Coarse-grained simulations of bacterial cell wall growth reveal that local coordination alone can be sufficient to maintain rod shape
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Edited by Joe Lutkenhaus, University of Kansas Medical Center, Kansas City, KS, and approved June 15, 2015 (received for review March 3, 2015)

Significance
The rod shape of walled bacteria is determined by the peptidoglycan (PG) sacculus, but how rod shape is maintained as cells grow remains a fundamental question in bacterial cell biology. We have developed a coarse-grained modeling method to study rod shape maintenance. Individual PG remodeling enzymes, including transglycosylases, transpeptidases, and endopeptidases, are for the first time, to our knowledge, explicitly modeled to explore how they can coordinate to remodel a sacculus several orders of magnitude larger than the enzymes themselves. Rather than requiring top-down regulation of new PG insertion sites, our work shows that local coordination of the PG remodeling enzymes within discrete complexes can be sufficient to maintain the integrity and rod shape of the sacculus.
Abstract
Bacteria are surrounded by a peptidoglycan (PG) cell wall that must be remodeled to allow cell growth. While many structural details and properties of PG and the individual enzymes involved are known, how the process is coordinated to maintain cell integrity and rod shape is not understood. We have developed a coarse-grained method to simulate how individual transglycosylases, transpeptidases, and endopeptidases could introduce new material into an existing unilayer PG network. We find that a simple model with no enzyme coordination fails to maintain cell wall integrity and rod shape. We then iteratively analyze failure modes and explore different mechanistic hypotheses about how each problem might be overcome by the macromolecules involved. In contrast to a current theory, which posits that long MreB filaments are needed to coordinate PG insertion sites, we find that local coordination of enzyme activities in individual complexes can be sufficient to maintain cell integrity and rod shape. We also present possible molecular explanations for the existence of monofunctional transpeptidases and glycosidases (glycoside hydrolases), trimeric peptide crosslinks, cell twisting during growth, and synthesis of new strands in pairs.
The cytoplasmic membrane of Gram-negative rod-shaped bacteria is surrounded by a peptidoglycan (PG) sacculus that protects the cell from internal turgor pressure, and its architecture determines the cell’s shape (1, 2). The sacculus is composed of long glycan strands crosslinked by peptides into a mesh-like network. The glycan repeating unit is a disaccharide of an N-acetylglucosamine and an N-acetylmuramic acid attached to a stem pentapeptide l-Ala–d-iGlu–m-A2pm–d-Ala–d-Ala. Crosslinks are formed between peptides on adjacent strands—most at the fourth (d-Ala) residues of the donors and the third (m-A2pm) residues of the acceptors. While it is now understood that, in Gram-negative cells, the glycan strands run parallel to the cell surface, how the strands are arranged in this plane is still debated. While recent atomic force microscopy images of purified sacculi were interpreted to indicate that glycan strands had random orientations (3), electron cryotomography of sacculi (4) and other evidence from both Gram-negative (1, 5) and -positive (6, 7) sacculi have, instead, pointed to a universal “circumferential” model, in which the stiff glycan strands are circumferentially around the rod and the flexible peptide crosslinks parallel to the rod’s long axis.
Cell growth requires that the sacculus be elongated without losing its integrity or characteristic shape. This remodeling process requires the presence of not only transglycosylases and transpeptidases, also known as penicillin-binding proteins (PBPs), to polymerize and crosslink new glycan strands into the existing network (2, 8) but also, endopeptidases to cleave covalent bonds to open space for the new material (9). How these small enzymes work together to maintain the order and rod shape of a sacculus three orders of magnitude larger is not understood. Burman and Park (10) proposed that lysis could be prevented if the activities of the hydrolases were temporally coordinated with those of the synthases. Koch (11) further argued that, to be successful, PG remodeling must follow a ‘two-for-one’ ‘make-before-break’ strategy, in which autolysins cleave crosslinks along an existing strand only after two new strands are fully crosslinked underneath (11). This basic idea was then revised in a ‘three-for-one’ model that posits that three new strands are crosslinked to the sacculus by trimeric crosslinks before the template strand is released (12). A detailed molecular mechanism, however, was not provided to explain how the enzymes could be coordinated to execute such temporally and spatially separated operations.
It has been shown that PG-remodeling enzymes are regulated by factors both above (outside) and below (inside) the sacculus. From above, in Escherichia coli, the two outer membrane lipoproteins LpoA and LpoB activate the two major bifunctional transglycosylase/transpeptidases PBP1A and PBP1B (13, 14). Below the sacculus and inner membrane, in some but not all rod-shaped cells, the cytoplasmic actin homolog MreB (15) is required for shape maintenance (16, 17). Based on fluorescence microscopy, MreB was first thought to form a helical track extending the length of the cell (17⇓⇓⇓⇓⇓⇓–24), leading to the idea that MreB might maintain rod shape by coordinating the insertion sites of new PG along a helical pattern across the cell. This view has been challenged, however, by more recent results. At least in the original images of E. coli (20, 25), extended MreB helices were shown to be artifacts of the fluorescent protein tag (26, 27). Later fluorescence studies reported that, instead of extended helices, MreB localizes in tight patches that, driven by PG synthesis, move circumferentially around the cell (28⇓–30). Even more recent studies reported that it forms filaments (31⇓⇓–34) driven around the cell by extracellular motors (32, 33) or curvature-based localization (35). Thus, the oligomeric form, driving force, and function of MreB remain unclear.
To explore different mechanistic models of sacculus growth, we have developed a computational method that allows us to vary potential properties of PG-remodeling enzymes and their coordination. PG is represented by a coarse-grained model with mechanical properties that are derived from all-atom molecular dynamics (MD) simulations of isolated glycan strands and peptides. Individual enzymes, including transglycosylases, transpeptidases, and endopeptidases, are explicitly represented and constrained to the sacculus surface to implicitly represent confinement within the thin periplasmic space. We first tested a simple model, in which there was no coordination among the enzymes (Remodeler 1.0), and showed that sacculus integrity and shape are lost quickly. We then explored different biophysically plausible hypotheses that we thought might alleviate problems and iterated this procedure until rod shape was maintained for many generations (Remodeler 1.1–1.13). We found that local (rather than long-range) spatial and temporal coordination of enzymes can be sufficient to maintain rod shape, providing an alternative to the need for extended cytoskeletal scaffolds.
