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Computational analysis reveals the therapeutic potential of Asiatic acid against the miRNA correlated differentially expressed genes of bipolar disorder

  • Bipolar disorder is a psychiatric condition that consists of recurring episodes of severe mood swings between depression and manic episodes. The diagnosis is generally based on clinical interviews and observations, but is often misdiagnosed as unipolar depression, leading to significant delays in treatment. However, the disorder's heterogeneous nature and overlap with other psychiatric conditions, such as schizophrenia, present challenges in its diagnosis and treatment. To address these challenges, this study aims to explore the gene targets of differentially expressed miRNA associated with differentially expressed genes and to find a suitable phytochemical through molecular docking studies. The altered expression level of miRNAs (either increased or decreased) and genes had been observed to play a crucial role in different psychiatric disorders, thus suggesting their potential as biomarkers. The data of patients with bipolar disorder was retrieved from the Gene expression omnibus and Sequence read archive. The differentially expressed genes and miRNAs were identified through DESeq2 post processing. The gene targets of the downregulated miRNA and the upregulated genes were compared to identify the main targets of bipolar disorder. Furthermore, the phytochemicals with neuro-protective properties were identified through a literature study. The drug likeness property of each phytochemical was evaluated on the basis of Lipinski's rule of 5, followed by a toxicity evaluation. Molecular docking studies were carried out using AutoDock to determine the best drug against bipolar disorder. Therefore, the present study targets key proteins overexpressed in patients with bipolar disorder to facilitate a multi-faceted treatment approach.

    Citation: Harshita Maheshwari, Maitreyi Pathak, Prekshi Garg, Prachi Srivastava. Computational analysis reveals the therapeutic potential of Asiatic acid against the miRNA correlated differentially expressed genes of bipolar disorder[J]. AIMS Molecular Science, 2024, 11(2): 99-115. doi: 10.3934/molsci.2024007

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  • Bipolar disorder is a psychiatric condition that consists of recurring episodes of severe mood swings between depression and manic episodes. The diagnosis is generally based on clinical interviews and observations, but is often misdiagnosed as unipolar depression, leading to significant delays in treatment. However, the disorder's heterogeneous nature and overlap with other psychiatric conditions, such as schizophrenia, present challenges in its diagnosis and treatment. To address these challenges, this study aims to explore the gene targets of differentially expressed miRNA associated with differentially expressed genes and to find a suitable phytochemical through molecular docking studies. The altered expression level of miRNAs (either increased or decreased) and genes had been observed to play a crucial role in different psychiatric disorders, thus suggesting their potential as biomarkers. The data of patients with bipolar disorder was retrieved from the Gene expression omnibus and Sequence read archive. The differentially expressed genes and miRNAs were identified through DESeq2 post processing. The gene targets of the downregulated miRNA and the upregulated genes were compared to identify the main targets of bipolar disorder. Furthermore, the phytochemicals with neuro-protective properties were identified through a literature study. The drug likeness property of each phytochemical was evaluated on the basis of Lipinski's rule of 5, followed by a toxicity evaluation. Molecular docking studies were carried out using AutoDock to determine the best drug against bipolar disorder. Therefore, the present study targets key proteins overexpressed in patients with bipolar disorder to facilitate a multi-faceted treatment approach.



    In order to describe the evolution of fecal-oral transmitted diseases in the Mediterranean regions, Capasso and Paveri-Fontana [1] proposed the following model

    {u(t)=au+cv,v(t)=bv+G(u), (1.1)

    where a,b,c are all positive constants, u(t) and v(t) denote the concentration of the infectious agent in the environment and the infective human population respectively. The coefficients a and b are the intrinsic decay rates of the infectious agent and the infective human population respectively, c represents the multiplication rate of the infectious agent due to the human infected population. The function G(u) stands for the force of infection of the human population due to the concentration of infectious agent. We assume that G(u) satisfies the two specific cases: (ⅰ) a monotone increasing function with constant concavity; (ⅱ) a sigmoidal function of bacterial concentration tending to some finite limit, and with zero gradient at zero. These two cases contain most of the features of forces of infection in real epidemics. For some epidemic, if the density of infectious agent is small, the force of infection of the humans will be weak and may tend to zero, and the function G will satisfy case (ⅱ). In this paper, we focus on such case, and assume that the function G:R+R+ satisfies:

    (G1) GC2(R+), G(0)=0, G(z)>0 for any z>0 and limzG(z)=1;

    (G2) there exists ξ>0 such that G"(z)>0 for z(0,ξ) and G"(z)<0 for z(ξ,).

    Denote

    θ=cG(0)ab.

    Under two specific cases stated above, the global dynamics of the cooperative system (1.1) has been described in detail in [2]. It follows from [2, Theorem 4.3] that the global dynamics of (1.1) under conditions (G1) and (G2) can be described as follows:

    (ⅰ) If θ<1 and G(z)z<abc for any z>0, then the trivial solution is the only equilibrium for problem (1.1) and it is globally asymptotically stable in R+×R+.

    (ⅱ) If θ>1, then problem (1.1) has only one nontrivial equilibrium point (u,v) in addition to (0,0) and it is globally asymptotically stable in R+×R+.

    (ⅲ) If θ<1 and G(z1)z1>abc for some z1>0, then problem (1.1) has three equilibrium points:

    E0=(0,0),E1=(K1,aK1c) and E2=(K2,aK2c),

    where 0<K1<K2 are the positive roots of G(z)abcz=0. In this case, E1 is a saddle point, E0 and E2 are stable nodes.

    In 1997, Capasso and Wilson [3] further considered spatial variation and studied the problem

    {ut=dΔuau+cv,(t,x)(0,+)×Ω,vt=bv+G(u),(t,x)(0,+)×Ω,u(t,x)=0,(t,x)(0,+)×Ω,u(0,x)=u0(x), v(0,x)=v0(x),xΩ, (1.2)

    where Ω is bounded. By some numerical simulation, they speculated that the dynamical behavior of system (1.2) is similar to the ODE case. To understand the dispersal process of epidemic from outbreak to an endemic, Xu and Zhao [4] studied the bistable traveling waves of (1.2) in xR.

    The epidemic always spreads gradually, but the works mentioned above are hard to explain this gradual expanding process. To describe such a gradual spreading process, Du and Lin [5] introduced the free boundary condition to study the invasion of a single species. They considered the problem

    {utduxx=u(abu),t>0, 0<x<h(t),ux(t,0)=0, u(t,h(t))=0,t>0,h(t)=μux(t,h(t)),t>0,h(0)=h0, u(0,x)=u0(x),0xh0, (1.3)

    and showed that (1.3) admits a unique solution which is well-defined for all t0 and spreading and vanishing dichotomy holds. Moreover, the criteria for spreading and vanishing are obtained: (ⅰ) for h0π2da, the species will spread; (ⅱ) for h0<π2da and given u0(x), there exists μ such that the species will spread for μ>μ, and the species will vanish for 0<μμ. Finally, they gave the spreading speed of the spreading front when spreading occurs. Since then, many problems with free boundaries and related problems have been investigated, see e.g. [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and their references.

    In 2016, Ahn et al. [23] considered (1.2) with monostable nonlinearity and free boundaries. They obtained the global existence and uniqueness of the solution and spreading and vanishing dichotomy. Furthermore, by introducing the so-called spatial-temporal risk index

    RF0(t)=G(0)cba+d(πh(t)g(t))2,

    they proved that: (ⅰ) if R0=cG(0)ab1, the epidemic will vanish; (ⅱ) if RF0(0)1, the epidemic will spread; (ⅲ) if RF0(0)<1, epidemic will vanish for the small initial densities; (ⅳ) if RF0(0)<1<R0, epidemic will spread for the large initial densities. Recently, Zhao et al. [24] determined the spreading speed of the spreading front of problem described in [23].

