Research article

Revisiting spontaneous silver nanoparticles formation: a factor influencing the determination of minimum inhibitory concentration values?

  • The present study gives evidence that silver nanoparticles (AgNPs) are spontaneously formed from Ag+ ions in Mueller-Hinton broth, which is frequently used as a standard cultivation medium for many types of bacteria. Silver ions often serve as a reference in the determination of minimum inhibitory concentration (MIC) values of engineered AgNPs. It is thus a question if the MIC values determined for engineered AgNPs are not influenced by the presence of spontaneously formed AgNPs. Furthermore, as shown here, the addition of augmented concentrations of selected amino acids, namely glutamic acid and glutamine, can change the growth and characteristic features of spontaneously formed AgNPs. For the sake of a direct comparison, the influence of the two selected amino acids on characteristics and MIC values determination of engineered AgNPs has been also investigated. The determined MIC values of all investigated systems (i.e., with and without the presence of engineered AgNPs) and their mutual comparison demonstrated that MIC values are slightly influenced by the actual composition of a cultivation medium for bacterial growth. On the other hand, the actual composition of a cultivation medium is crucial for the final characteristics of AgNPs. The changes in characteristic features of spontaneously formed as well as engineered AgNPs are most probably induced by the covalent bonding of amino acids to AgNPs surface which is proven by vibrational spectroscopic techniques.

    Citation: Karolína M. Šišková, Renáta Večeřová, Hana Kubičková, Magdaléna Bryksová, Klára Čépe, Milan Kolář. Revisiting spontaneous silver nanoparticles formation: a factor influencing the determination of minimum inhibitory concentration values?[J]. AIMS Environmental Science, 2015, 2(3): 607-622. doi: 10.3934/environsci.2015.3.607

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  • The present study gives evidence that silver nanoparticles (AgNPs) are spontaneously formed from Ag+ ions in Mueller-Hinton broth, which is frequently used as a standard cultivation medium for many types of bacteria. Silver ions often serve as a reference in the determination of minimum inhibitory concentration (MIC) values of engineered AgNPs. It is thus a question if the MIC values determined for engineered AgNPs are not influenced by the presence of spontaneously formed AgNPs. Furthermore, as shown here, the addition of augmented concentrations of selected amino acids, namely glutamic acid and glutamine, can change the growth and characteristic features of spontaneously formed AgNPs. For the sake of a direct comparison, the influence of the two selected amino acids on characteristics and MIC values determination of engineered AgNPs has been also investigated. The determined MIC values of all investigated systems (i.e., with and without the presence of engineered AgNPs) and their mutual comparison demonstrated that MIC values are slightly influenced by the actual composition of a cultivation medium for bacterial growth. On the other hand, the actual composition of a cultivation medium is crucial for the final characteristics of AgNPs. The changes in characteristic features of spontaneously formed as well as engineered AgNPs are most probably induced by the covalent bonding of amino acids to AgNPs surface which is proven by vibrational spectroscopic techniques.


    Stochastic nonlinear control is a hot topic because of its wide application in economic and engineering fields. The pioneer work is [1,2,3,4,5,6,7,8]. Specifically, [1,2,3] propose designs with quadratic Lyapunov functions coupled with weighting functions and [4,5,6,7,8] develop designs with quartic Lyapunov function, which are further developed by [9,10,11,12]. Recently, a class of stochastic systems (SSs) whose Jacobian linearizations may have unstable modes, has received much attention. Such systems are also called stochastic high-order nonlinear systems (SHONSs), which include a class of stochastic benchmark mechanical systems [13] as a special case. In this direction, [14] studies the state-feedback control with stochastic inverse dynamics; [15] develops a stochastic homogeneous domination method, which completely relaxes the order restriction required in [13], and investigates the output-feedback stabilization for SHONSs with unmeasurable states; [16] investigates the output-feedback tracking problem; [17] studies the adaptive state-feedback design for state-constrained systems. It should be emphasized that [13,14,15,16,17], only achieve stabilization in asymptotic sense (as time goes to infinity). However, many real applications appeals for prescribed-time stabilization, which permits the worker to prescribe the convergence times in advance.