Since dynamic processes like sacculus remodeling are difficult to show in static figures, at this point, we strongly encourage readers to watch Movies S1–S3. First, we show how we built the coarse-grained model of sacculus growth (Movie S1). Second, we present the various hypotheses that were tested to address shape loss problems (Movie S2). Third, we show a final model that shows how local enzyme coordination alone can support rod shape maintenance (Movie S3).
Methods
Coarse-Grained Sacculus Model.
Because the sacculus is a giant molecule (on the order of
Construction of the coarse-grained model. (A) A glycan strand represented by (Left) an atomic model (Center) coarse-grained with each bead representing one disaccharide (blue) attached by a peptide (red) and (Right) coarse-grained with each bead representing two disaccharides attached by an in-plane peptide. (B) Snapshots of a glycan strand in all-atom MD and coarse-grained simulations. In the latter, the strand was modeled as a chain of beads connected by springs. (C) Extension dependence of force on a peptide crosslink extracted from all-atom MD simulations (blue) and after fitting to a WLC model (red). (D) The starting sacculus comprises circumferential glycan strands crosslinked by longitudinal peptides. (E, Left) Crystal structures of a transglycosylase [3FWM (55)] in orange, a transpeptidase [3EQV (61)] in yellow, and an endopeptidase [2EX2 (91)] in gray. The enzymes are modeled as beads in E, Right. (Lower) Inner and (Upper) outer membranes are shown for context. (F1) Visual depiction of enzymatic activities. Transglycosylase (F2) initiates a new strand (green) and (F4, F7, and F9) elongates it. (F3 and F5) Endopeptidase cleaves peptide crosslinks. (F6 and F8) Transpeptidase crosslinks the new strand to the sacculus.
Glycan Mechanical Properties.
The development of all-atom force fields for PG (6, 36) allowed us to match the mechanical properties of the coarse-grained model to those exhibited by all-atom MD simulations. To calculate the stiffness of glycan, a fully solvated system of a 160-disaccharide strand without stem peptides was equilibrated for 6.6 ns using the software NAMD (37) (Fig. 1B). Histograms of distances between tetrasaccharides and bending angles were then extracted. MD simulations were then run on a coarse-grained strand, where adjacent beads were connected by springs of constant
where
A Langevin damping term was added to mimic water viscosity. We iteratively sampled parameters until the histograms extracted from coarse-grained simulations matched those of the all-atom simulations. The final coarse-grained parameters were
Peptide CrossLink Mechanical Properties.
To study the rigidity of peptide crosslinks, we determined the potential of mean force (PMF) as a function of end-to-end extension in all-atom MD adaptive biasing force (ABF) simulations. ABF is a quasi-equilibrium method, in which the biasing forces exerted on the two terminal (reaction) atoms are iteratively calculated as the positive gradient of the PMF, thus making the two atoms diffuse freely and allowing the full energy landscape to be assessed quickly (38, 39). We found that the peptide crosslink is better modeled as a worm-like chain (WLC) than a spring (i.e., the force is almost zero at small extension but then, increases dramatically at large extension) (Fig. 1C). We, therefore, fit the force vs. extension curve to the following formula:
where
Turgor Pressure.
The effect of turgor pressure was taken into account by adding to the total energy of the system (
Initial Sacculus Architecture.
The initial sacculus model was built with a cylindrical waist and two polar caps (Fig. 1D). The lengths of glycan strands were chosen uniformly randomly within a range from 10 to 20 tetrasaccharides. As the circumference of typical WT E. coli cells is ∼1,000 tetrasaccharides, to reduce the computational cost, most of our simulations began with a smaller sacculus of circumference 100 tetrasaccharides, so that each hoop (ring of end-to-end strands) consisted of 5–10 strands. To test the effects of size, we also ran simulations on sacculi two and four times larger in diameter (400 and 800 disaccharides, respectively, in circumference). To speed up simulations on the latter, the two polar caps were not included (Movie S4).
Note that the best visualization of the sacculus architecture to date has been of purified sacculi revealing a single layer of glycan strands that are circumferentially disordered, possibly a result of sacculus collapse in the absence of turgor pressure (4). Here, the initial sacculus was built with ordered glycan strands that became slightly disordered after the first relaxation step (Movie S1). Thus, our results apply only to this arrangement.
Enzyme Movement.
Generic transglycosylases, transpeptidases, and endopeptidases were modeled explicitly as individual beads (Fig. 1E). They were tethered to the sacculus to represent confinement within the thin periplasmic space (details in SI Appendix, Fig. S1A). To model diffusion, random forces were generated and exerted on the enzymes (details in SI Appendix).
It is known that the outer membrane lipoproteins LpoA and LpoB protrude down through the sacculus to interact with and activate the bifunctional transglycosylases PBP1A and PBP1B, which are partially embedded in the inner membrane (13, 14). Active transglycosylase–lipoprotein complexes, therefore, cannot cross through strands or crosslinks. To model this constraint, as a transglycosylase approached the edge of a hole in the network, a repulsive force was applied (details in SI Appendix, Fig. S1B).
We also modeled the interaction between enzymes and PG (SI Appendix, Fig. S1D). As a transglycosylase was elongating a new strand, the enzyme was linked to the strand tip by a spring of constant
Enzyme Tethering.
Because the bifunctional transglycosylase/transpeptidases PBP1A and PBP1B are the major synthases in E. coli (2), in our model, transglycosylase and transpeptidase were linked together by a spring-like force (SI Appendix, Fig. S1C). As the enzymes diffused, if the distance
Transglycosylation.
An inactive transglycosylase was activated with a probability of once every
Transpeptidation.
Transpeptidation was modeled to happen in two successive events: loading of a donor peptide and loading of an acceptor peptide/bond formation/transpeptidase release. An uncrosslinked peptide was loaded to a transpeptidase with a probability
Endopeptidation.
If during a time step, endopeptidase diffused across a peptide crosslink, the crosslink was cleaved with a probability of 0.1 (Fig. 1F).
Relaxation.
To relax sacculi after initial generation and during growth, we used a simple MD simulation of the coarse-grained model, in which inertia of the beads was ignored. Displacements were, therefore, simply linear functions of forces. The viscous drag coefficients of the enzymes were chosen as four times those of the PG beads (details in SI Appendix). To prevent system instability, we constrained the maximal displacement in any time step to 0.005 nm.
