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Research article Special Issues

Characterization of Lysobacter enzymogenes B25, a potential biological control agent of plant-parasitic nematodes, and its mode of action

  • Received: 09 December 2022 Revised: 16 February 2023 Accepted: 22 February 2023 Published: 01 March 2023
  • It is certainly difficult to estimate productivity losses due to the action of phytopathogenic nematodes but it might be about 12 % of world agricultural production. Although there are numerous tools to reduce the effect of these nematodes, there is growing concern about their environmental impact. Lysobacter enzymogenes B25 is an effective biological control agent against plant-parasitic nematodes, showing control over root-knot nematodes (RKN) such as Meloidogyne incognita and Meloidogyne javanica. In this paper, the efficacy of B25 to control RKN infestation in tomato plants (Solanum lycopersicum cv. Durinta) is described. The bacterium was applied 4 times at an average of concentration around 108 CFU/mL showing an efficacy of 50–95 % depending on the population and the pressure of the pathogen. Furthermore, the control activity of B25 was comparable to that of the reference chemical used. L. enzymogenes B25 is hereby characterized, and its mode of action studied, focusing on different mechanisms that include motility, the production of lytic enzymes and secondary metabolites and the induction of plant defenses. The presence of M. incognita increased the twitching motility of B25. In addition, cell-free supernatants obtained after growing B25, in both poor and rich media, showed efficacy in inhibiting RKN egg hatching in vitro. This nematicidal activity was sensitive to high temperatures, suggesting that it is mainly due to extracellular lytic enzymes. The secondary metabolites heat-stable antifungal factor and alteramide A/B were identified in the culture filtrate and their contribution to the nematicidal activity of B25 is discussed. This study points out L. enzymogenes B25 as a promising biocontrol microorganism against nematode infestation of plants and a good candidate to develop a sustainable nematicidal product.

    Citation: Sònia Martínez-Servat, Lola Pinyol-Escala, Oriol Daura-Pich, Marta Almazán, Iker Hernández, Belén López-García, Carolina Fernández. Characterization of Lysobacter enzymogenes B25, a potential biological control agent of plant-parasitic nematodes, and its mode of action[J]. AIMS Microbiology, 2023, 9(1): 151-176. doi: 10.3934/microbiol.2023010

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  • It is certainly difficult to estimate productivity losses due to the action of phytopathogenic nematodes but it might be about 12 % of world agricultural production. Although there are numerous tools to reduce the effect of these nematodes, there is growing concern about their environmental impact. Lysobacter enzymogenes B25 is an effective biological control agent against plant-parasitic nematodes, showing control over root-knot nematodes (RKN) such as Meloidogyne incognita and Meloidogyne javanica. In this paper, the efficacy of B25 to control RKN infestation in tomato plants (Solanum lycopersicum cv. Durinta) is described. The bacterium was applied 4 times at an average of concentration around 108 CFU/mL showing an efficacy of 50–95 % depending on the population and the pressure of the pathogen. Furthermore, the control activity of B25 was comparable to that of the reference chemical used. L. enzymogenes B25 is hereby characterized, and its mode of action studied, focusing on different mechanisms that include motility, the production of lytic enzymes and secondary metabolites and the induction of plant defenses. The presence of M. incognita increased the twitching motility of B25. In addition, cell-free supernatants obtained after growing B25, in both poor and rich media, showed efficacy in inhibiting RKN egg hatching in vitro. This nematicidal activity was sensitive to high temperatures, suggesting that it is mainly due to extracellular lytic enzymes. The secondary metabolites heat-stable antifungal factor and alteramide A/B were identified in the culture filtrate and their contribution to the nematicidal activity of B25 is discussed. This study points out L. enzymogenes B25 as a promising biocontrol microorganism against nematode infestation of plants and a good candidate to develop a sustainable nematicidal product.



    In 1903, Mittag-Leffler [22] provided the function Eσ(z) defined by

    Eσ(z)=j=0 zjΓ(σj+1),(σ,zC,R(σ)>0),

    where Γ is the gamma function and R means the real part.