    Inspired by the work [23], we want to study (1.2) with bistable nonlinearity and free boundaries. Meanwhile, we also want to consider the effect of the advection. In 2009, Maidana and Yang [25] studied the propagation of West Nile Virus from New York City to California. In the summer of 1999, West Nile Virus began to appear in New York City. But it was observed that the wave front traveled 187 km to the north and 1100 km to the south in the second year. Therefore, taking account of the advection movement has the greater realistic significance. Recently, there are some works considering the advection. In 2014, Gu et al. [26] was the first time to consider the long-time behavior of problem (1.3) with small advection. Then, the asymptotic spreading speeds of the free boundaries was given in [27]. For more general reaction term, Gu et al. [10] studied the long time behavior of solutions of Fisher-KPP equation with advection β>0 and free boundaries. For single equation with advection, there are many other works. For example, [28,29,30,31,32,33,34] and their references. Besides, there are also several works devoted to the system with small advection, such as, [35,36,37,38,39,40] and their references.

    Taking account of the effect of advection, we consider

    {ut=duxxβuxau+cv,t>0, g(t)<x<h(t),vt=bv+G(u),t>0, g(t)<x<h(t),u(t,x)=v(t,x)=0,t0, x=g(t) or x=h(t),g(0)=h0, g(t)=μux(t,g(t)),t>0,h(0)=h0, h(t)=μux(t,h(t)),t>0,u(0,x)=u0(x), v(0,x)=v0(x),h0<x<h0, (1.4)

    where we use the changing region (g(t),h(t)) to denote the infective environment of disease, where the free boundaries x=g(t) and x=h(t) represent the spreading fronts of epidemic. Since the diffusion coefficient of v is much smaller than that of u, we assume that the diffusion coefficient of v is zero. When u spreads into a new environment, some humans in the new environment may be infected. Hence, we can use (g(t),h(t)) to represent the habit of infective humans. We use I0(h0,h0) to denote the initial infective environment of epidemic. The initial functions u0(x) and v0(x) satisfy

    u0(x)X1(h0){u0(x)W2p(I0): u0(x)>0 for xI0, u0(x)=0 for xRI0},v0(x)X2(h0){v0(x)C2(I0): v0(x)>0 for xI0, v0(x)=0 for xRI0},

    where p>3. The derivation of the stefan conditions h(t)=μux(t,h(t)) and g(t)=μux(t,g(t)) can be found in [41,42]. In this paper, we always assume that G satisfies (G1)-(G2) and

    (G3) G(z) is locally Lipschitz in zR+, i.e., for any L>0, there exists a constant ρ(L)>0 such that

    |G(z1)G(z2)|ρ(L)|z1z2|,  z1,z2[0,L].

    Furthermore, we assume that 0<β<β with

    β={,θ<1,2d(cG(0)ba),θ>1.

    The rest of this paper is organized as follows. In Section 2, the global existence and uniqueness of solution, comparison principle and some results about the principal eigenvalue are given. Section 3 is devoted to the long time behavior of (u,v). We get a spreading and vanishing dichotomy and give the criteria for spreading and vanishing. Finally, we give some discussions in Section 4.

    Firstly, we prove the existence and uniqueness of the solution.

    Lemma 2.1. For any given (u0,v0)X1(h0)×X2(h0) and any α(0,1), there exists a T>0 such that problem (1.4) admits a unique solution

    (u,v,g,h)(W1,2p(ΩT)C1+α2,1+α(¯ΩT))×C1([0,T];L([g(t),h(t)]))×[C1+α2([0,T])]2, (2.1)

    moreover,

    uW1,2p(ΩT)+uC1+α2,1+α(¯ΩT)+gC1+α2([0,T])+hC1+α2([0,T])C, (2.2)

    where ΩT={(t,x)R2: 0tT, g(t)xh(t)}, C and T depend only on h0, α, u0W2p([h0,h0]) and v0.

    Proof. This proof can be done by the similar arguments in [43]. But there are some differences. Hence, we give the details. Let

    y=2xg(t)h(t)h(t)g(t),w(t,y)=u(t,(h(t)g(t))y+h(t)+g(t)2),

    and

    z(t,y)=v(t,(h(t)g(t))y+h(t)+g(t)2).

    Then problem (1.4) becomes

    {wtdA2wyy+(βAB)wy=aw+cz,0<t<T, 1<y<1,w(t,1)=w(t,1)=0,0t<T,w(0,y)=u0(h0y)w0(y),1<y<1, (2.3)
    {vt=bv+G(u),0<t<T, g(t)<x<h(t),v(t,g(t))=v(t,h(t))=0,0t<T,v(0,x)=v0(x),h0<x<h0, (2.4)

    and

    {g(t)=μAwy(t,1),0<t<T,h(t)=μAwy(t,1),0<t<T,g(0)=h0, h(0)=h0, (2.5)

    where

    A=A(g(t),h(t))=2h(t)g(t) and B=B(g(t),h(t),y)=h(t)+g(t)h(t)g(t)+yh(t)g(t)h(t)g(t).

    Denote g=μh0u0(h0) and h=μh0u0(h0). For 0<Th02(2+g+h), define

    T=[0,T]×[1,1],D1T={wC(T): w(0,y)=w0(y), w(t,±1)=0, ww0C(T)1},D2T={gC1([0,T]): g(0)=h0, g(0)=g, ggC([0,T])1},D3T={hC1([0,T]): h(0)=h0, h(0)=h, hhC([0,T])1}.

    It is easy to see that DTD1T×D2T×D3T is a complete metric space with the metric

    d((w1,g1,h1),(w2,g2,h2))=w1w2C(T)+g1g2C1([0,T])+h1h2C1([0,T]).

    For any given (w,g,h)DT, there exist some ξ1,ξ2(0,t) such that

    |g(t)+h0|+|h(t)h0|=|g(ξ1)|t+|h(ξ2)|tT(2+g+h)h02,

    which implies that

    2h0h(t)g(t)3h0,  t[0,T].

    Thus, A(g(t),h(t)) and B(g(t),h(t),y) are well-defined. By the definition of w, we have

    u(t,x)=w(t,2xg(t)h(t)h(t)g(t)). (2.6)

    Since |w(t,y)|w0L+1 for (t,y)T, we have

    |u(t,x)|w0L+1M1,  (t,x)[0,T]×[g(t),h(t)].

    Define

    ˜v0(x)={v0(x),x(h0,h0),0,xR(h0,h0) and tx:={tgx,x[g(T),h0) and x=g(tgx),0,x[h0,h0],thx,x(h0,h(T)] and x=h(thx).

    For u defined as (2.6) and any given x[g(T),h(T)], we consider the following ODE problem

    {vt=bv+G(u(t,x)),tx<t<T,v(tx,x)=˜v0(x). (2.7)

    By the similar arguments as the step 1 in the proof of [44, Lemma 2.3], it is easy to show that (2.7) admits a unique solution v(t,x) for t[tx,T1], where T1(0,h02(2+g+h)]. Hence, problem (2.4) has a unique solution v(t,x)C1([0,T1];L([g(t),h(t)])). By the continuous dependence of the solution on parameters, we can have

    vxL(ΩT1)C1.

    Then

    vxL(ΩT)vxL(ΩT1)C1,  TT1.

    For this v, we can get

    z(t,y)=v(t,(h(t)g(t))y+h(t)+g(t)2).

    For (w,g,h) and z obtained above, we consider the following problem

    {¯wtdA2¯wyy+(βAB)¯wy=aw+cz,0<t<T, 1<y<1,¯w(t,1)=¯w(t,1)=0,0t<T,¯w(0,y)=u0(h0y),1<y<1. (2.8)

    Applying standard Lp theory and the Sobolev imbedding theorem, we can have there exists T2(0,T1] such that (2.8) admits a unique solution ¯w(t,y) and

    ¯wW1,2p(T2)+¯wC1+α2,1+α(T2)C2,

    where C2 is a constant depending only on h0, α and u0W2p([h0,h0]). Then

    ¯wW1,2p(T)+¯wC1+α2,1+α(T)¯wW1,2p(T2)+¯wC1+α2,1+α(T2)C2,  TT2. (2.9)

    Define

    ¯g(t)=h0t0μA(g(τ),h(τ))¯wy(τ,1)dτ,¯h(t)=h0t0μA(g(τ),h(τ))¯wy(τ,1)dτ,

    then we have ¯g(0)=h0, ¯h(0)=h0,

    ¯g(t)=μA(g(t),h(t))¯wy(t,1), ¯h(t)=μA(g(t),h(t))¯wy(t,1),

    and hence

    ¯gCα2([0,T]), ¯hCα2([0,T])μh10C2C3. (2.10)

    Now, we can define the mapping F:DTC(T)×C1([0,T])×C1([0,T]) by

    F(w,g,h)=(¯w,¯g,¯h).