Rod Shape Characterization.
Several measures were used to characterize sacculus shape and structure. First, we quantified hole size, which is defined as the surface area covered by the hole (details in SI Appendix). Second, to quantify surface bulges, we calculated local radii by (i) determining a central line through the sacculus between the polar caps and then, (ii) calculating local radii as the average distance from the local PG beads to the central line (details in SI Appendix). Third, we developed measures of sacculus straightness, defined as the ratio of end-to-end length (shortest path) to the contour length of the central line, and surface roughness, defined as the ratio of standard deviation to the mean of the local radii.
Results
To uncover the basic principles of PG synthesis and rod shape maintenance, we first started with a very simple model (Remodeler 1.0), which failed to maintain the sacculus’ rod shape. We analyzed obvious causes of shape loss, implemented a hypothesis to fix the problems, and then, ran simulations again. This process of identifying problems and adding hypotheses to correct them was iterated until rod shape was maintained over multiple generations of growth (Remodeler 1.1–1.12). Finally, we identified the smallest set of these hypotheses capable of maintaining rod shape, forming our final model (Remodeler 1.13).
Initial Model—Remodeler 1.0.
Sacculus growth requires at least three types of enzymes: transglycosylases to synthesize new glycan strands, transpeptidases to crosslink them together, and endopeptidases to cleave existing peptide crosslinks to open space for new material (9, 42, 43). E. coli has other hydrolases that can modify PG, but none of them have been shown to be essential for rod shape maintenance (2, 44, 45). Thus, in our initial simulations, we simply introduced transglycosylases, transpeptidases, and endopeptidases onto the surface of the sacculus and modeled what happened as they diffused around performing their functions. We assumed that transglycosylase and transpeptidase exist as bifunctional enzymes and therefore, tethered them to each other but not to endopeptidase. Large holes, many more than 10 times the average hole size in the starting sacculus, developed almost immediately (Figs. 2A and 3 A and B). In our simulation conditions, the presence of uncoordinated enzymes alone was, therefore, insufficient to maintain rod shape.
Sacculus growth (new strands are shown in green; A) in the initial model; after adding (B) the multienzyme complex, (C) cleaved crosslink capture, (D) bend-induced termination, (E) fixed transglycosylase orientation, (F) crosslink before terminate, (G) tail hydrolysis, (H) paired transpeptidases, (I) peptide maturation, (J) crosslink–dependent processivity, (K) first crosslink always on the same side, (L) hole-dependent processivity, and (M) strands inserted in pairs hypotheses; and (N) the final model.
Structural characterization of grown sacculi. In the initial model, sacculi had many large holes, but they reduced in (A) size and (B) number as hypotheses were added. (C and D) Bulging was obvious in Remodeler 1.6 but reduced gradually toward the final model.
Multi-enzyme Complex Hypothesis—Remodeler 1.1.
Crosslink cleavage clearly must be coupled to the addition of new material for sacculus integrity to be preserved, as argued previously (10, 46). Indirect evidence for multi-enzyme complexes exists (23, 47⇓⇓⇓⇓–52), so that, in the next round of simulations, we tried tethering the transglycosylases to both the transpeptidases and endopeptidases to form trimeric complexes (details in SI Appendix, Fig. S1C). Large holes still formed quickly, leading to loss of sacculus integrity and rod shape (Figs. 2B and 3 A and B). Simply linking the enzymes into complexes was, therefore, insufficient to maintain rod shape.
Cleaved CrossLink Capture Hypothesis—Remodeler 1.2.
Analyzing the dynamics of hole formation, we found that, after a crosslink was cleaved, the two previously crosslinked strands sometimes moved apart because of turgor pressure before the transpeptidase crosslinked them to the new strand (SI Appendix, Fig. S2). Exploring ways this failure mode might be prevented, we increased the transglycosylation rate, so that peptide donors would appear more quickly. Unfortunately, quick transglycosylation created other problems, such as strand bending (discussed in below). We also tried accelerating the peptide loading rate (by increasing the reaction distance), but the problem remained. Implementing a ‘make-before-break’ strategy (described in detail in Final Modifications—Remodeler 1.13) did not solve the problem either, as this strategy only works if multiple strands are inserted concurrently. Finally, hypothesizing that endopeptidases might bind tightly to cleaved crosslinks until being competed off by transpeptidases, we lowered the probability of peptide release after crosslink cleavage to, on average, once every
Bend-Induced Termination Hypothesis—Remodeler 1.3.
At this stage, elongation and termination of glycan strands were modeled to be purely stochastic and independent of strand conformation. If a growing strand elongated but the distance between the two ends could not increase (for example, because the enzyme could not move forward and the nongrowing end had already been crosslinked), the strand bent (SI Appendix, Fig. S3). In real cells, however, the strand stiffness would affect the affinity of the enzyme, and strand bending would likely pry the tip out of the transglycosylase active site. Indeed, a flap was discovered in the structure of the Aquifex aeolicus transglycosylase PBP1A folding over the active-site cleft, presumably to prevent dissociation of the growing strand (53). Strand bending could pry the flap open, resulting in a higher termination probability. A bend-induced termination hypothesis was, therefore, tested, in which bending increased the probability of termination (details in SI Appendix). As a result, the formation of long bent strands was prevented. New strands, however, still aggregated in a few areas on the sacculus surface, introducing bumps (Fig. 2D).
Fixed Transglycosylase Orientation Hypothesis—Remodeler 1.4.
Each time a transglycosylase terminated and reinitiated a new strand, Brownian motion changed the complex’s orientation, in many cases leading to reversal of synthesis direction and consequently, local aggregation of new material (SI Appendix, Fig. S4). Decreasing the probability of initiation could decrease aggregation, since diffusion would move non-polymerizing enzyme complexes some distance before re-initiating, but this strategy would not prevent reversal of synthesis direction and, therefore, would still result in insertion of incomplete hoops and likely still cause bumps. Some recent experiments (28, 29) suggest that PBPs move processively around cells. Considering that transglycosylase could form a complex with other rod shape-determining proteins, such as MreB/C/D, RodA/Z, and LpoA/B (SI Appendix, Fig. S5) (54), we implemented a fixed transglycosylase orientation hypothesis that resisted deviation of the growing strands’ orientation from the circumferential direction (details in SI Appendix). As a result, new strands no longer aggregated. The enzyme complexes did not reverse direction, instead moving processively around the sacculus, as seen for MreB (28⇓–30). Sacculus growth was smoother, but large holes still developed gradually, giving rise to small distortions and loss of sacculus integrity (Figs. 2E and 3 A and B).