    Wiman [34] introduced the following generalized Mittag-Leffler function

    Eσ,μ(z)=j=0 zjΓ(σj+μ),(σ,μ,zC,[R(σ),R(μ)]>0).

    Prabhakar [25] introduced the following function Eρσ,μ(z) in the form

    Eρσ,μ(z)=j=0 (ρ)jΓ(μ+σj).zjj!,   (σ,μ,ρ,zC,[R(σ),R(μ),R(ρ)]>0).

    Later, Shukla and Prajapati [27] (see also [32]) defined another generalized Mittag-Leffler function

    Eρ,kσ,μ(z)=j=0 (ρ)kjΓ(μ+σj)zjj!,(σ,μ,ρ,zC,[R(σ),R(μ),R(ρ)]>0)

    where k(0,1)N and (ρ)kj=Γ(ρ+kj)Γ(ρ) is the generalized Pochhammer symbol defined as

    kkjkm=1(ρ+m1k)j if kN.

    Bansal and Prajapat [5] and Srivastava and Bansal [31] investigated geometric properties of the Mittag-Leffler function Eσ,μ(z), including starlikeness, convexity, and close-to-convexity (see [1,4,6,8,12,13,17,28,29]). In reality, the generalized Mittag-Leffler function Eσ,μ(z) and its extensions are still widely used in geometric function theory and in a variety of applications (see, for details, [2,3,7,16,24]).

    Let S(p) be the class of functions of the form

    f(z)=zp+j=p+1ajzj, (1.1)

    where f is holomorphic and multivalent in the open unit disk O={z:|z|<1}.

    Let f and F be two functions in S(p). Then the convolution (or Hadamard product), denoted by fF, is defined as

    (fF)(z)=zp+j=p+1ajdjzj=(Ff)(z),

    where f(z) is in (1.1) and F(z)=zp+j=p+1djzj.

    Let f(z) and h(z) be two analytic functions defined in O. The function f(z) is called subordinate to h(z), or h(z) is superordinate to f(z), denoted by f(z)h(z) and h(z)f(z), respectively, if there is a Schwarz function φ with φ(z)=0,|φ(z)|<1 and f(z)=h(φ(z)). If the function h is univalent in O, then the following equivalence is true if

    f(z)h(z)  (zO)f(0)=h(0) and f(O)h(O).

    Definition 1.1. ([18]). Let 0<q<1. Then [j]q! denotes the q-factorial, which is defined as follows:

    [j]q!={[j]q[j1]q[2]q[1]q,    j=1,2,3,1,    j=0

    where [j]q=1qj1q=1+j1m=1 qm and [0]q=0.

    Definition 1.2 ([18]). The q-generalized Pochhammer symbol [ρ]j,q, ρC, is given as

    [ρ]j,q=[ρ]q[ρ+1]q[ρ+2]q[ρ+j1]q,

    and the q-Gamma function is defined as

    Γq(ρ+1)=[ρ]qΓq(ρ) and Γq(1)=1.

    It follows that Γq(j+1)=[j]q!.

    Lately, many results have been given for some related special functions such as the Wright function [3] and multivalent functions (see [10,23,26]).

    Here, we propose a q-extension of specific extensions of the Mittag-Leffler function, motivated by the success of Mittag-Leffler function applications in physics, biology, engineering, and applied sciences. We generalize the Mittag-Leffler function given by Shukla and Prajapati [27] and obtain a new generalized q-Mittag-Leffler function.

    Now, we present a new generalized q-Mittag-Leffler function as follows

    Eρσ,μ(q;z)=z+j=2 (ρ)kjΓq(μ+σj)zjj!. (1.2)

    It is obvious that, when q1, the resulting function is the generalized Mittag-Leffler function, which is given by Shukla and Prajapati [27].