    Obviously, DT is a bounded and closed convex set of C(T)×C1([0,T])×C1([0,T]), F is continuous in DT, and (w,g,h) is a fixed point of F if and only if (w,v,g,h) solve (2.3), (2.4) and (2.5). By (2.9) and (2.10), we have F is compact and

    ¯ww0C(T)C2T1+α2,¯ggC([0,T])C3Tα2,¯hhC([0,T])C3Tα2.

    Therefore if we take Tmin{T2, C21+α2, C2α3}T3, then F maps DT into itself. It now follows from the Schauder fixed point theorem that F has a fixed point (w,g,h) in DT. Moreover, we have (w,v,g,h) solve (2.3), (2.4) and (2.5),

    wW1,2p(T)+wC1+α2,1+α(T)C2, vxL(ΩT)C1,  TT3.

    Define as before,

    u(t,x)=w(t,2xg(t)h(t)h(t)g(t)).

    Then (u,v,g,h) solve (1.4), and satisfies (2.1) and (2.2).

    In the following, we prove the uniqueness of (u,v,g,h). Let (ui,vi,gi,hi) (i=1,2) be the two solutions of problem (1.4) for T(0,T3] sufficiently small. Let

    wi(t,y)=ui(t,(hi(t)gi(t))y+hi(t)+gi(t)2).

    Then it is easy to see that (wi,vi,gi,hi) solve (2.3), (2.4) and (2.5). Denoting

    Ai=A(gi(t),hi(t)), Bi=B(gi(t),hi(t),y), W=w1w2, Z=z1z2, G=g1g2, H=h1h2,

    we can have

    {WtdA21Wyy+(βA1B1)Wy=aW+cZ                +(dA21dA22)w2yy+[(βA1B1)+(βA2B2)]w2y,0<t<T, 1<y<1,W(t,1)=W(t,1)=0,0t<T,W(0,y)=0,1<y<1,

    and

    {G=μA1Wy(t,1)+μ(A2A1)w2y(t,1),0<t<T,H=μA1Wy(t,1)+μ(A2A1)w2y(t,1),0<t<T,G(0)=0, H(0)=0. (2.11)

    Using the Lp estimates for parabolic equations and Sobolev imbedding theorem, we obtain

    WW1,2p(T)C4(ZC(T)+GC1([0,T])+HC1([0,T])), (2.12)

    where C4 depends on C2, C3 and the functions A and B. Next we should estimate z1z2C(T). For convenience, we define

    Hm(t)min{h1(t),h2(t)}, HM(t)max{h1(t),h2(t)},Gm(t)min{g1(t),g2(t)}, GM(t)max{g1(t),g2(t)},ΩGm,HMT[0,T]×[Gm(t),HM(t)].

    By direct calculations, we have

    z1(t,y)z2(t,y)C(T)= v1(t,(h1(t)g1(t))y+h1(t)+g1(t)2)v2(t,(h2(t)g2(t))y+h2(t)+g2(t)2)C(T) v1(t,(h1(t)g1(t))y+h1(t)+g1(t)2)v2(t,(h1(t)g1(t))y+h1(t)+g1(t)2)C(T)+v2(t,(h1(t)g1(t))y+h1(t)+g1(t)2)v2(t,(h2(t)g2(t))y+h2(t)+g2(t)2)C(T) v1(t,x)v2(t,x)C(ΩGm,HMT)+v2xL(ΩGm,HMT)(GC([0,T])+HC([0,T])). (2.13)

    Now we estimate |(v1v2)(t,x)| for any fixed (t,x)ΩGm,HMT. It will be divided into the following three cases.

    Case 1. x[h0,h0].

    Since (2.4) is equivalent to the following integral equation:

    v(t,x)=ebt[v0(x)+t0ebsG(u)(s,x)ds],

    we have

    v1(t,x)v2(t,x)= ebt[t0ebs(G(u1)G(u2))(s,x)ds].

    Then,

    |v1(t,x)v2(t,x)|ρ(M1)bu1u2C(ΩGm,HMT). (2.14)

    Case 2. x(h0,Hm(t)).

    In this case, there exist t1, t2(0,t) such that h1(t1)=h2(t2)=x. Without loss of generality, we may assume that 0t1t2. Then,

    v1(t,x)v2(t,x)= ebt[v1(t2,x)ebt2+tt2ebs(G(u1)G(u2))(s,x)ds].

    Thus,

    |v1(t,x)v2(t,x)||v1(t2,x)|+ρ(M1)bu1u2C(ΩGm,HMT).

    By (G1) and (G2), we can have that there exists γ such that G(z)γz for z0. Now we estimate v1(t2,x). Direct calculations give that

    v1(t2,x)=ebt2t2t1ebsG(u1)(s,x)dsγbmaxt[t1,t2]|u1(t,x)|=γbmaxt[t1,t2]|(u1u2)(t,x)|.

    Hence,

    |v1(t,x)v2(t,x)|γ+ρ(M1)bu1u2C(ΩGm,HMT). (2.15)

    Case 3. x[Hm(t),HM(t)].

    Without loss of generality, we assume that h2(t)<h1(t). In this case, there exists t1 such that h1(t1)=x. Then v1(t1,x)=0, u2(t,x)=v2(t,x)=0 for t[t1,t]. Hence, V(t,x)=v1(t,x) and

    v1(t,x)=ebttt1ebsG(u1)(s,x)dsγbmaxt[t1,t]|u1(t,x)|=γbmaxt[t1,t]|(u1u2)(t,x)|.

    Hence,

    |v1(t,x)v2(t,x)|γbu1u2C(ΩGm,HMT). (2.16)

    By (2.14), (2.15) and (2.16), we have

    v1v2C(ΩGm,HMT)C5u1u2C(ΩGm,HMT), (2.17)

    where C5 depends on b, ρ, M1 and γ. Now we estimate u1(t,x)u2(t,x)C(ΩGm,HMT).

    u1(t,x)u2(t,x)C(ΩGm,HMT)= w1(t,2xg1(t)h1(t)h1(t)g1(t))w2(t,2xg2(t)h2(t)h2(t)g2(t))C(ΩGm,HMT) w1(t,2xg1(t)h1(t)h1(t)g1(t))w2(t,2xg1(t)h1(t)h1(t)g1(t))C(ΩGm,HMT)+w2(t,2xg1(t)h1(t)h1(t)g1(t))w2(t,2xg2(t)h2(t)h2(t)g2(t))C(ΩGm,HMT) w1(t,y)w2(t,y)C(T)+C6(GC([0,T])+HC([0,T])), (2.18)

    where C6 only depends on h0 and w2xC(T3). By ¯W(0,y)=0 and Sobolev imbedding theorem, we have

    W(t,y)C(T)[W]Cα2,0(T)Tα2C7Tα2[W]Cα2,α(T)C8Tα2WW1,2p(T), (2.19)

    where C7 and C8 do not depend on T. By (2.12), (2.13), (2.17), (2.18) and (2.19), we can get

    WW1,2p(T)C9Tα2WW1,2p(T)+C10(GC1([0,T])+HC1([0,T])),

    where C9 depends on C4, C5 and C8; C10 depends on C1, C5 and C6. If Tmin{T3,(2C9)2α}T4,

    WW1,2p(T)2C10(GC1([0,T])+HC1([0,T])). (2.20)

    In the following, we estimate GC1([0,T]) and HC1([0,T]). Since G(0)=G(0)=0, we have

    GC1([0,T])= maxt[0,T]G(t)+maxt[0,T]G(t)maxξ[0,T]G(ξ)T+maxt[0,T]G(t) (1+T)maxt[0,T]G(t)G(0)(t0)α2Tα2=Tα2(1+T)[G]Cα2([0,T]).