CrossLink Before Terminate Hypothesis—Remodeler 1.5.
At this point, the main cause of hole development was that strand growth sometimes terminated soon after crosslink cleavage. While one of two released peptides became crosslinked to the terminated strand, the other released peptide became crosslinked to the next strand initiated, producing a small hole (SI Appendix, Fig. S5). If this phenomenon happened repeatedly in the same region, the hole enlarged. We hypothesized that there might, therefore, be a feedback mechanism that reduced the probability of transglycosylase termination while a released peptide was not yet crosslinked to the new strand and remained bound to endopeptidase (details in SI Appendix). After this crosslink before terminate hypothesis was added, the biggest problem was no longer hole formation (Fig. 3 A and B) but instead, the blockage of synthetic enzyme complexes by long, uncrosslinked glycan segments (Fig. 2F and SI Appendix, Fig. S6).
Tail Hydrolysis Hypothesis—Remodeler 1.6.
Because of the stochastic nature of the model, transglycosylases sometimes polymerized long initial tails that were never crosslinked to the network (SI Appendix, Fig. S6). These glycan tails did not contribute to the mechanical strength of the sacculus and caused two problems. First, they sometimes drifted above or below and then across existing strands and blocked movement of enzyme complexes (SI Appendix, Fig. S6) (active complexes cannot cross strands in any plane in our simulations, because the complexes presumably extend all the way from the inner to the outer membrane). Second, crosslinks sometimes formed between misoriented tails and new strands being inserted in the sacculus, creating a multilayered and distorted arrangement.
We reasoned that the cell might solve the tail problem in one of two ways: preventing tail formation or removing tails after they formed. The first strategy would require initiating transglycosylases to pause until the first PG units of the growing strand were crosslinked. This mechanism might be possible if transglycosylases were able to sense crosslink formation either directly or by interaction with transpeptidases. The second strategy would involve glycosidases that target PG tails but not crosslinked PG. E. coli does, in fact, have multiple glycosidases, but how they are regulated is not known (2, 45). If they were free to cleave any glycosidic bond, presumably the cell would lyse. Instead, these glycosidases might be tethered to the membranes and therefore, only able to cleave glycan tails that drift above or below the sacculus surface, leaving crosslinked PG intact. Since the first strategy required an unknown sensing mechanism among the enzymes, we decided to test the second strategy, implementing the tail hydrolysis hypothesis (details in SI Appendix). While adding this hypothesis solved the problem of enzyme blockage, as the sacculus grew longer, new glycan strands still sometimes became disordered, giving rise to bulges and loss of rod shape (Figs. 2G and 3 C and D).
Paired-Transpeptidases Hypothesis—Remodeler 1.7.
We found that, sometimes, instead of alternating between the two sides of a growing strand, successive crosslinks formed on the same side (SI Appendix, Fig. S7). As a result, the lengths of the glycan segments forming the opposing edges of a hole were unequal, causing the sacculus to buckle. Because only new peptides can serve as donors (1), crosslink polarity (donor on the left and acceptor on the right or vice versa) alternates along an inserted strand in a well-ordered network. At this point in our model, there was only one transpeptidase in each enzyme complex. Thus, to form all of the expected crosslinks, the transpeptidase would have to flip back and forth across the strand every other tetrasaccharide. This enzyme flip is unreasonable, since the transpeptidase in bifunctional enzymes is rigidly fused to the transglycosylase, which in turn, is clamped around the stiff growing strand.
Considering that a monofunctional transpeptidase, PBP2, is also required for rod shape maintenance (42), we next modeled the complex to have two transpeptidases, one monofunctional and one bifunctional, each restricted to making crosslinks on just one side (SI Appendix, Fig. S8A). Based on the crystal structure of the bifunctional enzyme PBP1B, this enzyme arrangement seemed reasonable, because it appears that the transpeptidase domain is rigidly fixed on one side of the emerging strand (55). The bifunctional transpeptidase was arbitrarily assigned to catalyze crosslinks on the left (looking down the growing strand from above in the direction of its growth), and the monofunctional transpeptidase was arbitrarily assigned to catalyze crosslinks on the right. Except for the assignment of sides, all other properties of the two transpeptidases in the simulation were equivalent. After implementation of this hypothesis, bulging was reduced (Fig. 3 C and D), and overall rod shape was maintained reasonably well through the first length doubling (Fig. 2H). To find out if rod shape could be maintained through additional generations, length-doubled sacculi were divided into halves, polar caps were added to the daughters, and additional growth was simulated. Rod shape deteriorated in the second generation (SI Appendix, Fig. S8B).
Peptide Maturation Hypothesis—Remodeler 1.8.
At this point, most of the defects were caused by the introduction of short strands with just one or two crosslinks. If a terminated strand was attached to the network by only one crosslink, it did not contribute to the mechanical strength of the sacculus. When this loose strand/tail failed to be removed quickly by glycosidases, it was later crosslinked to another new strand and became misoriented (SI Appendix, Fig. S8C).
One solution could be to have instant removal of uncrosslinked PG by glycosidases, a condition that would require colocalization of glycosidases in the synthesis complex and a regulation mechanism that allows them to remove only strands that have already been released by transglycosylases. Cells could solve this problem by at least two other different mechanisms: termination could be prevented until there were several crosslinks (discussed below), or the formation of crosslinks between loose tails and other strands could be prevented. For the second mechanism, either loose tails could be cleaved more quickly or their crosslinking could be delayed to allow more time for them to be cleaved. Considering the abundance of carboxypeptidases that target pentapeptides (56), we added into the model a peptide maturation hypothesis to prevent the loading of pentapeptides onto transpeptidases as acceptors (details in SI Appendix). Addition of this hypothesis solved the problem of loose tails being crosslinked into defective network patterns (Fig. 2I), hole size was well-controlled (Fig. 3 A and B), and bulging was nearly eliminated (Fig. 3 C and D), but other problems arising from quick termination remained.
