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University of Connecticut DigitalCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 9-4-2013 A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria Akeisha Belgrave [email protected] Follow this and additional works at: http://digitalcommons.uconn.edu/dissertations Recommended Citation Belgrave, Akeisha, "A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria" (2013). Doctoral Dissertations. 260. http://digitalcommons.uconn.edu/dissertations/260 A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria Akeisha Belgrave, PhD University of Connecticut, 2013 Rod-shaped bacteria grow in length without changing their width [1, 2]. A major feature of bacterial cell growth is the remodeling of the cell wall, the primary structure giving the bacterium its shape and structural integrity. In many rod-shaped bacteria, the process by which the cell wall gets larger as the cell grows involves severing bonds that link the existing cell wall material together in order to insert and bind new material. Coordination between severing and insertion is likely necessary in order to prevent the cell wall from rupturing due to the large pressure difference between the inside and outside of the cell. As more bacteria are becoming resistant to antibiotics, which typically target cell wall synthesis, understanding the regulatory mechanisms involved in bacterial growth and morphology may provide new paradigms for treating infections. Although there have been many hypotheses proposed about the mechanisms underlying the regulation of cell width, the actual process by which bacteria grow and maintain their shape remains unclear [3, 4]. To begin to address this question, we developed a simple mechanochemical model explaining the dependence of cell length and degree of crosslinking on the replication rate of rod-shaped bacteria [5]. In this model we observed that faster growing cells are less crosslinked and longer in length than slower growing cells, as has been experimentally measured. Since our model predicts that faster growing cells have weaker cell walls, a prediction of the model is that the fractional change in length of a cell upon osmotic shock should be greater for faster growing Akeisha Belgrave – University of Connecticut, 2013 cells. We tested this prediction and found good agreement between the model and experiments. Finally, in order to understand the consequences of crosslinking fraction and glycan strand length on the elastic properties of the cell wall, we developed a 2D percolation model of the peptidoglycan meshwork. By simulating the effects of a given stress on this network, we were able to define a constitutive relationship for the cell wall as a function of crosslink fraction and glycan strand length. This model allows us to observe how mechanical and biochemical properties can affect cell shape. A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria Akeisha Belgrave B.S., Norfolk State University, 2008 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Connecticut 2013 Copyright by Akeisha Belgrave 2013 APPROVAL PAGE Doctor of Philosophy Dissertation A Mechanochemical Study into the Growth and Shape Maintenance of Rod-shaped Bacteria Presented by Akeisha Belgrave, B.S. Major Advisor _________________________________________________________ Dr. Ann Cowan Associate Advisor ______________________________________________________ Dr. Charles Wolgemuth Associate Advisor ______________________________________________________ Dr. Ji Yu Associate Advisor ______________________________________________________ Dr. William Mohler Associate Advisor ______________________________________________________ Dr. Michael Blinov University of Connecticut 2013 Acknowledgements I would like to send great thanks to the wonderful people in my life that made my work a fighting success. I cannot say this journey was an easy one, but it was a life-changing and successful one. To Dr. Ann Cowan and Dr. Raquell Holmes, thank you both for all the assistance you gave to me from the start of the program to get me adjusted and educated in this area of studies. Coming in to a biomedical program with a background in Physics and Mathematics I wondered how I was going to catch up with the rest of the crowd understanding the concepts and terms needed to progress throughout this journey. I must say, being from another country, with no family here, you two felt like extended mothers, and I felt like I could come to you with anything whether it was related to work or just my life. To my co-workers and lab members, Eunji, Erika, Olena, Jing, Sofya, thank you guys for the fun times I needed to get a break when things got too stressful. Mike you are a life saver, thank you for your kind assistance in my experiments, thank you Dhruv (Mr. Happy Hour). Thank you for being there to assist when I needed you. Elnaz, Jing and Pilhwa thank you for helping me with any problems I encountered in my mathematical calculations. Thanks to Dr. Koen Visscher for assisting me in the final stages of my work with resources unavailable to me at the time. To my family, especially my mom, my sister and my Auntie Lynda, thank you for your support, you guys were always there to listen and motivate me when I felt like giving up. Special thanks to my close friends Anicia, Shari, DeAnn, Tamara, Vanessa, Sharon and Ryan for the support throughout this entire journey. You were always there when I needed you even from more than a thousand miles away. Now for the big thank you, to my advisor Dr. Charles Wolgemuth; the most down to earth, EXTREMELY smart person in my life through it all. Thank you for being so helpful and patient with me. Even when I would look at you with blank expression, you always found a way to explain the things in the best way. You were fun to work with, and your ability to know almost everything truly inspired me. I could not have learnt more in my life than from working under your guidance. This would not have been possible without you. Thank you. i Dedication This work is dedicated to my mother Adorial Maxwell-Hazell and most importantly to our heavenly father. Mom you fulfill your dreams even when times got tough and never gave up. You are truly a person of strength and inspiration. Thank you for your continued motivation and assistance when times seemed too difficult. ii Table of Contents Acknowledgments ……………………………………………………………………… i Dedication ……………………………………………………………………………….ii List of figures …………………………………………………………………………… iv Chapter 1: Introduction 1 1.1 General Bacterial Physiology …………………………………………………………2 - 3 1.2 Bacterial Cell Wall ……………………………………………………………………3 - 6 1.3 Bacterial Synthesis …………………………………………………………………....6 - 7 1.4 Bacterial Biosynthetic pathway ……………………………………………………..........7 1.5 Peptidoglycan Synthetic machinery ………………………………………….............8 -11 1.6 Bacterial Growth …………………………………………………………………….11 - 12 Chapter 2: Elasticity and biochemistry of growth relate replication rate to cell length and crosslink density in rod-shaped bacteria (Biophysical Journal) 2.1 Introduction……………………………………………………………………………….13 2.2 A Mechanochemical Model of Bacterial Morphology .............................................. 14 - 19 2.3 Results and Discussion ……………………………………………………………..20 – 25 Chapter 3: The relationship between replication rate and cell wall strain in Escherichia coli suggests mechanisms of cell wall synthesis 3.1 Introduction ………………………………………………………………………….26 - 28 3.2 Methods ……………………………………………………………………………...29 - 30 3.3 Results and Discussion……………………………………………………………....30 - 36 Chapter 4: The effects of peptidoglycan structure on the elasticity of the Gram-negative cell wall 4.1 Introduction ………………………………………………………………………....37 - 39 4.2 Modeling the Gram-negative cell wall………………………………………………39 - 43 4.3 Results and Discussion ……………………………………………………………...44 - 47 Chapter 5: Conclusions and suggestions for future research ……………………………....48 - 51 Bibliography …………………………………………………………………….………….52-59 iii List of Figures 1.1. Schematic representation of the structure of the bacterial cell wall in Gram-negative cells (e.g. E. coli) and Gram-positive cells (e.g. B. subtilis) …………………………………….11 1.2. Schematic representation of a rod-shaped bacterium and the bacterial PG network ….......12 1.3. Schematic representation of Penicillin binding protein (PBP) grouping. HMWs, LMWs, and -lactamases. The HMW group subdivided into Class A and Class B…………………….13 2.1 Schematic of bacterial growth .……………………………………………………………..25 2.2 Mean cell length increases with division rate consistent with experiments on E. coli while crosslinking decreases .................................................................................................................26 3.1. Schematic representation of the flow of water molecules due to changes in extracellular solute …………………………………………………………………………………………....35 3.2 The replication rate of E. coli MG1655 increases approximately linearly with temperature and the average cell length increases with the replication rate, whereas, the width shows a slight decrease ……………………………………………………………………………………..….36 3.3 Representative images of E. coli MG1655 cells before and after the addition of 500 mM sucrose and the fractional change in length and width increase with replication rate ..…..……37 4.1 Peptidoglycan percolation model for a gram-negative bacterium at 100% connectivity (i.e., Pp = Pg = 1), and 50% crosslink density (Pp = 0.5) and 96% mean glycan chain length (Pg = 0.96), within the range appropriate for E. coli …………………………………………………………………... 48 4.2 Elastic moduli of a single layer of peptidoglycan at varying crosslink densities and glycan chain length ………………………………………………………………………………….....49 iv Chapter 1 Introduction Bacteria are known to be the cause of many common illnesses (i.e. stomach ulcers, strep throat and many other bacterial infections). In general, these infections are treated using a range of antibiotics; some common ones being penicillin, tetracycline, streptomycin and azithromycin. Most antibiotics work by interfering with the composition of or the formation of the bacteria’s cell wall. Therefore, pharmaceutical companies have made this structure the major drug target to treat bacterial infections. The increasing prevalence of antibiotic-resistant bacteria requires new treatment methods and/or new locations for drug targeting. In order to increase the efficacy of these drugs, a more precise understanding of the structure and synthesis of the bacterial cell wall is needed. The bacterial cell wall, which is made of a meshed material known as peptidoglycan (PG), is the shape determining factor in bacteria. This elastic material protects the cell against mechanical stresses applied to the cell from the environment and also resists the large internal turgor pressure of the cell, which can exceed atmospheres of pressures and can easily rupture the cell membrane. Bacterial cells differ in size, shape, cellular and genetic makeup. The most common shapes are cocci (spherical), bacillus (straight-rod), or vibrio (curved rod) shaped. Not only do bacteria possess various shapes, but these shapes are maintained throughout many generations. The simple genetic makeup and reproducibility of shape and size throughout generations also make these organisms an ideal model for understanding growth and form in biology. 