Processing math: 100%
Commentary Topical Sections

A unified model for BAM function that takes into account type Vc secretion and species differences in BAM composition

  • Transmembrane proteins in the outer membrane of Gram-negative bacteria are almost exclusively β-barrels. They are inserted into the outer membrane by a conserved and essential protein complex called the BAM (for β-barrel assembly machinery). In this commentary, we summarize current research into the mechanism of this protein complex and how it relates to type V secretion. Type V secretion systems are autotransporters that all contain a β-barrel transmembrane domain inserted by BAM. In type Vc systems, this domain is a homotrimer. We argue that none of the current models are sufficient to explain BAM function particularly regarding type Vc secretion. We also find that current models based on the well-studied model system Escherichia coli mostly ignore the pronounced differences in BAM composition between different bacterial species. We propose a more holistic view on how all OMPs, including autotransporters, are incorporated into the lipid bilayer.

    Citation: Jack C. Leo, Dirk Linke. A unified model for BAM function that takes into account type Vc secretion and species differences in BAM composition[J]. AIMS Microbiology, 2018, 4(3): 455-468. doi: 10.3934/microbiol.2018.3.455

    Related Papers:

    [1] Fangming Cai, Jie Rui, Deguang Zhong . Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma. AIMS Mathematics, 2023, 8(12): 30992-31007. doi: 10.3934/math.20231586
    [2] Narcisse Batangouna . A robust family of exponential attractors for a time semi-discretization of the Ginzburg-Landau equation. AIMS Mathematics, 2022, 7(1): 1399-1415. doi: 10.3934/math.2022082
    [3] Javier Sánchez-Reyes, Leonardo Fernández-Jambrina . On the reach and the smoothness class of pipes and offsets: a survey. AIMS Mathematics, 2022, 7(5): 7742-7758. doi: 10.3934/math.2022435
    [4] Qi Liu, Anwarud Din, Yongjin Li . Some aspects of generalized von Neumann-Jordan type constant. AIMS Mathematics, 2021, 6(6): 6309-6321. doi: 10.3934/math.2021370
    [5] Dongsheng Xu, Xiangxiang Cui, Lijuan Peng, Huaxiang Xian . Distance measures between interval complex neutrosophic sets and their applications in multi-criteria group decision making. AIMS Mathematics, 2020, 5(6): 5700-5715. doi: 10.3934/math.2020365
    [6] Samia BiBi, Md Yushalify Misro, Muhammad Abbas . Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy. AIMS Mathematics, 2021, 6(8): 8625-8641. doi: 10.3934/math.2021501
    [7] Faiza Shujat, Faarie Alharbi, Abu Zaid Ansari . Weak $ (p, q) $-Jordan centralizer and derivation on rings and algebras. AIMS Mathematics, 2025, 10(4): 8322-8330. doi: 10.3934/math.2025383
    [8] Juyoung Jeong . The Schatten $ p $-quasinorm on Euclidean Jordan algebras. AIMS Mathematics, 2024, 9(2): 5028-5037. doi: 10.3934/math.2024244
    [9] Ai-qun Ma, Lin Chen, Zijie Qin . Jordan semi-triple derivations and Jordan centralizers on generalized quaternion algebras. AIMS Mathematics, 2023, 8(3): 6026-6035. doi: 10.3934/math.2023304
    [10] Cai-Yun Li, Chun-Gang Zhu . Construction of the spacelike constant angle surface family in Minkowski 3-space. AIMS Mathematics, 2020, 5(6): 6341-6354. doi: 10.3934/math.2020408
  • Transmembrane proteins in the outer membrane of Gram-negative bacteria are almost exclusively β-barrels. They are inserted into the outer membrane by a conserved and essential protein complex called the BAM (for β-barrel assembly machinery). In this commentary, we summarize current research into the mechanism of this protein complex and how it relates to type V secretion. Type V secretion systems are autotransporters that all contain a β-barrel transmembrane domain inserted by BAM. In type Vc systems, this domain is a homotrimer. We argue that none of the current models are sufficient to explain BAM function particularly regarding type Vc secretion. We also find that current models based on the well-studied model system Escherichia coli mostly ignore the pronounced differences in BAM composition between different bacterial species. We propose a more holistic view on how all OMPs, including autotransporters, are incorporated into the lipid bilayer.


    Let $ \Gamma \subset {\Bbb C} $ be a closed Jordan curve and let $ \Omega $ be the bounded component of $ {\Bbb C}\setminus \Gamma $. For any $ \lambda > 0 $, the set

    $ Γλ:={zΩ:dist(z,Γ)=λ}. $

    is called the constant distance boundary of $ \Gamma $. Meanwhile, let

    $ Ωλ:={zΩ:dist(z,Γ)>λ}. $

    Here $ \operatorname{dist}(z, \Gamma): = \inf\{|z- \zeta|: \zeta\in \Gamma\} $. In [2], Brown showed that for all but countable number of $ \lambda $, every component of $ \Gamma_ \lambda $ is a single point, or a simple arc, or a simple closed curve. It was also proved that $ \Gamma_ \lambda $ is a $ 1 $-manifold for almost all $ \lambda $ in [3]. Blokh, Misiurewiczch and Oversteegen generalised Brown's result in [1], they provided that for all but countably many $ \lambda > 0 $ each component of $ \Gamma_ \lambda $ is either a point or a simple closed curve. If $ \Gamma $ is smooth or having positive reach, $ \Gamma_ \lambda $ is called the offset of $ \Gamma $ in [6]. For $ \lambda $ within a positive reach, most nice properties are fulfilled by the $ \Gamma_ \lambda $. For instance, $ \Gamma_ \lambda $ shares the class of differentiability of the $ \Gamma $, since there is no ambiguity about the nearest point on $ \Gamma $ in such region. And points of $ \Gamma_ \lambda $ project onto $ \Gamma $ along the normal to $ \Gamma $ through such point.

    In Figure 1, we display three examples to show the relationship between $ \partial \Omega_ \lambda $ and $ \Gamma_ \lambda $. In the left two graphs, $ \Gamma $ is the outside polygon. The interior "curve" of (A) is $ \Gamma_ \lambda $, which is not a real curve. The interior curve of (B) is $ \partial \Omega_ \lambda $. In graph (C), the outer curve is $ \Gamma $ and the interior two curves make up $ \partial \Omega_ \lambda $. So, in general $ \Omega_ \lambda $ is not necessarily to be connected and its boundary $ \partial \Omega_ \lambda $ may not be equal with $ \Gamma_ \lambda $. However, it is not hard to see that $ \partial \Omega_ \lambda \subset \Gamma_ \lambda $. We would like to ask that what is the sufficient condition for $ \Gamma_ \lambda $ to be a Jordan curve and what is the sufficient condition for $ \partial \Omega_ \lambda = \Gamma_ \lambda $? These questions are studied in [7]. If $ \Gamma_ \lambda $ is a Jordan curve when $ \lambda $ is small enough, we find that, with a flatness condition, $ \Gamma_ \lambda $ is approaching to $ \Gamma $ in the sense of Hausdorff distance as $ \lambda $ goes to $ 0 $. This means that $ \Gamma_ \lambda $ is similar to $ \Gamma $ when $ \lambda $ is small enough. Thus we may expect $ \Gamma_ \lambda $ inherits the geometric properties of $ \Gamma $. This approximation property of $ \Gamma_ \lambda $ could be applied in the theory of integration. In another paper we are preparing for, we show that $ \int_{ \Gamma_ \lambda}f \to \int_{ \Gamma} f $ with some geometric restriction on $ \Gamma $.