CrossLink–Dependent Processivity Hypothesis—Remodeler 1.9.
As mentioned above, quick termination sometimes resulted in new strands being incorporated into the sacculus with only two crosslinks. Because of the effect of turgor pressure, these strands were pulled parallel to the long axis of the sacculus, and they sometimes blocked the movement of enzyme complexes (SI Appendix, Fig. S9).
Transglycosylase is processive in vitro (57), perhaps because of the presence of the flap folding over the glycan strand in the active-site cleft (53), but the factors that promote termination in vivo are unknown. Until now, termination had been modeled as a stochastic process with a constant probability at any time step, but this model would not produce the broad length distribution observed experimentally (58). A newly initiated, still–uncrosslinked glycan strand might easily remain in the transglycosylase active site, even as the enzyme experienced random collisions with other macromolecules, but once the nascent strand became crosslinked to the sacculus, thermal motions of the transglycosylase might disrupt its association with the now-immobilized strand.
Based on these considerations, we hypothesized that the enzyme processivity is high when the new strand is uncrosslinked but decreases with increasing numbers of crosslinks (details in SI Appendix). The average lengths of new strands before and after this change were 22 and 25 disaccharides, respectively, within the range of 21–33 disaccharides measured experimentally (58, 59). After addition of the crosslink–dependent processivity hypothesis, the defects caused by quick termination were reduced (Fig. 2J), but rod shape was still lost in the second generation (SI Appendix, Fig. S10).
First CrossLink Always on the Same Side Hypothesis—Remodeler 1.10.
At this point, since the activities of the two transpeptidases in the complex were independent, whether the first crosslink formed on the left or right was random (SI Appendix, Fig. S10). As a result, sometimes the last crosslink of one strand and the first crosslink on the next along a hoop were on the same side, producing a small pucker in the network that became progressively larger. We considered the enzymes’ crystal structures for a clue to how the first crosslink might be formed on a new strand. While the transpeptidase active site must be positioned farther into the periplasm to interact with the sacculus, the transglycosylase active site, being next to the transmembrane region, must be near the inner membrane (55, 60, 61). Thus, newly added disaccharides must move up from the inner membrane to the sacculus before they can be crosslinked, presumably passing directly from the transglycosylase to the transpeptidase active sites (55). We, therefore, reasoned that the first crosslink on a new strand is likely to be formed by the bifunctional enzyme and always be on the same side (Fig. 4A). This rule was introduced as the first crosslink always on the same side hypothesis, with the first crosslink always on the left after the arbitrary assignment made earlier for the bifunctional enzyme.
Addition of the first crosslink always on the same side hypothesis led to sacculus twisting. (A) Schematic depicting the first crosslink always on the same side hypothesis. A bifunctional transglycosylase (orange) transfers the strand to its associated transpeptidase domain (yellow) for transpeptidation (cyan arrow labeled 1st). Thus, the first crosslink is formed by the bifunctional transpeptidase. Only then does the monofunctional transpeptidase participate in transpeptidation (cyan arrow labeled 2nd). The crystal structures shown are of the E. coli bifunctional PBP1B (3FWM) and the Neisseria gonorrheae monofunctional PBP2 (3EQV). (B) Sacculus twisting caused by bias of start crosslinks on one side. (B1) The start sacculus with two markers (red) on the two polar caps. (B2) Before adding the first crosslink always on the same side hypothesis, no sacculus twisting occurred. (B3) Addition of the first crosslink always on the left caused left-handed twisting. (B4) Switching positions of the monofunctional and bifunctional transpeptidases to make first crosslink always on the right caused the sacculus to twist in a right-handed fashion.
The resulting sacculus surface was much smoother (Fig. 2K), but for the first time, the sacculus started twisting during growth (Fig. 4B and Movie S5). The left part of the sacculus always shifted backward and the right always shifted forward with respect to the growth direction of the new strand, resulting in obvious incremental rotation. To confirm that the new rotation was a result of the rule, we switched the positions and functions of the bifunctional and monofunctional transpeptidases, so that the first crosslink was always formed on the right side of the new strand. As expected, the sacculus now twisted in a right-handed fashion during growth (Fig. 4B and Movie S5). Sacculus twisting was also observed experimentally and in simulations where Wang et al. (31) speculated it was caused by the left-handed nature of MreB filaments guiding insertion sites. We speculate that it might also have been caused by biased first crosslinks in their simulations, a point that could be tested by counting the number of first crosslinks formed on the left and the right (Discussion). After introduction of this rule, sacculi grew more smoothly, but some holes still gradually expanded longitudinally (SI Appendix, Fig. S11).
Hole-Dependent Processivity Hypothesis—Remodeler 1.11.
Hole expansion arose from repeated termination in the same location. When an enzyme complex encountered a hole, sometimes it divided the hole by laying a strand across. If, however, the transglycosylase terminated before it crossed the hole (a rare but significant event), the loose tail was subsequently cleaved by glycosidases, and the hole expanded longitudinally (SI Appendix, Fig. S11). To reduce the risk of hole enlargement when a transglycosylase is in a large hole, cells might have a mechanism to increase processivity of the enzyme, reducing termination probability. This mechanism is biophysically plausible, since lipoproteins LpoA and LpoB must protrude through holes in the PG to activate PBP1A and PBP1B (SI Appendix, Fig. S12) (13, 14). Larger holes might allow stronger activation, perhaps by altering a bifunctional enzyme’s orientation or conformation in a way that increases precursor loading rates or reduces termination (3, 14, 54). We, therefore, hypothesized that the probability of transglycosylase loading PG precursors was increased and termination probability decreased, both by a factor of
Strands Inserted in Pairs Hypothesis—Remodeler 1.12.
When a ladder of crosslinks between two strands was cleaved, the opposing acceptor (old) peptide stems were in register, but the donor (new) peptide stems of the new inserting strand alternated sides. The process of crosslinking the new strand into the network, therefore, required that the two sides of the sacculus rotate one disaccharide forward or back with respect to each other to bring the donors and acceptors into register on both sides (SI Appendix, Fig. S13A). As a result, the sacculus twisted gradually as it grew (Fig. 4B and Movie S5). Wherever new strands terminated, small stresses and defects were produced, because the twist could not propagate all of the way around the hoop. While some of these defects were relaxed by later events, others grew, leading to shape loss in subsequent generations.