1 1.1 General Bacterial Physiology Bacteria have a fairly simple basic structure. The inside of the cell is known as the cytoplasm and contains water, proteins, and DNA. The cytoplasm is separated from the outside world by the inner (or cytoplasmic) membrane, a thin bi-layer membrane that is largely impermeable. Channels and pumps in the membrane control the flow of ions, nutrients and water into and out of the cell. The cell wall lies just outside the inner membrane. In addition to these three structures, some bacterial species also have a second bilayer membrane, the outer membrane that encloses the cell wall. Bacteria are often generically classified as being either Gram positive or Gram negative. This classification is based on a technique that uses a dye that turns the cells purple. After washing, however, only Gram-positive cells retain the dye, while Gram-negative cells fade to pink, thus giving them the corresponding names. Gram staining detects the peptidoglycan in the cell wall of bacteria [6]. Gram positive bacteria stain because they do not have an outer membrane and because they typically have thicker cell walls than Gram negatives [7].The two model organisms for these types of bacteria are Escherichia coli (Gram negative) and Bacillus subtilis (Gram positive). These two types of bacteria are categorized as bacillus due to their elongated rodshape. On average, E. coli is about 1.0 m wide and 2.0 – 4.0 m long [8, 9]. B. subtilis is about 0.8 m wide by 4.0 m long [10]. Non-pathogenic E.coli lives in the digestive tracts of humans and animals. The feces from these animals can then contaminate foods and water, leading to bacterial infections. B. subtilis are non-pathogenic bacteria that are normally found in soil and vegetation. These bacteria are known to contaminate food but seldom cause food poisoning. The manageable size of the genome and proteome of these bacteria and the availability of mutants 2 with altered morphologies make these two bacteria excellent choices for understanding cell shape at the molecular level and will be used throughout this work. 1.2 Bacterial Cell wall The primary component of the bacterial cell wall is a macromolecular structure known as the peptidoglycan (PG), which makes up 5-90% of the cell wall mass in bacteria [11]. As mentioned, this structure is located outside the cytoplasmic membrane. The PG is composed of glycan strands, which are polymeric chains of sugars and amino acids that are linked together via covalent glycosidic and peptide bonds [12]. Because these bonds are strong, the PG is a cohesive mechanical structure which maintains cell shape. The PG even retains the same shape and approximately the same size when it is isolated from the rest of the bacterium [13, 14, 15, 16, 17, 18, 19]. The peptidoglycan is inflated by the internal osmotic pressure of the cell and acts to protect the cell against externally-applied forces, thereby reducing the likelihood of cell lysis [20]. The internal osmotic gauge pressure (or turgor pressure) is on the order of hundreds of kilo pascals (kPa) to a few MPas, with Gram positive bacteria typically having larger turgor pressures (15 to 20 atm) than Gram negative (0.8 to 5 atm) respectively [21-24]. These large turgor pressures require cells to have strong cell walls to counteract this pressure [25]. The thickness of the cell wall under Gram staining defines whether a cell is Gram-positive or Gram-negative. The percentage of PG is higher in Gram-positive than in Gram-negative. In Gram-negative bacteria, the cell wall is a thin elastic monolayer (i.e. 1-9 nm) located within the periplasm, a viscous or gel-like media that makes up 40% of the cell volume and contains some of the important cellular enzymes[26, 27, 28,29,30] (Figure 1.1A). On the other hand, in Gram3 positive bacteria, the cell wall is a thicker, multi-layered structure (i.e. 30-100 nm) interwoven with teichoic acids and other proteins [30] (Figure 1.1B). Unlike-, Gram-negative bacteria, these bacteria do not possess an external membrane. The intact cell wall of Gram-negative bacteria can withstand pressures up to 5 atm and Gram-positives can withstand pressures up to 50 atm [2123]. A molecular view of the PG layer shows that it is a meshwork of polysaccharide (glycan) strands comprised of N-acetylmuramic acid (MurNAc) and N-acetylglucosamine (GlcNAc) monomers interconnected by peptide stems [16-32]. These glycan strands are then connected together by peptide stems of five D- and L-amino acids, including diaminopimelic acid that is attach to a lactyl group within the MurNAc monomer. The peptide stems are wrapped helically around the glycan strand in such a way that subsequent stems on a given strand are roughly perpendicular to each other [13, 19]. In Gram-positive bacteria there may exist an additional peptide bridge that connects the two peptide stems. The glycan strands are considered to be relatively stiff compared to the more flexible peptides [19, 33]. Despite some controversy, the glycan strands are generally thought to be circumferentially oriented about the cell [14]. Various methods have been used to measure glycan strand length and the degree of polymerization of the strands in the cell wall. The most widely used method is reversed-phase HPLC, which separates the products obtained after complete degradation of the PG with muramidases that degrade the glycan strands or amidases that cleave the attached peptides from the glycans [27, 34]. Exponentially growing, wild-type E.coli cells have a mean glycan length (mGCL) of 25-35 disaccharide units, but the glycan strands can be up to 80 disaccharide units in 4 length [12, 34, 27, 35]. In B. subtilis a mean glycan length (mGCL) of 1,300 disaccharides was found [36]. The peptide stems are of a fixed length but not all of these stems are crosslinked to neighboring glycan strands. The degree of crosslinking of E.coli has been measured to be around 20% [31]. Due to the location of the peptide stems on the glycan strands, the number of possible peptide stems is a function of glycan chain length (GCL). Lower degrees of crosslinking presumably lead to weaker cell walls. Therefore, it is expected that the Young’s modulus (i.e. a measure of the stiffness of an elastic material) of the cell wall will be lower in cells with lower degrees of crosslinking. 1.3 Bacterial Synthesis Imagine yourself inflating a cylindrically-shaped balloon. The balloon will have an appreciable pressure filling the latex structure causing it to take shape. Now imagine that, by some strange means, chunks of the latex material from random points along the balloon were cleaved while maintaining constant internal pressure. These chunks are then replaced by larger pieces of latex. One would likely expect this mechanism to produce a balloon that is both longer and wider. However, the method just described is a fairly good analogy for the processes that occur within the cell wall of a growing rod-shaped bacterium. Bacterial growth requires synthases to attach new PG material to the existing PG. In concert with these synthases, hydrolases work to cleave the existing PG to allow insertion of new PG units. Therefore, bacterial biosynthesis involves (1) 5 the severing of existing PG material; (2) insertion of newly synthesized PG units; and (3) recrosslinking of peptide stems. 1.4 Bacterial Biosynthetic pathway Synthesis of the cell wall begins in the cytoplasm with the synthesis of UDP-activated precursors of the MurNAc-pentapeptide and GlcNAc monomers. The UDPMurNAc-pentapeptide is then anchored to a membrane carrier molecule undecaprenylphosphate, also known as bacteprenol. This complex forms the Lipid I precursor. The attachment of GlcNAc to the Lipid I precursor at the inner side of the cytoplasmic membrane forms a subunit (lipid II) that is then flipped across the inner membrane [12, 31]. During cell wall turnover, autolysins (which are bacterial enzymes) are directed to the site of PG digestion and growth to sever both the glycosidic bonds as well as the pentapeptide bridges [12]. The carrier bacteprenol is then released from Lipid II, and the breaks in the existing PG layer are filled by insertion of the two remaining monomers by glycosidase enzymes. Transglycosidase enzymes then catalyze the formation of new glycosidic bonds between the new disaccharide unit and the existing PG layer. New glycan strands in the PG are then crosslinked through the formation of peptide bridges by transpeptidases, leading to a strong cell wall. Studies suggest that many rod-shaped bacteria (including E. coli and B. subtilis) elongate by inserting material along the lateral cell wall with little to no insertion at the poles [16-20, 37]. 