    Figure 1.  In (A) and (B) $ \lambda = \frac{xy}{2} $. In (C) $ \lambda > \frac{zw}{2} $.

    Given two points $ x, y\in \Gamma $, denotes $ \Gamma(x, y) $ by the subarc of $ \Gamma $ containing and connecting $ x $ and $ y $ which has a smaller diameter, or, to be either subarc when both have the same diameter. Let $ \ell_{x, y} $ be the infinite line through $ x $ and $ y $, let

    $ ζΓ(x,y)=1|xy|sup{dist(z,x,y):zΓ(x,y)}. $

    Following definition can be introduced.

    Definition 1.1. [7] A Jordan curve $ \Gamma $ is said to have the $ (\zeta, r_0) $-chordal property for a certain $ \zeta > 0 $ and $ r_0 > 0 $, if

    $ sup{ζΓ(x,y):x,yΓand|xy|r0}ζ. $

    Also, denote

    $ ζΓ=limr0sup{ζΓ(x,y):x,yΓ and |xy|r}. $

    These quantities are used to measure the local deviation of the subarcs from their chords. It is not hard to see that $ \Gamma $ is smooth if and only if $ \zeta_ \Gamma = 0 $. Therefore all smooth curves have the $ (\zeta, r_0) $-chordal property. Moreover, if a piecewise smooth Jordan curve only has corner points then it has the $ (\zeta, r_0) $-chordal property. However if a piecewise smooth Jordan curve has a cusp point then it does not have the $ (\zeta, r_0) $-chordal property.

    Theorem 1.1. [7] Let $ \Gamma $ be a Jordan curve in $ {\Bbb R}^2 $. If $ \Gamma $ has the $ (1/2, r_0) $-chordal property for some $ r_0 > 0 $, then $ \Gamma $ has the level Jordan curve property i.e., there exists $ \lambda_0 > 0 $ such that $ \Gamma_ \lambda $ is a Jordan curve for each $ \lambda < \lambda_0 $.

    This theorem provides us a method to verify whether $ \Gamma_ \lambda $ is a Jordan curve. As we have seen in Figure 1, even through $ \Gamma $ is a simple Jordan curve, $ \Gamma_ \lambda $ varies greatly. Based on this theorem, the authors of [7] also studied the quasi-circle property of $ \Gamma_ \lambda $. However, we are interested in the limit behaviour of $ \Gamma_ \lambda $ as $ \lambda $ approaching to $ 0 $. The $ (\zeta, r_0) $-chordal property of Jordan curves is an essential condition in the proof of the main theorem, we show that $ \Gamma_ \lambda $ converges to $ \Gamma $ if $ \Gamma $ has the $ (1/2, r_0) $-chordal property.

    The other parts of the paper will be organized as follows: In Section 2, we investigate some basic properties of constant distance boundary of Jordan curves. We prove that if $ \Gamma $ and $ \Gamma_ \lambda $ are Jordan curves then there exist at least three points of $ \Gamma $ which have distance $ \lambda $ from $ \Gamma_ \lambda $. Also, we find out the relation between $ \Gamma_{ \lambda+\mu} $ and $ (\Gamma_\mu)_ \lambda $. This relation will be used in the proof of our main theorem. Section 3 is devoted to prove our main result, Theorem 3.1. The definition and some basic properties of Hausdorff distance, $ d_H(\cdot, \cdot) $, are introduced firstly. We show that under the $ (1/2, r_0) $-chordal property of $ \Gamma $, the upper and lower bounds of $ d_H(\Gamma, \Gamma_ \lambda) $ are obtained. Consequently, the main theorem can be obtained.

    In this section, we investigate several fundamental properties according to the $ (\zeta, r_0) $-chordal property of Jordan curve $ \Gamma $. In this paper, we always assume that $ \lambda > 0 $ and that $ \Gamma_ \lambda $ is non-empty. First we introduce a notation which will be used frequently through the paper. For each $ x\in \Gamma $, define

    $ Γxλ:={yΓλ:|xy|=λ}. $

    And for any $ y\in \Gamma_ \lambda $, define

    $ Γy:={xΓ:|xy|=λ}. $

    In [5], the so called $ \lambda $-parallel set of $ \Gamma $ is introduced. The definition is the following,

    $ Ωpλ:={zΩ:dist(z,Γ)<λ}. $

    Recall that we already have the set

    $ Ωλ={zΩ:dist(z,Γ)>λ}. $

    We have seen in Figure 1 that $ \partial \Omega_ \lambda $ is a proper subset of $ \Gamma_ \lambda $ and Theorem 1.1 states that if $ \Gamma $ has the $ (1/2, r_0) $-chordal property for some $ r_0 > 0 $ then $ \partial \Omega_ \lambda = \Gamma_ \lambda $ whenever $ \lambda $ is small enough. However, the identical of $ \Gamma_ \lambda $ and $ \partial \Omega_ \lambda^p $ can be obtained directly without the $ (1/2, r_0) $-chordal property.

    Proposition 2.1. $ \Gamma_ \lambda = \partial \Omega_ \lambda^p $.

    Proof. According to the continuity of the distance function, the relation of $ \partial \Omega_ \lambda^p \subset \Gamma_ \lambda $ is obvious.

    Let $ z\in \Gamma_ \lambda $. Then there exists $ x\in \Gamma^z $. Consider an arbitrary point $ y $ on the segment $ (x, z) $. We know that $ \operatorname{dist}(y, \Gamma)\le |x-y| < \lambda $. Thus $ y\in \Omega_ \lambda^p $. Since the point $ z $ is the limit of points along the segment $ [x, z] $, we know that $ z\in \overline{ \Omega_ \lambda^p} $. Therefore we have $ \Gamma_ \lambda \subset \partial \Omega_ \lambda^p $.

    In the above proof, $ [x, z] $ stands for the line segment connecting points $ x $ and $ z $, while $ (x, z) $ is $ [x, z]\setminus \{x, z\} $.

    Proposition 2.2. Let $ x, y\in \Gamma_ \lambda $ be different points and let $ x'\in \Gamma^x $ and $ y'\in \Gamma^y $. If the two segments $ [x, x'] $ and $ [y, y'] $ intersect at $ p $, i.e., $ [x, x']\cap [y, y'] = \{p\} $ then $ x' = y' = p $.

    Proof. If $ x'\not = y' $ then $ \{p\} = (x, x')\cap(y, y') $. We have

    $ |xp|+|px|=λ and |yp|+|py|=λ. $

    Since

    $ |xp|+|py||xy|λ. $

    It follows that

    $ |xp||yp|. $

    Then

    $ |xy||xp|+|py|λ. $

    Because of $ |x'-y|\ge \lambda $, we know that $ |x'-y| = |x'-p|+|p-y| = \lambda $. This means that the points $ y $, $ p $ and $ x' $ are collinear, i.e., $ p\in (y, x') $. However $ p\in (y, y') $, this is impossible unless $ x' = y' $. Therefore, we must have $ x' = y' = p $.

