. (2014) sRNAbench: profiling of small RNAs and its sequence variants in single or multi-species high-throughput experiments. Methods Next Gener Seq 1: 21-31.
    [44] Kozomara A, Birgaoanu M, Griffiths-Jones S (2019) MiRBase: From microRNA sequences to function. Nucleic Acids Res 47: D155-D162. doi: 10.1093/nar/gky1141
    [45] Karere GM, Glenn JP, Vandeberg JL, et al. (2010) Identification of baboon microRNAs expressed in liver and lymphocytes. J Biomed Sci 17: 1-8. doi: 10.1186/1423-0127-17-54
    [46] Karere GM, Glenn JP, VandeBerg JL, et al. (2012) Differential microRNA response to a high-cholesterol, high-fat diet in livers of low and high LDL-C baboons. BMC Genomics 13: 320. doi: 10.1186/1471-2164-13-320
    [47] Costa B (2017)  sRNA-workflow Documentation. Available from: https://readthedocs.org/projects/srna-workflow/downloads/pdf/master/.
    [48] Agarwal V, Bell GW, Nam JW, et al. (2015) Predicting effective microRNA target sites in mammalian mRNAs. Elife 4: 1-38. doi: 10.7554/eLife.05005
    [49] Georgakilas G, Vlachos IS, Zagganas K, et al. (2016) DIANA-miRGen v3.0: Accurate characterization of microRNA promoters and their regulators. Nucleic Acids Res 44: D190-D195. doi: 10.1093/nar/gkv1254
    [50] Braza-Boïls A, Mari-Alexandre J, Gilabert J, et al. (2014) MicroRNA expression profile in endometriosis: Its relation to angiogenesis and fibrinolytic factors. Hum Reprod 29: 978-988. doi: 10.1093/humrep/deu019
    [51] Laudanski P, Charkiewicz R, Kuzmicki M, et al. (2013) MicroRNAs expression profiling of eutopic proliferative endometrium in women with ovarian endometriosis. Reprod Biol Endocrinol 11: 78. doi: 10.1186/1477-7827-11-78
    [52] Bashti O, Noruzinia M, Garshasbi M, et al. (2018) miR-31 and miR-145 as Potential Non-Invasive Regulatory Biomarkers in Patients with Endometriosis. Cell J 20: 84-89.
    [53] Adammek M, Greve B, Kassens N, et al. (2013) MicroRNA miR-145 inhibits proliferation, invasiveness, and stem cell phenotype of an in vitro endometriosis model by targeting multiple cytoskeletal elements and pluripotency factors. Fertil Steril 99: 1346-1355.e5. doi: 10.1016/j.fertnstert.2012.11.055
    [54] Yang H, Kong W, He L, et al. (2008) MicroRNA expression profiling in human ovarian cancer: miR-214 induces cell survival and cisplatin resistance by targeting PTEN. Cancer Res 68: 425-433. doi: 10.1158/0008-5472.CAN-07-2488
    [55] Obata K, Morland SJ, Watson RH, et al. (1998) Frequent PTEN/MMAC mutations in endometrioid but not serous or mucinous epithelial ovarian tumors. Cancer Res 58: 2095-2097.
    [56] Sanchez G (2013) Las instituciones de ciencia y tecnología en los procesos de aprendizaje de la producción agroalimentaria en Argentina. El Sist argentino innovación Inst Empres y redes El desafío la creación y apropiación Conoc 1-49.
    [57] Welsh P, Doolin O, McConnachie A, et al. (2012) Circulating 25OHD, dietary vitamin D, PTH, and Calcium Associations with Incident Cardiovascular Disease and Mortality: The MIDSPAN Family Study. J Clin Endocrinol Metab 97: 4578-4587. doi: 10.1210/jc.2012-2272
    [58] Wright KR, Mitchell B, Santanam N (2017) Redox regulation of microRNAs in endometriosis-associated pain. Redox Biol 12: 956-966. doi: 10.1016/j.redox.2017.04.037
    [59] Snowdon J, Zhang X, Childs T, et al. (2011) The microRNA-200 family is upregulated in endometrial carcinoma. PLoS One 6: e22828. doi: 10.1371/journal.pone.0022828
    [60] Braza-Boïls A, Salam S, Josep M, et al. (2015) Peritoneal fluid modifies the microRNA expression profile in endometrial and endometriotic cells from women with endometriosis. Hum Reprod 30: 2292-2302. doi: 10.1093/humrep/dev204
    [61] Pateisky P, Pils D, Szabo L, et al. (2018) hsa-miRNA-154-5p expression in plasma of endometriosis patients is a potential diagnostic marker for the disease. Reprod Biomed Online 37: 449-466. doi: 10.1016/j.rbmo.2018.05.007
    [62] Wang L, Huang W, Fang X, et al. (2016) Analysis of Serum microRNA Profile by Solexa Sequencing in Women with Endometriosis. Reprod Sci 23: 1359-1370. doi: 10.1177/1933719116641761
    [63] Hawkins SM, Creighton CJ, Han DY, et al. (2011) Functional microRNA involved in endometriosis. Mol Endocrinol 25: 821-832. doi: 10.1210/me.2010-0371
    [64] Ohlsson Teague EMC, Print CG, Hull ML (2009) The role of microRNAs in endometriosis and associated reproductive conditions. Hum Reprod Update 16: 142-165. doi: 10.1093/humupd/dmp034
    [65] Nap AW (2012) Theories on the Pathogenesis of Endometriosis. Endometr Sci Pract 2014: 42-53. doi: 10.1002/9781444398519.ch5
    [66] Laganà AS, Garzon S, Götte M, et al. (2019) The pathogenesis of endometriosis: Molecular and cell biology insights. Int J Mol Sci 20: 1-42. doi: 10.3390/ijms20225615
    [67] Marquardt RM, Kim TH, Shin JH, et al. (2019) Progesterone and estrogen signaling in the endometrium: What goes wrong in endometriosis? Int J Mol Sci 20: 3822. doi: 10.3390/ijms20153822
    [68] Kashima H, Wu RC, Wang Y, et al. (2015) Laminin C1 expression by uterine carcinoma cells is associated with tumor progression. Gynecol Oncol 139: 338-344. doi: 10.1016/j.ygyno.2015.08.025
    [69] Zheng Y, Khan Z, Zanfagnin V, et al. (2016) Epigenetic Modulation of Collagen 1A1: Therapeutic Implications in Fibrosis and Endometriosis1. Biol Reprod 94: 1-10. doi: 10.1095/biolreprod.115.138115
    [70] Bhagwat SR, Chandrashekar DS, Ruchi K, et al. (2013) Endometrial Receptivity: A Revisit to Functional Genomics Studies on Human Endometrium and Creation of HGEx-ERdb. PLoS One 8: e58419. doi: 10.1371/journal.pone.0058419
    [71] Afshar Y, Hastings J, Roqueiro D, et al. (2013) Changes in Eutopic Endometrial Gene Expression During the Progression of Experimental Endometriosis in the Baboon, Papio Anubis1. Biol Reprod 88: 1-9. doi: 10.1095/biolreprod.112.104497
    [72] Rock JA (1995) The revised American Fertility Society classification of endometriosis: reproducibility of scoring.63: 1108-1110.
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