6 1.5 Peptidoglycan Synthetic machinery In order to maintain the integrity of the cell, cleavage and insertion of material in the cell wall must be well coordinated. Although internal pressure plays a role in cell shape by inflating the cell wall, the detailed shape of the bacterium is regulated by specific proteins within the periplasm that coordinate their behavior to influence cell shape. As mentioned in the previous section, hydrolases are required to cleave existing PG material while synthases are required to attach the new PG material to the existing cell wall material. In this section I will identify the known key regulatory proteins and their function during bacterial cell wall synthesis. The main PG-synthesizing enzymes, the penicillin-binding proteins, or PBPs, involved in the last stages of PG synthesis have been identified [8, 38-39, 40-42]. PBPs can be either high molecular weight (HMW) or low molecular weight (LMW) or -lactamases [43]. HMW PBPs can be further divided into class A or B enzymes made up of either monofunctional or multifunctional PBPs (Figure 1.3) [43, 44]. The HMW PBPs, PBP2 and PBP3 are monofunctional Class B enzymes found in E.coli. These proteins are anchored to the cytoplasmic membrane and are known to be responsible for the insertion of newly synthesized PG precursors into the cell wall. Localization studies using fluorescent microscopy identified newly incorporated PG precursors in live cells and showed that these enzymes are involved in PG synthesis [45]. Apart from the previously mentioned monofunctional enzymes PBP2 and PBP3, there also exist enzymes with bifunctional properties. Some bifunctional enzymes with both transglycosylase and transpeptidase activity are PBP1A, PBP1B, and PBP1C. It was shown in [46] that deletion of either of the HMW class A enzymes 7 PBP1A or PBP1B in E.coli cells was lethal to the cell and hence both of them are required for growth. Along with the PBPs, specific hydrolases are required to cleave covalent bonds to allow for insertion of new precursors. Cell wall synthesis requires the cooperation of glycosidases that work to polymerize the precursors that form the glycan strands. This process is followed by crosslinkage of these strands through the formation of peptide bridges by transpeptidases. The Cterminus of the HMW PBPs is the penicillin-binding domain, which is responsible for crosslinking the PG layer, while the N-terminus has transglycosylase activity making it important in the elongation of glycan strands (transglycosylation) and the formation of peptide crosslinks (transpeptidation) [1]. The bacterial cytoskeleton plays an important role in directing bacterial cell wall synthesis. The integral membrane proteins RodA and FtsW have a high degree of similarity to each other and are involved in the translocation of the lipid-linked PG precursors [47, 48]. These membrane proteins have been shown to interact with the transpeptidases PBP2 and PBP3 [49] and may be responsible for directing the PG precursors to the transpeptidases for insertion into the wall. RodA, FtsW and the actin-like rod-shape determining protein MreB have been shown to be required for cell elongation by being part of the pathway for insertion of new PG units along the lateral walls of in E. coli [40] and B. subtilis [50]. The absence of these proteins in rod-shaped bacteria cells leads to spherical cells. MreB was first believed to be organized as a helix located beneath the cytoplasmic membrane that spanned the length of the cell [50]; however, recent experiments have shown that this is an artifact [51]. MreC and MreD reside outside the inner membrane and depletion of these two proteins in rod-shaped cells also leads to a spherical shape. In E. coli [52], MreC interacts with both MreB and MreD forming a multiprotein complex. The 8 depletion of the multiprotein complex MreBCD highlights its importance in maintaining cell width since depletion led to increased cell width [52]. 1.6 Bacterial Growth The work presented here describes research into the mechanisms underlying the synthesis of the bacterial cell wall that occurs when bacteria are actively replicating. The changes in length and/or size of the bacterium during this process could be defined as growth. However, it is important to note that in the literature bacterial growth often refers to the process of replication. That is, the bacterial growth rate often denotes the time rate of change of the number of bacterial cells, as opposed to the change in size of an individual bacterium. Throughout this dissertation, I will try to avoid this ambiguity by referring to the change in the number of bacteria as the replication rate and the time rates of change of length and width, etc. will be specifically stated as such. In regard to bacterial replication, it is also important to note that there are four phases of growth: the lag phase, the log phase, the stationary phase, and the death phase. The lag phase occurs just after a bacterial culture is diluted. This is a transient period during which the culture does not grow in size. Following this period, the cells begin to replicate and divide, with a fairly constant time between one replication and the next. Therefore, the number of cells in the population increases exponentially with time. During this phase, if the concentration of cells is plotted as a function of time on a semi-log plot, a straight line is obtained. Hence the name “log phase”. The 9 slope of this line is the replication rate, which can vary depending on the strain of bacteria and other environmental factors of the growth medium and can range from minutes up to days. During the stationary phase, the size of the population remains constant even though some cells start to divide, while some die. The death phase which follows the stationary phase occurs when the death of the cells exceeds the formation of new cells. Previous research suggests that cell morphology is correlated with the rate that cells replicate and divide. It is known that cells that replicate faster have longer average cell lengths [53, 54-56]. Concurrently, as cells replicate faster there is a decrease in the degree of crosslinking [5]. As previously mentioned, it is commonly believed that these synthetic reactions (i.e. severing of peptide stems and/or glycan strands, insertion of new disaccharides and re-crosslinking of the peptide stems) occur simultaneously due to the efficient functioning of a multi-enzyme complex. Throughout our research we attempted to explain this dependence on the replication rate by developing a simple mechanochemical model for the mechanics and biochemistry of PG synthesis. In the next chapter we explain our model in full detail by including results published in a recent paper [5] in the Biophysical Journal. This model predicts that the strain in the cell wall should increase with replication rate. Chapter 3 describes the experiments that test this prediction. In Chapter 4, we develop a computational model that predicts how glycan length and the degree of peptide crosslinkage affect the mechanical properties of the PG layer in a 2D mechanochemical model. Lastly, in Chapter 5 we conclude with a summary of this work and suggestions for future directions. 10 Figures (A) (B) Figure. 1.1: Schematic representation of the structure of the bacterial cell wall in (A) Gram-negative cells (e.g. E. coli) that have a single layer (5-7 nm) of PG within the periplasmic space which lies between the cytoplasmic and outer membrane, and (B) Gram-positive cells (e.g. B. subtilis) that have a multi-layered (20-25 nm) of PG on the external side of the cytoplasmic membrane. 11 Figure 1.2: Schematic representation of a rod-shaped bacterium and the bacterial PG network. The network consists of a repeated chain of MurNAc-pentapeptide monomers and GlcNAc monomers forming a disaccharide unit. These chains are then crosslinked by the pentapeptide stems, thereby forming the PG. 12 High-molecularweight (HMW) PBPs Class A: HMW Low-molecularweight (LMW) PBPs -lactamases Class B: HMW Mono or Multifunctional PBPs Figure 1.3. Penicillin binding proteins (PBPs) are involved in the last stages of PG synthesis. These proteins are classified into three separate groups, HMWs, LMWs, and -lactamases. The HMW group is subdivided into Class A and Class B. 13 Chapter 2 Elasticity and biochemistry of growth relate replication rate to cell length and crosslink density in rod-shaped bacteria (Biophysical Journal, Volume 104, Issue 12, 2607-2611, 18 June 2013) 2.1 Introduction In this chapter we describe a simple, yet quantitative, model of the synthesis of the cell wall of rod-shaped bacteria that combines the fundamental physics and biochemical kinetics involved in this process. Our principle hypothesis is that the length of a bacterium is defined by the total amount of material in the cell wall and how much that material is stretched by the turgor pressure. Since the amount of material in the cell wall is due to the rate that new material is inserted in and removed from the cell wall, the biochemical kinetics of synthesis dictates the amount of material in the wall; however, these synthetic reactions can depend on the mechanical state of the cell wall. We, therefore, propose a mechanochemical model that explains the dependence of cell length and crosslinking on the replication rate. Our model shows good agreement with existing experimental data and provides further evidence that cell wall synthesis is mediated by multi-enzyme complexes; however, our results suggest that these synthesis complexes only mediate glycan insertion and crosslink severing, while re-crosslinking is performed independently. 14 2.2 A Mechanochemical Model of Bacterial Morphology As previously described, the bacterial cell wall is made of a covalently linked, elastic material known as peptidoglycan (PG), which is a network of glycan strands crosslinked by flexible pentapeptide stems [3,4,5]. In order to insert new material into the PG, existing material must be cleaved [20]. Therefore, synthesis of the PG involves (a) severing of existing peptide bonds and/or glycan strands; (b) insertion of new dissacharide units; and (c) re-crosslinking by formation of new peptide bonds [25]. These biochemical processes are driven by enzymes, such as the penicillin binding proteins (PBPs). Recent evidence suggests that some aspects of cell wall synthesis are processive and that the enzymes that drive synthesis may be localized in a multienzyme complex [4]. However, how biochemistry and biophysics conspire to change the length of the cell without changing the width remains enigmatic, even though this process has received a fair amount of scrutiny in the last ten years [12, 14-16, 31]. For example, computational modeling was used to investigate the effect of insertion and severing on the morphology of rodshaped bacteria but did not directly consider the biochemical reactions that drive PG synthesis [12]. Another group posited that the mechano-chemistry of insertion leads to dynamics that attempt to minimize the free energy of the cell wall without describing how this dynamics could arise from biochemical kinetics [31]. A recent model also examined processive synthesis of cell wall growth, but did not directly model how crosslink density, elasticity and replication rate affect cell length [14]. While efforts to understand bacterial cell width control are ongoing, an equally puzzling phenomenon has received less attention. Bacterial cell morphology is tightly coupled to the rate 15 that cells replicate and divide. It is well known that growth conditions that increase the bacterial replication rate (which is inversely proportional to the doubling time) also lead to populations of cells that have a longer average length [25, 26, 31, 32]. In other words, cells that divide faster are longer. How does faster replication lead to longer cell length? To address this question, we take an alternative approach to modeling bacterial morphology and construct a simple model for the mechanics and chemistry of PG synthesis. We focus on the fundamental physics and biochemical kinetics in order to elucidate general principles of rodshaped growth. Specifically, we address the two experimental observations that cell length increases as the replication rate of the bacteria increases, while the fraction of