    This proposition tells us that two such segments $ [x, x'] $ and $ [y, y'] $ could only intersect at the end points.

    Proposition 2.3. Let $ x\in \Gamma $ and $ y\in \Gamma_ \lambda^x $. If $ z\in (x, y) $ such that $ |x-z| = \mu $ for some $ 0 < \mu < \lambda $ then $ z\in \Gamma_\mu^x $.

    Proof. Since $ y\in \Gamma_ \lambda^x $ and $ |x-z| = \mu $, we have $ |y-z| = \lambda-\mu $ and $ \operatorname{dist}(z, \Gamma)\le|x-z| = \mu $. Suppose that $ \operatorname{dist}(z, \Gamma) < \mu $, there exists $ t\in \Gamma $ such that $ |z-t| = \operatorname{dist}(z, \Gamma) < \mu $. Then $ |y-t|\le |y-z|+|z-t| < \lambda-\mu+\mu = \lambda $. It follows that $ \operatorname{dist}(y, \Gamma) < \lambda $. This contradicts to the fact that $ y\in \Gamma_ \lambda^x \subset \Gamma_ \lambda $. Thus $ \operatorname{dist}(z, \Gamma) = \mu $ and then $ z\in \Gamma_\mu^x $.

    In the proofs of the above three propositions, the set $ \Gamma $ is not necessarily to be a Jordan curve. So these properties are correct for any compact subset of $ {\Bbb C} $. In the following context, we assume that $ \Gamma $ is a Jordan curve. The Lemma 4.2 of [7] states that if $ \Gamma_ \lambda $ is a Jordan curve and if there exist distinct $ x, y\in \Gamma^z_ \lambda $ for some $ z\in \Gamma $, then the subarc $ \Gamma_ \lambda(x, y) $ is a circular arc of the circle centred at $ z $ and with radius $ \lambda $, which denoted by $ \gamma(z, \lambda) $.

    Lemma 2.1. If $ \Gamma $ and $ \Gamma_ \lambda $ are Jordan curves then there exist at least three points of $ \Gamma $ which all have distance $ \lambda $ from $ \Gamma_ \lambda $.

    Proof. Suppose that there is no point on $ \Gamma $ has distance $ \lambda $ from $ \Gamma_ \lambda $. It means that for any $ p\in \Gamma $ the distance $ \operatorname{dist}(p, \Gamma_ \lambda)\not = \lambda $. It is clear that $ \operatorname{dist}(p, \Gamma_ \lambda) < \lambda $ is incorrect. Thus $ \operatorname{dist}(p, \Gamma_ \lambda) > \lambda $ for all $ p\in \Gamma $. It follows that for a fixed point $ q\in \Gamma_ \lambda $ we know that $ |p-q| > \lambda $ for all $ p\in \Gamma $. Therefore $ \operatorname{dist}(q, \Gamma) > \lambda $. This contradicts the fact that $ q\in \Gamma_ \lambda $.

    Suppose that there is only one point $ x\in \Gamma $ which has distance $ \lambda $ from $ \Gamma_ \lambda $. Therefore $ \operatorname{dist}(x, \Gamma_ \lambda) = \lambda $ and $ \operatorname{dist}(p, \Gamma_ \lambda) > \lambda $ for any $ p\in \Gamma $ when $ p\not = x $. Thus for arbitrary $ q\in \Gamma_ \lambda $, we have $ |q-p| > \lambda $ when $ p\not = x $. It implies that $ |q-x| = \lambda $. Then $ \Gamma_ \lambda \subset \Gamma_ \lambda^x $. So $ \Gamma_ \lambda $ is a circular arc of the circle with center at $ x $ and with radius $ \lambda $, i.e., $ \Gamma_ \lambda \subset \gamma(x, \lambda) $. Because $ \Gamma_ \lambda $ is a Jordan curve, we must have $ \Gamma_ \lambda = \gamma(x, \lambda) $. Therefore $ \Gamma $ is the union of $ \{x\} $ and a certain subset of circle $ \gamma(x, 2 \lambda) $. In other words, $ \Gamma $ is separated by $ \Gamma_ \lambda $ into two parts. This contradicts the fact that $ \Gamma $ is a Jordan curve.

    Suppose that there are only two points $ x, y\in \Gamma $ which have distance $ \lambda $ from $ \Gamma_ \lambda $. It means that $ \operatorname{dist}(x, \Gamma_ \lambda) = \lambda = \operatorname{dist}(y, \Gamma_ \lambda) $ and $ \operatorname{dist}(p, \Gamma_ \lambda) > \lambda $ for any $ p\in \Gamma $ when $ p\not = x, y $. Similar to the one point case, we know that $ \Gamma_ \lambda \subset \Gamma_ \lambda^x\cup \Gamma_ \lambda^y $. Since $ \Gamma_ \lambda $ is a Jordan curve, there are three situations we should consider.

    (i) If $ |x-y| = 2 \lambda $ then $ \Gamma_ \lambda $ lies in the two tangential circles $ \gamma(x, \lambda) $ and $ \gamma(y, \lambda) $. Because $ \Gamma_ \lambda $ is a Jordan curve, it could only contained in one circle. Then $ x $ and $ y $ are separated by this circle which contradicts that fact that $ \Gamma $ is a Jordan curve.

    (ii) If $ |x-y| < 2 \lambda $ then $ \Gamma_ \lambda $ is the curve looks like number eight which enclose $ x $ and $ y $ at the inside area. While $ \Gamma\setminus\{x, y\} $ is in the outside area otherwise $ \Gamma = \{x, y\} $. The both situations contradict the fact that $ \Gamma $ is a Jordan curve.

    (iii) If $ |x-y| > 2 \lambda $ then $ \Gamma_ \lambda $ lies in one of the disjoint two circles $ \gamma(x, \lambda) $ and $ \gamma(y, \lambda) $. Therefore $ x $ and $ y $ are not connected which contradicts that fact that $ \Gamma $ is a Jordan curve.

    By the above analysis we finished the proof.

    The constant distance boundary $ \Gamma_ \lambda $ of $ \Gamma $ will be a Jordan curve under specific conditions (see Theorem 1.1). Thus we can consider the constant distance boundary of $ \Gamma_ \lambda $, denoted by $ (\Gamma_ \lambda)_\mu $ if which is non-empty. Naturally we will investigate the relationship between $ (\Gamma_ \lambda)_\mu $ and $ \Gamma_{ \lambda+\mu} $.

    Lemma 2.2. Let $ \lambda_0 > 0 $. Suppose that $ \Gamma_ \lambda $ is a Jordan curve for each $ \lambda < \lambda_0 $. Then for $ 0 < \mu < \lambda < \lambda_0 $ we have

    $ Γλ(Γμ)λμ. $

    Proof. Since $ 0 < \mu < \lambda < \lambda_0 $, it follows from Proposition 2.1 that $ \Omega_\mu^p \subset \Omega_ \lambda^p $. For any $ y\in \Gamma_ \lambda $, there is $ x\in \Gamma $ such that $ |x-y| = \lambda $. This means that $ y\in \Gamma_ \lambda^x $. Let $ z $ be a point of segment $ [x, y] $ such that $ |x-z| = \mu $. By Proposition 2.3, we conclude that $ z\in \Gamma_\mu^x $, i.e., $ z\in \Gamma_\mu $.