    By (2.11), we have

    [G]Cα2([0,T])=C11[[Wy(t,1)]Cα2,0([0,T])+(GC1([0,T])+HC1([0,T]))[w2y(t,1)]Cα2([0,T])],

    where C11 depends on μ, A and h0. It follows from the proof of [45, Theorem 1.1] that we have

    [Wy(t,y)]Cα2,0(T)C12[Wy(t,y)]Cα2,α(T)C13WW1,2p(T),

    where C12 and C13 do not depend on T. Therefore, we have

    GC1([0,T])C14Tα2(1+T)(GC1([0,T])+HC1([0,T])), (2.21)

    where C14 depends on C2, C10, C11 and C13. Similarly, there exists C15 such that

    HC1([0,T])C15Tα2(1+T)(GC1([0,T])+HC1([0,T])). (2.22)

    It follows from (2.21) and (2.22) that

    GC1([0,T])+HC1([0,T])=C16Tα2(1+T)(GC1([0,T])+HC1([0,T]))12(GC1([0,T])+HC1([0,T]))

    if Tmin{T4, 1, (4C16)2α}T5, where C16=C14+C15. Hence, G=H=0 for TT5. It follows from (2.20) that W=0. This implies that u1u2. By (2.17), we have v1v2. The uniqueness is obtained.

    Then it follows from the arguments in [23] that we can get the following estimates.

    Lemma 2.2. Let (u,v,g,h) be a solution of problem (1.4) defined for t(0,T0], where T0(0,+). Then there exist M1, M2 and M3 independent of T0 such that

    (ⅰ) 0<u(t,x)M1, 0<v(t,x)M2 for t(0,T0] and x[g(t),h(t)].

    (ⅱ) 0<g(t), h(t)M3 for t(0,T0].

    Just like the proof of [37, Theorem 3.2], we can obtain the global existence and uniqueness.

    Theorem 2.3. The solution exists and is unique for all t>0.

    Then, we exhibit the following comparison principle, which can be proven by the similar argument in [23,Lemma 2.5].

    Theorem 2.4. Assume that

    ¯g, ¯hC1([0,+)), ¯u(t,x), ¯v(t,x)C(¯D)C1,2(D),¯u(0,x)X1(h0), ¯v(0,x)X2(h0)

    with

    D:={(t,x)R2: 0<t<, ¯g(t)<x<¯h(t)},

    and (¯u,¯v,¯g,¯h) satisfies

    {¯utd¯uxxβ¯uxa¯u+c¯v,t>0, ¯g(t)<x<¯h(t),¯vtb¯v+G(¯u),t>0, ¯g(t)<x<¯h(t),¯u(t,¯g(t))=¯u(t,¯h(t))=0,t0,¯v(t,¯g(t))=¯v(t,¯h(t))=0,t0,¯g(0)h0, ¯g(t)μ¯ux(t,¯g(t)),t>0,¯h(0)h0, ¯h(t)μ¯ux(t,¯h(t)),t>0,¯u(0,x)u0(x), ¯v(0,x)v0(x),h0<x<h0.

    Then the solution (u,v,g,h) of the free boundary problem (1.4) satisfies

    h(t)¯h(t), g(t)¯g(t),  t0,
    u(t,x)¯u(t,x), v(t,x)¯v(t,x),  t0, g(t)xh(t).

    Remark 2.5. The pair (¯u,¯v,¯g,¯h) in Theorem 2.4 is usually called an upper solution of problem (1.4). Similarly, we can define a lower solution by reversing all the inequalities in the suitable places.

    In the following part, we consider the following eigenvalue problem

    {λϕ=dϕxxβϕxaϕ+cG(0)bϕ,l<x<l,ϕ(l)=ϕ(l)=0. (2.23)

    Denote by λ0(l) the principal eigenvalue of problem (2.23) with some fixed l.

    Lemma 2.6. λ0(l) has the following form:

    λ0(l)=β24d+dπ24l2(cG(0)ba).

    Proof. We choose β to be small and determine it later. By a simple calculation, we can achieve the characteristic equation

    dμ2βμ+λa+cG(0)b=0, (2.24)

    and let μi (i=1,2) be the roots of (2.24). Then the solution of (2.23) is

    ϕ(x)=c1eμ1x+c2eμ2x,

    where c1 and c2 will be determined later. Since ϕ(l)=ϕ(l)=0, we can derive that

    Δ=β24d(λa+cG(0)b)<0.

    In fact, if Δ=β24d(λa+cG(0)b)0, we have ϕ0, which is a contradiction. Hence, (2.24) has two complex roots:

    μ1=β+i4d(λa+cG(0)b)β22d, μ2=βi4d(λa+cG(0)b)β22d.

    Then

    ϕ(x)= c1eβ2dx[cos4d(λa+cG(0)b)β22dx+isin4d(λa+cG(0)b)β22dx]+c2eβ2dx[cos4d(λa+cG(0)b)β22dxisin4d(λa+cG(0)b)β22dx].

    By ϕ(l)=ϕ(l)=0, we have c1=c2 and

    4d(λa+cG(0)b)β22dl=π2+kπ,  kN.

    When k=0, λ attain its minimum, we have

    λ0(l)=β24d+dπ24l2(cG(0)ba),

    and the corresponding eigenfunction ϕ(x)=eβ2dxcos(π2lx).

    Then we have the following properties about λ0(l).

    Lemma 2.7. The following assertions hold:

    (ⅰ) λ0(l) is continuous and strictly decreasing in l,

    liml0λ0(l)=, limlλ0(l)=β24d(cG(0)ba).

    (ⅱ) If cG(0)ab>1 and 0<β<2d(cG(0)ba), then there exists

    l=2dπ/4d(cG(0)ba)β2

    such that λ0(l)=0. Furthermore, λ0(l)>0 for 0<l<l, and λ0(l)<0 for l>l.

    (ⅲ) If cG(0)ab1, then λ0(l)>β24d(cG(0)ba)>0.

    Proof. By the expression of λ0(l) in Lemma 2.6, the proof of lemma is obvious. We omit it here.

    Firstly, we give the definitions of spreading and vanishing of the disease:

    Definition 3.1. We say that vanishing happens if

    hg< and limt(u(t,)C([g(t),h(t)])+v(t,)C([g(t),h(t)]))=0,

    and spreading happens if

    hg= and lim supt(u(t,)C([g(t),h(t)])+v(t,)C([g(t),h(t)]))>0.

    Then, we give the following lemmas.

    Lemma 3.2. Let (u,v,g,h) be the solution of (1.4). If hg<, then there exists a constant C>0 such that

    u(t,)C1([g(t),h(t)])C,  t>1. (3.1)

    Moreover,

    limtg(t)=limth(t)=0. (3.2)

    Proof. We can use the method in [46, Theorem 2.1] to get (3.1). Then the proof of (3.2) can be done as [16,Theorem 4.1].

    Lemma 3.3. Let d, μ and h0 be positive constants, wC1+α2,1+α([0,)×[g(t),h(t)]) and g, hC1+α2([0,)) for some α>0. We further assume that w0(x)X1(h0). If (w,g,h) satisfies

    {wtdwxxβwxaw,t>0, g(t)<x<h(t),w(t,x)=0,t0, xg(t),w(t,x)=0,t0, xh(t),g(0)=h0, g(t)μwx(t,g(t)),t>0,h(0)=h0, h(t)μwx(t,h(t)),t>0,w(0,x)=w0(x),0,h0<x<h0, (3.3)

    and

    limtg(t)=g>, limtg(t)=0,limth(t)=h<, limth(t)=0,w(t,)C1([g(t),h(t)])M,  t>1

    for some constant M>0. Then

    limtmaxg(t)xh(t)w(t,x)=0.

    Proof. It can be proved by the similar arguments in [16,Theorem 4.2].

    By above Lemmas 3.2 and 3.3, we can derive the following result.

    Theorem 3.4. If hg<, then

    limt(u(t,)C([g(t),h(t)])+v(t,)C([g(t),h(t)]))=0.

    Proof. Firstly, we can use the method in the proof of [46,Theorem 2.1] to get

    uC1+α2,1+α([0,)×[g(t),h(t)])+gC1+α2([0,))+hC1+α2([0,))C.

    Recall that u satisfies (3.3). By Lemmas 3.2 and 3.3, we can get limtu(t,)C([g(t),h(t)])=0.

    Noting that v(t,x) satisfies

    vt=bv+G(u), t>0, g(t)<x<h(t)

    and G(u)0 uniformly for x[g(t),h(t)] as t, we have limtv(t,)C([g(t),h(t)])=0.

    Lemma 3.5. If G(z)z<abc for any z>0, then hg<.