peptides in the PG that are crosslinked decreases [53, 54, 55, 56]. Bacterial cell length is a consequence of two of things, the total amount of material in the cell wall and how much that material is stretched by the turgor pressure. The amount of PG that is in the cell wall is controlled by the rates that new PG is inserted and old PG is removed. There is then a net insertion rate that is equal to the insertion rate minus the removal rate. This net insertion rate times the doubling time sets how much cell wall material a given cell has. The PG is elastic and gets stretched by the turgor pressure. The Young’s modulus of the cell wall will depend on how crosslinked the PG is. To construct a mathematical model that incorporates these features, we begin by considering a single growing Gram negative cell with a single layer of PG. While conclusive evidence on the orientation of the glycan strands is lacking, the predominant view favors alignment of the glycan 16 strands about the circumference of the cell with the peptide chains oriented parallel with the long axis of the cell [28, 57] (Fig. 1). As mentioned above, the cell length is a function of the amount of material and how stretched it is. Therefore, we consider the glycan strands to be hoops about the cell circumference with each hoop defining a cross-section (Fig. 1B). We further consider that the cell maintains a fixed number of disaccharides, 4N, per cross-section. The total amount of material is then proportional to the total number of cross-sections, which we denote as X. Each disaccharide in a glycan strand has one peptide stem; however, the peptide stems rotate ~90º per disaccharide [19]. Therefore, we assume that only 1-in-4 peptides are available to crosslink any two cross-sections, which implies that there are only N possible crosslinking sites between each cross-section. If the average fraction of bound crosslinks per cross-section is and each bound peptide chain acts like a linear elastic spring with spring constant k and rest length a, the turgor pressure P will induce a strain between cross-sections given by R2 P akN 0 (2.1) where R is the radius of the bacterium. Here we have assumed that the elastic stress in the cell wall equilibrates fast compared to the biochemical reactions, which is a standard assumption [59, 60]. The total length of the growing cell is then equal to the number of cross-sections times the distance between cross-sections, L = aX(1 + ). 17 To complete our model, we need to describe the biochemical reactions that occur during cell wall synthesis (Fig. 1). Because our simple description of the cell wall ignores glycan strand length, we only consider the kinetics of peptide severing and crosslinking, along with insertion of new material. We define the rate constants of severing and crosslinking to be koff and kon, respectively. New glycan monomers are inserted between existing cross-sections at a net rate RI (which includes insertion and turnover). Insertion can occur in one of two ways, the new disaccharide can be inserted either with or without crosslinking it to a neighboring strand. We, therefore, define f to be the fraction of new monomers that are inserted and crosslinked simultaneously. The fraction of peptide bonds then obeys the kinetic equation, t koff kon 1 fRI N (2.2) The physics and kinetics just described hold for the crosslinking and strain between any two cross-sections. Insertion of new disaccharide units between existing cross-sections also leads to an increase in the number of cross-sections. Since there are 4N disaccharides per cross-section, the time rate of change of the cross-sections is X t RI X 4 N . To finish our description of a single growing cell, we need to define the local insertion rate, RI. It is important to note that this insertion rate is the rate that new material is inserted between a given cross-section. The cell-level insertion rate is then XRI. Bacterial cell wall synthesis involves the production of precursor molecules and enzymes in the cytoplasm. The precursor molecules are then transported across the inner membrane in a lipid-bound form and are then incorporated into the existing PG structure by enzymes, such as the PBPs [12]. Transport of the 18 precursor molecules across the membrane is believed to occur at a sufficient rate to match the precursor production rate [12]. We, therefore, expect that the transport rate should be proportional to the concentration of precursor molecules times the surface area of the cell, and the local insertion rate (which is the insertion rate per length) should is proportional to the flux of precursors out of the cytoplasm, which depends on the density of the precursor molecules, the cell circumference and the number of available insertion sites. Recent experiments and modelling have shown that ribosome production, protein production, and the replication rate of bacteria are tightly coupled [61]. Indeed, these results suggest that the total number of molecules of a given protein in the cell should be proportional to et, where is the replication rate [61]. Note that our model makes no assumption on what sets the replication rate. This rate is likely determined by the nutrient capacity of the organism in the environment [61]. The cell volume is proportional to XN2, and the cell circumference is proportional to N. Therefore, the flux of precursor molecules out of the cytoplasm is p0et/XN, where p0 is a constant that is proportional to the initial concentration of the precursors. We assume that new strands can only be inserted between cross-sections at locations that are not currently crosslinked. Therefore, the insertion rate should be RI = K0p0et(1-)/X , where K0 is a rate constant. The elongation of the cell is then dictated by the coupled set of equations for the number of cross-sections and the fraction of bound crosslinks: t X K 0 p0e 1 t 4N fK p et 1 koff kon 1 0 0 t NX (2.3) The model described so far considered a single growing cell that does not divide. For cells that are growing in length while replicating and dividing, the exponential rate of division can balance 19 elongation and lead to a well-defined average length of the cells in the population. This average length can be determined by the total length defined above divided by the total number of cells, L L M aX 1 M , where M = M0et is the total number of bacteria. 2.3 Results and Discussion We begin by examining the behavior of the model for a single growing cell that does divide. It is straightforward to show that for timescales much longer than the replication rate, the elongation model (Eqs. 2.3) predicts that the number of cross-sections and the crosslink fraction should be K 0 p0et 1 X 4N k 4f on kon koff (2.4) Therefore, the length of the cell grows exponentially in time as L a 1 X aK 0 p0 et R 2 P 1 1 4N akN (2.5) This result is consistent with the finding that when septation is blocked (such as by inhibiting FtsZ, the protein that initiates the cytokinetic ring) the length of rod-shaped bacteria increases exponentially in time [62]. 20 For a population of bacteria, the initial amount of precursor molecules, p0, is proportional to the initial number of cells, M0. Therefore, our model predicts that the average length of a population of cells is R 2 P K L 1 1 N akN (2.6) where K = aK0p0/4M0. Note that the ratio p0/M0 reflects the average number of precursor molecules per cell. Therefore, the factor K is a constant that depends on the average behavior of the population and is not dependent on the initial conditions. Experimental data suggests that the average cell length of E. coli and B. subtilis increases with the division rate [54-56], while the degree of crosslinking decreases [53]. (The width also increases, but modeling that effect is outside the scope of our simple model.) The force balance equation (Eq. 2.1) gives that the strain in the cell wall is only a function of the turgor pressure and the fraction of cross-links. Therefore, at constant turgor pressure, the average length of the cell only depends on the degree of crosslinking, which can be determined by Eq. 2.4. If severing and/or crosslinking of the peptides during cell wall synthesis are mediated by a multi-enzyme synthesis complex, then we would expect that the rates koff and kon, respectively, would be proportional to the insertion rate and would, therefore, depend, directly or indirectly, on the growth rate. We, therefore, consider four scenarios: (A) severing and crosslinking are independent of insertion (koff and kon are constant); (B) severing or (C) crosslinking is linked to 21 insertion (koff or kon, respectively, are linearly functions of ); and (D) both rates are proportional to the division rate. From Eqs. 2.4, 2.6, one way to get an increase in length with the division rate is if koff alone is dependent on (i.e., Scenario B)(Fig. 2A). This naturally leads to the result that the degree of cross-linking will decrease with the division rate (Fig. 2B). This mechanism suggests that severing and insertion are linked together during synthesis; however, crosslinking is independent. Basically as the cell grows faster, severing occurs more rapidly while the rate of recrosslinking remains fixed. Consequently, faster replicating cells have a lower fraction of bound peptides, and there is a larger strain in the existing peptide crosslinks. Another mechanism that would also work is if kon = A-B, where A and B are positive constants. This mechanism assumes that peptide crosslinking decreases as the replication rate increases, which suggests that peptide crosslinking does not occur simultaneously with insertion. We favor the first mechanism, as it seems more likely that a bacterium would sever old crosslinks in order to insert new material, rather than inserting material at locations that have not yet been crosslinked. In order to validate this model, we set koff = kac, where ka and c are constants, and compare the results of Eq. 2.6 to the experimental measurements of E. coli length versus division rate given in [54,55] (Fig. 2A). Our model depends on four parameters, a dimensionless pressure, R2P/akN, and the ratios kon/ka, c/ka, and 4f/ka. We find good agreement between our model and the experimental results when R2P/akN ~ 0.3, kon/ka ~ 3.5, c/ka ~ 0.5, and 4f/ka ~ 0.4. These 22 parameters suggest that is on order between 0.7-0.8. Since our model only considers the peptides that are parallel to the long axis of the cell (i.e., half the total peptide chains), we predict that the degree of crosslinking is around 35-40%, which is consistent with experimental measurements in E. coli [53]. In addition, our model predicts that the strain in the cell wall should be approximately 50%, which is consistent with the decrease in length that is observed upon osmotic shock [63, 