    Corresponding to the function Eρσ,μ(q;z) in (1.2), we establish the following generalized q-Mittag-Leffler function Eρσ,μ(p,q;z) in multivalent functions S(p), as given below

    Eρσ,μ(p,q;z)=zp+j=p+1 (ρ)k(jp)Γq(μ+σ(jp))zj(jp)!. (1.3)

    Again, using the new function (1.3), we define the following function:

    Gρσ,μ(p,q;z):=zpΓq(μ)Eρσ,μ(p,q;z)=zp+j=p+1 Γq(μ)(ρ)k(jp)Γq(μ+σ(jp))zj(jp)!. (1.4)

    Definition 1.3. For fS(p), we define the new linear operator Aμ,ρ;kσ;p,qf(z):S(p)S(p) by

    Aμ,ρ;kσ;p,qf(z)=Gρσ,μ(p,q;z)f(z)=zp+j=p+1 χjajzj, (1.5)

    where χj=Γq(μ)(ρ)kjΓq(μ+σj)j!.

    We now define a subclass Qμ,ρ;kσ;q(M,N;τ,p) of the family S(p) using the multivalent linear operator in (1.5) and the subordination concept.

    Definition 1.4. Let Aμ,ρ;kσ;p,qf(z) be an operator in (1.5). A function f(z)S(p) is said to be in the class Qμ,ρ;kσ;q(M,N;τ,p) if satisfies the following subordination condition:

    1pτ(z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)τ)1+Mz1+Nz,  (zO) (1.6)

    or equivalently

    z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)p+(pN+(MN)(pτ))z1+Nz,  (zO)

    and

    |z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)pNz(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)[pN+(MN)(pτ)]|<1, (1.7)

    where 1M<N1, 0τ<p, and pN.

    Remark 1.1. Some well-known special classes of the class Qμ,ρ;kσ;q(M,N;τ,p) can be obtained by choosing the values of the parameters ς,μ,ρ;τ,k,p,q, M, and N.

    (1) Q0,0,10,1(M,N;τ,p)=Sp(M,N;τ,p) was provided by Aouf [2].

    (2) Q0,0,10,1(M,N;0,p)=Sp(M,N;p) was provided by Goel and Sohi [16].

    In this work, we introduce a new subclass of multivalent functions Qμ,ρ;kσ;q(M,N;τ,p) defined by the new linear operator Aμ,ρ;kσ;p,qf(z). And we study some geometric properties for the class Qμ,ρ;kσ;q(M,N;τ,p) such as the coefficient estimates, convexity and convex linear combination. Finally, the radius theorems associated with the generalized Srivastava-Attiya integral operator will be investigated.

    The first theorem in this section presents the necessary and sufficient condition for the function f(z) in (1.1) belong to the class Qμ,ρ;kσ;q(M,N;τ,p).

    Theorem 2.1. A function f(z) is in the class Qμ,ρ;kσ;q(M,N;τ,p) if and only if

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj|aj|(MN)(pτ), (2.1)

    where 1M<N1, 0τ<p, and pN.

    Proof. Assume that the condition (2.1) is true. Then by (1.7), we have

    |z(Aμ,ρ;kσ;p,qf(z))pAμ,ρ;kσ;p,qf(z)||Nz(Aμ,ρ;kσ;p,qf(z))[(MN)(pτ)+pN]Aμ,ρ;kσ;p,qf(z)|=|j=p+1(jp)χjajzj||(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj|(MN)(pτ)+j=p+1[(1+N)(jp)+((MN)(pτ))]χj|aj|0.

    By maximum modulus theorem [11], we get f(z)Qμ,ρ;kσ;q(M,N;τ,p).

    Conversely, suppose that f(z)Qμ,ρ;kσ;q(M,N;τ,p). Then

    |z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)pNz(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)[pN+(MN)(pτ)]|=|j=p+1(jp)χjajzj(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj|<1.

    Since R(z)|z|, we get

    R{j=p+1(jp)χjajzj(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj}<1.

    Taking z1, we have

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj|aj|(MN)(pτ).

    This completes the proof.

    Theorem 2.2. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p). Then f1f2Qμ,ρ;kσ;q(M,N;τ,p), where

    τ1=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1, (2.2)

    where χ1=Γq(μ)(ρ)kΓq(μ+ς).