    Proof. Direct calculations yield

    ddth(t)g(t)(u(t,x)+cbv(t,x))dx= h(t)g(t)(ut+cbvt)dx= h(t)g(t)(duxxβuxau+cbG(u))dx= dμ(h(t)g(t))+h(t)g(t)(au+cbG(u))dx.

    Integrating from 0 to t gives

    h(t)g(t)(u(t,x)+cbv(t,x))dx= h0h0(u0(x)+cbv0(x))dxdμ(h(t)g(t))+dμ2h0+t0h(s)g(s)(au+cbG(u))dxds.

    Since u0, v0 and G(u)abcu for u0, we have

    h(t)g(t)μdh0h0(u0(x)+cbv0(x))dx+2h0<.

    Letting t, we have hg<.

    Lemma 3.6. Assume that G(z1)z1>abc for some z1>0. If λ0(h0)>0 holds, then vanishing will happen provided that u0 and v0 are sufficiently small.

    Proof. We prove this result by constructing the appropriate upper solution. Let ϕ be the corresponding eigenfunction of λ0(h0). Since λ0(h0)>0, we can choose some small δ such that

    δβh0δ22d(2+δ)+34λ01(1+δ)2>0.

    Set

    σ(t)=h0(1+δδ2eδt), t0,¯u(t,x)=εeδtϕ(xh0σ(t))eβ2d(1h0σ(t))x, t0, σ(t)xσ(t),¯v(t,x)=(G(0)b+λ04c)h20σ2¯u, t0, σ(t)xσ(t).

    Direct computations yield

    ¯utd¯uxx+β¯ux+a¯uc¯v= ¯u(δϕϕxh0σσ2+βh0x2dσσ2)dεeδteβ2d(1h0σ)x[ϕ(h0σ)2+2ϕh0σβ2d(1h0σ)+ϕ(β2d)2(1h0σ)2]+βεeδteβ2d(1h0σ)x[ϕh0σ+ϕβ2d(1h0σ)]+a¯uc(G(0)b+λ04c)h20σ2¯u= ¯u(δϕϕxh0σσ2+βh0x2dσσ2)+εeδteβ2d(1h0σ(t))x[h20σ2(dϕ+βϕ)+ϕβ24d(1h20σ2)]+a¯uc(G(0)b+λ04c)h20σ2¯u ¯u(δβh02dσσ+34λ0h20σ2)+(1h20σ2)(β24d¯u+a¯u)> ¯u[δβh0δ22d(2+δ)+34λ01(1+δ)2]> 0,

    and

    ¯vt+b¯vG(¯u)= (G(0)b+λ04c)2h20σσ3¯u+(G(0)b+λ04c)h20σ2(¯ut+b¯u)G(ξ)¯u (G(0)b+λ04c)2h20σ2δ22+δ¯u+(G(0)b+λ04c)h20σ2[δβh0δ22d(2+δ)+b]¯uG(ξ)¯u= ¯u{(G(0)b+λ04c)h20σ2[δβh0δ22d(2+δ)]+G(0)h20σ2[12δ2b(2+δ)]G(ξ)+λ0h204cσ2(b2δ22+δ)}B

    for all t>0 and σ(t)<x<σ(t), where ξ(0,¯u). Let

    ε=δ2h0(1+δ2)2μmin{1ϕ(h0)eβ2dδh0,1ϕ(h0)eβ4dδh0}.

    Since ¯uεeβ2dh0δ, we can choose δ to be sufficiently small such that B>0. Noting that

    σ(t)=h0δ22eδt, ¯ux(t,σ(t))=εeδtϕ(h0)h0σeβ2d(σ(t)h0),¯ux(t,σ(t))=εeδtϕ(h0)h0σeβ2d(h0σ(t)),

    then we have

    {¯utd¯uxxβ¯uxa¯u+c¯v,t>0, σ(t)<x<σ(t),¯vtb¯v+G(¯u),t>0, σ(t)<x<σ(t),¯u(t,σ(t))=¯u(t,σ(t))=0,t0,¯v(t,σ(t))=¯v(t,σ(t))=0,t0,σ(0)h0, σ(t)μ¯ux(t,σ(t)),t>0,σ(0)h0, σ(t)μ¯ux(t,σ(t)),t>0.

    If u0 and v0 are sufficiently small such that

    u0(x)εϕ(x1+δ/2)eβδx2d(2+δ),  x[h0(1+δ/2),h0(1+δ/2)]

    and

    v0(x)(G(0)b+λ04c)1(1+δ/2)2εϕ(x1+δ/2)eβδx2d(2+δ),  x[h0(1+δ/2),h0(1+δ/2)],

    then

    u0(x)¯u(0,x), v0(x)¯v(0,x),  x(h0,h0).

    Applying Theorem 2.4 gives that h(t)σ(t) and g(t)σ(t). Hence, hg2h0(1+δ)<. By Theorem 3.4, we have limt(u(t,)C([g(t),h(t)])+v(t,)C([g(t),h(t)]))=0.

    By Lemma 3.6, we can derive the following corollary directly.

    Corollary 3.7. Assume that G(z1)z1>abc for some z1>0, then the following statements holds:

    (ⅰ) If cG(0)ab<1, then vanishing will happen for u0 and v0 sufficiently small.

    (ⅱ) If cG(0)ab>1 and h0<l, then vanishing will happen for u0 and v0 sufficiently small.

    Lemma 3.8. Assume that G(z1)z1>abc for some z1>0 and cG(0)ab>1. If h0>l, then spreading will happen.

    Proof. Let ϕ be the corresponding eigenfunction of λ0(h0). Since cG(0)ab>1 and h0>l, we have λ0(h0)<0. Then we construct a suitable lower solution. Since

    cG(0)b+λ04=β24d+dπ24l2+a3λ04>0,

    we can define

    u_(t,x)=ϵϕ(x), t0, h0xh0,v_(t,x)=(G(0)b+λ04c)ϵϕ(x), t0, h0xh0.

    Direct computations yield

    u_tdu_xx+βu_x+au_cv_= ϵ(dϕxx+βϕx+aϕcG(0)bϕλ04ϕ)=34λ0ϵϕ<0,

    and

    v_t+bv_G(u_)=ϵϕ(G(0)G(ξ)+bλ04c)

    for all t>0 and h0<x<h0, where ξ(0,u_). We can choose ϵ small enough such that

    G(0)G(ξ)+bλ04c0, ϵϕ(x)u0(x), (G(0)b+λ04c)ϵϕ(x)v0(x).

    Then

    {u_tdu_xxβu_xau_+cv_,t>0, h0<x<h0,v_tbv_+G(u_),t>0, h0<x<h0,u_(t,h0)=u_(t,h0)=0,t0,v_(t,h0)=v_(t,h0)=0,t0,0μu_x(t,h0), 0μu_x(t,h0),t>0,u_(0,x)u(0,x), v_(0,x)v(0,x),h0<x<h0.

    It follows from Remark 2.5 that u(t,x)u_(t,x) in [0,)×[h0,h0]. Hence,

    limtu(t,)C([g(t),h(t)])ϵϕ(x)>0.

    By Theorem 3.4, we have hg=.

    Lemma 3.9. Assume that G(z1)z1>abc for some z1>0 and cG(0)ab>1. If h0<l, then hg= provided that u0 and v0 are sufficiently large.

    Proof. We first note that there exists T>l such that λ0(T)<0.

    Inspired by the argument of [8,proposition 5.3], we consider

    {dφ(12+T+1)φ=˜λ0φ,0<x<1,φ(0)=φ(1)=0. (3.4)

    It is well-known that the first eigenvalue ˜λ0 of (3.4) is simple and the corresponding eigenfunction φ can be chosen positive in [0,1) and φL(1,1)=1. Moreover, one can easily see that ˜λ0>0 and φ(x)<0 in (0,1]. We extend φ to [1,1] as an even function. Then clearly

    {dφ(12+T+1)sgn(x)φ=˜λ0φ,1<x<1,φ(1)=φ(1)=0.

    Now we construct a suitable lower solution to (1.4). Define

    η(t)=t+ϱ, 0tT,u_(t,x)={m(t+ϱ)kφ(xt+ϱ),0tT, η(t)<x<η(t),0,0tT, |x|η(t),

    where the constants ϱ, m, k are chosen as follows:

    0<ϱmin{1,h20}, k˜λ0+a(T+1), m(T+1)k2μmin{φ(1),φ(1)}.