    Now we have $ \operatorname{dist}(y, \Gamma_\mu)\le |y-z| = \lambda-\mu $. If the equality holds then $ y\in (\Gamma_\mu)_{ \lambda-\mu} $. If $ \operatorname{dist}(y, \Gamma_\mu) < |y-z| $ then there exists $ z'\in \Gamma_\mu $ such that $ \operatorname{dist}(y, \Gamma_\mu) = |y-z'| < |y-z| = \lambda-\mu $. Because of $ z'\in \Gamma_\mu $, there must exists $ x'\in \Gamma $ such that $ |z'-x'| = \mu $. Therefore $ \operatorname{dist}(y, \Gamma)\le |y-x'|\le |y-z'|+|z'-x'| < \lambda-\mu+\mu = \lambda $ which contradicts to the fact of $ y\in \Gamma_ \lambda $. Therefore we must have $ \operatorname{dist}(y, \Gamma_\mu) = |y-z| = \lambda-\mu $ which means $ y\in (\Gamma_\mu)_{ \lambda-\mu} $. It follows that $ \Gamma_ \lambda \subset (\Gamma_\mu)_{ \lambda-\mu} $.

    In Lemma 2.2 even though we assume that the sets $ \Gamma_ \lambda $ and $ \Gamma_\mu $ are Jordan curves, but $ \Gamma_\mu $ does not necessarily satisfy the $ (1/2, r_0) $-chordal property, thus the set $ (\Gamma_\mu)_{ \lambda-\mu} $ probably is not a Jordan curve (see Theorem 1.1).

    Corollary 2.1. Let $ \Gamma $ be a Jordan curve and has level Jordan curve property for some $ \lambda_0 > 0 $. If $ \Gamma_\mu $ has $ (1/2, r_0) $-chordal property for a $ \mu < \lambda_0 $, then $ \Gamma_ \lambda = (\Gamma_\mu)_{ \lambda-\mu} $ when $ 0 < \mu < \lambda < \lambda_0 $ and $ \lambda-\mu < \delta $ for some $ \delta > 0 $.

    Proof. By the assumption of level Jordan curve property of $ \Gamma $, we know that $ \Gamma_ \lambda $ and $ \Gamma_\mu $ are Jordan curves if $ 0 < \mu < \lambda < \lambda_0 $. Because of Lemma 2.2 we have

    $ Γλ(Γμ)λμ. $

    By Theorem 1.1 and by the assumption of $ \Gamma_\mu $ has $ (1/2, r_0) $-chordal property, we know that the curve $ \Gamma_\mu $ has level Jordan curve property for some $ \delta > 0 $. Thus its constant distance boundary $ (\Gamma_\mu)_{ \lambda-\mu} $ is a Jordan curve when $ \lambda-\mu < \delta $. Then both $ \Gamma_ \lambda $ and $ (\Gamma_\mu)_{ \lambda-\mu} $ are Jordan curves, it implies that $ \Gamma_ \lambda = (\Gamma_\mu)_{ \lambda-\mu} $.

    In Corollary 2.1, the $ (1/2, r_0) $-chordal property of $ \Gamma_\mu $ is crucial, because it is a necessary condition for the set $ (\Gamma_\mu)_{ \lambda-\mu} $ to be a Jordan curve, i.e., the curve $ \Gamma_\mu $ has level Jordan curve property. So far, we only know that if $ \Gamma_\mu $ has $ (1/2, r_0) $-chordal property then $ (\Gamma_\mu)_{ \lambda-\mu} $ is a Jordan curve when $ \lambda-\mu < \delta $ for some $ \delta > 0 $.

    In this section, we study the limit behaviour of $ \Gamma_ \lambda $ as $ \lambda $ tends to $ 0 $. All the limits are considered in the sense of Hausdorff distance. For the convenience of readers, we briefly introduce the concept and some elementary properties of Hausdorff distance, which can be found in [4].

    Definition 3.1. Let $ X $ and $ Y $ be two non-empty subsets of $ {\Bbb C} $. The Hausdorff distance of $ X $ and $ Y $, denoted by $ d_H(X, Y) $, is defined by

    $ dH(X,Y):=max{supxXinfyY|xy|,supyYinfxX|xy|}. $

    Denote by

    $ d(X,Y):=supxXdist(x,Y)andd(Y,X):=supyYdist(y,X) $

    the distance from $ X $ to $ Y $ and $ Y $ to $ X $ respectively. We could rewrite

    $ dH(X,Y)=max{d(X,Y),d(Y,X)}. $ (3.1)

    Note that $ d(X, Y)\not = d(Y, X) $ usually happens.

    For non-empty subsets $ X $ and $ Y $ of $ {\Bbb C} $, we know that

    $ d(X,Y)=0xX,dist(x,Y)=0xX,x¯YX¯Y. $

    Here $ \overline Y $ is the closure of $ Y $ in $ {\Bbb C} $. We summarize these equivalence relations in the following proposition.

    Proposition 3.1. Let $ X $ and $ Y $ be two non-empty subsets of $ {\Bbb C} $. Then $ d(X, Y) = 0 $ if and only if $ X \subset \overline Y $. Furthermore, $ d_H(X, Y) = 0 $ if and only if $ \overline X = \overline Y $.

    The triangle inequality is true not only for $ d_H $ but also for $ d $. That is for any compact subsets $ A $, $ B $ and $ C $ of $ {\Bbb C} $ we have

    $ d(A,B)d(A,C)+d(C,B). $ (3.2)

    We left the proof of (3.2) for interested readers as an exercise.

    Denote by $ \Pi $ the set of compact subsets of $ {\Bbb C} $. Federer shows in [4] that $ (\Pi, d_H) $ is a complete metric space. According to our consideration, $ \Gamma $ is a Jordan curve, so it is compact. By the definition of constant distance boundary, $ \Gamma_ \lambda $ is compact as well. Thus we have $ \Gamma, \Gamma_ \lambda \in \Pi $. Observe that $ \Gamma_0 = \Gamma $, so we want to know whether the limit of $ \Gamma_ \lambda $, in $ (\Pi, d_H) $, is $ \Gamma $ or not as $ \lambda $ approaching to zero. The first proposition we obtained is the following.

    Proposition 3.2. If there exists $ L\in\Pi $ such that $ \lim_{ \lambda\to 0} \Gamma_ \lambda = L $ then $ L \subset \Gamma $.

    Proof. If $ \lim_{ \lambda\to 0} \Gamma_ \lambda = L $ then $ \lim_{ \lambda\to 0}d_H(\Gamma_ \lambda, L) = 0 $. It follows that $ \lim_{ \lambda\to 0}d(L, \Gamma_ \lambda) = 0 $. We know that $ d(\Gamma_ \lambda, \Gamma) = \lambda $, since

    $ d(Γλ,Γ)=supxΓλdist(x,Γ)=λ. $

    It follows from (3.2) that $ d(L, \Gamma)\le d(L, \Gamma_ \lambda)+d(\Gamma_ \lambda, \Gamma) $. Letting $ \lambda $ tends to $ 0 $ implies that $ d(L, \Gamma) = 0 $, thus $ L \subset \overline \Gamma $. By the compactness of $ \Gamma $, we have that $ L \subset \overline \Gamma = \Gamma $.