    Proof. We will show that τ1 is the largest satisfying

    j=p+1 ((1+N)(jp)+(MN)(pτ1))χj(MN)(pτ1)aj,1aj,21. (2.3)

    Since f1,f2Qμ,ρ;kσ;q(M,N;τ,p), by the condition (2.1) and the Cauchy-Schwarz inequality, we get

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,1aj,21. (2.4)

    From (2.3) and (2.4), we observe that

    aj,1aj,2[((1+N)(jp)+(MN)(pτ))χj](pτ1)[((1+N)(jp)+(MN)(pτ1))χj](pτ).

    From (2.4), it is necessary to prove

    (MN)(pτ)((1+N)(jp)+(MN)(pτ))χj[((1+N)(jp)+(MN)(pτ))χj](pτ1)[((1+N)(jp)+(MN)(pτ1))χj](pτ). (2.5)

    Furthermore, from the inequality (2.5) it follows that

    τ1p(jp)(1+N)(MN)(pτ)2χj[((1+N)(jp)+(MN)(pτ1))χj]2(MN)2(pτ)2χj.

    Now, set

    E(j)=p(jp)(1+N)(MN)(pτ)2χj[((1+N)(jp)+(MN)(pτ1))χj]2(MN)2(pτ)2χj.

    We observe that the function E(j) is increasing for jN. Putting j=1, we have

    τ1=E(1)=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1.

    This completes the proof.

    Theorem 2.3. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p) of forms given in (1.1) with aj,1 and aj,2, respectively. Then

    w(z)=zp+j=p+1(a2j,1+a2j,2)zjQμ,ρ;kσ;q(M,N;τ,p),

    where

    η=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1.

    Proof. By Theorem 2.1, we have

    j=p+1 [((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2a2j,sj=p+1 [((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,s]21, (s=1,2).

    From the above inequality, we obtain

    j=p+1 12[((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2(a2j,1+a2j,2)1.

    Therefore, the largest η can be obtained such that

    ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)12[((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2.

    That is,

    ηp2(jp)(1+N)(MN)(pτ)2χ1[((1+N)(jp)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    Now, set

    E(j)=p2(jp)(1+N)(MN)(pτ)2χ1[((1+N)(jp)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    We observe that the function E(j) is increasing for jN. Putting j=1, we have

    η=E(1)=p2(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    This completes the proof.

    Theorem 2.4. Let f1,f2Qμ,ρ;kσ;q(M,N;τ,p). Then for γ[0,1], the function F(z)=(1γ)f1+γf2 belongs to the class Qμ,ρ;kσ;q(M,N;τ,p).

    Proof. Since the functions f1 and f2 belong to the class Qμ,ρ;kσ;q(M,N;τ,p),

    F(z)=(1γ)f1+γf2=zp+j=p+1ηjzj,

    where ηj=(1γ)aj,1+γaj,2.

    By (2.1), we observe that

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj[(1γ)aj,1+γaj,2]=(1γ)j=p+1 ((1+N)(jp)+(MN)(pτ))χjaj,1+γj=p+1 ((1+N)(jp)+(MN)(pτ))χjaj,2(1γ)(MN)(pτ)+γ(MN)(pτ).

    Hence F(z)Qμ,ρ;kσ;q(M,N;τ,p).

    Theorem 2.5. Let fs(z)=zp+j=p+1aj,szj be in the class Qμ,ρ;kσ;q(M,N;τ,p) for s=1,2,,m. Then the function P(z)=ms=1sfs, where ms=1s=1, is also in the class Qμ,ρ;kσ;q(M,N;τ,p).

    Proof. By Theorem 2.1, we have

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,s1.

    Since

    P(z)=ms=1sfs=ms=1s(zp+j=p+1aj,szj)=zp+j=p+1(ms=1saj,s)zj,
    j=p+1((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)ms=1saj,s1.

    Thus P(z)Qμ,ρ;kσ;q(M,N;τ,p).

    In this section, we investigate radii of multivalent starlikeness, multivalent convexity, and multivalent close-to-convex for the function f(z) in the class Qμ,ρ;kσ;q(M,N;τ,p) with the generalized integral operator of Srivastava-Attiya.

    Jung et al. [19] introduced an integral operator with one parameter as follows:

    Iδ(f)(z):=2δzΓ(δ)z0 (log(zv) )δ1f(v)dv=z+j=2 (2j+1)δajzj(δ>0;fS).

    In 2007, Srivastava and Attiya [30] investigated a new integral operator, which is called Srivastava-Attiya operator, given by

    Ju,mf(z)=z+j=1(1+uj+u)δajzj.

    Many studies are concerned with the study of the operator of Srivastava-Attiya (see [9,14,15,20]).

    Mishra and Gochhayat [21] (also [33]) provided a fractional differintegral operator Jmu,pf(z):S(p)S(p) which is called a generalized of Srivastava-Attiya integral operator, defined by

    Jmu,pf(z)=zp+j=p+1(p+uj+u)δajzj. (3.1)

    Theorem 3.1. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent starlike of order τ in |z|r1, where

    r1=infjp+1{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)(j2p+τ)(p+u)δ}. (3.2)

    Proof. According to the definition of a starlike function in [28], we have

    |z(Jmu,pf(z))Jmu,pf(z)p|pτ, (3.3)
    |z(Jmu,pf(z))Jmu,pf(z)p|=|j=p+1(jp)(p+uj+u)δajzjj=p+1(p+uj+u)δajzj|j=p+1(jp)(p+uj+u)δaj|z|jj=p+1(p+uj+u)δaj|z|j.

    By (3.2), we have

    j=p+1(j2p+τ)(p+u)δaj|z|j(pτ)(j+u)δ1.

    By (2.1) in Theorem 2.1, it is clear that

    (j2p+τ)(p+u)δ(pτ)(j+u)δ|z|j((1+N)(jp)+(MN)(pτ))χj(MN)(pτ).

    Therefore,

    |z|{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)(j2p+τ)(p+u)δ}1j.

    This completes the proof.

    Theorem 3.2. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent convex of order τ in |z|r2, where

    r2=infjp+1{((1+N)(jp)+(MN)(pτ))χjp(j+u)δ(MN)[j(j2p+τ)](p+u)δ}. (3.4)

    Proof. To verify (3.4), it is necessary to prove

    |(1+z(Jmu,pf(z))(Jmu,pf(z)))p|pτ,

    but the result is obtained by repeating the steps in Theorem 3.1.

    Corollary 3.1. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent close-to-convex of order τ in |z|r3, where

    r3=infj1{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)j(p+u)δ}. (3.5)

    In this work, we established and investigated a new generalized Mittag-Leffler function, which is a generalization of q-Mittag-Leffler function defined by Shukla and Prajapati [27]. Also, we studied some of the geometric properties of a certain subclass of multivalent functions. In addition, we introduced radius theorem using a generalized Srivastava-Attiya integral operator. Since the Mittag-Leffler function is of importance, it is related to a wide range of problems in mathematical physics, engineering, and the applied sciences. The results obtained in this article may have many other applications in special functions.

    The authors express many thanks to the Editor-in-Chief, handling editor, and the reviewers for their outstanding comments that improve our paper.

    The authors declare that they have no competing interests concerning the publication of this article.



    Conflict of interest



    The use of L. enzymogenes B25 as a biocontrol agent is subjected to a patent application. BLG, CF, IH, LPE, MA, ODP, and SMS were employed by Futureco Bioscience S.A.

    Author contributions:



    BLG and SMS conceived the work, contributed to the design and interpretation of the results and did the main writing of the article. CF, IH, LPE, and MA partially wrote specific sections. IH, LPE, MA, and ODP performed the experiments presented in the work and analyzed the data. CF supervised the work and revised the manuscript. All the authors have read the manuscript and agreed to its content.

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