    For the prescribed-time control, [18,19] design time-varying feedback to solve the regulation problems of normal-form nonlinear systems; [20] considers networked multi-agent systems; [21,22] investigates the prescribed-time design for linear systems in controllable canonical form; [23] designs the output-feedback controller for uncertain nonlinear strict-feedback-like systems. It should be noted that the results in [18,19,20,21,22,23] don't consider stochastic noise. For SSs, [24] is the first paper to address the stochastic nonlinear inverse optimality and prescribed-time stabilization problems; The control effort is further reduced in [25]; Recently, [26] studies the prescribed-time output-feedback for SSs without/with sensor uncertainty. It should be noted that [24,25,26] don't consider SHONSs. From practical applications, it is important to solves the prescribed-time control problem of SHONSs since it permits the worker to set the convergence times in advance.

    Motivated by the above discussions, this paper studies the prescribed-time design for SSs with high-order structure. The contributions include:

    1) This paper proposes new prescribed-time design for SHONSs. Since Jacobian linearizations of such a system possibly have unstable modes, all the prescribed-time designs in [24,25,26] are invalid. New design and analysis tools should be developed.

    2) This paper develops a more practical design than those in [13,14,15,16,17]. Different from the designs in [13,14,15,16,17] where only asymptotic stabilization can be achieved, the design in this paper can guarantee that the closed-loop system is prescribed-time mean-square stable, which is superior to those in [13,14,15,16,17] since it permits the worker to prescribe the convergence times in advance without considering the initial conditions.

    The remainder of this paper is organized as follows. In Section 2, the problem is formulated. In Section 3, the controller is designed and the stability analysis is given. Section 4 uses an example to explain the validity of the prescribed-time design. The conclusions are collected in Section 6.

    Consider the following class of SHONSs

    $ dx1=xp2dt+φT1(x)dω,
    $
    (2.1)
    $ dx2=udt+φT2(x)dω,
    $
    (2.2)

    where $ x = (x_{1}, x_{2})^{T} \in R^{2} $ and $ u \in R $ are the system state and control input. $ p\geq1 $ is an odd integral number. The functions $ \varphi_{i}: R^{2}\rightarrow R^{m} $ are smooth in $ x $ with $ \varphi_{i}(0) = 0 $, $ i = 1, 2 $. $ \omega $ is an $ m $-dimensional independent standard Wiener process.

    The assumptions we need are as follows.

    Assumption 1. There is a constant $ c > 0 $ such that

    $ |φ1(x)|c|x1|p+12,
    $
    (2.3)
    $ |φ2(x)|c(|x1|p+12+|x2|p+12).
    $
    (2.4)

    We introduce the function:

    $ μ(t)=(Tt0+Tt)m,t[t0,t0+T),
    $
    (2.5)

    where $ m\geq 2 $ is an integral number. Obviously, the function $ \mu(t) $ is monotonically increasing on $ [t_{0}, t_{0}+T) $ with $ \mu(t_{0}) = 1 $ and $ \lim\limits_{t\rightarrow t_{0}+T}\mu(t) = +\infty $.

    Our control goal is to design a prescribed-time state-feedback controller, which guarantees that the closed-loop system has an almost surely unique strong solution and is prescribed-time mean-square stable.

    In the following, we design a time-varying controller for system (2.1)–(2.2) step by step.

    Step 1. In this step, our goal is to design the virtual controller $ x_{2}^{*} $.

    Define $ V_{1} = \frac{1}{4}\xi_{1}^{4} $, $ \xi_{1} = x_{1} $, from (2.1), (2.3) and (2.5) we have

    $ LV1(ξ1)=ξ31xp2+32ξ21|φ1|2ξ31xp2+32c2ξp+31ξ31(xp2xp2)+ξ31xp2+32c2μpξp+31.
    $
    (3.1)

    Choosing

    $ x2=μ(2+32c2)1/pξ1μα1ξ1,
    $
    (3.2)

    which substitutes into (3.1) yields

    $ LV1(ξ1)2μpξp+31+ξ31(xp2xp2),
    $
    (3.3)

    where $ \alpha_{1} = (2+\frac{3}{2}c^{2})^{1/p} $.