64]. From Eq. 1, the effective Young’s modulus for the cell wall is E = akN/2Rl, where l is the thickness of the PG, which is ~4.5 nm for E. coli [26, 58]. Measurements of the Young’s modulus in E. coli suggest that E ~ 25 MPa [26, 58]. Therefore, our model predicts that the turgor pressure is P ~ 3lE/5RMPa, which falls in the range of a number of experimentally-based estimates [58, 63]. As described, our model assumes that the number of disaccharide subunits about the circumference of the cell is fixed. This assumption may not be entirely valid as cell width also increases with the replication rate. However, cell width is less affected by cell growth than cell length. Using a linear function to fit the previous data on cell length and width as a function of replication rate that was reported in [55] suggests that L = L0(1+0.42) and W = W0(1+0.19). Therefore, a 42% change in length only corresponds with a 19% change in cell width (which changes by approximately 50% over the range of replication rates in the experimental data, whereas the length changes by a factor of 2). In addition, it is not clear how much of this change in width is due to an increase in the number of disaccharides about the circumference and how much is due to additional strain in the cell wall due to a reduction in the PG elasticity. It is then likely that the number of disaccharides about the cell circumference (i.e., N) does not change significantly with the replication rate. 23 Here we have developed a simple model for bacterial cell wall synthesis that couples the biochemistry of synthesis with the biomechanics of the cell wall. The model reproduces the observation that the average cell length in a population increase roughly linearly with the replication rate of the bacteria. This result is a consequence of the increase in the number of unbound peptides. Decreasing the number of bound peptides does two things. First, it reduces the Young’s modulus for the cell, and, therefore, the turgor pressure strains the cell wall more. Second, it increases the number of sites where new strands can be inserted, which increases the rate that new material is incorporated into the cell wall. In order for the model to predict an increase in unbound peptides requires that severing and insertion are coupled and independent of re-crosslinking. This result is not consistent with the 3-for-1 model of synthesis that suggests that all three processes occur simultaneously [12]. 24 Figures Figure 2.1. Schematic of bacterial growth. (A) A bacterium of initial length, L0, is inflated by the turgor pressure, P. (B) The glycan strands are assumed to form circumferential hoops about the cell that are connected by peptide chains (thin lines) that are strained an amount ε by the turgor pressure. (C) Cell wall synthesis includes severing and recrosslinking of peptides and insertion of new disaccharide units, which build new cross-sections between existing ones (D). (E) Insertion of new material, therefore, leads to lengthening of the bacterium. 25 Figure 2.2. Mean cell length only increases with division rate for Scenario B (solid red line), which is consistent with the experiments on E. coli (Data taken from (54) (blue circles) and (55) (green squares)). Scenarios C (dashed line) and D (dotted line) led to decreases in length with division rate. (B) The degree of crosslinking decreases for Scenario B (solid line), but increases for the other Scenarios. The parameter values are as given in the text. 26 Chapter 3 The relationship between replication rate and cell wall strain in Escherichia coli suggests mechanisms of cell wall synthesis 3.1 Introduction Bacterial cells must be able to withstand large, rapid environmental changes, such as a sudden change in the concentrations of extracellular solutes. Changes in the external solute concentrations, known as osmotic shocks or stresses, can cause a rapid flow of water into or out of the cell through osmosis, which consequently causes the cell’s volume to change. In hypotonic environments the concentration of solutes is higher inside the cell, which causes a flow of water molecules into the cytoplasm of the cell due to the osmotic pressure difference, as shown in (Figure 3.1A). Consequently, the cell inflates. The increase in cytoplasmic volume stretches the cell wall and increases the hydrostatic pressure in the cell. The hydrostatic pressure difference between the inside and outside of the cell increases and eventually becomes equal to the osmotic pressure, at which point the flow of water into the cell ceases (Figure 3.1B). Note, that the solute concentration inside and outside of the cell does not become equal. On the other hand, if the external solute concentration becomes more concentrated (hypertonic), then water molecules will move out of the cytoplasm of the cell, causing the cell to shrink (Figure 3.1C). 27 In the preceding chapter, we developed a simple model for bacterial cell wall synthesis that explains the observation that cells that replicate faster are, on average, longer. The model suggests that severing and insertion of new material are processes that are directly coordinated with the replication rate of the cell; however, crosslinking of the peptide chains on newly inserted strands occurs at a rate that is independent of the replication rate. Therefore, as the replication rate increases, the rate that the peptide bonds are severed increases while the rate that they are re-crosslinked remains constant. Consequently, the degree of crosslinking is predicted to decrease. The reason that the model then predicts that faster replicating cells are longer is then two-fold. First, in a Gram-negative bacterium that has a single layered cell wall, new glycan strands can only be inserted in between strands that are not already cross-linked. Therefore, fewer crosslinks provide more sites for insertion of new strands. Hence, the rate that new strands are inserted increases. In addition, because there are fewer cross-links, the cell wall is predicted to be weaker and, therefore, is stretched more by the internal turgor pressure. That is, the model predicts that if the turgor pressure were to be reduced by a constant amount, cells that replicate faster would undergo a larger fractional change in length than cells that were in conditions where they replicate slower. In this chapter, we test the predictions of our model by using hypertonic solutions to osmotically shock wild-type E. coli (MG1655) cells. Our experiments show that the fractional change in length of E. coli cells increases with the division rate as predicted by our model. We also observe that the fractional change in width does not depend as strongly on the replication rate, which is consistent with the assumption in our model that the number of glycan monomers about the circumference of the cell is roughly constant. These findings, therefore, validate our model 28 for lateral cell wall synthesis in Gram-negative bacteria, which then provides a basis for developing a deeper understanding of the mechanisms underlying bacterial morphology. 3.2 Methods Strains and Media Wild-type K12 strain of E. coli (MG1655) was used in all experiments. The bacteria were grown to mid-to-late log phase (OD600 of 0.45-0.65) in LB medium (Sigma) at either 23, 33, 37 or 42°C while shaking at 150 rpm. Measurement of the Replication Rate Cells grown at a given temperature were washed twice with fresh LB medium and then resuspended to a dilution of 1:10 in 1 mL of fresh LB. The cultures were then shaken at 150 rpm at the original temperature. Every 15 minutes for up to eight hours, 1.5 mL of the culture was micropipetted into a cuvette and the OD600 was measured using a photospectrometer. The replication rate was then determined from the linear region of the semi-log plot of OD600 versus time. Sample Preparation for Osmotic Shock Experiments Overnight 5 mL cultures of E. coli were grown in waterbaths at the desired temperature while shaking. Prior to imaging, these cultures were diluted 1:10 in fresh LB media and grown to midlog phase (OD600 = .45-.65). Wheat Germ Agglutinin (WGA) conjugated to Alexa-Fluor 488 29 (Invitrogen) was added to 1 mL of the culture at a final concentration of 25 g/ml, in order to label the cell wall. The samples were incubated with the dye for 20 minutes at the growth temperature, before being pelleted and resuspended in fresh media. We followed the method described in [64] to construct tunnel slides by adhering coverslips to glass slides using two sided tape. 1% PEI solution was flowed into the tunnel slides and allowed to settle, before flowing in 50 l of the WGA labeled E. coli. The slides were then incubated for 10 minutes while the cells adhered to the PEI, and again washed with LB to remove any unbound cells. Microscopy The tunnel slide with WGA-Alexa 488 stained cells were imaged every 20 seconds for 20 minutes using DIC and epifluorescence on a Zeiss Axio-Observer V. At 5 minutes, 50 L of 500 mM sucrose in LB was flowed into the tunnel slide, inducing hyperosmotic conditions for the E coli. This procedure was done for cells grown at each temperature. The image processing software, Image J [65] was then used to measure the length and width before and after the shock. We only measured cells that were clearly in focus before and after the shock. The average measurements from three time points before (or after) the osmotic shock were used to measure the initial (and final) lengths and widths. 