    Let

    tx:={t1x,x[η(T),ϱ) and x=η(t1x),0,x[ϱ,ϱ],t2x,x(ϱ,η(T)] and x=η(t2x)

    and

    v_0(x)={ε2+ε2cos(πϱx),ϱxϱ,0,|x|>ϱ,

    where we choose ε small enough such that

    v_0(x)v0(x),  x(ϱ,ϱ).

    Then we define

    v_(t,x)=ebt(ttxebτG(u_(τ,x))dτ+v_0(x)), txtT, η(t)xη(t).

    Direct computations yield

    u_tdu_xx+βu_x+au_cv_ m(t+ϱ)k+1[kφ+x2t+ϱφ+dφt+ϱφa(t+ϱ)φ] m(t+ϱ)k+1[kφ+(12+T+1)sgn(x)φ+dφa(T+1)φ] m(t+ϱ)k+1[dφ+(12+T+1)sgn(x)φ+˜λ0φ]=0,

    and

    v_t+bv_G(u_)=0, 0<tT, η(t)<x<η(t).

    For x[ϱ,ϱ], we have tx=0. Then

    v_(0,x)=v_0(x)v0(x),  x[ϱ,ϱ].

    Moreover,

    η(t)+μu_x(t,η(t))=12t+ϱ+μm(t+ϱ)k+12φ(1)0,  t(0,T),η(t)μu_x(t,η(t))=12t+ϱμm(t+ϱ)k+12φ(1)0,  t(0,T).

    If u0 is sufficiently large such that u_(0,x)=mϱkφ(xϱ)u0(x) for x[ϱ,ϱ], then we have

    {u_tdu_xxβu_xau_+cv_,0<tT, η(t)<x<η(t),v_tbv_+G(u_),0<tT, η(t)<x<η(t),u_(t,x)=v_(t,x)=0,0tT, xη(t),u_(t,x)=v_(t,x)=0,0tT, xη(t),η(t)μu_x(t,η(t)),0<tT,η(t)μu_x(t,η(t)),0<tT,u_(0,x)u0(x), v_(0,x)v0(x),η(0)<x<η(0).

    Noting that η(0)=ϱh0, we can use Remark 2.5 to conclude that h(t)η(t) and g(t)η(t) in [0,T]. Specially, we obtain h(T)η(T)=T+ϱ>T and g(T)<T. Then

    (l,l)(T,T)(g(t),h(t)),  tT.

    Hence, we have hg=+ by Lemma 3.8.

    Next, we present the sharp criteria on initial value, which separates spreading and vanishing.

    Theorem 3.10. For some γ>0 and ω1 and ω2 in X(h0), let (u,v,g,h) be a solution of (1.4) with (u0,v0)=γ(ω1,ω2), then the following statements holds:

    (ⅰ) Assume that cG(0)ab<1. If G(z)z<abc for any z>0, then vanishing will happen. If G(z1)z1>abc for some z1>0, then vanishing will happen for u0 and v0 sufficiently small.

    (ⅱ) Assume that cG(0)ab>1 and 0<β<2d(cG(0)ba). If G(z)z<abc for any z>0, then vanishing will happen. If G(z1)z1>abc for some z1>0, then the following will hold:

    (a) If h0>l, then spreading will happen; (b) If h0<l, then there exists γ(0,) such that spreading occurs for γ>γ, and vanishing happens for 0<γγ.

    Proof. This theorem follows from Lemma 3.5, Corollary 3.7, Lemmas 3.8 and 3.9. The conclusion (b) can be proven by the same arguments in [23,Theorem 4.3].

    Finally, we give the asymptotic behavior of (1.4) when spreading happens.

    Theorem 3.11. Assume that cG(0)ab>1, 0<β<2d(cG(0)ba) and G(z1)z1>abc for some z1>0. If hg=, then

    (u_(x),v_(x))lim inft(u(t,x),v(t,x))lim supt(u(t,x),v(t,x))(u,v)

    for xR, where (u_(x),v_(x)) will be given in the proof.

    Proof. We denote by (u(t),v(t)) the solution of (1.1) with

    u(0)=u0L([h0,h0]) and v(0)=v0L([h0,h0]).

    Applying the comparison principle gives

    (u(t,x),v(t,x))(u(t),v(t)) for t>0 and g(t)xh(t).

    Since cG(0)ab>1, limt(u(t),v(t))=(u,v). Hence,

    lim supt(u(t,x),v(t,x))(u,v) uniformly for xR.

    By Lemma 2.7, we can find some L>l such that λ0(L)<0, where λ0(L) is the principal eigenvalue of problem (2.23) with l=L and ϕ(x) is the corresponding eigenfunction. For such L, it follows from hg= that there exists TL such that

    [L,L][g(t),h(t)], tTL.

    Let (u_(t,x),v_(t,x))=δ(ϕ(x),(G(0)b+λ04c)ϕ(x)), then we can choose small δ such that

    {u_tdu_xx+βu_x+au_cv_0,t>TL, L<x<L,v_t+bv_G(u_)0,t>TL, L<x<L,u_(t,x)=v_(t,x)=0,tTL, x=L or x=L,u_(TL,x)u(TL,x), v_(TL,x)v(TL,x),L<x<L.

    Applying the comparison principle gives that

    (u(t,x),v(t,x))δ(ϕ(x),(G(0)b+λ04c)ϕ(x)), tTL, LxL.

    We extend δ(ϕ(x),(G(0)b+λ04c)ϕ(x)) to (u_(x),v_(x)) by defining

    (u_(x),v_(x))={δ(ϕ(x),(G(0)b+λ04c)ϕ(x)),LxL,0,x<L or x>L.

    Then we have lim inft(u(t,x),v(t,x))(u_(x),v_(x)) for xR.

    In this paper, we have dealt with a partially degenerate epidemic model with free boundaries and small advection. At first, we obtain the global existence and uniqueness of the solution. Then the effect of small advection is considered. We have proved that the results is similar to that in [20,23] under the condition 0<β<β. But we should explain that, for the case that cG(0)ab>1 and β2d(cG(0)ba), the criteria for spreading and vanishing is hard to get by using the results of eigenvalue problem to construct the suitable upper and lower solution. We will study it in the future. When spreading occurs, the precise long-time behavior also needs a further consideration.

    In order to study the spreading of disease, the asymptotic spreading speed of the spreading fronts is one of the most important subjects. To estimate the precise asymptotic spreading speed, we need to study the corresponding semi-wave problem or some other new technique. This may be not an easy task and deserves further study. We will consider it in another paper.

    Due to the advection term, we find that the spreading barrier l becomes larger if we increase the size of β for β(0,β). This means that if β(0,β), the more lager the size of advection is, the more difficult the disease will spread. This result may provide us a suggestion in controlling and preventing the disease. It may be an effective measure to make the infectious agents move along a certain direction by artificial means.

    We are very grateful to the anonymous referee for careful reading and helpful comments which led to improvements of our original manuscript. The first author was supported by FRFCU (lzujbky-2017-it55) and the second author was partially supported by NSF of China (11731005, 11671180).

    The authors declare there is no conflict of interest.


    Acknowledgments



    We would like to acknowledge the support provided by the Bioinformatics tools, software & databases in the completion of the work. We would also like to thank Amity University Uttar Pradesh, Lucknow Campus, where all the benchwork of the present study was conducted.

    Conflict of interest



    The authors declare no conflict of interest.