    Step 2. In this step, our goal is to design the actual controller $ u $.

    Define $ \xi_{2} = x_{k}-x_{2}^{*} $, from (3.2) we get

    $ ξ2=x2+μα1ξ1.
    $
    (3.4)

    By using (2.1)–(2.2) and (3.4) we get

    $ dξ2=(u+mTμ1+1/mα1ξ1+μα1xp2)dt+(φT2+μα1φT1)dω.
    $
    (3.5)

    We choose the following Lyapunov function

    $ V2(ξ1,ξ2)=V1(ξ1)+14ξ42.
    $
    (3.6)

    It follows from (3.3), (3.5)–(3.6) that

    $ LV22μpξp+31+ξ31(xp2xp2)+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+32ξ22|φ2+μα1φ1|2.
    $
    (3.7)

    By using (3.4) we have

    $ ξ31(xp2xp2)|ξ1|3|ξ2|(xp12+xp12)|ξ1|3|ξ2|((β1+1)μp1αp11ξp11+β1ξp12)=(β1+1)μp1αp11|ξ1|p+2|ξ2|+β1|ξ1|3|ξ2|p,
    $
    (3.8)

    where

    $ β1=max{1,2p2}.
    $
    (3.9)

    By using Young's inequality we get

    $ (β1+1)μp1αp11|ξ1|p+2|ξ2|13μpξp+31+13+p(3+p3(2+p))(2+p)((β1+1)αp11)p+3μ3ξp+32
    $
    (3.10)

    and

    $ β1|ξ1|3|ξ2|p13μpξ3+p1+pp+3(3+p9)3/pβ(3+p)/p1μ3ξp+32.
    $
    (3.11)

    Substituting (3.10)–(3.11) into (3.8) yields

    $ ξ31(xp2xp2)23μpξp+31+(pp+3(p+39)3/pβ(p+3)/p1+1p+3(p+33(p+2))(p+2)((β1+1)αp11)p+3)μ3ξp+32.
    $
    (3.12)

    From (2.3), (2.4), (3.2) and (3.4) we have

    $ |φ2+μα1φ1|22|φ2|2+2μ2α21|φ1|24c2(|x1|1+p+|x2|1+p)+2c2α21μ2|x1|1+pβ2μ1+p|ξ1|p+1+4c22p|ξ2|1+p,
    $
    (3.13)

    where

    $ β2=4c2(1+2pαp+11)+2c2α21.
    $
    (3.14)

    By using (3.13) we get

    $ 32ξ22|φ2+μα1φ1|213μpξ3+p1+(23+p(p+33(1+p))(1+p)/2(32β2)(3+p)/2+6c22p)μ3(1+p)/2ξp+32.
    $
    (3.15)

    It can be inferred from (3.7), (3.12) and (3.15) that

    $ LV2μpξp+31+ξ32(u+mTμ1+1/mα1ξ1+μα1xp2)+β3μ3(p+1)/2ξp+32,
    $
    (3.16)

    where

    $ β3=p3+p(3+p9)3/pβ(3+p)/p1+13+p(p+33(p+2))(p+2)((β1+1)αp11)3+p+23+p(p+33(1+p))(1+p)/2(32β2)(3+p)/2+6c22p.
    $
    (3.17)

    We choose the controller

    $ u=mTμ1+1/mα1ξ1μα1xp2(1+β3)μ3(p+1)/2ξp2,
    $
    (3.18)

    then we get

    $ LV2μpξp+31μ3(p+1)/2ξp+32.
    $
    (3.19)

    Now, we describe the main stability analysis results for system (2.1)–(2.2).

    Theorem 1. For the plant (2.1)–(2.2), if Assumption 1 holds, with the controller (3.18), the following conclusions are held.

    1) The plant has an almost surely unique strong solution on $ [t_{0}, t_{0}+T) $;

    2) The equilibrium at the origin of the plant is prescribed-time mean-square stable with $ \lim\limits_{t\rightarrow t_{0}+T} E|x|^{2} = 0 $.