    Proposition 3.2 states that if the limit of $ \Gamma_ \lambda $ exists in $ \Pi $ then it must be a subset of $ \Gamma $. But we still cannot confirm whether this limit is a proper subset of $ \Gamma $ or equal to $ \Gamma $. While if $ \Gamma $ has the $ (1/2, r_0) $-chordal property, we obtain the following result.

    Lemma 3.1. Suppose that $ \Gamma $ has $ (1/2, r_0) $-chordal property and that $ \lambda\le r_0/2 $. Then $ \lambda\le d_H(\Gamma_ \lambda, \Gamma)\le (2\sqrt{5}+1) \lambda $.

    Proof. Because that $ \Gamma $ has the $ (1/2, r_0) $-chordal property, we may assume that $ \Gamma_ \lambda $ in consideration is a Jordan curve. Recall that $ d(\Gamma_ \lambda, \Gamma) = \lambda $. By (3.1), we already have

    $ dH(Γλ,Γ)=max{d(Γλ,Γ),d(Γ,Γλ)}λ. $

    Because of $ d(\Gamma, \Gamma_ \lambda)\ge \lambda $ we assume that $ d(\Gamma, \Gamma_ \lambda) > \lambda $, otherwise $ d_H(\Gamma_ \lambda, \Gamma) = \lambda $.

    Now suppose that there exists a $ w\in \Gamma $ such that $ \operatorname{dist}(w, \Gamma_ \lambda) > \lambda $. By Lemma 2.1 there are at least three points of $ \Gamma $ which have distance $ \lambda $ from $ \Gamma_ \lambda $. So we can choose a subarc $ \Gamma(x, y) $ of $ \Gamma $ such that $ w\in \Gamma(x, y) $ and $ d(p, \Gamma_ \lambda)\ge \lambda $ for all $ p\in \Gamma(x, y) $, especially, the equality holds only when $ p\in \{x, y\} $. The reason is that if there is a $ z\in \Gamma(x, y)\setminus \{x, y\} $ such that $ d(z, \Gamma_ \lambda) = \lambda $ then one of the two subarcs $ \Gamma(x, z) $ or $ \Gamma(z, y) $ contains $ w $. Thus only need to replace $ \Gamma(x, y) $ by this subarc. By the compactness of $ \Gamma_ \lambda $, there exist $ x'\in \Gamma_ \lambda^x $ and $ y'\in \Gamma_ \lambda^y $.

    (i) Consider the case when $ x' = y' = q $. It is not hard to know that $ |x-y|\le |x-q|+|y-q| = 2 \lambda $ and thus $ |x-y|\le r_0 $. By the $ (1/2, r_0) $-chordal property of $ \Gamma $, we obtain that $ \operatorname{dist}(p, \ell_{x, y})\le 1/2|x-y|\le \lambda $ for every $ p\in \Gamma(x, y) $. The straight line $ \ell_{x, y} $ separates the complex plane into two parts, which denoted by $ {\Bbb C}^R $ and $ {\Bbb C}^L $.

    Firstly, we assume that $ \Gamma\cap {\Bbb C}^R $ and $ \Gamma\cap {\Bbb C}^L $ are non-empty. Let $ p_0\in {\Bbb C}^R\cap \Gamma(x, y) $ such that

    $ dist(p0,x,y)=max{dist(p,x,y):pCRΓ(x,y)}. $

    Similarly, let $ p_1\in {\Bbb C}^L\cap \Gamma(x, y) $ such that

    $ dist(p1,x,y)=max{dist(p,x,y):pCLΓ(x,y)}. $

    Construct straight lines $ \ell_{p_0} $ and $ \ell_{p_1} $ pass through $ p_0 $ and $ p_1 $ respectively and parallel to $ \ell_{x, y} $. Thus the arc $ \Gamma(x, y) $ is bounded in the strip region between $ \ell_{p_0} $ and $ \ell_{p_1} $ which has width at most $ 2 \lambda $. It is needed to explain that $ \Gamma(x, y) $ may only at one side of the line $ \ell_{x, y} $. Thus $ p_0 $ or $ p_1 $ may does not exist. However, $ \Gamma(x, y) $ can not be a straight line otherwise $ x' $ and $ y' $ must be different. Therefore, at least, one of $ p_0 $ or $ p_1 $ must exists. Then the mentioned strip region now is between $ \ell_{p_1} $ and $ \ell_{x, y} $ if $ p_0 $ does not exist, while the strip region is between $ \ell_{p_0} $ and $ \ell_{x, y} $ if $ p_1 $ does not exist. In these cases, the width of the strip region is at most $ \lambda $.

    Construct straight lines $ \ell_x $ and $ \ell_y $ pass through $ x $ and $ y $ respectively and perpendicular to $ \ell_{x, y} $. Choose $ x' $ and $ x'' $ on $ \ell_x\cap \Gamma(x, y) $ such that

    $ |xx|=max{|st|:s,txΓ(x,y)}. $

    Because $ \Gamma(x, y) $ is bounded in the strip region with width at most $ 2 \lambda $, we must have $ |x'-x''|\le 2 \lambda\le r_0 $. Thus the $ (1/2, r_0) $-chordal property implies that the arc $ \Gamma(x', x'') $ is bounded in a strip region which has width at most $ 2 \lambda $. By the similar argument for $ \ell_y $, we obtain that $ \Gamma(x, y) $ is bounded in a rectangular with width $ 2 \lambda $ and length $ 4 \lambda $. We denote this rectangular by $ \Delta $. Thus $ |w-x|\le \operatorname{diam} \Delta = 2 \sqrt 5 \lambda $. Here $ \operatorname{diam} \Delta $ is the diameter of $ \Delta $. So $ |w-q|\le |w-x|+|x-q|\le 2 \sqrt 5 \lambda+ \lambda = (2\sqrt 5+1) \lambda $. This implies that $ \operatorname{dist}(w, \Gamma_ \lambda)\le (2\sqrt 5+1) \lambda $.

    (ii) Consider the case when $ x'\not = y' $. For every $ q\in \Gamma_ \lambda(x', y') $ there exists $ p\in \Gamma^q $.

    If $ p\in \Gamma(x, y) $ the selection condition of $ \Gamma(x, y) $ implies that $ p\in \{x, y\} $. We can see that if $ p = x $ then replace $ x' $ by $ q $, also denoted by $ x' $, and if $ p = y $ then replace $ y' $ by $ q $, also denoted by $ y' $. Choose another point $ q'\in \Gamma_ \lambda(x', y') $ and continuous the above process, finally we must have that $ x' = y' $. Then repeat the proof of case (i), we also have $ \operatorname{dist}(w, \Gamma_ \lambda)\le (2\sqrt 5+1) \lambda $.