    [1] Jain A, Mitra P (2023) Bipolar disorder. Available from: https://www.ncbi.nlm.nih.gov/books/NBK558998/.
    [2] Hilty DM, Leamon MH, Lim RF, et al. (2006) A review of bipolar disorder in adults. Psychiatry (Edgmont) 3: 43-55.
    [3] Zhang N, Hu G, Myers TG, et al. (2019) Protocols for the analysis of microRNA expression, biogenesis, and function in immune cells. Curr Protoc Immunol 126: e78. https://doi.org/10.1002/cpim.78
    [4] Filipowicz W, Bhattacharyya SN, Sonenberg N (2008) Mechanisms of post-transcriptional regulation by microRNAs: are the answers in sight?. Nat Rev Genet 9: 102-114. https://doi.org/10.1038/nrg229
    [5] Anjum A, Jaggi S, Varghese E, et al. (2016) Identification of differentially expressed genes in RNA-seq data of arabidopsis thaliana: A compound distribution approach. J Comput Biol 23: 239-247. https://doi.org/10.1089/cmb.2015.0205
    [6] Rodriguez-Esteban R, Jiang X (2017) Differential gene expression in disease: a comparison between high-throughput studies and the literature. BMC Med Genomics 10: 59. https://doi.org/10.1186/s12920-017-0293-y
    [7] Pfaffenseller B, da Silva Magalhães PV, De Bastiani MA, et al. (2016) Differential expression of transcriptional regulatory units in the prefrontal cortex of patients with bipolar disorder: potential role of early growth response gene 3. Transl Psychiatry 6: e805. https://doi.org/10.1038/tp.2016.78
    [8] Machado-Vieira R, Manji HK, Zarate CA, et al. (2009) The role of lithium in the treatment of bipolar disorder: convergent evidence for neurotrophic effects as a unifying hypothesis. Bipolar disord 11: 92-109. https://doi.org/10.1111/j.1399-5618.2009.00714.x
    [9] Qureshi NA, Al-Bedah AM (2013) Mood disorders and complementary and alternative medicine: A literature review. Neuropsychiatr Dis Treat 2013: 639-658. https://doi.org/10.2147/NDT.S43419
    [10] Clough E, Barrett T (2016) The gene expression omnibus database. Statistical genomics . New York: Humana Press 93-110. https://doi.org/10.1007/978-1-4939-3578-9_5
    [11] Leinonen R, Sugawara H, Shumway M (2011) International Nucleotide Sequence Database Collaboration. The sequence read archive. Nucleic Acids Res : D19-D21. https://doi.org/10.1093/nar/gkq1019
    [12] FastQC: A quality control tool for high throughput sequence data. Available from: http://www.bioinformatics.babraham.ac.uk/projects/fastqc
    [13] Jalili V, Afgan E, Gu Q, et al. (2020) The Galaxy platform for accessible, reproducible and collaborative biomedical analyses: 2020 update. Nucleic Acids Res 48: 8205-8207. https://doi.org/10.1093/nar/gkaa554
    [14] Friedländer MR, Mackowiak SD, Li N, et al. (2012) miRDeep2 accurately identifies known and hundreds of novel microRNA genes in seven animal clades. Nucleic Acids Res 40: 37-52. https://doi.org/10.1093/nar/gkr688
    [15] Kim D, Langmead B, Salzberg SL (2015) HISAT: A fast spliced aligner with low memory requirements. Nat Methods 12: 357-360. https://doi.org/10.1038/nmeth.3317
    [16] Mackowiak SD (2011) Identification of novel and known miRNAs in deep-sequencing data with miRDeep2. Curr Protoc Bioinformatics . https://doi.org/10.1002/0471250953.bi1210s36
    [17] Liao Y, Smyth GK, Shi W (2014) featureCounts: An efficient general purpose program for assigning sequence reads to genomic features. Bioinformatics 30: 923-930. https://doi.org/10.1093/bioinformatics/btt656
    [18] Love MI, Huber W, Anders S (2014) Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biol 15: 550. https://doi.org/10.1186/s13059-014-0550-8
    [19] Riffo-Campos ÁL, Riquelme I, Brebi-Mieville P (2016) Tools for sequence-based miRNA target prediction: What to choose?. Int J Mol Sci 17: 1987. https://doi.org/10.3390/ijms17121987
    [20] Venny—An interactive tool for comparing lists with Venn's diagrams. Available from: https://bioinfogp.cnb.csic.es/tools/venny/index.html
    [21] Molinspiration cheminformatics. Available from: https://molinspiration.com/
    [22] Banerjee P, Eckert OA, Schrey AK, et al. (2018) ProTox-II: A webserver for the prediction of toxicity of chemicals. Nucleic Acids Res 46: W257-W263. https://doi.org/10.1093/nar/gky318
    [23] Rizvi SMD, Shakil S, Haneef M (2013) A simple click by click protocol to perform docking: AutoDock 4.2 made easy for non-bioinformaticians. EXCLI J 12: 831-857.
    [24] Pacifico R, Davis RL (2017) Transcriptome sequencing implicates dorsal striatum-specific gene network, immune response and energy metabolism pathways in bipolar disorder. Mol Psychiatry 22: 441-449. https://doi.org/10.1038/mp.2016.94
    [25] Pai S, Li P, Killinger B, et al. (2019) Differential methylation of enhancer at IGF2 is associated with abnormal dopamine synthesis in major psychosis. Nat Commun 10: 2046. https://doi.org/10.1038/s41467-019-09786-7
    [26] Hu J, Xu J, Pang L, et al. (2016) Systematically characterizing dysfunctional long intergenic non-coding RNAs in multiple brain regions of major psychosis. Oncotarget 7: 71087-71098. https://doi.org/10.18632/oncotarget.12122
    [27] Xu Z, Adilijiang A, Wang W, et al. (2019) Arecoline attenuates memory impairment and demyelination in a cuprizone-induced mouse model of schizophrenia. Neuroreport 30: 134-138. https://doi.org/10.1097/WNR.0000000000001172
    [28] Suresh P, Raju AB (2013) Antidopaminergic effects of leucine and genistein on shizophrenic rat models. Neurosciences 18: 235-241.
    [29] Lin JC, Lee MY, Chan MH, et al. (2016) Betaine enhances antidepressant-like, but blocks psychotomimetic effects of ketamine in mice. Psychopharmacology (Berl) 233: 3223-3235. https://doi.org/10.1007/s00213-016-4359-x
    [30] Ben-Azu B, Aderibigbe AO, Omogbiya IA, et al. (2018) Morin pretreatment attenuates schizophrenia-like behaviors in experimental animal models. Drug Res (Stuttg) 68: 159-167. https://doi.org/10.1055/s-0043-119127
    [31] Kumar G, Patnaik R (2016) Exploring neuroprotective potential of Withania somnifera phytochemicals by inhibition of GluN2B-containing NMDA receptors: An in silico study. Med Hypotheses 92: 35-43. https://doi.org/10.1016/j.mehy.2016.04.034
    [32] Chen W, Qi J, Feng F, et al. (2014) Neuroprotective effect of allicin against traumatic brain injury via Akt/endothelial nitric oxide synthase pathwaymediated anti-inflammatory and anti-oxidative activities. Neurochem Int 68: 28-37. https://doi.org/10.1016/j.neuint.2014.01.015
    [33] Zhu HT, Bian C, Yuan JC, et al. (2014) Curcumin attenuates acute inflammatory injury by inhibiting the TLR4/MyD88/NF-κB signaling pathway in experimental traumatic brain injury. J Neuroinflammation 11: 59. https://doi.org/10.1186/1742-2094-11-59
    [34] Krishnamurthy RG, Senut MC, Zemke D, et al. (2009) Asiatic acid, a pentacyclic triterpene from Centella asiatica, is neuroprotective in a mouse model of focal cerebral ischemia. J Neurosci Res 87: 2541-2550. https://doi.org/10.1002/jnr.22071
    [35] Chandrasekaran K, Mehrabian Z, Spinnewyn B, et al. (2001) Neuroprotective effects of bilobalide, a component of the Ginkgo biloba extract (EGb 761), in gerbil global brain ischemia. Brain Res 922: 282-292. https://doi.org/10.1016/S0006-8993(01)03188-2
    [36] Zhao J, Kobori N, Aronowski J, et al. (2006) Sulforaphane reduces infarct volume following focal cerebral ischemia in rodents. Neurosci Lett 393: 108-112. https://doi.org/10.1016/j.neulet.2005.09.065