3.3 Results and Discussion In order to determine whether the elastic strain in the cell wall is larger for cells that replicate faster, we needed a set of growth conditions that would produce different replication rates. The 30 previous data on the dependence of cell length as a function of replication rate showed that the average length in a population of cells is not strongly dependent on the specifics of the growth conditions. Over a large range of different conditions, including changes in media, temperature, and nutrients, average cell length followed the same dependence on replication rate [54, 55]. We, therefore, chose to use growth temperature to control the replication rate, as this was easily adjustable and does not alter the nutrient availability or solute concentrations in the media. The replication rate of bacteria is known to depend on temperature. At low temperatures, increasing the temperature leads to higher replication rates, whereas at high temperatures, the replication rate decreases with increases in the temperature. That is, there is an optimal temperature at which the growth rate of a bacterial strain is most rapid. For K12 serotype E. coli, this optimal temperature is around 42ºC [66]. Therefore, we chose to use four temperatures between 23ºC and 42ºC to examine the relationship between elastic strain in the cell wall and replication rate. To begin, we determined the replication rates of E. coli strain MG1655, which we denoted as , at 23ºC, 27ºC, 33ºC and 42ºC. We found that the replication rate increases roughly linearly in this temperature range from 1.35 hr-1 to 1.81 hr-1 (Figure 3.2A). At 37 ºC, we found a replication rate of 1.61 hr-1, which is comparable to what has been measured previously [55]. Next, we measured the average length and width of cells that were grown at these temperatures. We used DIC microscopy to examine the morphology of cells that were grown to mid-log phase at each temperature. Consistent with previous measurements, we found that the average length of cells in a population increases roughly linearly with the replication rate (Figure 3.2B). Fitting this data to a line, we found a slope of 0.9 m/hr, which agrees well with the value 0f 0.97 m/hr that was reported previously [54]. Interestingly, we found that the width decreased slightly 31 between the slowest replication rate (at 23ºC) and the rate at 37ºC. The width was approximately constant for the three largest growth rates (Figure 3.2B). This finding is different than what has been previously reported [67, 55]; however, the range of replication rates that we investigated covered only the high range of what was used in the previous reports. Indeed, the few data points from the other reports that were in the same range that we considered also show a small decrease in width at fast replication rates [55]. Next, we osmotically shocked the cells by washing in LB medium that contained 500 mM sucrose. Because sucrose is unable to enter the cell, the addition of sucrose in the medium increases the external osmolarity of the medium by an amount that is roughly proportional to the concentration of sucrose [68]. Therefore, upon the addition of sucrose medium, the cells shrink due to the effective drop in the internal turgor pressure (Figure 3.3A-D). We also find that cells grown at higher temperatures shrink more on average than cells grown at lower temperatures (Figure 3.3A-D). However, since cells grown at higher temperatures are also longer (Figure 3.2B), the larger change in length with higher replication rate does not confirm the predictions of our model. A cell that is stretched by the turgor pressure has an unstretched length and width, which we can denote as L0 and W0, respectively. The turgor stretches these to a final length and width Lf and Wf. The elastic strains in the cell wall are given by (Lf - L0)/L0 and (Wf - W0)/W0. These strains should be proportional to the turgor pressure, and inversely related to the stiffness of the cell wall. Therefore, decreasing the turgor pressure by addition of sucrose causes the elastic strains in the cell wall to change. If the length and width after the shock are L and W, we can define the fractional changes in length and width as (Lf - L)/L and (Wf - W)/W, respectively. These should 32 be a measure of the total strain in the cell wall and are also expected to be inversely proportional to the cell wall stiffness in the axial and circumferential directions, respectively. We, therefore, measured the fractional changes in length and width of cells grown to mid-late log phase at the four temperatures that were used to determine the replication rates and average lengths and widths. We found that the fractional change in length increased linearly with the replication rate, going from 0.129 ± 0.001 at 23ºC to 0.191 ± 0.001 at 42ºC (Figure 3.3E) with a p-value of 3.1 E -12 between those two temperatures. We also measured the fractional change in width, which also increases with replication rate; however, the fractional change in width seems to plateau at high replication rates and is approximately a factor of two smaller than the fractional change in length (Figure 3.3B). Here we have shown that the elastic strain in the cell wall of E. coli is dependent on the replication rate of the bacteria. This result was predicted by the mechanochemical model for cell wall synthesis that we developed in Chapter 2. The primary hypotheses of this model were that cell wall synthesis in Gram-negative bacteria requires new material to be inserted between uncrosslinked glycan strands, and that the rate of crosslinking new strands is independent of the replication rate, whereas insertion and severing of material are proportional to the replication rate. Therefore, our experimental results provide evidence that these mechanisms are involved in establishing and maintaining the morphology of Gram-negative rod-shaped bacteria. However, the results that have been described so far only provide qualitative validation of the model predictions. Because we find the same dependence in average cell length with replication rate as was used to determine the parameters in our model (See Chapter 2), it is possible to use the 33 mechanochemical model to predict quantitatively what is expected for the fractional change in length. Equations 2.1 and 2.4 can be used to write the strain in the cell wall as R 2 P kon koff akN kon 4 f (0.1) Using the parameter values previously determined, the strain is then predicted to obey 1.2 0.3 3.5 0.4 (3.1) where is in units of hr-1. This equation predicts that the total strain in the cell wall should vary roughly linearly with respect to replication rate for values of less than about 5, and for values of in this range, the slope of the strain versus replication rate should be approximately 0.1 hr. Since our osmotic shock experiments do not necessarily measure the total strain in the cell wall, but rather measure the change in length for a fixed change in turgor pressure, it is better to compare the model prediction for the slope to the experimental data. In our experiments, the fractional change in length varied by 0.06 while the replication rate varied by 0.46 hr-1, which gives a slope of 0.13hr, which agrees extremely well with the model prediction. It is important to note that no further adjustment of the model parameters was necessary to get this agreement between the model and the data. Therefore, our experimental results provide strong evidence to support the mechanochemical model. Our finding that the fractional change in length increases with the replication rate contrasts a recent report that examined the strain in the cell wall of the Gram-positive bacterium B.subtilis using osmotic shock [69]. This previous work observed no dependence of the fractional change 34 in length with replication rate. It is likely that the difference in our results from the previous work reflects the difference in cell wall synthesis between Gram-negative and Gram-positive cells. Figures (A) (B) (C) Figure 3.1. (A) In a hypotonic environment, the extracellular solute concentration is low. There is then a tendency for water molecules to move into the cell from outside (this, tendency is the osmotic pressure), which causes the intracellular hydrostatic pressure to increase. (B) In an isotonic environment, the osmotic pressure is balanced by the hydrostatic pressure, and, consequently, there is no net flow of water molecules into or out of the cell. (C) In hypertonic environments, the extracellular solute concentration is high, and water flows out of the cell, decreasing the internal hydrostatic pressure. As a result, the cell shrinks. 35 (A) (B) Figure 3.2. (A) The replication rate of E. coli MG1655 increases approximately linearly with temperature. (B) Average cell length (blue diamonds) and width (red circles) as a function of the replication rate. The average cell length increases with the replication rate, whereas, the width shows a slight decrease. Error bars show the standard error of the mean (SEM). 36 A B C D E Figure 3.3. Representative images of E. coli MG1655 cells before (A,C) and after (B,D) the addition of 500 mM sucrose. Panels A,B show a cell grown at 23°C, and panels C,D are a cell grown at 42°C. (E) The fractional change in length (blue diamonds) and width (red circles) increase with replication rate. The fractional change in length varies approximately linearly, as is predicted by the mechanochemical model. The fractional change in width is always smaller (by approximately a factor of 2) than the fractional change in length and does not show a linear increase. Error bars denote the SEMs of the measurements. 37 Chapter 4 The effects of peptidoglycan structure on the elasticity of the Gram-negative cell wall 4.1 Introduction The peptidoglycan of bacterial cells is an elastic macromolecular material which serves as the stress bearing structure of the cell and provides the mechanical framework that defines the bacterium’s shape. Although this structure provides rigidity for the cells, the elastic properties of the PG also allow the cell to expand or contract under varying environmental conditions. Furthermore, the specific elastic properties of the cell may be very important in determining cellular morphology, as applied force has been shown to affect cell shape and how force is distributed through the cell wall depends on the elastic properties of the material. Despite the fact that many of the structural details of the PG are known, how PG elasticity depends on factors such as glycan strand length and crosslink density is completely unknown. Gram-negative cell walls are monolayered structures comprised predominantly of a repeated chain of sugars (glycan strands) crosslinked by peptide stems forming a covalently linked meshwork. This layer wraps around the entire circumference of the cell and spans the entire cell length, forming roughly hemispherical caps at the cell end. The peptide chains that are created by binding peptide stems on neighboring glycan strands are believed to be more flexible than the relatively stiff glycan strands [19, 33]; however, the evidence that supports this belief is largely circumstantial. Early EM data suggested that the glycan strands were arranged circumferentially, 38 with the peptide stems oriented along the long axis of the rod-shaped cell [12, 26, 70-72]. In addition, AFM experiments suggest that the elastic modulus of the cell wall along the long axis is between a factor of 2 to 3 less than the circumferential modulus [26], and hyperosmotic shock experiments show a roughly two fold difference between the fractional changes in length and width (See Chapter 3). Combining these two observations then suggests that the peptides are 2-3 times less stiff than the glycan strands, which would also imply that the PG layer is an anisotropic elastic material. In this chapter we investigate the elasticity of a single layer of PG as a function of the glycan strand length and the degree of crosslinking. Reversed HPLC has been used frequently to measure glycan strand lengths giving values of 25-35 disaccharide units for