    Proof. By using Young's inequality we get

    $ 14μξ41μpξp+31+β4μ3,
    $
    (3.20)
    $ 14μξ42μ3(p+1)/2ξp+32+β4μ3,
    $
    (3.21)

    where

    $ β4=p1p+3(3+p4)4/(p1)(14)(3+p)/(p1).
    $
    (3.22)

    It can be inferred from (3.19)–(3.21) that

    $ LV214μξ4114μξ42+2β4μ3=μV2+2β4μ3.
    $
    (3.23)

    From (2.1), (2.2) and (3.18), the local Lipschitz condition is satisfied by the plant. By (3.23) and using Lemma 1 in [24], the plant has an almost surely unique strong solution on $ [t_{0}, t_{0}+T) $, which shows that conclusion 1) is true.

    Next, we verify conclusion 2).

    For each positive integer $ k $, the first exit time is defined as

    $ ρk=inf{t:tt0,|x(t)|k}.
    $
    (3.24)

    Choosing

    $ V=ett0μ(s)dsV2.
    $
    (3.25)

    From (3.23) and (3.25) we have

    $ LV=ett0μ(s)ds(LV2+μV2)2β4μ3ett0μ(s)ds.
    $
    (3.26)

    By (3.26) and using Dynkin's formula we get

    $ EV(ρkt,x(ρkt))=Vn(t0,x0)+E{ρktt0LV(x(τ),τ)dτ}Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ.
    $
    (3.27)

    Using Fatou Lemma, from (3.27) we have

    $ EV(t,x)Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ,t[t0,t0+T).
    $
    (3.28)

    By using (3.25) and (3.28) we get

    $ EV2ett0μ(s)ds(Vn(t0,x0)+2β4tt0μ3eτt0μ(s)dsdτ),t[t0,t0+T).
    $
    (3.29)

    By using (3.6) and (3.29) we obtain

    $ limtt0+TE|x|2=0.
    $
    (3.30)

    Consider the prescribed-time stabilization for the following system

    $ dx1=x32dt+0.1x21dω,
    $
    (4.1)
    $ dx2=udt+0.2x1x2dω,
    $
    (4.2)

    where $ p_{1} = 3 $, $ p_{2} = 1 $. Noting $ 0.2x_1x_2\leq 0.1(x_{1}^{2}+x_{2}^{2}) $, Assumption 1 is satisfied.

    Choosing

    $ μ(t)=(11t)2,t[0,1),
    $
    (4.3)

    According to the design method in Section 3, we have

    $ u=3μ1.5x11.5μx3257μ6(x2+1.5μx1)3
    $
    (4.4)

    In the practical simulation, we select the initial conditions as $ x_{1}(0) = -6 $, $ x_{2}(0) = 5 $. Figure 1 shows the responses of (4.1)–(4.4), from which we can obtain that $ \lim\limits_{t\rightarrow t_{0}+T} E|x_{1}|^{2} = \lim\limits_{t\rightarrow t_{0}+T} E|x_{2}|^{2} = 0 $, which means that the controller we designed is effective.

    Figure 1.  The responses of closed-loop system (4.1)–(4.4).

    This paper proposes a new design method of prescribed-time state-feedback for SHONSs. the controller we designed can guarantee that the closed-loop system has an almost surely unique strong solution and the equilibrium at the origin of the closed-loop system is prescribed-time mean-square stable. The results in this paper are more practical than those in [13,14,15,16,17] since the design in this paper permits the worker to prescribe the convergence times in advance without considering the initial conditions.

    There are some related problems to investigate, e.g., how to extend the results to multi-agent systems [27], impulsive systems [28,29,30] or more general high-order systems [31,32,33,34].

    This work is funded by Shandong Provincial Natural Science Foundation for Distinguished Young Scholars, China (No. ZR2019JQ22), and Shandong Province Higher Educational Excellent Youth Innovation team, China (No. 2019KJN017).

    The authors declare there is no conflict of interest.

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