    [37] Lakstygal AM, Kolesnikova TO, Khatsko SL, et al. (2019) DARK classics in chemical neuroscience: Atropine, scopolamine, and other anticholinergic deliriant hallucinogens. ACS Chem Neurosci 10: 2144-2159. https://doi.org/10.1021/acschemneuro.8b00615
    [38] Huang SS, Tsai MC, Chih CL, et al. (2001) Resveratrol reduction of infarct size in Long-Evans rats subjected to focal cerebral ischemia. Life Sci 69: 1057-1065. https://doi.org/10.1016/S0024-3205(01)01195-X
    [39] Leiderman E, Zylberman I, Zukin SR, et al. (1996) Preliminary investigation of high-dose oral glycine on serum levels and negative symptoms in schizophrenia: An open-label trial. Biol Psychiatry 39: 213-215. https://doi.org/10.1016/0006-3223(95)00585-4
    [40] Hannan MA, Rahman MA, Sohag AAM, et al. (2021) Black cumin (Nigella sativa L.): A comprehensive review on phytochemistry, health benefits, molecular pharmacology, and safety. Nutrients 13: 1784. https://doi.org/10.3390/nu13061784
    [41] Yadav M, Jindal DK, Dhingra MS, et al. (2018) Protective effect of gallic acid in experimental model of ketamine-induced psychosis: Possible behaviour, biochemical, neurochemical and cellular alterations. Inflammopharmacology 26: 413-424. https://doi.org/10.1007/s10787-017-0366-8
    [42] Mukherjee PK, Kumar V, Mal M, et al. (2007) In vitro acetylcholinesterase inhibitory activity of the essential oil from Acorus calamus and its main constituents. Planta Med 73: 283-285. https://doi.org/10.1055/s-2007-967114
    [43] Azimi A, Ghaffari SM, Riazi GH, et al. (2016) α-Cyperone of Cyperus rotundus is an effective candidate for reduction of inflammation by destabilization of microtubule fibers in brain. J Ethnopharmacol 194: 219-227. https://doi.org/10.1016/j.jep.2016.06.058
    [44] Alhebshi AH, Gotoh M, Suzuki I (2013) Thymoquinone protects cultured rat primary neurons against amyloid β-induced neurotoxicity. Biochem Biophys Res Commun 433: 362-367. https://doi.org/10.1016/j.bbrc.2012.11.139
    [45] Fuentes RG, Arai MA, Sadhu SK, et al. (2015) Phenolic compounds from the bark of Oroxylum indicum activate the Ngn2 promoter. J Nat Med 69: 589-594. https://doi.org/10.1007/s11418-015-0919-3
    [46] Rayan NA, Baby N, Pitchai D (2011) Costunolide inhibits proinflammatory cytokines and iNOS in activated murine BV2 microglia. Front Biosci (Elite Ed) 3: 1079-1091. https://doi.org/10.2741/e312
    [47] Khanra R, Dewanjee S, Dua TK, et al. (2015) Abroma augusta L. (Malvaceae) leaf extract attenuates diabetes induced nephropathy and cardiomyopathy via inhibition of oxidative stress and inflammatory response. J Transl Med 13: 6. https://doi.org/10.1186/s12967-014-0364-1
    [48] Gray NE, Magana AA, Lak P, et al. (2018) Centella asiatica: Phytochemistry and mechanisms of neuroprotection and cognitive enhancement. Phytochem Rev 17: 161-194. https://doi.org/10.1007/s11101-017-9528-y
    [49] Sarkar T, Salauddin M, Chakraborty R (2020) In-depth pharmacological and nutritional properties of bael (Aegle marmelos): A critical review. J Agric Food Res 2: 100081. https://doi.org/10.1016/j.jafr.2020.100081
    [50] Okugawa H, Ueda R, Matsumoto K, et al. (1995) Effect of α-santalol and β-santalol from sandalwood on the central nervous system in mice. Phytomedicine 2: 119-126. https://doi.org/10.1016/S0944-7113(11)80056-5
    [51] BIOVIA Discovery Studio. Available from: https://www.3ds.com/products/biovia/discovery-studio
    [52] Judd LL, Akiskal HS (2003) The prevalence and disability of bipolar spectrum disorders in the US population: Re-analysis of the ECA database taking into account subthreshold cases. J Affect Disord 73: 123-131. https://doi.org/10.1016/s0165-0327(02)00332-4
    [53] Osby U, Brandt L, Correia N, et al. (2001) Excess mortality in bipolar and unipolar disorder in Sweden. Arch Gen Psychiatry 58: 844-850. https://doi.org/10.1001/archpsyc.58.9.844
    [54] Taguchi YH, Wang H (2018) Exploring microRNA biomarker for amyotrophic lateral sclerosis. Int J Mol Sci 19: 1318. https://doi.org/10.3390/ijms19051318
    [55] Fame RM, MacDonald JL, Dunwoodie SL, et al. (2016) Cited2 regulates neocortical layer II/III generation and somatosensory callosal projection neuron development and connectivity. J Neurosci 36: 6403-6419. https://doi.org/10.1523/JNEUROSCI.4067-15.2016
    [56] Yang C, Zhang K, Zhang A, et al. (2022) Co-expression network modeling identifies specific inflammation and neurological disease-related genes mRNA modules in mood disorder. Front Genet 13: 865015. https://doi.org/10.3389/fgene.2022.865015
    [57] Ng PH, Kim GD, Chan ER, et al. (2020) CITED2 limits pathogenic inflammatory gene programs in myeloid cells. FASEB J 34: 12100-12113. https://doi.org/10.1096/fj.202000864R
    [58] Adnan G, Rubikaite A, Khan M, et al. (2020) The GTPase Arl8B plays a principle role in the positioning of interstitial axon branches by spatially controlling autophagosome and lysosome location. J Neurosci 40: 8103-8118. https://doi.org/10.1523/JNEUROSCI.1759-19.2020
    [59] Bagshaw RD, Callahan JW, Mahuran DJ (2006) The Arf-family protein, Arl8b, is involved in the spatial distribution of lysosomes. Biochem Biophys Res Commun 344: 1186-1191. https://doi.org/10.1016/j.bbrc.2006.03.221
    [60] Boeddrich A, Haenig C, Neuendorf N, et al. (2023) A proteomics analysis of 5xFAD mouse brain regions reveals the lysosome-associated protein Arl8b as a candidate biomarker for Alzheimer's disease. Genome Med 15: 50. https://doi.org/10.1186/s13073-023-01206-2
    [61] NUDT4 nudix hydrolase 4 [Homo sapiens (human)] (2024). Available from: https://www.ncbi.nlm.nih.gov/gene/11163#summary
    [62] Hua LV, Green M, Warsh JJ, et al. (2001) Molecular cloning of a novel isoform of diphosphoinositol polyphosphate phosphohydrolase: A potential target of lithium therapy. Neuropsychopharmacology 24: 640-651. https://doi.org/10.1016/S0893-133X(00)00233-5
    [63] Ding L, Liu T, Ma J (2023) Neuroprotective mechanisms of Asiatic acid. Heliyon 9: e15853. https://doi.org/10.1016/j.heliyon.2023.e15853
    [64] Zheng CJ, Qin LP (2007) Chemical components of Centella asiatica and their bioactivities. J Chin Integr Med 5: 348-351.
    [65] Rao KGM, Rao SM, Rao SG (2006) Centella asiatica (L.) leaf extract treatment during the growth spurt period enhances hippocampal CA3 neuronal dendritic arborization in rats. Evid-Based Complement Alternat Med 3: 349-357. https://doi.org/10.1093/ecam/nel024
    [66] Subathra M, Shila S, Devi MA, et al. (2005) Emerging role of Centella asiatica in improving age-related neurological antioxidant status. Exp Gerontol 40: 707-715. https://doi.org/10.1016/j.exger.2005.06.001
    [67] Lu CW, Lin TY, Pan TL, et al. (2021) Asiatic acid prevents cognitive deficits by inhibiting calpain activation and preserving synaptic and mitochondrial function in rats with kainic acid-induced seizure. Biomedicines 9: 284. https://doi.org/10.3390/biomedicines9030284
    [68] Kato T, Kato N (2000) Mitochondrial dysfunction in bipolar disorder. Bipolar Disord 2: 180-190. https://doi.org/10.1034/j.1399-5618.2000.020305.x
    [69] Ding H, Xiong Y, Sun J, et al. (2018) Asiatic acid prevents oxidative stress and apoptosis by inhibiting the translocation of α-synuclein into mitochondria. Front Neurosci 12: 431. https://doi.org/10.3389/fnins.2018.00431
    [70] Krishnamurthy RG, Senut MC, Zemke D, et al. (2009) Asiatic acid, a pentacyclic triterpene from Centella asiatica, is neuroprotective in a mouse model of focal cerebral ischemia. J Neurosci Res 87: 2541-2550. https://doi.org/10.1002/jnr.22071
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