the mean glycan chain length (GCL) in exponentially growing E. coli cells [12, 27]. The mean glycan chain length varies also for different bacterial species; ranging from 50-250 disaccharide units in Bacilli (B. subtilis, B. licheniformis and B. cereus [73-75], about 18 disaccharide units in Staphylococcus aureas (S. aureas) [73] and less than 10 disaccharide units for H. pylori [76]. It would seem reasonable that the longer the glycan strands, the stiffer the elasticity of the cell wall, which presumably would be favorable for the bacterium. However, there is evidence that glycan strands are synthesized and inserted with lengths longer than the GCL and are then cut enzymatically to shorter lengths. Taking this together with the fact that different species of bacteria have different GCLs might suggest that glycan strand length is a factor that is regulated for some purpose. However, it is not clear what that purpose is nor even how changing GCL affects the elasticity of the cell wall. In addition, the peptide stems are of a fixed length, but not all of these stems are crosslinked to neighboring glycan strands. The degree of crosslinking of E. coli has been measured and varies from 40% to 60% [35, 53]. Due to the location of the peptide stems on the glycan strands, the number of possible peptide stems is a function of glycan chain 39 length (GCL). Lower degrees of crosslinking presumably lead to weaker cell walls. Therefore, it is expected that the Young’s modulus of the cell wall is lower in cells with lower degrees of crosslinking. Why, then, does bacterial cell wall synthesis seem to regulate GCL and the degree of crosslinking? Here we make the assumption that these features are important for determining the elastic properties of the cell wall. It is likely that the elastic properties of the PG are crucial for creating and maintaining the shape of the cell. Therefore, determining how GCL and the degree of crosslinking affect the elasticity of the PG is necessary for understanding bacterial morphology. To address this question, we simulate the PG network and determine the strain that is induced by a range of stresses. Our simulations estimate the physiological range of GCL and degree of crosslinking that are necessary to prevent cell lysis and also suggest that these parameters may be tuned to optimize the anisotropy of the bacterial cell wall. 4.2 Modeling the Gram-negative cell wall The disaccharide units that build the peptidoglycan structure of the cell wall are around 1.03 nm in length and the stretched peptide chains are on order of 4 nm [28, 77 -81]. Therefore, the effective area of the subunit is around 5 nm2. The total surface area of E. coli or B. subtilis is around 5 - 10 m2. A single layered cell wall for E. coli requires roughly 106 disaccharides and the multi-layered cell wall of B. subtilis requires at least ten times more subunits than E.coli. While computational simulations can be performed to simulate these structures [82], it is computationally expensive and difficult to determine how changes in the physical structure and/or biochemistry affect the gross behavior of the cell wall. 40 To overcome this large discrepancy in length scales, we constructed a discrete model of a patch of peptidoglycan that is on the order of 0.01 m2 in size (a patch of 50x50 disaccharide units). Each disaccharide was treated as a particle that can bind to other disaccharides through Hookean spring-like interactions that represent either the glycosidic or the peptide bonds (Figure 4.1). The spring stiffnesses were estimated from mechanical measurements of the Young’s modulus of bacteria, which have been carried out in E. coli [59], B. subtilis [83], and other bacteria [83-84]. In addition, we used that the glycan strands are roughly three times stiffer than the peptide chains [32, 26, 19, 33]. In essence, this model is conceptually similar to the computational model used by Huang and coworkers [59, 81, 84]; however, we use realistic sizes for the disaccharides and use our simulations to extract continuum-level parameters, such as the Young’s moduli along the longitudinal and circumferential directions. In order to construct the PG structure for a single layer of a patch of cell wall, which was then used to determine the elastic moduli as a function of the GCL and degree of crosslinking, we used a percolation type analysis, such as has been used to study many problems in randomly connected networks [87]. We considered a 50x50 rectangular array of nodes. Each node represents a potential disaccharide unit. In principle, any disaccharide unit can be connected to any of its nearest neighbors in the y-direction. This direction represents the orientation of the glycan strands, and a connection between nearest neighbors in the y-direction represent glycosidic bonds. We define a probability Pg that sets the probability that any disaccharide unit is connected to the neighbor above it. For each node, we use a random number generator to select a number between 0 and 1. If the number is less than or equal to the probability, then we connect the disaccharide to its upward neighbor. Otherwise, those nodes are not connected. A 41 similar procedure is used to define peptide crosslinks that connect the strands along the xdirection. For the peptides, we define a separate probability Pp that defines the probability that a crosslink is formed. However, as mentioned in previous chapters, the peptide stems rotate by approximately 90 degrees per disaccharide unit [59]. Therefore, only every 4th disaccharide can be connected by a peptide crosslink to its neighbor to the right. For every 4th node along a given column, we generate another random number. If this number is less than or equal to Pp, then we connect the node to its rightward neighbor. These connections then define a randomly generated network that has a GCL that is equal to the inverse of (1 – Pg) and a degree of crosslinking = Pp. The network is only a viable model for the cell wall if there are paths that connect at least one node on each edge to at least one node on each of the other edges. In the language of percolation theory, this represents an infinite cluster [87]. If an infinite cluster does not exist, then the cell wall will lyse for any applied force. Therefore, we only consider networks that include an infinite cluster. In addition, any nodes that are not connected back to the infinite cluster are removed from the network. Two representative networks are shown in Figure 4.1. To understand the effect of force on the material properties of these simulated PG networks, we treat the connections as Hookean springs with spring constants that were determined as described above. Because the disaccharides are molecular-sized and are surrounded by fluid, we assume that the movement of the disaccharides obeys overdamped dynamics, with a drag coefficient ; i.e., if the position of a node is defined as X, then the time rate of change of the position obeys dX/dt) = F, where F is the sum of the forces that act on that node and we only consider motion in the x-y plane.We started our simulations from the unstretched state and then applied a fixed constant 42 force directed along the positive x-direction to each node on the right side of the array, and an equal and opposite force to the nodes on the left side of the array. Likewise, we applied a force directed along the positive y-direction to the nodes on the top edge of the array and an equal and opposite force to the bottom edge nodes. We then solve for the dynamics of each node and run our simulations until the network reaches equilibrium. Using the equilibrium conformation, we then calculated the stress and the strain. The xx component of the stress tensor was calculated as xx = Fx/Lg, where Fx was the total force that we applied to the right side of the network, and Lg was the projected length along the y-axis of the right side of the stretched network. The yy component was found in a similar manner using the total force applied to the top of the network and the projected length along the x-axis of the top side of the stretched network. In order to calculate the strain, we first computed the displacement vector for each node, u, which is the difference between the stretched position of the node and its initial position. The strain tensor is related to derivatives of the displacement vector. For example, the xx component of the strain tensor depends on the derivative of the x-component of the displacement with respect to the x-direction. To compute this derivative using our networks, we took the average value of the x displacement of the nodes on the right edge and subtracted them from the average value of the x-component of the displacements for the nodes on the left edge. The difference between these two averages was then divided by the unstretched distance between those two sides. The resulting quantity is defined as uxx. In a similar way, we calculate the other derivatives, such as the derivative of the x-component of the displacement along the y direction, 43 uxy, which is calculated using the difference between the average value of the x-component of the displacement along the top edge and the average value of the x-component of the displacement along the bottom edge divided by the unstretched length of the top edge. The strain tensor is then given by ij 1 uij u ji uilu jl 2 (4.1) 4.3 Results and Discussion In order to determine how the elastic properties of the bacterial cell wall depend on GCL and the degree of crosslinking, we simulated random networks with a range of different GCLs and degrees of crosslinking (as described in Sec. 4.2). We independently varied Pp (which is equivalent to the degree of crosslinking) and Pg (which is a probability that is related to the GCL) from 0 to 1 in increments 0.05. For each combination of Pp and Pg, we constructed ten networks, and, for each network, we applied a range forces to the edges in the x- and y-directions, as described in Sec. 4.2. The nodes were then moved according to the overdamped dynamic equation and the equilibrium stress and strain was computed for every force combination. All simulations were carried out in MATLAB. As is well known from percolation theory [87], only certain values of the probabilities will lead to networks that contain an infinite cluster. The consequence of this for the bacterial cell wall is that the PG will only be a single connected macromolecule if the GCL and degree of crosslinking are sufficiently large. The networks that we simulate reflect the structure that is commonly 44 assumed for the PG. We find that we only get infinite clusters when Pg > 0.6 and Pp > 0.25. These values correspond to a minimum GCL of 3 and a degree of crosslinking equal to 12%. In addition, though, the value of Pg