Loading [MathJax]/jax/output/SVG/jax.js
Research article

Genomic characterization of a multidrug-resistant uropathogenic Escherichia coli and evaluation of Echeveria plant extracts as antibacterials

  • These two authors contributed equally
  • Received: 27 October 2023 Revised: 22 December 2023 Accepted: 10 January 2024 Published: 17 January 2024
  • Uropathogenic Escherichia coli (UPEC) is the most common bacterial agent associated with urinary tract infections, threatening public health systems with elevated medical costs and high morbidity rates. The successful establishment of the infection is associated with virulence factors encoded in its genome, in addition to antibacterial resistance genes, which could limit the treatment and resolution of the infection. In this sense, plant extracts from the genus Echeveria have traditionally been used to treat diverse infectious diseases. However, little is known about the effects of these extracts on bacteria and their potential mechanisms of action. This study aims to sequence a multidrug-resistant UPEC isolate (UTI-U7) and assess the multilocus sequence typing (MLST), virulence factors, antimicrobial resistance profile, genes, serotype, and plasmid content. Antimicrobial susceptibility profiling was performed using the Kirby-Bauer disk diffusion. The antibacterial and anti-adherent effects of the methanol extracts (ME) of Echeveria (E. craigiana, E. kimnachii, and E. subrigida) against UTI-U7 were determined. The isolate was characterized as an O25:H4-B2-ST2279-CH40 subclone and had resistant determinants to aminoglycosides, β-lactams, fluoroquinolones/quinolones, amphenicols, and tetracyclines, which matched with the antimicrobial resistance profile. The virulence genes identified encode adherence factors, iron uptake, protectins/serum resistance, and toxins. Identified plasmids belonged to the IncF group (IncFIA, IncFIB, and IncFII), alongside several prophage-like elements. After an extensive genome analysis that confirmed the pathogenic status of UTI-U7 isolate, Echeveria extracts were tested to determine their antibacterial effects; as an extract, E. subrigida (MIC, 5 mg/mL) displayed the best inhibitory effect. However, the adherence between UTI-U7 and HeLa cells was unaffected by the ME of the E. subrigida extract.

    Citation: Ana M. Castañeda-Meléndrez, José A. Magaña-Lizárraga, Marcela Martínez-Valenzuela, Aldo F. Clemente-Soto, Patricia C. García-Cervantes, Francisco Delgado-Vargas, Rodolfo Bernal-Reynaga. Genomic characterization of a multidrug-resistant uropathogenic Escherichia coli and evaluation of Echeveria plant extracts as antibacterials[J]. AIMS Microbiology, 2024, 10(1): 41-61. doi: 10.3934/microbiol.2024003

    Related Papers:

    [1] Changgui Gu, Ping Wang, Tongfeng Weng, Huijie Yang, Jos Rohling . Heterogeneity of neuronal properties determines the collective behavior of the neurons in the suprachiasmatic nucleus. Mathematical Biosciences and Engineering, 2019, 16(4): 1893-1913. doi: 10.3934/mbe.2019092
    [2] Miguel Lara-Aparicio, Carolina Barriga-Montoya, Pablo Padilla-Longoria, Beatriz Fuentes-Pardo . Modeling some properties of circadian rhythms. Mathematical Biosciences and Engineering, 2014, 11(2): 317-330. doi: 10.3934/mbe.2014.11.317
    [3] Ying Li, Zhao Zhao, Yuan-yuan Tan, Xue Wang . Dynamical analysis of the effects of circadian clock on the neurotransmitter dopamine. Mathematical Biosciences and Engineering, 2023, 20(9): 16663-16677. doi: 10.3934/mbe.2023742
    [4] Yanqin Wang, Xin Ni, Jie Yan, Ling Yang . Modeling transcriptional co-regulation of mammalian circadian clock. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1447-1462. doi: 10.3934/mbe.2017075
    [5] Zhenzhen Shi, Huidong Cheng, Yu Liu, Yanhui Wang . Optimization of an integrated feedback control for a pest management predator-prey model. Mathematical Biosciences and Engineering, 2019, 16(6): 7963-7981. doi: 10.3934/mbe.2019401
    [6] Ya'nan Wang, Sen Liu, Haijun Jia, Xintao Deng, Chunpu Li, Aiguo Wang, Cuiwei Yang . A two-step method for paroxysmal atrial fibrillation event detection based on machine learning. Mathematical Biosciences and Engineering, 2022, 19(10): 9877-9894. doi: 10.3934/mbe.2022460
    [7] E.V. Presnov, Z. Agur . The Role Of Time Delays, Slow Processes And Chaos In Modulating The Cell-Cycle Clock. Mathematical Biosciences and Engineering, 2005, 2(3): 625-642. doi: 10.3934/mbe.2005.2.625
    [8] Jun Zhou . Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences and Engineering, 2016, 13(4): 857-885. doi: 10.3934/mbe.2016021
    [9] Hussein Obeid, Alan D. Rendall . The minimal model of Hahn for the Calvin cycle. Mathematical Biosciences and Engineering, 2019, 16(4): 2353-2370. doi: 10.3934/mbe.2019118
    [10] Ryotaro Tsuneki, Shinji Doi, Junko Inoue . Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators. Mathematical Biosciences and Engineering, 2014, 11(1): 125-138. doi: 10.3934/mbe.2014.11.125
  • Uropathogenic Escherichia coli (UPEC) is the most common bacterial agent associated with urinary tract infections, threatening public health systems with elevated medical costs and high morbidity rates. The successful establishment of the infection is associated with virulence factors encoded in its genome, in addition to antibacterial resistance genes, which could limit the treatment and resolution of the infection. In this sense, plant extracts from the genus Echeveria have traditionally been used to treat diverse infectious diseases. However, little is known about the effects of these extracts on bacteria and their potential mechanisms of action. This study aims to sequence a multidrug-resistant UPEC isolate (UTI-U7) and assess the multilocus sequence typing (MLST), virulence factors, antimicrobial resistance profile, genes, serotype, and plasmid content. Antimicrobial susceptibility profiling was performed using the Kirby-Bauer disk diffusion. The antibacterial and anti-adherent effects of the methanol extracts (ME) of Echeveria (E. craigiana, E. kimnachii, and E. subrigida) against UTI-U7 were determined. The isolate was characterized as an O25:H4-B2-ST2279-CH40 subclone and had resistant determinants to aminoglycosides, β-lactams, fluoroquinolones/quinolones, amphenicols, and tetracyclines, which matched with the antimicrobial resistance profile. The virulence genes identified encode adherence factors, iron uptake, protectins/serum resistance, and toxins. Identified plasmids belonged to the IncF group (IncFIA, IncFIB, and IncFII), alongside several prophage-like elements. After an extensive genome analysis that confirmed the pathogenic status of UTI-U7 isolate, Echeveria extracts were tested to determine their antibacterial effects; as an extract, E. subrigida (MIC, 5 mg/mL) displayed the best inhibitory effect. However, the adherence between UTI-U7 and HeLa cells was unaffected by the ME of the E. subrigida extract.