    Suppose that $ p\in \Gamma(y, x) $. Here $ \Gamma(y, x) = \Gamma\setminus \Gamma(x, y) $. Because the segment $ [p, q] $ does not intersect with $ \Gamma_ \lambda(y', x') $. Then $ \Gamma_ \lambda(y', x') $ is enclosed by the union of arcs $ L: = \Gamma(x, y)\cup [y, y']\cup \Gamma_ \lambda(x', y')\cup [x', x] $. It implies that for every $ q'\in \Gamma_ \lambda(y', x') $ there must exists $ p'\in \Gamma^{q'} $ such that $ p'\in \Gamma(x, y) $. If this is not the case then $ p'\in \Gamma(y, x) $, and then $ [p', q'] $ intersects $ L $ which is impossible. Repeat the analysis of the case when $ p\in \Gamma(x, y) $ for $ p'\in \Gamma(x, y) $, it follows that $ \operatorname{dist}(w, \Gamma_ \lambda)\le (2\sqrt 5+1) \lambda $.

    In the above analysis, we have considered all the possible situations. As a conclusion, we obtain that $ d(\Gamma, \Gamma_ \lambda)\le (2\sqrt 5+1) \lambda $. Therefore we have the inequalities $ \lambda\le d_H(\Gamma_ \lambda, \Gamma)\le (2\sqrt 5+1) \lambda $.

    In Lemma 3.1, the condition which $ \Gamma $ has $ (1/2, r_0) $-chordal property is crucial, because of the $ (1/2, r_0) $-chordal property the curve $ \Gamma $ has level Jordan property and then the upper bound of $ d_H(\Gamma_ \lambda, \Gamma) $ can be decided. However this condition is rigorous, we should consider the questions for curves without this restriction in the future work.

    Theorem 3.1. If Jordan curve $ \Gamma $ has $ (1/2, r_0) $-chordal property then $ \lim_{ \lambda\to 0} \Gamma_ \lambda = \Gamma $ in $ (\Pi, d_H) $.

    Proof. By Lemma 3.1, we immediately obtain that $ d_H(\Gamma_ \lambda, \Gamma)\le (2\sqrt 5+1) \lambda $ when $ 2 \lambda\le r_0 $. It implies that $ \lim_{ \lambda\to 0} \Gamma_ \lambda = \Gamma $.

    This theorem provides us a sufficient condition for $ \Gamma $ such that its constant distance boundaries converging to itself. Now let $ \lambda $ take discrete values $ \{\frac{1}{n}\}_{n = 1}^\infty $, we have the following corollary.

    Corollary 3.1. Let $ \Gamma $ be a Jordan curve and has level Jordan curve property for some $ \lambda_0 > 0 $. If $ \Gamma_{\frac{1}{n}} $ has $ (1/2, r_0) $-chordal property when $ 1/n < \lambda_0 $. Then the limit $ \lim_{n\to \infty} \Gamma_{\frac{1}{n}} $ exists.

    Proof. The Corollary 2.1 implies that $ \Gamma_{\frac{1}{n}} = (\Gamma_{\frac{1}{m}})_{{\frac{1}{n}}-{\frac{1}{m}}} $ when $ 0 < {\frac{1}{m}} < {\frac{1}{n}} < \lambda_0 $ and $ {\frac{1}{n}}-{\frac{1}{m}} < \delta $ for some $ \delta > 0 $. By Lemma 3.1, it follows that

    $ dH(Γ1m,(Γ1m)1n1m)(25+1)(1n1m), $

    when $ {\frac{1}{n}}-{\frac{1}{m}} < r_0/2 $. Therefore when $ {\frac{1}{n}}-{\frac{1}{m}} < \min\{ \delta, r_0/2\} $, we obtain that

    $ dH(Γ1m,Γ1n)=dH(Γ1m,(Γ1m)1n1m)(25+1)(1n1m). $

    Therefore $ \{ \Gamma_{\frac{1}{n}}\} $ is a Cauchy sequence in $ (\Pi, d_H) $, and then the limit $ \lim_{n\to \infty} \Gamma_{\frac{1}{n}} $ exists.

    The first author is supported by NSFC (12101453).

    The authors declared that they have no conflicts of interest to this work.

    [1] Klauser T, Pohlner J, Meyer TF (1993) The secretion pathway of IgA protease-type proteins in Gram-negative bacteria. Bioessays 15: 799–805. doi: 10.1002/bies.950151205
    [2] Pohlner J, Halter R, Beyreuther K, et al. (1987) Gene structure and extracellular secretion of Neisseria gonorrhoeae IgA protease. Nature 325: 458–462. doi: 10.1038/325458a0
    [3] Klauser T, Pohlner J, Meyer TF (1990) Extracellular transport of cholera toxin B subunit using Neisseria IgA protease beta-domain: conformation-dependent outer membrane translocation. EMBO J 9: 1991–1999.
    [4] Nicolay T, Vanderleyden J, Spaepen S (2015) Autotransporter-based cell surface display in Gram-negative bacteria. Crit Rev Microbiol 41: 109–123. doi: 10.3109/1040841X.2013.804032
    [5] Henderson IR, Navarro-Garcia F, Desvaux M, et al. (2004) Type V protein secretion pathway: the autotransporter story. Microbiol Mol Biol Rev 68: 692–744. doi: 10.1128/MMBR.68.4.692-744.2004
    [6] Fan E, Chauhan N, Udatha DB, et al. (2016) Type V secretion systems in bacteria. Microbiol Spectr 4.
    [7] Guerin J, Bigot S, Schneider R, et al. (2017) Two-partner secretion: combining efficiency and simplicity in the secretion of large proteins for bacteria-host and bacteria-bacteria interactions. Front Cell Infect Mi 7: 148. doi: 10.3389/fcimb.2017.00148
    [8] Bassler J, Alvarez BH, Hartmann MD, et al. (2015) A domain dictionary of trimeric autotransporter adhesins. Int J Med Microbiol 305: 265–275. doi: 10.1016/j.ijmm.2014.12.010
    [9] Linke D, Riess T, Autenrieth IB, et al. (2006) Trimeric autotransporter adhesins: variable structure, common function. Trends Microbiol 14: 264–270. doi: 10.1016/j.tim.2006.04.005
    [10] Salacha R, Kovacic F, Brochier-Armanet C, et al. (2010) The Pseudomonas aeruginosa patatin-like protein PlpD is the archetype of a novel Type V secretion system. Environ Microbiol 12: 1498–1512.
    [11] Casasanta MA, Yoo CC, Smith HB, et al. (2017) A chemical and biological toolbox for Type Vd secretion: Characterization of the phospholipase A1 autotransporter FplA from Fusobacterium nucleatum. J Biol Chem 292: 20240–20254. doi: 10.1074/jbc.M117.819144
    [12] Leo JC, Oberhettinger P, Schutz M, et al. (2015) The inverse autotransporter family: intimin, invasin and related proteins. Int J Med Microbiol 305: 276–282. doi: 10.1016/j.ijmm.2014.12.011
    [13] Wu T, Malinverni J, Ruiz N, et al. (2005) Identification of a multicomponent complex required for outer membrane biogenesis in Escherichia coli. Cell 121: 235–245. doi: 10.1016/j.cell.2005.02.015
    [14] Knowles TJ, Scott-Tucker A, Overduin M, et al. (2009) Membrane protein architects: the role of the BAM complex in outer membrane protein assembly. Nat Rev Microbiol 7: 206–214. doi: 10.1038/nrmicro2069
    [15] Malinverni JC, Werner J, Kim S, et al. (2006) YfiO stabilizes the YaeT complex and is essential for outer membrane protein assembly in Escherichia coli. Mol Microbiol 61: 151–164. doi: 10.1111/j.1365-2958.2006.05211.x
    [16] Sklar JG, Wu T, Gronenberg LS, et al. (2007) Lipoprotein SmpA is a component of the YaeT complex that assembles outer membrane proteins in Escherichia coli. P Natl Acad Sci USA 104: 6400–6405. doi: 10.1073/pnas.0701579104
    [17] Robert V, Volokhina EB, Senf F, et al. (2006) Assembly factor Omp85 recognizes its outer membrane protein substrates by a species-specific C-terminal motif. PLoS Biol 4: e377. doi: 10.1371/journal.pbio.0040377
    [18] Noinaj N, Kuszak AJ, Gumbart JC, et al. (2013) Structural insight into the biogenesis of beta-barrel membrane proteins. Nature 501: 385–390. doi: 10.1038/nature12521
    [19] Gu Y, Li H, Dong H, et al. (2016) Structural basis of outer membrane protein insertion by the BAM complex. Nature 531: 64–69. doi: 10.1038/nature17199
    [20] Han L, Zheng J, Wang Y, et al. (2016) Structure of the BAM complex and its implications for biogenesis of outer-membrane proteins. Nat Struct Mol Biol 23: 192–196. doi: 10.1038/nsmb.3181
    [21] Albrecht R, Schutz M, Oberhettinger P, et al. (2014) Structure of BamA, an essential factor in outer membrane protein biogenesis. Acta Crystallogr D Biol Crystallogr 70: 1779–1789. doi: 10.1107/S1399004714007482
    [22] Bakelar J, Buchanan SK, Noinaj N (2016) The structure of the beta-barrel assembly machinery complex. Science 351: 180–186. doi: 10.1126/science.aad3460
    [23] Iadanza MG, Higgins AJ, Schiffrin B, et al. (2016) Lateral opening in the intact beta-barrel assembly machinery captured by cryo-EM. Nat Commun 7: 12865. doi: 10.1038/ncomms12865
    [24] Noinaj N, Kuszak AJ, Balusek C, et al. (2014) Lateral opening and exit pore formation are required for BamA function. Structure 22: 1055–1062. doi: 10.1016/j.str.2014.05.008
    [25] Gatzeva-Topalova PZ, Warner LR, Pardi A, et al. (2010) Structure and flexibility of the complete periplasmic domain of BamA: the protein insertion machine of the outer membrane. Structure 18: 1492–1501. doi: 10.1016/j.str.2010.08.012
    [26] Gatzeva-Topalova PZ, Walton TA, Sousa MC (2008) Crystal structure of YaeT: conformational flexibility and substrate recognition. Structure 16: 1873–1881. doi: 10.1016/j.str.2008.09.014
    [27] Knowles TJ, Jeeves M, Bobat S, et al. (2008) Fold and function of polypeptide transport-associated domains responsible for delivering unfolded proteins to membranes. Mol Microbiol 68: 1216–1227. doi: 10.1111/j.1365-2958.2008.06225.x
    [28] Lee J, Xue M, Wzorek JS, et al. (2016) Characterization of a stalled complex on the beta-barrel assembly machine. P Natl Acad Sci USA 113: 8717–8722. doi: 10.1073/pnas.1604100113
    [29] Schiffrin B, Calabrese AN, Higgins AJ, et al. (2017) Effects of periplasmic chaperones and membrane thickness on BamA-catalyzed outer-membrane protein folding. J Mol Biol 429: 3776–3792. doi: 10.1016/j.jmb.2017.09.008
    [30] Hohr AIC, Lindau C, Wirth C, et al. (2018) Membrane protein insertion through a mitochondrial beta-barrel gate. Science 359.
    [31] Jain S, Goldberg MB (2007) Requirement for YaeT in the outer membrane assembly of autotransporter proteins. J Bacteriol 189: 5393–5398. doi: 10.1128/JB.00228-07
    [32] Sauri A, Soprova Z, Wickstrom D, et al. (2009) The Bam (Omp85) complex is involved in secretion of the autotransporter haemoglobin protease. Microbiology 155: 3982–3991. doi: 10.1099/mic.0.034991-0
    [33] Lehr U, Schutz M, Oberhettinger P, et al. (2010) C-terminal amino acid residues of the trimeric autotransporter adhesin YadA of Yersinia enterocolitica are decisive for its recognition and assembly by BamA. Mol Microbiol 78: 932–946. doi: 10.1111/j.1365-2958.2010.07377.x
    [34] Oberhettinger P, Leo JC, Linke D, et al. (2015) The inverse autotransporter intimin exports its passenger domain via a hairpin intermediate. J Biol Chem 290: 1837–1849. doi: 10.1074/jbc.M114.604769
    [35] Albenne C, Ieva R (2017) Job contenders: roles of the beta-barrel assembly machinery and the translocation and assembly module in autotransporter secretion. Mol Microbiol 106: 505–517. doi: 10.1111/mmi.13832
    [36] Pavlova O, Peterson JH, Ieva R, et al. (2013) Mechanistic link between beta barrel assembly and the initiation of autotransporter secretion. P Natl Acad Sci USA 110: E938–E947. doi: 10.1073/pnas.1219076110