at which the percolation transition occurs depends on Pp. Therefore, it is not possible to have a GCL of 3 and a degree of crosslinking equal to 12%. The white region in Figure 4.2 shows the values for 1/Ng and = Pp where infinite clusters are not observed (Here Ng is the average number of disaccharides per glycan strand (i.e., the GCL), which is also equal to 1/(1-Pg)). This region represents combinations of GCL and that should be inaccessible to bacteria with a single-layered PG and shows that smaller values of GCL require larger degrees of crosslinking, which is consistent with the observations that E. coli has GCLs in the range of 25-30 and a degree of crosslinking around 20%, whereas S. aureus has a GCL of 3-10 and a degree of 93% [31]. For networks that produced infinite clusters, we then determined the relationship between the stress and strain. By deforming the networks using various forces applied to the edges (as described in Sec. 4.2), we computed the diagonal components of the stress tensor as a function of the components of the strain tensor. In general we find that both of these components of the stress depend on the xx and yy components of the strain; however, they were not strongly dependent on the xy component of the strain. It is likely that the xy dependence of the strain was not important since we only applied expansive forces in our simulations and did not apply any shear forces. We chose to only use expansive forces, because we were most interested in the response of the cell wall to the turgor pressures. We found that both components of the stress depended nonlinearly on both diagonal components of the strain. Therefore, we fit the stress vs. strain curves to the following quadratic functions: 45 (4.2) The Young’s modulus of a material defines the linear relationship between a given component of the stress and the same component of the strain. When the stress depends nonlinearly on the strain, the Young’s modulus, therefore, depends on the strain. A general expression for the Young’s modulus is then that the modulus along a given direction Eij = dij/dij. In order to obtain a single value for the Young’s moduli of the PG in the longitudinal (peptide) and circumferential (glycan) directions, Ep and Eg, repectively, we used the expressions given in Eq. 4.2 to define the Young’s moduli around a strain of 0.2 (which is approximately the strain that is observed in our hyperosmotic shock experiments (Chapter 3). Using this formalism, we found that the longitudinal elastic modulus (i.e, the modulus along the peptide bonds) decreases with decreases in the GCL and the degree of crosslinking (Figure 4.2A). This result is somewhat expected as it states that as the PG network is degraded it should get softer. The dependence of the Young’s modulus along the circumferential direction on the GCL and degree of crosslinking was more surprising. While, in general, decreasing either of these factors lead to smaller values of Eg, we found that for large value of the GCL (for glycan binding probabilities near 1, Pg > 0.95) that the maximum value of Eg was located near = 0.7 (Figure 4.2B). This implies that when the GCL is greater than 20 disaccharides, decreasing the degree of crosslinking from 1 to 0.7 leads to increases in the stiffness of the cell wall in the circumferential direction. Because the longitudinal stiffness decreases in the same regime, this 46 suggests that the anisotropy of the cell wall is optimized for values of = 0.7 (which corresponds to a degree of crosslinking around 35%).. Since E. coli has a GCL of 25-35 disaccharides and a degree of crosslinking around30%, this suggests that the biochemical kinetics of synthesis may be regulated to optimize the anisotropy of the cell wall. It may, then, be that this anisotropy is important for maintaining the width of the rod-shaped cell. However, the work presented here only suggests this possibility. More work is needed to validate this hypothesis. 47 Figures (A) (B) Figure 4.1. Peptidoglycan percolation models for a gram-negative bacterium at (A) 100% connectivity (i.e., Pp = Pg = 1), and (B) 50% crosslink density (Pp = 0.5) and 96% mean glycan chain length (Pg = 0.96), which is within the range appropriate for E. coli [10, 32, 18, 64]. (Red) represents glycosidic bonds and black represents peptide bonds. 48 (A) (B) Figure 4.2. Elastic moduli of a single layer of peptidoglycan at varying crosslink densities and glycan chain length. (A).Elastic modulus, Ep decreases with either a decrease in degree of crosslinkage or a decrease in glycan chain length. Ep increases with an increase in peptide crosslinkage, Ψ for large values of Pg. (Dotted lines) (B) Elastic modulus, Eg shows peak at intermediate values of peptide crosslinkage, Ψ for longer glycan strands. Eg increases when peptide crosslinkage, Ψ is around 0.7 for large values of Pg. (Dotted lines) 49 Chapter 5 Conclusions and suggestions for future research At face value, the ability of a rod-shaped bacterial cell to grow in length without changing its width does not seem that difficult or interesting. The bacterial cell is just like a balloon, where the elastic cell wall material is analogous to the latex and is inflated by the turgor pressure. For many years, the scientific community held this belief. Then, in the early part of this century, the discovery that bacteria possess cytoskeletal proteins structurally homologous to actin that are necessary for proper cell shape led to the realization that the creation and regulation of bacterial morphology is much more complex than we had imagined. Research over the past 15 years has now clearly shown that we still don’t understand much about how even a simple rod-shaped bacterium controls its width. In addition, we now also know that bacterial morphology can be influenced by external forces. The work presented in this dissertation sought to begin to develop a more mechanistic picture of bacterial cell wall synthesis. We took the hypothesis that bacterial morphology is a consequence of the interplay between the biochemical kinetics of synthesis and the biophysical interactions due to the material properties of the PG and the expansive force from the turgor pressure. Working from this hypothesis, we started simple and aimed to understand the long standing observation that E. coli cells that replicate faster are also longer. While this observation is well known and has been reported by many groups, no explanation for why it happens has been proposed. To investigate the mechanisms that could lead to this phenomenon, we considered three of the basic biochemical processes involved in cell wall synthesis: severing of existing crosslinks, insertion of new glycan strands, and re-crosslinking of the PG. We simplified our 50 analysis by considering a simple structure for the PG where a fixed number of glycan monomers are arranged circumferentially. We then developed a model based on the biochemical kinetics of severing and crosslinking of the peptides. We included in this model the biophysics of the stretching of the cell wall by the turgor pressure. We found that this simple model could only reproduce the observation that faster replicating cells are longer if the severing and insertion rates were proportional to the replication rate and the crosslinking rate was independent of the replication rate. Comparing the model to the observed dependence of the length on replication rate allowed us to predict values of the biochemical kinetic rates and turgor pressure inside the cells. In addition, the model predicts that the PG in faster growing cells is weaker and therefore should be stretched more by the turgor pressure. Next, we tested the predictions of our mechanochemical model by using hyperosmotic shock to effectively reduce the turgor pressure inside the cells. We shocked E. coli cells that were grown at different temperatures and, therefore, had different replication rates. We found that the fractional change in length was linearly proportional to the replication rate, as predicted by the model. Interestingly, without changing the parameters in our model from what we estimated from the dependence of length on replication rate, we were able to accurately predict the dependence of the fractional change in length on the replication rate. These results provide good support for our mechanochemical model. While our experiments provide credence to our mechanochemical model, the model itself is too simple to address larger questions in bacterial morphology. Our assumption that the number of glycan monomers about the circumference of the cell is fixed does not allow us to investigate the biochemical mechanisms involved with glycan strand elongation and glycan chain severing, which are likely important for determining bacterial morphology. 51 To investigate this further, we constructed a discrete model of a growing patch of the bacterial cell wall, where we were able to vary the average number of disaccharides per glycan strand and the fraction of crosslinked peptides. We constructed random networks using a percolation type analysis and represented glycosidic bonds and peptide chains as Hookean springs. Our model predicts the values of GCL and degree of crosslinking that are required to create a single macromolecular PG structure, which explains why bacterial cells with shorter GCL are typically more crosslinked. Then by applying a series of forces to the edges of this simulated PG patch, we were able to compute the dependence of the effective Young’s moduli as a function of glycan chain length and the degree of crosslinking. We found that the Young’s modulus along the longitudinal axis of the cell should decrease with decreases in GCL or degree of crosslinking. Interestingly, though, we found that the circumferential Young’s modulus has a peak at around 35% peptide crosslinking when the GCL is larger than 20 disaccharides. Therefore, for PG structures with GCLs over 20, there is a degree of crosslinking that optimizes the elastic anisotropy of the cell wall. Since E. coli has a GCL of 25-35 disaccharides and a crosslinking fraction of around 20-30%, it may be that these values are regulated to maintain cell wall anisotropy. The work presented here, therefore, lays the groundwork for a more mechanistic understanding of bacterial morphology and will be very beneficial to the advancement of this field towards understanding how bacteria create and maintain their shapes. Our mathematical model for cell wall synthesis may also aid in developing novel antibiotic treatments. Completing the discrete model by additionally simulating the insertion of newly synthesized material would allow us to get a mode detailed outlook at the biochemical reactions in the 52 bacteria cell. 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