Based on studies of peptide acceptor:donor ratios, Burman and Park proposed decades ago that new glycan strands are inserted in pairs (10). Two later studies contradicted these results, suggesting instead that new strands are inserted one at a time (62, 63). Later, PBP1A and PBP1B were found to form homodimers in vivo (64, 65), and PBP1B can dimerize at high concentrations and produce pairs of crosslinked strands in vitro (66), again suggesting that new strands might be synthesized in pairs. If new strands were inserted in pairs, all of the new donor peptide stems on both sides would be in register with the old acceptor stems without strand shifting or sacculus twisting (SI Appendix, Fig. S13A).
We, therefore, implemented a strands inserted in pairs mode. A second bifunctional enzyme was added to each complex, so that two new strands could be added simultaneously. The rules governing the processivity of each transglycosylase remained the same (still dependent on crosslink numbers and hole size), and initiation and termination of the two transglycosylases remained stochastic and independent. Insertion of a pair of strands requires crosslinking both between them and on either side. Because our model assumed that the transglycosylases were bifunctional enzymes, as, for example, either PBP1A or PBP1B of E. coli, the positions and orientations of the two associated transpeptidases were assumed to be fixed within the complex (Fixed Transglycosylase Orientation Hypothesis—Remodeler 1.4) and they were expected to form crosslinks of a single polarity. If the two bifunctional enzymes formed their crosslinks on the left, for instance, a third transpeptidase was required to form the remaining row of crosslinks of opposite polarity on the right, forming a triple-transpeptidase complex. Because only the transpeptidase function was needed, we modeled it as a monofunctional transpeptidase, such as PBP2 of E. coli. Because the first crosslink always on the same side hypothesis was now irrelevant, it was removed from the model, allowing all three transpeptidases to act independently (Fig. 5A).
Remodeler 1.13. (A, Upper) Schematic of a synthetic complex in the final model: two bifunctional transglycosylase/transpeptidase enzymes (bi-Gtase/bi-Tpase), a monofunctional transpeptipase (mono-Tpase), and an endopeptidase (Edase). (A, Lower) The projection on the sacculus surface shows how three transpeptidases (visualized as ellipses with donor domains in green and acceptor domains in blue) could be oriented to create all needed crosslinks tethering two new strands (green) to old strands (blue). (B) Plots of (Upper) sacculus straightness and (Lower) surface roughness show that, in the final model, rod shape was maintained well (Left) through multiple growth generations and (Right) with varying sacculus diameter.
Rod shape was now maintained through several rounds of growth and division (SI Appendix, Fig. S13B). To quantitatively characterize rod shape, we calculated sacculus straightness and surface roughness for both the single- and paired-strand insertion modes (SI Appendix, Fig. S13C). While sacculus straightness declined over successive generations in the single-strand insertion mode, it remained constant in the paired-strand insertion mode. Also, the sacculus surface became rougher more quickly in the single-strand insertion mode.
Final Modifications—Remodeler 1.13.
Having implemented many hypotheses in the model that finally resulted in maintenance of rod shape over multiple generations, we wondered whether any of the hypotheses were redundant. As previously noted, the start on left hypothesis was removed in the paired-strand insertion mode. To find out if other hypotheses were required, we removed each from the model and checked if rod shape was still maintained. Repeating this process for all hypotheses, we found two more that were no longer needed. First, introduction of the crosslink–dependent processivity hypothesis prevented defects caused by quick termination, rendering the peptide maturation hypothesis redundant. Second, the crosslink before terminate hypothesis could be removed in the paired-strand insertion mode, because the paired transglycosylases almost never terminated simultaneously. As a result, even when one transglycosylase terminated right after crosslink cleavage, the other continued, crosslinking one of the newly freed peptide stems onto its growing strand and holding the terminated transglycosylase nearby, so that when the terminated transglycosylase reinitiated (a stochastic but usually quick event), it was well positioned to capture the second newly-freed peptide stem before it moved away.
Observing the paired-strand insertion mode, we realized that two other hypotheses, cleaved crosslink capture and bend-induced termination, could be re-implemented with more likely molecular mechanisms. Decades ago, it was shown that, in addition to dimeric crosslinks, trimeric crosslinks can also form (59). A typical dimeric crosslink is formed at the fourth d-Ala residue of the donor and the third A2pm residue of the acceptor peptide, but the A2pm residue on the donor peptide can simultaneously serve as an acceptor for another crosslink, resulting in a trimeric crosslink. Based on this possibility, the ‘make-before-break’ strategy (11) was elaborated into a ‘three-for-one’ model, in which formation of a trimeric crosslink hooking three pre-fabricated strands into the sacculus preceded cleavage of the original dimeric crosslink (12). Building on these ideas, we re-implemented the cleaved crosslink capture hypothesis as follows. If endopeptidase bound a peptide crosslink, the exposed A2pm group on the donor peptide became accessible as an acceptor for transpeptidation. Subsequent formation of a trimeric crosslink then immediately triggered cleavage of the original crosslink, releasing the acceptor peptide. We called this mechanism trimeric crosslink–triggered endopeptidation (SI Appendix, Fig. S14A).
Since the transglycosylase domains of the bifunctional enzymes are located immediately adjacent to the membrane, approximately eight disaccharides are likely added to the strand before it reaches the transpeptidation site (movie 3 in ref. 55). We, therefore, reasoned that, during elongation, when Brownian motion elevated the growing strand, transpeptidase might capture it, clearing the transglycosylase active site for another lipid II precursor and thereby, facilitating translocation in a ratchet-like fashion (SI Appendix, Fig. S13B). We, therefore, replaced the bend-induced termination hypothesis with a transpeptidation-facilitated translocation hypothesis specifying that the probability of translocation was low, once every
With these final changes, rod shape was still maintained well through multiple generations (Fig. 5B and SI Appendix, Fig. S14C). To test the effects of size, we ran simulations on sacculi two and four times larger in diameter (Movie S4). Similar to our previous results, an early model (Remodeler 1.1) quickly lost integrity and rod shape, but the final model (Remodeler 1.13) maintained rod shape well (Fig. 5B and Movie S4), suggesting that size does not fundamentally alter the basic challenges of maintaining integrity and rod shape during growth.
Reviewing the hypotheses in the final model underscores the importance of local spatial and temporal coordination among the enzymes as well as a house-keeping activity. Based on our exploration of rules and parameters, we suggest that the enzymes likely exist in a complex (multi-enzyme complex), in which two bifunctional transglycosylases with fixed orientation (fixed transglycosylase orientation) synthesize two new strands concomitantly (paired-strand insertion), two bifunctional transpeptidases form crosslinks on one side, and one monofunctional transpeptidase forms crosslinks on the other (triple transpeptidase). Next, the enzymes act in a temporally coordinated fashion, as, for instance, if endopeptidation is regulated by formation of trimeric crosslinks (trimeric crosslink–triggered endopeptidation) and transglycosylation is regulated by transpeptidation (transpeptidation-facilitated translocation). Termination of transglycosylase is likely not purely stochastic but rather, regulated, for instance, by crosslinkage of the growing strand (crosslink–dependent processivity) and hole size (hole-dependent processivity). Finally, glycosidases that target uncrosslinked glycan tails may be needed to clean up long tails (tail hydrolysis).
Discussion
Here, we have developed a coarse-grained model to explore how the cell wall-hydrolytic and -synthetic enzymes of bacterial cells could be arranged and coordinated to maintain rod shape and cell integrity while allowing growth. We acknowledge that we could not fully explore parameter space, and therefore, the results are limited to certain simulation conditions. They do, nevertheless, reveal the following principles of cell morphogenesis. First, the work presents new hypotheses about cell wall elongation mechanisms that can now be tested experimentally. Second, we find that rod shape maintenance does not necessarily require long cytoskeletal filament scaffolds (as has been previously argued) but could, instead, proceed with just local coordination of the enzymes within individual, randomly diffusing complexes. Third, methodologically, the work shows how coarse-grained simulations can be used to explore complex biological processes, such as cell wall growth, at the level of individual enzymes. The complete computer code used is attached in Datasets S1–S4 to facilitate exploration of new ideas by others. We expect the same method to be useful to study many related questions, including, for example, how Gram-positive cell walls might be elongated, how perturbed cells recover rod shape, how bacteria divide, and how cells engulf forespores.
Challenges of Rod Shape Maintenance.
Large holes frequently formed in our early models (Figs. 2 A, B, and F and 3 A and B), suggesting that the primary challenge that the cell might face while incorporating new material into its wall is to preserve integrity. The next challenge might be to maintain enzymes’ processivity, because failing to do so can lead to aggregation of new material and stalled growth (Fig. 2 C–F). Finally, the cell might face the challenge of preserving the regular arrangement of glycan strands, because losing this regularity can lead to bulges (Figs. 2 G–J and 3 C and D).
How Is Endopeptidation Regulated?
While previous models have suggested that PG-remodeling enzymes might be coordinated to prevent lysis during growth, the mechanistic details and long-term consequences have not been tested (10⇓–12). Our results suggest that the enzymes likely act in a complex and that their activities are coordinated for peptides released from crosslink cleavage to be recaptured in new crosslinks. We found that either a cleaved crosslink capture scheme or the formation of trimeric crosslinks before endopeptidation could prevent lysis.
Are Strands Inserted in Pairs?
Left-handed twisting of growing cells was observed experimentally and hypothesized to be caused by left-handed chirality of MreB (31). Later reversal in the twisting handedness was reported in cells expressing different transpeptidases (69). We found that, if new strands are inserted alone, coupling of enzymes causing the first crosslink to be formed always on one side resulted in sacculus twisting, and switching the first crosslink side also switched the twisting handedness (Movie S5), thus providing an alternative explanation for the phenomenon based simply on control at the enzyme level. Insertion of pairs of strands avoids peptide registration problems, however, and could reduce defects. While insertion of any even number of strands would provide these advantages, insertion of more than two parallel strands seems unlikely, because the maximal extension of crosslinks is ∼5 nm, and each transglycosylase is ∼4 nm wide [estimated from E. coli PBP1B, Protein Data Bank ID code 3FWM (55)]. It has been thought (see below) that enzyme complexes are associated with the cytoskeletal protein MreB. If this assumption is true, from movies of fluorescently tagged MreB foci circling around and around cells circumferentially (30), one could theoretically count foci per cell and estimate how many strands must be emerging from each focus to double the cell length in a given time. It will be interesting to test this carefully to determine whether strands are actually inserted in pairs.
Why Are Monofunctional Transpeptidases Essential?
Another idea that emerged from this work is a possible explanation for why the monofunctional transpeptidase PBP2 is essential for rod shape maintenance (42), although the bifunctional enzyme PBP1A can synthesize and crosslink new PG into sacculi in vitro (67). It has been suggested that the bifunctional transpeptidase first hooks the new strand to an old strand before the monofunctional transpeptidase crosslinks the new strand to another old strand (8). Our results suggest that the monofunctional transpeptidase is required to form crosslinks on one side of growing strands, since the bifunctional transpeptidase probably only forms crosslinks on the other. Within the parameter space tested, existence of PBP2 in the same complex with the other enzymes is needed to maintain rod shape (Movie S6). Consistent with our simulations, fluorescently tagged monofunctional transpeptidase PBP2a of Bacillus subtilis moves circumferentially around cells (28, 29), and PBP2 of E. coli was shown by bacterial two-hybrid and chemical crosslinking assays to interact with PBP1A in vitro and in the cell (52). A recent study reported, however, that fluorescently-tagged PBP2 of E. coli moves with a diffusive behavior, which is not affected by A22 treatment (70), while the same treatment causes fluorescently-tagged Caulobacter crescentus PBP2 to lose its spiral localization pattern (71). Thus, additional study is needed to resolve these discrepancies.
PG Turnover.
Cells release PG during growth (72, 73), but the mechanism of PG turnover is not clear. It was proposed in the ‘three-for-one’ model that PG turnover is a result of a PG remodeling mechanism that replaces one existing strand with three new strands (12). E. coli has many glycosidases, but puzzlingly, none of them have been shown to be essential (2, 45). Our simulations suggest a different explanation for how PG is released: glycosidases cleave loose uncrosslinked PG tails. Interestingly, while this manuscript was under revision, Cho et al. (68) published evidence identifying Slt as a hydrolase that removes uncrosslinked PG strands, thus providing strong experimental support for the tail hydrolysis hypothesis.
Structure, Movement, and Function of MreB.
As detailed in the Introduction, previous studies have disagreed about whether MreB forms extended filaments or not (17⇓⇓⇓⇓⇓⇓–24, 27⇓⇓⇓⇓⇓⇓⇓–35, 71, 74). While there is agreement that MreB structures move, the force that drives them remains unclear. Kim et al. concluded that, in analogy to actin, MreB filaments move forward by treadmilling (75). Subsequent papers concluded that MreB movement is driven by PG synthesis (28⇓–30). Most recently, it was suggested that MreB is moved by unidentified periplasmic motors (32, 33), or curvature-based localization (35). Here, we have shown how transpeptidation might move synthetic complexes forward through a ratchet-like mechanism: crosslink formation captures new strands in an elevated position, moving the transglycosylase forward and clearing its active site for additional polymerization, while existing crosslinks and glycan strand stiffness prevent regression (SI Appendix, Fig. S14B).
Finding that coordination of enzyme activities over a distance is not necessary, our simulations reveal three roles that MreB could fill. First, together with MreC, MreD, RodA, and/or RodZ, MreB might nucleate a transmembrane scaffold that gathers the PBPs into a complex. Second, if MreB forms short filaments, they might have a preferred curvature that matches the curvature of the sacculus, causing the filaments to align circumferentially and act as rudders orienting the enzyme complexes (SI Appendix, Fig. S5A). Third, it was recently proposed that association of MreB with both cytoplasmic and periplasmic enzymes ensures efficient delivery of PG precursors to sites of PG incorporation in the periplasm (76). In our model, direct transfer of PG precursors from the cytoplasm to transglycosylases facilitates processivity and rapid re-initiation. Consistent with these ideas, the putative bridging proteins MreC, MreD, RodA, and RodZ have been shown to be essential for rod shape maintenance (21, 74, 77⇓⇓⇓⇓⇓–83) and interact with both cytoplasmic (84) and periplasmic enzymes (85, 86). Also, treatment with A22 to depolymerize MreB dramatically reduces PG synthesis in the periplasm (87) and results in shorter glycan strands (88).
Comparison with Previous Coarse-Grained Modeling.
Coarse-grained simulation of the sacculus was pioneered by Huang and coworkers to study changes in cell morphology in response to modifications of the sacculus architecture (89), as well as mechanisms for rod shape maintenance during growth (30, 31, 35, 90). Coarse-grained models are defined by what salient features they attempt to capture as well as what they leave out. Thus, it is not surprising that, when comparing the work by Huang and coworkers and our work, there are major differences in both the methods and conclusions because of the different questions being asked. While Huang and coworkers (30, 31, 35, 90) focused on reproducing the sacculus’ physical nature with the assumption that MreB determines where new PG is inserted, our work is rooted in the enzymatic construction of the sacculus, exploring potential molecular mechanisms of PG synthesis.
As a result, Huang and coworkers (30, 31, 35, 90) modeled the incorporation of each new PG strand into the sacculus as a single event, in which peptide crosslinks along a path were cleaved, a new strand was inserted, and new crosslinks were formed all at once before any relaxation of the sacculus occurred. This model reduced the computational cost but prevented exploration of the properties and coordination of PG remodeling enzymes, which was the main point of our simulations; our simulations modeled separate enzymatic steps individually in detail.
In addition, we used all-atom MD simulations to derive the mechanical properties of glycan strands and peptide crosslinks. The spring constant for glycan used by Huang and coworkers (30, 31, 35, 89, 90) (50 pN/nm) was 100 times smaller than our all-atom MD-derived parameter (5,570 pN/nm). The small spring constant in the model by Huang and coworkers (30, 31, 35, 90) caused new glycan strands to stretch about 6% (our estimate) after being inserted into the sacculus. This stretching likely contributed to the gradual increase in sacculus radius that they observed during growth (90). To address that problem, Huang and coworkers assumed that inserted strands were prestretched by 10% (90) and later, speculated that MreB could provide such force (31), but additional study is needed to address the following questions. First, because MreB is cytoplasmic and does not contact glycan strands directly, this pre-stretching force would presumably be delivered by the transglycosylase pulling on the strand tip. Would the enzyme be able to exert an ∼10-pN force without disengaging the glycan strand tip from the transglycosylase active site during translocation? Second, how could MreB pull PBPs forward to stretch strands if MreB movement is driven by the PBPs as claimed (30)? Third, would stretching cause new strands to align parallel to the MreB filament? If so, how could this alignment be reconciled with the claim that MreB forms left-handed helical filaments, whereas glycan strands have a right-handed orientation (31)? In contrast, the large spring constant that we derived for glycan strands from all-atom MD modeling resulted in very little stretching for new strands, and relaxing sacculus during each enzymatic event preserved sacculus radius, thus, obviating the need for pre-stretching.
Regarding peptide crosslinks, Huang and coworkers (30, 31, 89, 90) modeled them as springs, but the value reported for the spring constant
To date, the modeling by Huang and coworkers (30, 31, 35, 90) has mostly focused on different ways that MreB might guide where new PG is inserted, resulting in different claims, namely that rod shape maintenance requires either uniformly distributed insertion insensitive to local fluctuations (30, 31, 90) or insertion specially targeted to local surface deformations (35). While either of these roles for MreB may be involved in maintaining rod shape, we have shown here that rod shape can be maintained with coordination on a much smaller scale. We have assumed that enzymes diffuse randomly and that initiation of new strands occurs stochastically. In this case, we find that local coordination is necessary to prevent local defects caused by new PG incorporation, resulting in processive movement of enzymes along the cell’s circumference. We find that this can be sufficient to maintain rod shape. However, much additional work will be needed to test this experimentally.
Acknowledgments
The authors thank Catherine Oikonomou for revising the manuscript for clarity.
Footnotes
- ↵1To whom correspondence should be addressed. Email: jensen{at}caltech.edu.
Author contributions: L.T.N., M.B., and G.J.J. designed research; L.T.N. performed research; L.T.N. and J.C.G. contributed new reagents/analytic tools; L.T.N., J.C.G., M.B., and G.J.J. analyzed data; and L.T.N., J.C.G., M.B., and G.J.J. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504281112/-/DCSupplemental.
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