    [37] Noinaj N, Gumbart JC, Buchanan SK (2017) The beta-barrel assembly machinery in motion. Nat Rev Microbiol 15: 197–204. doi: 10.1038/nrmicro.2016.191
    [38] Arnold T, Zeth K, Linke D (2010) Omp85 from the thermophilic cyanobacterium Thermosynechococcus elongatus differs from proteobacterial Omp85 in structure and domain composition. J Biol Chem 285: 18003–18015. doi: 10.1074/jbc.M110.112516
    [39] Koenig P, Mirus O, Haarmann R, et al. (2010) Conserved properties of polypeptide transport-associated (POTRA) domains derived from cyanobacterial Omp85. J Biol Chem 285: 18016–18024. doi: 10.1074/jbc.M110.112649
    [40] Bos MP, Grijpstra J, Tommassen-van Boxtel R, et al. (2014) Involvement of Neisseria meningitidis lipoprotein GNA2091 in the assembly of a subset of outer membrane proteins. J Biol Chem 289: 15602–15610. doi: 10.1074/jbc.M113.539510
    [41] Volokhina EB, Beckers F, Tommassen J, et al. (2009) The beta-barrel outer membrane protein assembly complex of Neisseria meningitidis. J Bacteriol 191: 7074–7085. doi: 10.1128/JB.00737-09
    [42] Anwari K, Webb CT, Poggio S, et al. (2012) The evolution of new lipoprotein subunits of the bacterial outer membrane BAM complex. Mol Microbiol 84: 832–844. doi: 10.1111/j.1365-2958.2012.08059.x
    [43] Paramasivam N, Habeck M, Linke D (2012) Is the C-terminal insertional signal in Gram-negative bacterial outer membrane proteins species-specific or not? BMC Genomics 13: 510. doi: 10.1186/1471-2164-13-510
    [44] Volokhina EB, Grijpstra J, Beckers F, et al. (2013) Species-specificity of the BamA component of the bacterial outer membrane protein-assembly machinery. PLoS One 8: e85799. doi: 10.1371/journal.pone.0085799
    [45] Webb CT, Heinz E, Lithgow T (2012) Evolution of the beta-barrel assembly machinery. Trends Microbiol 20: 612–620. doi: 10.1016/j.tim.2012.08.006
    [46] Iqbal H, Kenedy MR, Lybecker M, et al. (2016) The TamB ortholog of Borrelia burgdorferi interacts with the beta-barrel assembly machine (BAM) complex protein BamA. Mol Microbiol 102: 757–774. doi: 10.1111/mmi.13492
    [47] Selkrig J, Mosbahi K, Webb CT, et al. (2012) Discovery of an archetypal protein transport system in bacterial outer membranes. Nat Struct Mol Biol 19: 506–510. doi: 10.1038/nsmb.2261
    [48] Gruss F, Zahringer F, Jakob RP, et al. (2013) The structural basis of autotransporter translocation by TamA. Nat Struct Mol Biol 20: 1318–1320. doi: 10.1038/nsmb.2689
    [49] Josts I, Stubenrauch CJ, Vadlamani G, et al. (2017) The structure of a conserved domain of TamB reveals a hydrophobic beta taco fold. Structure 25: 1898–1906. doi: 10.1016/j.str.2017.10.002
    [50] Shen HH, Leyton DL, Shiota T, et al. (2014) Reconstitution of a nanomachine driving the assembly of proteins into bacterial outer membranes. Nat Commun 5: 5078. doi: 10.1038/ncomms6078
    [51] Stubenrauch C, Belousoff MJ, Hay ID, et al. (2016) Effective assembly of fimbriae in Escherichia coli depends on the translocation assembly module nanomachine. Nat Microbiol 1: 16064. doi: 10.1038/nmicrobiol.2016.64
    [52] Kang'ethe W, Bernstein HD (2013) Charge-dependent secretion of an intrinsically disordered protein via the autotransporter pathway. P Natl Acad Sci USA 110: E4246–E4255. doi: 10.1073/pnas.1310345110
    [53] Norell D, Heuck A, Tran-Thi TA, et al. (2014) Versatile in vitro system to study translocation and functional integration of bacterial outer membrane proteins. Nat Commun 5: 5396. doi: 10.1038/ncomms6396
    [54] Heinz E, Stubenrauch CJ, Grinter R, et al. (2016) Conserved features in the structure, mechanism, and biogenesis of the inverse autotransporter protein family. Genome Biol Evol 8: 1690–1705. doi: 10.1093/gbe/evw112
    [55] Heinz E, Lithgow T (2014) A comprehensive analysis of the Omp85/TpsB protein superfamily structural diversity, taxonomic occurrence, and evolution. Front Microbiol 5: 370.
    [56] Heinz E, Selkrig J, Belousoff MJ, et al. (2015) Evolution of the Translocation and Assembly Module (TAM). Genome Biol Evol 7: 1628–1643. doi: 10.1093/gbe/evv097
    [57] Remmert M, Biegert A, Linke D, et al. (2010) Evolution of outer membrane beta-barrels from an ancestral beta beta hairpin. Mol Biol Evol 27: 1348–1358. doi: 10.1093/molbev/msq017
    [58] Remmert M, Linke D, Lupas AN, et al. (2009) HHomp-prediction and classification of outer membrane proteins. Nucleic Acids Res 37: W446–W451. doi: 10.1093/nar/gkp325
    [59] Kleinschmidt JH (2015) Folding of beta-barrel membrane proteins in lipid bilayers-Unassisted and assisted folding and insertion. BBA-Biomembranes 1848: 1927–1943. doi: 10.1016/j.bbamem.2015.05.004
    [60] Kleinschmidt JH (2003) Membrane protein folding on the example of outer membrane protein A of Escherichia coli. Cell Mol Life Sci 60: 1547–1558. doi: 10.1007/s00018-003-3170-0
    [61] Shahid SA, Bardiaux B, Franks WT, et al. (2012) Membrane-protein structure determination by solid-state NMR spectroscopy of microcrystals. Nat Methods 9: 1212–1217. doi: 10.1038/nmeth.2248
    [62] Junker M, Besingi RN, Clark PL (2009) Vectorial transport and folding of an autotransporter virulence protein during outer membrane secretion. Mol Microbiol 71: 1323–1332. doi: 10.1111/j.1365-2958.2009.06607.x
    [63] Sikdar R, Peterson JH, Anderson DE, et al. (2017) Folding of a bacterial integral outer membrane protein is initiated in the periplasm. Nat Commun 8: 1309. doi: 10.1038/s41467-017-01246-4
    [64] Ieva R, Skillman KM, Bernstein HD (2008) Incorporation of a polypeptide segment into the beta-domain pore during the assembly of a bacterial autotransporter. Mol Microbiol 67: 188–201.
    [65] Grin I, Hartmann MD, Sauer G, et al. (2014) A trimeric lipoprotein assists in trimeric autotransporter biogenesis in enterobacteria. J Biol Chem 289: 7388–7398. doi: 10.1074/jbc.M113.513275
    [66] Ishikawa M, Yoshimoto S, Hayashi A, et al. (2016) Discovery of a novel periplasmic protein that forms a complex with a trimeric autotransporter adhesin and peptidoglycan. Mol Microbiol 101: 394–410. doi: 10.1111/mmi.13398
    [67] Leo JC, Grin I, Linke D (2012) Type V secretion: mechanism(s) of autotransport through the bacterial outer membrane. Philos Trans R Soc Lond B Biol Sci 367: 1088–1101. doi: 10.1098/rstb.2011.0208
    [68] Alvarez BH, Gruber M, Ursinus A, et al. (2010) A transition from strong right-handed to canonical left-handed supercoiling in a conserved coiled-coil segment of trimeric autotransporter adhesins. J Struct Biol 170: 236–245. doi: 10.1016/j.jsb.2010.02.009
    [69] Leo JC, Lyskowski A, Hattula K, et al. (2011) The structure of E. coli IgG-binding protein D suggests a general model for bending and binding in trimeric autotransporter adhesins. Structure 19: 1021–1030.
    [70] Mikula KM, Leo JC, Lyskowski A, et al. (2012) The translocation domain in trimeric autotransporter adhesins is necessary and sufficient for trimerization and autotransportation. J Bacteriol 194: 827–838. doi: 10.1128/JB.05322-11
  • This article has been cited by:

    1. Xin Wei, Zhi-Ying Wen, Constant Distance Boundaries of the t-Quasicircle and the Koch Snowflake Curve, 2023, 43, 0252-9602, 981, 10.1007/s10473-023-0301-6
  • Reader Comments
  • © 2018 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(6918) PDF downloads(802) Cited by(12)

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog