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Bearing in mind that a thorough knowledge of structure can lead to an understanding of function, it is important to elucidate the detailed molecular structures of bacterial surfaces. There have been few recent studies of the physical properties of murein sacculi (12, 18, 20), and yet this cell wall component is of paramount importance to the integrity of the cell. It must have high tensile strength and at the same time be highly permeable to nutrients, waste products, and secreted proteins. Even large macromolecules such as DNA (in genetic exchange, such as transformation) and S-layer proteins can pass through (1, 17). The murein of a gram-negative bacterium is composed of one to three layers of peptidoglycan (12, 19) and is bonded via lipoprotein moieties to the outer membrane; these can be either covalently or electrostatically attached to the peptidoglycan (1). Our recent work on the elasticity and thickness of Escherichia coli and Pseudomonas aeruginosa murein sacculi has helped define several physical properties of this intriguing cell wall fabric (20), but the work did not elucidate the network's porosity. Since 35 to 50% of the peptide stems of peptidoglycan are cross-linked at a given time in the cell's growth cycle (5, 7), this bonding between neighboring glycan strands should be important not only for the fabric's ability to withstand physical strain but also for its porosity. Indeed, it could even be possible for this limited cross-linking in combination with the finite length of each glycan strand to form peptidoglycan arrangements such that holes of various kinds occur within the network. Because of the dynamic nature of cell growth and peptidoglycan turnover, these holes could be transient and difficult to find, especially since they would be at nanoscale dimensions. Yet, the question of holes in these networks is of fundamental importance, and here we resort to computer modeling to predict their existence and distribution. A knowledge of the shape and size distributions of such holes, as well as the variances in these distributions, can be relevant for understanding the passage of molecules through the peptidoglycan layer, the elasticity of the layer, and, possibly, aspects of the mechanics of cell division. In this note, we show the expected distribution of holes deduced from the distribution of glycan strand lengths (5, 7, 9) and discuss their possible consequences for the viability of the bacterium.

Peptidoglycan chemical structure and initial discussion of holes in a monomolecular thick peptidoglycan network. There can be two types of cross-linkages occurring between adjacent glycan strands in a peptidoglycan network: cross-links that occur somewhere in the middle region of a glycan strand and those that occur at the terminus of a strand. Since each peptidoglycan strand possesses a screw axis, the peptide stems are helically arranged so that they protrude at ~90° to one another (7, 11). Accordingly, only a maximum of 50% of the stems can be cross-linked in a horizontal plane. At that point, the network exhibits maximum cross-linking. Frequently, each cross-link involves a covalent bond between a 3-amino-acid stem and a 4-amino-acid stem (L-Gla-D-Glu-m-A₂pm-D-Ala---m-A₂pm-D-Glu-L-Ala) or between two 4-amino-acid stems (L-Ala-D-Glu-m-A₂pm-D-Ala---m-A₂pm [D-Ala]-D-Glu-L-Ala), each emanating from adjacent N-acetylmuramyl moieties on neighboring glycan strands (i.e., a septapeptide or an octapeptide cross-link). Whenever a cross-link involves a 4-amino-acid stem (L-Ala-D-Glu-m-A₂pm-D-Ala) and a 5-amino-acid stem (L-Ala-D-Glu-m-A₂pm-D-Ala-D-Ala; i.e., the terminal D-Ala has not been cleaved by the D,D-carboxypeptidase), the linkage is referred to as a nonapeptide cross-link. The parts of the peptide stems that proceed beyond each cross-link are not stressed by the bacterium's turgor pressure. If all glycan strands were much longer than the typical septapeptide, octapeptide, or nonapeptide cross-links and if the maximum possible number of these cross-links had been produced in a peptidoglycan network, i.e., if 50% of the peptide stems are cross-linked, then all the holes in the network would be essentially the same size. This size is the area bounded by two successive cross-links and the two intervening sections of glycan strands, as shown in Fig. 1A. Figure 1B shows a representation of a network which is maximally cross-linked, thus possessing the smallest possible holes. We have not shown the possibility that these smallest holes become approximately hexagonal when the network is stretched (10)we are not immediately concerned with what shape the holes adopt in a bacterium, since it is possible that the cell might exert local forces, thereby additionally distorting its peptidoglycan network (following Koch, we call these smallest holes tesserae [i.e., units having four corners]). In reality, however, the glycan strands are not very much longer than the cross-links but range from 2 to over 30 disaccharides long (5, 7, 9, 14), so it is possible that one of the intervening sections of a glycan strand bounding a tessera appears broken. Instead of one continuous glycan strand, the network at this point is composed of the ends of two different glycan strands. In this case, two adjacent tesserae are actually connected, thereby creating an aggregate (of tesserae) formed by two connected tesserae (Fig. 1C shows four such connected tesserae). An assumption will be made that there are minimal gaps between the ends of two abutting glycan strands, as shown in Fig. 1C. This assumption is justified by the observation that, in E. coli and P. aeruginosa, 50% of the pentapeptide strands form cross-links. If gaps existed between the ends of abutting glycan chains, then some unlinked peptapeptide chains lying in the plane of the network would lack another pentapeptide partner from an adjacent strand with which to form a cross-link. Finally, Fig. 1D and E show how the networks of Fig. 1B and C might behave at higher temperatures, where high-energy thermal oscillations give rise to distortions of the network. The network containing the aggregate (Fig. 1C) undergoes even larger distortions than that containing only tesserae because of the proximity of many glycan strand ends, which lead to the opening of a large hole in the lattice. It will be shown below that, in a monomolecular thick peptidoglycan network, it is possible to deduce the general form of the distribution of the holes, their general shape, and their alignment to one another from a knowledge of the distribution of glycan strand lengths.

View larger version (39K): [in this window] [in a new window]   FIG. 1.   (A) Diagram of a portion of a murein sacculus. The hexagons represent the N-acetylglucosamine and the N-acetylmuramic acid (Mur) groups. Two complete cross-links are shown attached to the latter. One is a nonapeptide link (left), while the other is an octapeptide link (right). Circles containing  or × indicate unlinked pentapeptide groups, attached to Mur groups and oriented out of or into the plane. (B and D) Diagram of infinitely long glycan strands (horizontal) maximally cross-linked via nonapeptide or octapeptide chains. A tessera is shown shaded. The dots represent the sugar groups along the glycan strands. (D) Effect of temperature-driven fluctuations in this elastic system. (C and E) The peptidoglycan network of A with six glycan strand ends shown (arrows) giving rise to an aggregate of four tesserae (shaded). The hexagonal structure is not represented here. (E) Effect of temperature-driven fluctuations in this elastic system. The arrow points to a small, attached but isolated peptidoglycan remnant that is relatively free to move. Note the possibility of opening a large hole in the sacculus.

Mathematical models and techniques. To explore the size distribution of holes in a peptidoglycan network formed from a set of glycan strands exhibiting a distribution of lengths, we considered trying to assemble the glycan strands in some manner, possibly random. This proved difficult, because it became apparent that when choosing a particular strand and a place to insert it, we had to be aware of the existing structure of the local network; otherwise, the fabric could be left with large gaps between the ends of successive strands. The use of Monte Carlo techniques to anneal a structure containing such gaps could be very time-consuming and would not necessarily solve the problem. This procedure of inserting glycan strands into a preexisting peptidoglycan network is performed by bacteria, as they (somehow) determine exactly where each piece should go, but we do not know the rules they are following. Accordingly, we solved this problem by cutting otherwise very long glycan strands so that the procedure resulted in the experimentally observed glycan strand length distribution. Cutting the strands of a network is a standard technique in the simulation of percolation and has been applied in a biological context (references 15 and 16 and references therein). That application, however, was without any analogue of the requirement here, i.e., that the resulting glycan chain length distribution must be in accord with experimental results. In that application, also, all bonds could be cut, leading to the possibility that sections of the network could be separated from the main body (15) (Fig. 1). Our model does not permit this, and we comment on this below.

View larger version (63K): [in this window] [in a new window]   FIG. 2.   (A) Diagram of infinitely long glycan strands (horizontal) maximally cross-linked via nonapeptide or octapeptide chains. A tessera is shown shaded. (B) The peptidoglycan system of panel A. The locations where the glycan strands may be cut are indicated by ×s. (C) Construction of an aggregate according to rule B1. The glycan chain has been cut in two places (dark arrows), and the three tesserae thus connected are shaded. (D) Illustration of rule B2. With the glycan chain cut at the dark arrow, the two ×s on either side of the cut (light arrows) have been removed to show that those sites may not be cut. The solid circles represent sugar groups along the glycan strands.

Creation of holes. The following procedure was used to create a set of aggregates possessing a size distribution D_s(s). First, we selected the total number of aggregates we wanted produced and then selected a size, s, in accord with the probability, D_s(s), that such a size would occur in the distribution. An aggregate of size s was then created as follows. We assumed, for simplicity, that the two glycan strands bordering a tessera could be cut at four sites, and their locations are identified in Fig. 2B. We selected one of these sites randomly but in accord with rules B1 and B2 (see below) and, by cutting there, attached a second tessera to the first tessera. The two-tessera aggregate thus created possessed five sites at which it was possible to cut. We then randomly chose one of the two tessera as well as a site on one of its glycan strands at which to cut. By cutting at that site, a third tessera was attached to the growing aggregate. This procedure is shown in Fig. 2C, where the two cuts that create the three-tessera aggregate are indicated. The procedure terminated when sufficient cuts had been performed to create an s-tessera aggregate. The resulting structure was then placed anywhere convenient on the maximally cross-linked peptidoglycan network.

(i) Rules B1 and B2. Rules B1 and B2 dealt with the procedures used to select sites on the glycan chains at which to cut. We began by imposing no conditions on which sites (Fig. 2B) we could cut at (rule B1) but with the condition that, as the aggregates were constructed, they were not cut in such a way as to cause a section of the network to become separated from the remainder of the network. For reasons to be described below, we also introduced rule B2. This states that, if a cut was made at a site, then cuts could not be made at the sites nearest to it along the same glycan strand and belonging to the same tessera. This is shown in Fig. 2D, where the cut is indicated and the two sites where cuts were not permitted in any future operation are also shown.

(ii) Rule H. There are many ways to mimic what a bacterium might be doing without suggesting that this is the method that it actually uses. In the mathematical model used here, we moved the aggregates on the plane of the network subject to an interaction, V, which determined their average spatial distribution. We are unaware of how a bacterium actually controls the distribution of glycan strand ends. Our intent was simply to discover whether it does so, and we achieved this by introducing the effective interaction, V, which was parameterized by an interaction strength, V₀, between pairs of aggregates. If V₀ was positive (negative), then the interaction between pairs of aggregates was repulsive (attractive). Choosing a V₀ equal to 0 would result in a random distribution, as long as the density of aggregates was not too high (i.e., as long as the aggregates could find sufficient room between the other aggregates to move around). It should be clear that the aggregates do not rotatethey undergo only lateral movement. Since the bacterium probably does not create glycan chain lengths by using our mathematical procedure, our interaction, V, might not be easily identified with parameters of the procedure actually used by the bacterium. The analytical form used for V is given in the Appendix.

Results. Figure 3A shows glycan strand length distributions with 25% of the network covered by aggregates and utilizing rule B1 and a gaussian distribution with s₀ equal to 4 and  equal to 2. The graph shows the distribution of glycan strand lengths, D_L(L), for V₀s of 0, 2, and 5, with an insert showing the distribution of aggregates for a V₀ of 5. These values were chosen with the intention of obtaining a glycan strand length distribution similar to that obtained by Obermann and Höltje (14). However, although D_L(L) exhibits a local maximum at values of L corresponding to those observed by those authors when V₀ is equal to 5, there is a maximum at the shortest length, an L of 2. The maximum at the shortest glycan chain lengths is absent in the results of Obermann and Höltje. The reason for this was easily discovered. There are two ways of creating glycan chain lengths in our system. The first is controlled by the distribution of aggregates on the network, and this gives rise to a local maximum at the larger value of L. However, the large numbers of glycan strands with an L of 2 arise inside aggregates. They are created when cuts are made at adjacent sites, along the same glycan strand, on either side of a cross-link (Fig. 1C and E and 2C). Although we have no proof that we will necessarily obtain a maximum in D_L(L) for an L of 2, we have been unable to find a counterexample that yields a maximum in accord with the results of Obermann and Höltje without another maximum at an L of 2. Obermann and Höltje (14) report that the number of strands of length (L) 2 is near zero. Because of this, we introduced rule B2 (see above), which states that if a cut was made at a site, then cuts could not be made at the sites nearest to it along the same glycan strand and belonging to the same unit (Fig. 2D).

View larger version (74K): [in this window] [in a new window]   FIG. 3.   DL(L), the fraction of glycan strands possessing length L, as a function of glycan strand length L, for different values of repulsive potential V0. The dashed lines indicate a V0 of 0 (short dashes) and a V0 of 2. The solid lines indicate a V0 of 5. At an L of 61, the curve indicates the sum of all values of DL(L) for an L of >60. The insets show typical distributions of aggregates for a V0 of 5. (A) Use of rule B1. Ds(s) is gaussian, with s0 equal to 4 and  equal to 2. (B to D) Use of rule B2 to eliminate short glycan strands. Ds(s) = 1 for s = s0 and Ds(s) = 0 for s  s0; otherwise all aggregates possess the same number of tesserae. (B) s0 = 2; (C) s0 = 3; (D) s0 = 5.

Conclusions. We have modeled the distribution of holes in the murein sacculus (peptidoglycan network) of gram-negative bacteria as represented by E. coli and P. aeruginosa. By requiring that the length distribution of glycan strands be in accord with the results of others (5, 7, 9, 14), we conclude that all the holes are slits running essentially perpendicular to the axis of the glycan strands (Fig. 2D and 3B, C, and D). If the slits are not too large, then our results are in accord with what has been observed by Beveridge and Jack (2) and Demchik and Koch (6) regarding the permeability of bacterial walls. In addition, there appear to be some possible advantages to a distribution of slits as described here.

View larger version (35K): [in this window] [in a new window]   FIG. 4.   (A) Turgor pressure-stretched peptidoglycan network such that the holes formed by the tesserae become hexagons. The cross-links at two peptide stem junctions are being cut at the regions marked by ×s to form a slit aggregate comprising three tesserae. (B) The network of A with the slit shown. It can be seen that although the slit is ~6 nm long, the lateral dimension will remain ~2 nm.