    1. Introduction

    Wide-type fruit flies, Drosophila melanogaster, might be the most extensively studied organism in circadian rhythm research. The researches of endogenous activity rhythm on Drosophila generally involve two different kinds of clock genes, called period (per, for short) [14,10] and timeless (tim, for short) [20,27]. Their encoded proteins, PER and TIM, bind to each other [5,0,27,29].

    PER protein and per mRNA cycle in a 24-hour period [7]. When PER protein is at a high level, per mRNA expression is repressed, suggesting that PER is an inhibitor of per mRNA accumulation [7]. The expression of per and tim genes is regulated by dCLOCK and CYC, and PER inhibits the transcription of per and tim by inactivating dCLOCK and CYC [1,3,18]. This negative feedback, introduced by PER inhibiting its own mRNA transcription, is the basis of many classic theoretical models of circadian rhythms [6,17,15,19].

    An alternative way to study circadian rhythms is based on a positive feedback, introduced by PER phosphorylation being an activator to PER [26]. Phosphorylation of PER is operated by a double-time gene encoded kinase, DOUBLE-TIME (DBT, for short) [13,16]. As suggested by the dbt mutants phenotypes, PER phosphorylation might be precluded to its degradation. PER and TIM stimulate transcription of per and tim genes by activating dClOCK [2]. Experimental results suggest that per mRNA is stabilized by PER/TIM dimers [24], and PER is stabilized by dimerization with TIM [13,16].

    The idea that PER phosphorylation introduces a positive feedback in PER accumulation can be expressed in a model of three-dimensional ordinary differential equations [26] (see (1) below). In [26], by imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the three-dimensional model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations (see (2) below). Then they used the powerful phase plane portraiture to study the simplified two-dimensional model. In this paper, we explore the original three-dimensional model directly. It is shown that the circadian rhythms occur if the model possesses a unique equilibrium which is unstable. Furthermore, we deeply investigate how circadian rhythms are affected by several model parameters, including mRNA translation, mRNA degradation, monomer phosphorylation, protein proteolysis, association of PER/TIM protein and equilibrium constant for dimerization. The results help to explain some former-observed phenomena of circadian rhythms. In particular, our numerical results extremely agree with those given in [26], indicating that their reduction work is greatly reasonable.


    2. Mechanism and mathematical model

    In this section, we restate the model proposed by Tyson et al. [26]. The molecular mechanism for the circadian rhythm in Drosophila is summarized in Figure 1. Here the total PER (monomer + dimer) degradation rate does not increase proportionally with the total PER concentration's increasing.

    Figure 1. A simple molecular mechanism for the circadian clock in Drosophila. Redrawn from [26]. PER and TIM proteins are synthesized in the cytoplasm, where they may be destroyed by proteolysis or they may combine to form relatively stable heterodimers. Heteromeric complexes are transported into the nucleus, where they inhibit transcription of per and tim mRNA. Here it is assumed that PER monomers are rapidly phosphorylated by DBT and then degraded. Dimers are assumed to be poorer substrates for DBT.

    The mechanism in Figure 1 could be translated into a set of six differential equations, for per and tim mRNAs, PER and TIM monomers, and PER/TIM dimers in the cytoplasm and nucleus. Such a complicated set of equations could not efficiently illustrate the importance of positive feedback in the reaction mechanism. So by noticing that PER and TIM messages and proteins followed roughly similar time courses in vivo, Tyson et al. [26] lumped them into a single pool of clock proteins. In addition, they assumed that the cytoplasmic and nuclear pools of dimeric protein were in rapid equilibrium. Then they established the following differential equations for [mRNA]=M, [monomer]=P1, and [dimer]=P2:

    {dMdt=vm1+(P2/Pcrit)2kmM,dP1dt=vpMkp1P1JP+P1+rP2kp3P12kaP21+2kdP2,dP2dt=kaP21kdP2kp2P2JP+P1+rP2kp3P2. (1)

    Here monomer was assumed to be phosphorylated more quickly than dimer, i.e., kp1kp2. The parameter r determined the inhibition of dimer to monomer phosphorylation. For convenience, in this paper we follow them by taking r=2.

    In their work, it was further assumed that the dimerization reactions were fast (ka and kd are large) such that monomers and dimers were always in equilibrium with each other. Then, by equilibrium conditions: P2=KeqP21, Keq=ka/kd, they obtained the reduced two-dimensional system:

    {dMdt=vm1+(Pt(1q)/(2Pcrit))2kmM,dPtdt=vpMkp1Ptq+kp2PtJP+Ptkp3Pt, (2)

    where Pt=P1+2P2=[total protein], kp1=kp1kp2kp1, and

    q=q(Pt)=21+1+8KeqPt.

    Two widely concerned points of circadian rhythms are whether the endogenous rhythms exist and how long the periods are. Since the mechanism has already been translated into mathematical models, attentions are drawn to examine the existence of periodic orbits and calculate the periods. In their work, system (2) has been thoroughly analyzed. In this paper, we try to study system (1). A typical oscillating solution of system (1) is illustrated in Figure 2, where the corresponding parameter values are chosen from Table 1.

    Figure 2. Numerical solution of (1). Parameter values are chosen as in Table 1. We take ka=106 and kd=ka/Keq.
    Table 1. Parameter values suitable for circadian rhythm of wild-type fruit flies.
    NameValueUnits Ea/RTDescription
    vm1 Cmh6Maximum rate of synthesis of mRNA
    km0.1 h14First-order rate constant for mRNA degradation
    vp0.5 CpCmh6Rate constant for translation of mRNA
    kp110 Cph6 Vmax for monomer phosphorylation
    kp20.03 Cph6 Vmax for dimer phosphorylation
    kp30.1 h16First-order rate constant for proteolysis
    Keq200 C1p-12Equilibrium constant for dimerization
    Pcrit0.1 Cp6Dimer concen at the half-maximum transcription rate
    JP0.05 Cp-16Michaelis constant for protein kinase (DBT)
    This table is adapted from Tyson et al. [26]. Parameters Cm and Cp represent characteristic concentrations for mRNA and protein, respectively. Ea is the activation energy of each rate constant (necessarily positive) or the standard enthalpy change for each equilibrium binding constant (may be positive or negative). The parameter values are chosen to ensure temperature compensation of the wild-type oscillator.
     | Show Table
    DownLoad: CSV

    3. Method and result

    It is well-known that for higher dimensional ordinary differential equations, there is no so-called Poincaré-Bendixson theory: any limit set is a limit cycle if it contains no steady state. So, in order to use the powerful phase plane analysis tools, Tyson et al. [26] reduced (1) into (2) by imposing some assumptions. Fortunately, we observe that (1) is a three-dimensional competitive system in some sense [8,0,22,23]. For n-dimensional competitive ordinary differential equations, the dynamics is co-dimensional one. Every limit set lies on a Lipschitz manifold with one dimension lower, and this manifold is homeomorphic to an (n1)-dimensional Euclidean space [28]. As for a three-dimensional competitive system, though there is no phase plane, one has a two-dimensional Lipschitz manifold (qualitatively exists but is unknown), where recurrent motions of system lie in. As a result, any limit set for such a system consists of either limit cycle, or steady state, or steady states connected with homoclinic or heteroclinic orbits. Suppose that all forward orbits for a three-dimensional competitive system are bounded and the system has a unique steady state. Then by the Perron-Frobenius theory, the linearized matrix at the steady state has a negative eigenvalue. The system has a one-dimensional stable manifold which is a strictly monotone curve [21], so it also rules out the third choice for limit set. For more details, please see [23] or the Appendix. We summarize the above discussion into the following theorem which can be found in [30]:

    Theorem 3.1. Suppose (1) has a unique steady state E. If the linearized matrix of (1) at E has one negative eigenvalue and two positive real part eigenvalues, then (1) has at least one stable limit cycle.

    Numerical calculation suggests that (1) has a unique equilibrium in a large region of parameter values. However, limit cycles do not exist all the time. According to Theorem 3.1, when either Keq or ka is small, (1) has no limit cycles but a unique equilibrium (see Table 2), and the equilibrium appears to be a global stable steady state.

    Table 2. Equilibrium of (1) and corresponding eigenvalues of its Jacobian matrix vary with Keq and ka.
    KeqkaEquilibrium1Eigenvalues
    200106 (10.00,0.05,0) {50.20,0.40,0.1}
    103 (10.00,0.05,6×106) {50.19,0.40,0.1}
    1 (8.62,0.10,0.04){25.96,0.01±0.11i}
    103 (1.38,0.04,0.24) {164.97,0.11±0.41i}
    106 (1.36,0.04,0.25) {1.47×105,0.12±0.42i}
    15106 (10.00,0.05,0) {50.20,0.40,0.1}
    103 (10.00,0.05,6×106) {50.19,0.40,0.1}
    1 (9.60,0.08,0.10) {30.77,0.03±0.08i}
    102 (5.09,0.08,0.10) {63.98,0.66±0.28i}
    103 (5.03,0.08,0.10) {417.94,1.43,0.54}
    106 (5.02,0.08,0.10) {3.9×105,1.57,0.52}
    1106 (10.00,0.05,0) {50.20,0.40,0.1}
    103 (10.00,0.05,6×106) {50.19,0.40,0.1}
    1 (10.00,0.05,2×103) {46.81,1.13,0.10}
    103 (10.00,0.05,3×103) {1240,28.12,0.10}
    106 (10.00,0.05,3×103) {1.2×106,28.49,0.10}
    1 Those zeros in equilibrium terms are actually very small positive numbers. Other parameter values are as given in Table 1.
     | Show Table
    DownLoad: CSV

    Comparing with the two-dimensional system (2), there are two more parameters ka and kd in system (1). Considering the equilibrium condition, Keq=ka/kd, one only needs to detect how periods of (1) are influenced by ka. As shown in Table 3 and Figure 3E, if we take Keq=200, periodic orbits occur when ka is larger than a critical value ka=0.9. In that region, as ka goes up, the period starts with a rapid decline, and then becomes quite insensitive. At first, we guess that the period is decreasing when ka is sufficiently large, but numerical calculations tell that it is not the case. In fact, the period even has a tendency to increase when ka is larger than 2.9×106 (see Table 3). The similar situations are observed with Keq=15 (see Table 4).

    Table 3. Period of endogenous rhythms of wild-type flies varies as ka (Keq=200) varies.
    ka0.0010.10.80.9110100
    Periodnonenonenone72.4463.1050.8932.51
    ka500100050001045×1041055×105
    Period28.6126.9024.8624.5424.2724.2424.21
    ka1062×1062.5×1062.9×1063×106
    Period24.2124.2124.2124.3024.44
    Periodic oscillations happen when ka is larger than the bifurcation value ka=0.9. Other parameter values are as given in Table 1.
     | Show Table
    DownLoad: CSV
    Figure 3. Relation between the oscillator period of (1) and some parameter values. In each diagram, other parameter values are chosen as in Table 1 and ka=106, and periodic oscillations occur only when the correlate parameter is in the interval [a,b]. In case A, a=0.2 and b=1.4; in case B, a=0.02 and b=0.44; in case C, a=7 and b=46; in case D, a=0 and b=0.4; in case E, a=0.9 and b=; in case F, a1=a2=4, b1=570 and b2=588. For the convenience of numerical integration, curve (1) is shown only with Keq40 in case F. As for 4Keq40, a decreasing period is suggested by curve (2) with increasing Keq. Particularly, on curve (1) the period maintains 24.2-25.2 when the parameter Keq varies in the interval [c,d]=[50,460].
    Table 4. Period of endogenous rhythms of perL mutant varies as ka (Keq=15) varies.
    ka0.0010.11.11.2210100500
    Periodnonenonenone57.1955.6741.3430.9829.21
    ka1000200050001041057×1057×1057×105
    Period28.9428.8028.7128.6728.6528.6529.2030.37
    Periodic oscillations occur when ka is beyond the bifurcation value ka=1.2. Other parameter values are as in Table 1.
     | Show Table
    DownLoad: CSV

    Based on Tables 3 and 4, one can choose a suitable value of ka to calculate the rhythms for wild-type and mutant flies. The numerical results are presented in Table 5, where temperature compensation is found in wild-type flies but not in perL mutant flies.

    Table 5. Period of the endogenous rhythms of wild-type and mutant flies based on (1).
    Genotype KeqTempPeriodGenotype kp1 kp2Period
    Wild type2452024.2 dbt+(1×)100.0324.2
    2002524.2 dbt+(2×)150.0624.3
    1643024.2 dbt+(3×)200.0925.7
    perL18.42026.5 dbtS100.317.6
    15.02528.7 dbt+100.0324.2
    12.33030.4 dbtL100.00325.1
    To simplify the integration, we take ka=106 for wild-type flies and ka=5000 for mutant flies. Other conditions are as in Table 6.
     | Show Table
    DownLoad: CSV

    Table 5 is due to the original three-dimensional system (1). As a comparison, we state Table 6, which is cited from [26] and based on the reduced two-dimensional system (2). Clearly, one can see that Table 5 and Table 6 are almost the same, which indicates that the reduction in [26] is greatly reasonable from this perspective.

    Table 6. Period of the endogenous rhythms of wild-type and mutant flies based on (2).
    Genotype KeqTempPeriodGenotype kp1 kp2Period
    Wild type2452024.2 dbt+(1×)100.0324.2
    2002524.2 dbt+(2×)150.0624.4
    1643024.2 dbt+(3×)200.0925.7
    perL18.42026.5 dbtS100.317.6
    15.02528.7 dbt+100.0324.2
    12.33030.5 dbtL100.00325.2
    This table is copied out of Tyson et al. [26]. It is assumed that each parameter k varies with temperature according to k(T)=k(298)exp{εa(1298/T)}, with values for k(298) and εa=Ea/(0.592kcalmol1) given in Table 1. The dbt+(n×) means n copies of the wild-type allele.
     | Show Table
    DownLoad: CSV

    In the next section, we will see more about the relation between circadian rhythms and parameters of (1).


    4. Discussion

    In the actual experiment, parameters of the circadian rhythms models are hard to be measured, or even unmeasurable. Parameter values in Table 1 have been chosen to yield a period close to 24-hours and ensure temperature compensation of the wild-type oscillator. The parameter values are arbitrary. Other combinations of parameter values may also yield circadian oscillations with possibly different periods.

    It is significant to study how parameters of (1) affect its periodic oscillations. The numerical results are given in Figure 3, where the following parameters are considered: mRNA translation, mRNA degradation, monomer phosphorylation, protein proteolysis, association of PER/TIM protein and equilibrium constant for dimerization.

    As shown in Figure 3A, periodic oscillation disappears when the protein synthesis rate vp is below a critical value. That coincides with the truth that the inhibiting effect of protein synthesis may eventually suppress the circadian rhythmicity [11,4,25]. Moreover, the period decreases when the protein synthesis rate is greater than a certain value. That matches the observations of anisomycin in the mollusk Bulla [12].

    The PER/TIM complex formation plays a key role in the model. Circadian rhythm is markedly affected by the dimerization reaction, precisely in the model, by the association rate constant ka and dissociation rate constant kd. In Figure 3E the period decreases as ka increases, which coincides with the suggestion that the heterodimeric dimerization is attenuated in the long-period perL mutant [5]. Here the attenuation is probably due to the competition of PER homodimeric complexes [0]. Note again that it always has kd=ka/Keq in this paper. The results of Figure 3F imply that circadian rhythm occurs only when kd is in a bounded range, and the period can be recognized as an increasing function of the dimer disassociation rate.

    In Figure 3B we show how the oscillation is affected by mRNA synthesis. Periodic rhythm requires the mRNA synthesis rate km to be bounded, implying that the oscillation may be destroyed if mRNA synthesizes either too slow or too fast. Moreover, the period of oscillation becomes shorter as the mRNA synthesis rate goes up. The qualitatively similar results (Figure 3D) are detected when we consider the proteins proteolysis rate kp3, except that periodic oscillation happens even if there is no proteins proteolysis.

    According to Theorem 3.1, in Figure 4 we inspect the dependence of oscillations on parameters Keq and kp1. A U-shape region is found, whose boundary is almost the same as the locus of Hopf bifurcation in [26] (see Figure 4 in [26]). Within that region the system exhibits periodic oscillations, and in the outside area the system exhibits no limit cycle but a stable steady state. Figure 3C and 3F help to investigate the variation of period in this U-shape region. In Figure 3C, periodic oscillation requests that kp1 is neither too small nor too large, which means that protein monomers are sufficient but not too unstable. In Figure 3F, periodic oscillation vanishes when Keq is smaller than a critical value, which implies that the proteins tend to dimerize. Meanwhile, when Keq varies within a quite large region beyond a certain value, the period of (1) remains virtually unchanged, suggesting that the wild-type oscillation has temperature compensation (see also Table 5). Furthermore, when Keq decreases in a large region the period increases, which agrees with the consensus that perL mutant introduces a longer period for the perL-encoded protein to reduce its tendency to form dimers [5,10]. By Table 5, one can also tell that perL mutant loses temperature compensation.

    Figure 4. Two-parameter (Keq and kp1) bifurcation diagram for system (???). Here Keq and kp1 are allowed to vary, and other parameter values are fixed as in Table 1. We take ka=106. Periodic oscillations happen only within the U-shape region bounded by the two curves. Outside this region the system evolves toward a stable steady state. We note that for any Keq one can find a kp1 such that oscillations happen, which differs from the boundedness requirement of Keq as in Figure 3F.

    Appendix. The concentrations of mRNA, monomers and dimers are naturally nonnegative. We therefore focus on the first orthant R3+={(M,P1,P2):M0,P10,P20}. It is easy to see that R3+ is a positively invariant set of system (1), i.e., any solution ((M(t),P1(t),P2(t)) of system (1) through a point in the first orthant lies in it when t0.

    Let a>vm/km,  b>vpa/kp3,  c=Keqb2 and B(a,b,c)={(M,P1,P2):0Ma, 0P1b, 0P2c}. Denote (f1,f2,f3) the vector field of (1). By estimating the sign of the vector field at vertexes on the boundary of B(a,b,c), one has

    {f1(a,P1,P2)=vm1+(P2/Pcrit)2kma<vmkma<0,f2(M,b,P2)=vpMkp1bJP+b+rP2kp3b2kab2+2kdP2<vpakp3b<0,f3(M,P1,c)=kaP21kdckp2cJP+P1+rckp3c<kp2cJP+P1+rckp3c<0.

    The vector field for (1) on the boundary of B(a,b,c) is shown in Figure 5, which implies that B(a,b,c) is positively invariant. Note that for any point in R3+, one can find such (a,b,c) satisfying that box B(a,b,c) contains the point. It follows immediately that any forward solution of system (1) is bounded. We summarize the above discussion into the following proposition:

    Figure 5. The vector field for (1) on the boundary of B(a,b,c).

    Proposition 1. For any a>vm/km,  b>vpa/kp3 and c=Keqb2, B(a,b,c) is positively invariant for (1), that is, all forward solutions for (1) are bounded.

    From Proposition 1, there are at least one steady state in B(a,b,c). Suppose that the steady state E is unique and there is no zero real part eigenvalue for the linearized matrix at E. Then the equilibrium E is either locally asymptotically stable, or has a two-dimensional unstable manifold. The following arguments show that the latter case provides the existence of limit cycles.

    By computing the Jacobian matrix of (1), one has

    Df=(0++0+),

    where '''' represents that the entry is strictly negative and ''+'' means strict positivity. According to [23], the system is competitive with respect to the cone K={(M,P1,P2)R3:M0,P10,P20}. By applying the theory on competitive systems in [23], we have

    Theorem A. Suppose (1) has a unique steady state E. If the linearized matrix of (1) at E has one negative eigenvalue and two positive real part eigenvalues, then (1) has at least one stable limit cycle.

    Therefore, in order to study the oscillations for (1), one only needs to discuss its steady state and the local stability of the steady state.



    Acknowledgments



    We would like to thank to Programa de Fomento y Apoyo a Proyectos de Investigación (PROFAPI) 2022 of the Universidad Autónoma de Sinaloa grant number PRO_A2_006 for the financial support of this work. Also, we thank to M.Sc. Yesmi Ahumada, Dr. Gabriela López, Dr. Samuel López and Dr. Carolina Murúa for technical assessment during antibacterial assays, extracts preparation and adherence assays, respectively.

    Conflict of interest



    All authors declare no conflicts of interest in this paper.

    Author contributions



    Marcela Martínez, Patricia García, Francisco Delgado, Aldo Clemente and Rodolfo Bernal contributed to the conceptualization and design of the study. Ana Castañeda, Antonio Magaña and Patricia García performed the experiments, organized and analyzed data. Ana Castañeda and Antonio Magaña wrote the first draft of the article. Patricia García, Francisco Delgado and Rodolfo Bernal revised the draft and corrected the sections of the manuscript. Marcela Martínez, Aldo Clemente and Rodolfo Bernal administrated the project. All authors contributed to manuscript revision, read, and approved the submitted version.

    [1] Flores-Mireles AL, Walker JN, Caparon M, et al. (2015) Urinary tract infections: epidemiology, mechanisms of infection and treatment options. Nat Rev Microbiol 13: 269-284. https://doi.org/10.1038/nrmicro3432
    [2] Raeispour M, Ranjbar R (2018) Antibiotic resistance, virulence factors and genotyping of uropathogenic Escherichia coli strains. Antimicrob Resist Infect Control 7: 118. https://doi.org/10.1186/s13756-018-0411-4
    [3] Klein RD, Hultgren SJ (2020) Urinary tract infections: microbial pathogenesis, host–pathogen interactions and new treatment strategies. Nat Rev Microbiol 18: 211-226. https://doi.org/10.1038/s41579-020-0324-0
    [4] Wiles TJ, Kulesus RR, Mulvey MA (2008) Origins and virulence mechanisms of uropathogenic Escherichia coli. Exp Mol Pathol 85: 11-19. https://doi.org/10.1016/j.yexmp.2008.03.007
    [5] Mobley HLT, Donnenberg MS, Hagan EC (2009) Uropathogenic Escherichia coli. EcoSal Plus 3. https://doi.org/10.1128/ecosalplus.8.6.1.3
    [6] McLellan LK, Hunstad DA (2016) Urinary tract infection: pathogenesis and outlook. Trends Mol Med 22: 946-957. https://doi.org/10.1016/j.molmed.2016.09.003
    [7] Kaper JB, Nataro JP, Mobley HLT (2004) Pathogenic Escherichia coli. Nat Rev Microbiol 2: 123-140. https://doi.org/10.1038/nrmicro818
    [8] Bien J, Sokolova O, Bozko P (2012) Role of uropathogenic Escherichia coli virulence factors in development of urinary tract infection and kidney damage. Int J Nephrol 2012: 681473. https://doi.org/10.1155/2012/681473
    [9] Croxen MA, Law RJ, Scholz R, et al. (2013) Recent advances in understanding enteric pathogenic Escherichia coli. Clin Microbiol Rev 26: 822-880. https://doi.org/10.1128/CMR.00022-13
    [10] Terlizzi ME, Gribaudo G, Maffei ME (2017) UroPathogenic Escherichia coli (UPEC) infections: virulence Factors, bladder responses, antibiotic, and Non-antibiotic antimicrobial strategies. Front Microbiol 8: 1566. https://doi.org/10.3389/fmicb.2017.01566
    [11] Zeng Q, Xiao S, Gu F, et al. (2021) Antimicrobial resistance and molecular epidemiology of uropathogenic Escherichia coli isolated from female patients in Shanghai, China. Front Cell Infect Microbiol 11: 653983. https://doi.org/10.3389/fcimb.2021.653983
    [12] Sanchez G V, Babiker A, Master RN, et al. (2016) Antibiotic resistance among urinary isolates from female outpatients in the United States in 2003 and 2012. Antimicrob Agents Chemother 60: 2680-2683. https://doi.org/10.1128/AAC.02897-15
    [13] Wojnicz D, Kucharska AZ, Sokół-Łętowska A, et al. (2012) Medicinal plants extracts affect virulence factors expression and biofilm formation by the uropathogenic Escherichia coli. Urol Res 40: 683-697. https://doi.org/10.1007/s00240-012-0499-6
    [14] Lee J-H, Cho HS, Joo SW, et al. (2013) Diverse plant extracts and trans-resveratrol inhibit biofilm formation and swarming of Escherichia coli O157:H7. Biofouling 29: 1189-1203. https://doi.org/10.1080/08927014.2013.832223
    [15] Reyes Santiago PJ, Islas Luna M de los Á, González Zorzano O, et al. Echeveria. Manual del perfil diagnóstico del género Echeveria en México (2011).
    [16] Reyes R, Sánchez-Vázquez ML, Merchant Larios H, et al. (2005) Calcium (hydrogen-1-malate) hexahydrate on Echeveria gibbiflora leaves and its effect on sperm cells. Arch Androl 51: 461-469. https://doi.org/10.1080/014850190944474
    [17] Martínez Ruiz MG, Gómez-Velasco A, Juárez ZN, et al. (2013) Exploring the biological activities of Echeveria leucotricha. Nat Prod Res 27: 1123-1126. https://doi.org/10.1080/14786419.2012.708662
    [18] López-Angulo G, Montes-Avila J, Díaz-Camacho SP, et al. (2019) Chemical composition and antioxidant, α-glucosidase inhibitory and antibacterial activities of three Echeveria DC. species from Mexico. Arab J Chem 12: 1964-1973. https://doi.org/10.1016/j.arabjc.2014.11.050
    [19] Olivas-Quintero S, Bernal-Reynaga R, Lopez-Saucedo C, et al. (2022) Bacteriostatic effect of Echeveria extracts on diarrheagenic E. coli pathotypes and non-cytotoxicity on human Caco-2 cells. J Infect Dev Ctries 16: 147-156. https://doi.org/10.3855/jidc.15125
    [20] Bauer AW, Kirby WM, Sherris JC, et al. (1966) Antibiotic susceptibility testing by a standardized single disk method. Am J Clin Pathol 45: 493-496.
    [21] (2018) CLSIPerformance standards for antimicrobial disk susceptibility tests. CLSI standard M02 . Wayne, PA: Clinical and Laboratory Standards Institute; 2018.
    [22] Magaña-Lizárraga JA, Gómez-Gil B, Rendón-Maldonado JG, et al. (2022) Genomic profiling of antibiotic-resistant Escherichia coli isolates from surface water of agricultural drainage in North-Western Mexico: Detection of the international high-risk lineages ST410 and ST617. Microorganisms 10: 662. https://doi.org/10.3390/microorganisms10030662
    [23] Martin M (2011) Cutadapt removes adapter sequences from high-throughput sequencing reads. EMBnet.journal 17: 10. https://doi.org/10.14806/ej.17.1.200
    [24] Bankevich A, Nurk S, Antipov D, et al. (2012) SPAdes: a new genome assembly algorithm and its applications to single-cell sequencing. J Comput Biol 19: 455-477. https://doi.org/10.1089/cmb.2012.0021
    [25] Gurevich A, Saveliev V, Vyahhi N, et al. (2013) QUAST: quality assessment tool for genome assemblies. Bioinformatics 29: 1072-1075. https://doi.org/10.1093/bioinformatics/btt086
    [26] Hasman H, Saputra D, Sicheritz-Ponten T, et al. (2014) Rapid whole-genome sequencing for detection and characterization of microorganisms directly from clinical samples. J Clin Microbiol 52: 139-146. https://doi.org/10.1128/JCM.02452-13
    [27] Larsen M V, Cosentino S, Lukjancenko O, et al. (2014) Benchmarking of methods for genomic taxonomy. J Clin Microbiol 52: 1529-1539. https://doi.org/10.1128/JCM.02981-13
    [28] Bosi E, Donati B, Galardini M, et al. (2015) MeDuSa: a multi-draft based scaffolder. Bioinformatics 31: 2443-2451. https://doi.org/10.1093/bioinformatics/btv171
    [29] Darling ACE, Mau B, Blattner FR, et al. (2004) Mauve: multiple alignment of conserved genomic sequence with rearrangements. Genome Res 14: 1394-1403. https://doi.org/10.1101/gr.2289704
    [30] Rissman AI, Mau B, Biehl BS, et al. (2009) Reordering contigs of draft genomes using the Mauve aligner. Bioinformatics 25: 2071-2073. https://doi.org/10.1093/bioinformatics/btp356
    [31] Aziz RK, Bartels D, Best A, et al. (2008) The RAST Server: Rapid annotations using subsystems technology. BMC Genomics 9: 1-15. https://doi.org/10.1186/1471-2164-9-75
    [32] Wirth T, Falush D, Lan R, et al. (2006) Sex and virulence in Escherichia coli: an evolutionary perspective. Mol Microbiol 60: 1136-1151. https://doi.org/10.1111/j.1365-2958.2006.05172.x
    [33] Beghain J, Bridier-Nahmias A, Le Nagard H, et al. (2018) ClermonTyping: an easy-to-use and accurate in silico method for Escherichia genus strain phylotyping. Microb genomics 4. https://doi.org/10.1099/mgen.0.000192
    [34] Clermont O, Dixit OVA, Vangchhia B, et al. (2019) Characterization and rapid identification of phylogroup G in Escherichia coli, a lineage with high virulence and antibiotic resistance potential. Environ Microbiol 21: 3107-3117. https://doi.org/10.1111/1462-2920.14713
    [35] Roer L, Johannesen TB, Hansen F, et al. (2018) CHTyper, a web tool for subtyping of extraintestinal pathogenic Escherichia coli based on the fumc and fimh alleles. J Clin Microbiol 56. https://doi.org/10.1128/JCM.00063-18
    [36] Weissman SJ, Johnson JR, Tchesnokova V, et al. (2012) High-resolution two-locus clonal typing of extraintestinal pathogenic Escherichia coli. Appl Environ Microbiol 78: 1353-1360. https://doi.org/10.1128/AEM.06663-11
    [37] Joensen KG, Tetzschner AMM, Iguchi A, et al. (2015) Rapid and easy in silico serotyping of Escherichia coli isolates by use of whole-genome sequencing data. J Clin Microbiol 53: 2410-2426. https://doi.org/10.1128/JCM.00008-15
    [38] Larsen M V, Cosentino S, Rasmussen S, et al. (2012) Multilocus sequence typing of total-genome-sequenced bacteria. J Clin Microbiol 50: 1355-1361. https://doi.org/10.1128/JCM.06094-11
    [39] Zankari E, Allesøe R, Joensen KG, et al. (2017) PointFinder: a novel web tool for WGS-based detection of antimicrobial resistance associated with chromosomal point mutations in bacterial pathogens. J Antimicrob Chemother 72: 2764-2768. https://doi.org/10.1093/jac/dkx217
    [40] Bortolaia V, Kaas RS, Ruppe E, et al. (2020) ResFinder 4.0 for predictions of phenotypes from genotypes. J Antimicrob Chemother 75: 3491-3500. https://doi.org/10.1093/jac/dkaa345
    [41] Joensen KG, Scheutz F, Lund O, et al. (2014) Real-time whole-genome sequencing for routine typing, surveillance, and outbreak detection of verotoxigenic Escherichia coli. J Clin Microbiol 52: 1501-1510. https://doi.org/10.1128/JCM.03617-13
    [42] Malberg Tetzschner AM, Johnson JR, Johnston BD, et al. (2020) In silico genotyping of Escherichia coli isolates for extraintestinal virulence genes by use of whole-genome sequencing data. J Clin Microbiol 58. https://doi.org/10.1128/JCM.01269-20
    [43] Carattoli A, Zankari E, García-Fernández A, et al. (2014) In silico detection and typing of plasmids using plasmidfinder and plasmid multilocus sequence typing. Antimicrob Agents Chemother 58: 3895-3903. https://doi.org/10.1128/AAC.02412-14
    [44] Carattoli A, Hasman H (2020) PlasmidFinder and in silico pMLST: Identification and typing of plasmid replicons in whole-genome sequencing (WGS). Methods Mol Biol 2075: 285-294. https://doi.org/10.1007/978-1-4939-9877-7_20
    [45] Arndt D, Grant JR, Marcu A, et al. (2016) PHASTER: a better, faster version of the PHAST phage search tool. Nucleic Acids Res 44: W16-W21. https://doi.org/10.1093/nar/gkw387
    [46] Zhou Z, Alikhan N-F, Mohamed K, et al. (2020) The EnteroBase user's guide, with case studies on Salmonella transmissions, Yersinia pestis phylogeny, and Escherichia core genomic diversity. Genome Res 30: 138-152. https://doi.org/10.1101/gr.251678.119
    [47] Zhou Z, Alikhan N-F, Sergeant MJ, et al. (2018) GrapeTree: visualization of core genomic relationships among 100,000 bacterial pathogens. Genome Res 28: 1395-1404. https://doi.org/10.1101/gr.232397.117
    [48] Letunic I, Bork P (2021) Interactive Tree Of Life (iTOL) v5: an online tool for phylogenetic tree display and annotation. Nucleic Acids Res 49: W293-W296. https://doi.org/10.1093/nar/gkab301
    [49] Ramírez K, Burgueño-Roman A, Castro-del Campo N, et al. (2019) In vitro invasiveness and intracellular survival of Salmonella strains isolated from the aquatic environment. Water Environ J 33: 633-640. https://doi.org/10.1111/wej.12436
    [50] Yousefipour M, Rezatofighi SE, Ardakani MR (2023) Detection and characterization of hybrid uropathogenic Escherichia coli strains among E. coli isolates causing community-acquired urinary tract infection. J Med Microbiol 72. https://doi.org/10.1099/jmm.0.001660
    [51] Barrios-Villa E, Cortés-Cortés G, Lozano-Zaraín P, et al. (2018) Adherent/invasive Escherichia coli (AIEC) isolates from asymptomatic people: new E. coli ST131 O25:H4/H30-Rx virotypes. Ann Clin Microbiol Antimicrob 17: 42. https://doi.org/10.1186/s12941-018-0295-4
    [52] Zakaria AS, Edward EA, Mohamed NM (2022) Pathogenicity islands in uropathogenic Escherichia coli clinical isolate of the globally disseminated O25:H4-ST131 pandemic clonal lineage: first report from Egypt. Antibiotics 11: 1620. https://doi.org/10.3390/antibiotics11111620
    [53] Munkhdelger Y, Gunregjav N, Dorjpurev A, et al. (2017) Detection of virulence genes, phylogenetic group and antibiotic resistance of uropathogenic Escherichia coli in Mongolia. J Infect Dev Ctries 11: 51-57. https://doi.org/10.3855/jidc.7903
    [54] Paramita RI, Nelwan EJ, Fadilah F, et al. (2020) Genome-based characterization of Escherichia coli causing bloodstream infection through next-generation sequencing. PLoS One 15: e0244358. https://doi.org/10.1371/journal.pone.0244358
    [55] Badger-Emeka LI, Kausar N, Estrella E, et al. (2022) A three-year look at the phylogenetic profile, antimicrobial resistance, and associated virulence genes of uropathogenic Escherichia coli. Pathogens 11: 631. https://doi.org/10.3390/pathogens11060631
    [56] Tantoso E, Eisenhaber B, Kirsch M, et al. (2022) To kill or to be killed: pangenome analysis of Escherichia coli strains reveals a tailocin specific for pandemic ST131. BMC Biol 20: 146. https://doi.org/10.1186/s12915-022-01347-7
    [57] Sharon BM, Nguyen A, Arute AP, et al. (2020) Complete genome sequences of seven uropathogenic Escherichia coli strains isolated from postmenopausal women with recurrent urinary tract infection. Microbiol Resour Announc 9: e00700-20. https://doi.org/10.1128/MRA.00700-20
    [58] Sarwar F, Rasool MH, Khurshid M, et al. (2022) Escherichia coli Isolates harboring bla ndm variants and 16s methylases belonging to clonal complex 131 in Southern Punjab, Pakistan. Microb Drug Resist 28: 623-635. https://doi.org/10.1089/mdr.2021.0315
    [59] Alqasim A, Abu Jaffal A, Alyousef AA (2018) Prevalence of multidrug resistance and extended-spectrum β-lactamase carriage of clinical uropathogenic Escherichia coli isolates in Riyadh, Saudi Arabia. Int J Microbiol 2018: 3026851. https://doi.org/10.1155/2018/3026851
    [60] Bevan ER, Jones AM, Hawkey PM (2017) Global epidemiology of CTX-M β-lactamases: temporal and geographical shifts in genotype. J Antimicrob Chemother 72: 2145-2155. https://doi.org/10.1093/jac/dkx146
    [61] Ali I, Rafaque Z, Ahmed I, et al. (2019) Phylogeny, sequence-typing and virulence profile of uropathogenic Escherichia coli (UPEC) strains from Pakistan. BMC Infect Dis 19: 620. https://doi.org/10.1186/s12879-019-4258-y
    [62] Sepp E, Andreson R, Balode A, et al. (2019) Phenotypic and molecular epidemiology of ESBL-, AmpC-, and carbapenemase-producing Escherichia coli in Northern and Eastern Europe. Front Microbiol 10: 2465. https://doi.org/10.3389/fmicb.2019.02465
    [63] Fuga B, Sellera FP, Cerdeira L, et al. (2022) WHO critical priority Escherichia coli as one health challenge for a post-pandemic scenario: genomic surveillance and analysis of current trends in Brazil. Microbiol Spectr 10: e0125621. https://doi.org/10.1128/spectrum.01256-21
    [64] Yao M, Zhu Q, Zou J, et al. (2022) Genomic characterization of a uropathogenic Escherichia coli ST405 isolate harboring blaCTX-M-15-Encoding IncFIA-FIB plasmid, blaCTX-M-24-encoding inci1 plasmid, and phage-like plasmid. Front Microbiol 13: 845045. https://doi.org/10.3389/fmicb.2022.845045
    [65] Chandran SP, Sarkar S, Diwan V, et al. (2017) Detection of virulence genes in ESBL producing, quinolone resistant commensal Escherichia coli from rural Indian children. J Infect Dev Ctries 11: 387-392. https://doi.org/10.3855/jidc.8574
    [66] Bodendoerfer E, Marchesi M, Imkamp F, et al. (2020) Co-occurrence of aminoglycoside and β-lactam resistance mechanisms in aminoglycoside- non-susceptible Escherichia coli isolated in the Zurich area, Switzerland. Int J Antimicrob Agents 56: 106019. https://doi.org/10.1016/j.ijantimicag.2020.106019
    [67] Altayb HN, Elbadawi HS, Alzahrani FA, et al. (2022) Co-Occurrence of β-Lactam and aminoglycoside resistance determinants among clinical and environmental isolates of Klebsiella pneumoniae and Escherichia coli: A Genomic Approach. Pharmaceuticals 15. https://doi.org/10.3390/ph15081011
    [68] Mirzaii M, Jamshidi S, Zamanzadeh M, et al. (2018) Determination of gyrA and parC mutations and prevalence of plasmid-mediated quinolone resistance genes in Escherichia coli and Klebsiella pneumoniae isolated from patients with urinary tract infection in Iran. J Glob Antimicrob Resist 13: 197-200. https://doi.org/10.1016/j.jgar.2018.04.017
    [69] Yakout MA, Ali GH (2022) A novel parC mutation potentiating fluoroquinolone resistance in Klebsiella pneumoniae and Escherichia coli clinical isolates. J Infect Dev Ctries 16: 314-319. https://doi.org/10.3855/jidc.15142
    [70] Tayh G, Nagarjuna D, Ben Sallem R, et al. (2021) Identification of virulence factors among ESBL-producing Escherichia coli clinical isolates from Gaza strip, Palestine. J Microbiol Biotechnol food Sci 11: e2865. https://doi.org/10.15414/jmbfs.2865
    [71] Bunduki GK, Heinz E, Phiri VS, et al. (2021) Virulence factors and antimicrobial resistance of uropathogenic Escherichia coli (UPEC) isolated from urinary tract infections: a systematic review and meta-analysis. BMC Infect Dis 21: 753. https://doi.org/10.1186/s12879-021-06435-7
    [72] Luna-Pineda VM, Ochoa S, Cruz-Córdova A, et al. (2019) Infecciones del tracto urinario, inmunidad y vacunación. Bol Med Hosp Infant Mex 75: 67-78. https://doi.org/10.24875/BMHIM.M18000011
    [73] Ballesteros-Monrreal MG, Arenas-Hernández MM, Enciso-Martínez Y, et al. (2020) Virulence and resistance determinants of uropathogenic Escherichia coli strains isolated from pregnant and Non-pregnant women from two states in Mexico. Infect Drug Resist 13: 295-310. https://doi.org/10.2147/IDR.S226215
    [74] Ballesteros-Monrreal MG, Arenas-Hernández MMP, Barrios-Villa E, et al. (2021) Bacterial morphotypes as important trait for uropathogenic E. coli diagnostic; a virulence-phenotype-phylogeny study. Microorganisms 9: 2381. https://doi.org/10.3390/microorganisms9112381
    [75] Sora VM, Meroni G, Martino PA, et al. (2021) Extraintestinal pathogenic Escherichia coli: virulence factors and antibiotic resistance. Pathogens 10: 1355. https://doi.org/10.3390/pathogens10111355
    [76] Ahumada-Santos YP, Soto-Sotomayor ME, Báez-Flores ME, et al. (2016) Antibacterial synergism of Echeveria subrigida (B. L. Rob & Seaton) and commercial antibiotics against multidrug resistant Escherichia coli and Staphylococcus aureus. Eur J Integr Med 8: 638-644. https://doi.org/10.1016/j.eujim.2016.08.160
    [77] Kim DY, Lee JC (2017) Adherence assay of uropathogenic Escherichia coli in vivo and in vitro. Urogenit Tract Infect 12: 122-129. https://doi.org/10.14777/uti.2017.12.3.122
    [78] Cozens D, Read RC (2012) Anti-adhesion methods as novel therapeutics for bacterial infections. Expert Rev Anti Infect Ther 10: 1457-1468. https://doi.org/10.1586/eri.12.145
    [79] Hassan MHA, Elwekeel A, Afifi N, et al. (2021) Phytochemical constituents and biological activity of selected genera of family Crassulaceae: a review. S Afr J Bot 141: 383-404. https://doi.org/10.1016/j.sajb.2021.05.016
  • microbiol-10-01-003-s001.xlsx
  • This article has been cited by:

    1. Shuang Chen, Jinqiao Duan, Ji Li, Dynamics of the Tyson–Hong–Thron–Novak circadian oscillator model, 2021, 420, 01672789, 132869, 10.1016/j.physd.2021.132869
    2. Alessio Franci, Marco Arieli Herrera-Valdez, Miguel Lara-Aparicio, Pablo Padilla-Longoria, Synchronization, Oscillator Death, and Frequency Modulation in a Class of Biologically Inspired Coupled Oscillators, 2018, 4, 2297-4687, 10.3389/fams.2018.00051
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2871) PDF downloads(274) Cited by(0)

Article outline

Figures and Tables

Figures(2)  /  Tables(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog