Residues in M. tuberculosis FtsZ that interact with residues in another subunit or with other cell division proteins.
\r\n\t- Traditionally accepted topics related to global health security,
\r\n\t- The impact of human activities and climate change on “planetary health”,
\r\n\t- The impact of global demographic changes and the emergence chronic health conditions as international health security threats.
\r\n\t- A theme dedicated to the COVID-19 Pandemic,
\r\n\t- Novel considerations, including the impact of social media and more recent technological developments on international health security.
\r\n\tThe goal of this book cycle is to provide a comprehensive compendium that will be able to stand on its own as an authoritative source of information on international health security.
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Miller",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10624.jpg",keywords:"Threats, Monitoring, Food Security, Emerging Infections, Transmission, Geopolitics, Climate Change, Cyber Health Security, COVID-19, Novel Coronavirus, Pandemic, Coronavirus",numberOfDownloads:173,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"August 20th 2020",dateEndSecondStepPublish:"November 5th 2020",dateEndThirdStepPublish:"January 4th 2021",dateEndFourthStepPublish:"March 25th 2021",dateEndFifthStepPublish:"May 24th 2021",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"An Associate Professor of Surgery at Temple University School of Medicine and a Chair of the Department of Research and Innovation, St. Luke's University Health Network. A member of multiple editorial boards and co-author of over 550 publications.",coeditorOneBiosketch:"An Associate Professor of Surgery & Integrative Medicine at Northeast Ohio Medical University and Cardiothoracic Surgeon at the Summa Health Care System. A prolific writer and presenter, with multiple books, hundreds of peer-reviewed articles, and innumerable presentations around the world.",coeditorTwoBiosketch:"A CEO of the INDUSEM Health and Medicine Collaborative, Global Executive Director. of the American College of Academic International Medicine (ACAIM) and head of the World Academic Council of Emergency Medicine.",coeditorThreeBiosketch:"A Director of Research in the Department of Emergency Medicine at Nazareth Hospital in Philadelphia, USA, and co-chief editor of the International Journal of Critical Illness and Injury Science. 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His academic experience over the last two decades includes consistent achievements and innovative strategies that have led to the creation of organizations, publication of landmark papers, and commendation with prestigious citations and honors for his works that have impacted academic medicine globally. He has played a defining role in founding and building internationally recognized interdisciplinary indexed journals. He is the CEO of the INDUSEM Health and Medicine Collaborative and heads the World Academic Council of Emergency Medicine (WACEM). Additionally, he serves as the Global Executive Director for the American College of Academic International Medicine (ACAIM). 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He completed residencies in Emergency Medicine and Internal Medicine at the State University of New York (SUNY) Downstate Medical Center (2010) where he served as Chief Resident for Research. He completed fellowships in Pulmonary Medicine at the University of Pittsburgh Medical Center (2013) and Critical Care Medicine at the National Institutes of Health (2014). He is active in the American College of Academic International Medicine, and is co-chief editor of the International Journal of Critical Illness and Injury Science. 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The replication and segregation of genetic material occur in karyokinesis. In cytokinesis, the cytoplasm and its contents increase in quantity and get partitioned between two daughter cells with the development of the septum between them and formation of new cell membrane and cell wall in the sacculus. The process of septation is guided by the cytokinetic protein, FtsZ [1, 2], which is the bacterial structural homologue of mammalian β-tubulin [3]. Polymeric FtsZ, as the inner membrane bound FtsZ ring at the mid-cell site [1, 2], guides septation. A large number of cell division proteins interact with FtsZ at the median (reviewed in [4, 5]). The duration of the whole process of cell division varies from bacterium to bacterium, with different strains of Escherichia coli completing cell division in 18–55 min [6], Bacillus subtilis taking 120 min [7], whereas Mycobacterium smegmatis taking 3 h [8], M. tuberculosis 18 h in vivo [9] and 24 h in vitro [10], and M. leprae completing it in 13.5 days once [11]. Accordingly, since the septation process takes only a part of the whole cell division duration, the polymerisation and depolymerisation dynamics of FtsZ of each bacterium need to be different in different bacteria to suit the septation duration in the bacterium. The required modifications in the polymerisation and depolymerisation dynamics of the cytokinetic protein to suit the time span of septation need to be effected through specific changes in the amino acid residues of FtsZ. Such changes, which can influence polymerisation and depolymerisation dynamics of FtsZ, can be vividly discerned from the evolutionarily purposeful placement of specific amino acid residues at specific structural locations on the protein. Here, we have tried to point out this particular aspect of the structure-function correlation of FtsZ from mycobacteria with that of FtsZ from other mycobacterial species and bacterial genera.
The biochemical identity of FtsZ was first established through the demonstration of the GTP binding and GTPase activities of Escherichia coli FtsZ (E. coli FtsZ) [12, 13]. Both the authors showed the presence of a peptide motif (105 GGGTGTG 111), similar to the tubulin signature motif (140 GGGTGSG 147), and postulated that it would form a phosphate-binding loop like in β-tubulin [14]. The GGGTGTG motif is crucial for GTP binding as its change to SGGTGTG in the FtsZ84 mutant of the temperature-sensitive E. coli (ftsZ84) strain markedly reduced GTP binding ability (by crosslinking) [13] and converted FtsZ into an ATPase in vitro [12]. Tubulin and FtsZ share poor primary sequence homology of 10–18%, except in two sequence regions (amino acids 95–175 in β-tubulin and 65–135 in E. coli FtsZ, and 305–350 in β-tubulin and 255–300 in E. coli FtsZ) where they show clear sequence homology of 85–87 and 51–78%, respectively [15]. The first region contains β-tubulin/FtsZ signature motif, flanked by identical looking secondary structural elements [15]. However, the overall three-dimensional structures of FtsZ and β-tubulin are almost similar [16, 17], which is reflected in the GTP-dependent polymerisation of E. coli FtsZ in vitro [18, 19] like β-tubulin polymerisation [20].
The crystal structures of the FtsZ proteins of Methanococcus jannaschii [16], Thermotoga maritima [21], Bacillus subtilis, Pseudomonas aeruginosa, and Aquifex aeolicus [22] essentially showed independently folding N- and C-terminal domains arranged around a central helix. The N-terminal domain contains GTP-binding site and an incomplete GTPase active site. During the course of polymerisation, the T7 loop of one monomer is supplied in trans to the GTP-binding pocket of the next, thus forming the active site for GTP hydrolysis [23, 24, 25, 26, 27, 28]. From the co-crystal structure of SulA-FtsZ, it was found that SulA, a cell division inhibitor [29], binds the T7 loop surface of FtsZ, thereby blocking polymer formation as a part of SOS response [30]. Thus, the T7 loop seems to be critically required for FtsZ polymerisation, thereby carrying the potential to be an antibacterial inhibitor target.
In all the solved structures of FtsZ of M. jannaschii [16], P. aeruginosa [22], A. aeolicus [22], T. maritima [21], and of B. subtilis [31], the end part of the C-terminus has been found to be unstructured and containing variable residues. It has been found to be a platform for the interaction of a large number of cell division proteins, ZipA [32], FtsA [33], EzrA [34], SepF [35, 36], ZapD [37], and with FtsZ itself [4, 5, 38].
GTP-binding residues are best conserved between FtsZ and β-tubulin [39]. Monomeric FtsZ binds GTP tightly with a Kd of 5 μM [40]. The FtsZ84 (G105S) mutant protein has an impaired GTPase activity in vitro [12]. In other examples, while the mutant FtsZ6460 (G109S) showed nil GTPase activity, FtsZ9124 (P203L) possessed reduced GTPase activity, and both the mutants failed to polymerise in vitro [41]. Extensive work has been carried out on the characterisation of FtsZ mutants in vitro and in vivo [42, 43, 44]. Based on a model structure of E. coli FtsZ, which was built on the crystal structure of M. jannaschii FtsZ [16], the mutants were found to be primarily of three classes [43]. (i) On the front and back surface of protofilament and few on the top and bottom surfaces not in contact with GTP, which were termed as benign mutants as all of them could complement ftsZ84 strain at non-permissive temperature. Few examples of these mutants are A70T, A81V/F268C (Z100), D158A, D158N, D187A, F268C (Z114), D269A, and D299A. (ii) GTP contact mutants, e.g., N43D, D45A, D45N, D209A, did not complement ftsZ84 and could polymerise only in the presence of DEAE-dextran. (iii) Mutants, D86K, D96A, E238A, S245F, and E250A, where the mutations were thought to be on the lateral surface of protofilaments and therefore believed to play role in inter-protofilament interactions. These mutants could polymerise in vitro, but none of them could complement ftsZ84 at non-permissive temperature.
Further studies showed that the mutations on the top surface of E. coli FtsZ model (e.g., G21K, L68C\\D\\W, F182C) have far less disturbing effect on the complementation (in plate and liquid culture) and GTPase activity of FtsZ compared to those of the mutations on the bottom surface (e.g. D96A, N207C, D209A\\C\\K, F210A) [44]. Interestingly, one top surface mutant Q47K that binds GTP at more than 1:1 ratio and having around 30-fold less GTPase activity, compared to that of wild-type FtsZ, failed to complement cell division, both in plate and in liquid culture [44]. The E83 and R85 in the helix H3 bend and lateral residues of E. coli FtsZ were found to be important for the polymerisation, GTPase activity, and cellular viability [45]. E93R substitution in E. coli FtsZ induces bundling of protofilaments, reduces GTPase activity, and impairs bacterial cytokinesis [46]. Similarly, R191 of B. subtilis FtsZ was found to be required for polymerisation [47].
The residue D212G of the T7 loop is conserved among all FtsZ sequences known [48] and all the mutations D212A\\N\\C, D212G (FtsZ2) impaired GTPase activity [42, 43, 44]. The location of FtsZ2 mutation was later validated by the crystallography data that SulA binds to T7 loop region of FtsZ and a mutation in T7 loop might prevent binding of SulA to block FtsZ polymerisation [30]. There was no other structural abnormality of FtsZ2 to bind GTP and therefore could polymerise in the presence of DEAE-dextran in vitro [19] or co-polymerise with wild-type FtsZ [25].
The C-terminal variable (CTV) region residues in E. coli FtsZ and B. subtilis FtsZ were found to mediate electrostatic interactions to facilitate lateral association of the FtsZ protofilaments to form polymers in vitro and in vivo [39] and for interaction with other proteins such as ZapD ([49], reviewed in [4, 5]).
FtsZ has been found to exist mostly as dimers [50, 51]. FtsZ assembly process leading to polymerisation has also been found to involve a dimer nucleus [52]. Among the wide variety of FtsZ structures visualised by various methods till date, the simplest one is the protofilament constituted by FtsZ subunits stacked one above the other with a diameter of 5 nm, with each subunit placed at 4.3 nm apart [53]. While straight protofilaments were favoured by high concentration of GTP or by GTPase inhibition, curved conformation was triggered by GTP hydrolysis or in the presence of GDP [54]. Examination of the FtsZ polymers using atomic force microscopy (AFM) showed that individual protofilaments can fragment and re-anneal on a surface [55]. Also, individual protofilaments were found to have tendency to form bundles in the presence of GTP and singular curved protofilaments were observed in the presence of GDP-AlF3 [55]. This observation might have implications on the assembly dynamics of FtsZ ring on the inner cell membrane during bacterial cell division.
The E. coli FtsZ-Q47K mutant, which does not support cell division [44], formed bundles and rings in yeast cytoplasm [56]. They found that the double mutant E. coli FtsZ-Q47K-D86K, which carries a lateral mutation (indicated in bold letters) formed long linear cables but did not assemble into a ring. In a separate observation, the authors have reported that FtsZ cables laterally assemble to form bundles in yeast cytoplasm. Thus, these observations indicated that the lateral contacts in FtsZ are important in vivo for FtsZ polymeric ring formation. Interestingly, it was found that the C-terminal unstructured tail, or region equivalent to it, is completely dispensable for in vitro and/or in vivo polymerisation of E. coli FtsZ [57, 58], P. aeruginosa FtsZ [30], and B. subtilis FtsZ [34].
In in vivo studies, FtsZ rings constituted by FtsZ polymers have been observed at the mid-cell site using fluorescence microscopy [1, 2] and super resolution 3D-structured illumination microscopy (3D-SIM) [59]. In vivo studies in E. coli [60] and B. subtilis [61] have shown oscillation of FtsZ in a helical pattern throughout the length in the presence of FtsZ ring in E. coli [60] and later constricting to the median as the Z-ring form in B. subtilis [61].
Like in the case of FtsZ from other bacterial systems [50, 51, 52], M. tuberculosis FtsZ exists as a dimer [62] and the assembly process leading to polymerisation has also been found to involve a dimer nucleus [63]. Like in the case of FtsZ from other bacterial systems, the T7 loop of one monomer is supplied in trans to the GTP-binding pocket of the next, thus forming the active site for GTP hydrolysis [64]. In a FRET-based system, M. tuberculosis FtsZ was found to take about 60–100 s to reach polymerisation saturation, about 10 times slower compared to E. coli FtsZ [63]. At steady state also, subunit turnover and GTPase activity were about 8–10 times slower than those of E. coli FtsZ [63]. FRAP experiments showed that M. tuberculosis FtsZ has a slower recovery than E. coli FtsZ in yeast cytoplasm also [56], which is consistent with the slower polymerisation of M. tuberculosis FtsZ [65]. Thus, the FtsZ polymerisation and assembly dynamics of FtsZ of M. tuberculosis, which divides once in 18–24 h [9, 10], are much slower than those of the FtsZ of E. coli, the different strains of which divide once in 18–55 min [6].
The N-terminal domain of the FtsZ proteins of all mycobacterial species is highly conserved (Figure 1). In spite of this conservation, conspicuous drastic differences exist at specific amino acid locations. A typical example is the glaring presence of T172 in M. leprae FtsZ as the lone exception in lieu of A172 in the FtsZ proteins of all the other mycobacterial species. M. leprae FtsZ was found to be polymerisation-lethargic in vitro, even in the presence of DEAE-dextran and under a variety of other conditions [66]. However, interestingly, change of T172 to A172, as it exists in M. tuberculosis FtsZ, showed dramatic polymerisation as in the case of M. tuberculosis FtsZ [66]. Conversely, the reciprocal replacement of A172 with T172 in M. tuberculosis FtsZ, as it exists in M. leprae FtsZ, completely abolished the polymerisation potential of M. tuberculosis FtsZ. These observations showed the crucial nature of A172 residue for polymerisation of FtsZ of all mycobacterial species that have generation time of 24 h or less. On the contrary, the change of A172 to T172, exclusively in M. leprae FtsZ, seems to be an evolution-driven modification, probably to dramatically tone down FtsZ polymerisation rate, as M. leprae divides only once in 13.5 days [11]. The positioning of T172, which can wield such dramatic influence on FtsZ polymerisation, is well justified to be in the N-terminal domain that is known to be important for polymerisation, as found in E. coli FtsZ [21, 57]. Modelling of M. leprae FtsZ and M. tuberculosis FtsZ showed that probably the presence of Thr, which is a hydrogen-bonding and branched residue, at 172 position in the T6 loop might have imposed rigidity on the T6 loop-H10 helix-T7 loop segment and thereby affecting the polymerisation potential of M. leprae FtsZ [66]. Conversely, at position 172, the presence of the non-branched residue Ala that does not engage in hydrogen bonding might not impose rigidity on the T6 loop-H10 helix-T7 loop, thereby facilitating polymerisation of M. tuberculosis FtsZ. The link of T6 loop, which contains the 172 residue, to the T7 loop, which might be crucial for polymerisation in vitro [66], via the H10 helix, supports this possibility. In view of these observations, it is needless to state that there have to be cellular factors that might enable polymerisation of M. leprae FtsZ in a very slow manner suiting the slow generation time of the bacterium when M. leprae divides inside human cells. The D84 and D94 of M. tuberculosis FtsZ are equivalent to D86 and D96 of E. coli FtsZ, respectively [42], and both the residues form salt bridge at the dimer interface [62]. The D94 and N22 form salt bridge with R181 and E136, respectively, in the other subunit [62]. Such interactions might be necessary for the protofilament association during M. tuberculosis FtsZ polymerisation as found necessary for the E. coli FtsZ polymerisation [42]. While G103 was found to be involved in GTP binding [67], C155 has also been found to play an important role in the assembly of M. tuberculosis FtsZ into protofilaments during polymerisation [68].
Homology comparison of the amino acid sequences in the N-terminal domain of FtsZ of various mycobacterial species.
The C-terminal stretch of FtsZ protein has been found to have structural and functional roles in different bacterial systems (reviewed in [4, 5]). The crystal structure of M. tuberculosis FtsZ did not contain the co-ordinates for the 66 residues at the C-terminal portion [62], indicating its unstructured nature like in the case of the C-terminal stretch of the full-length FtsZ of M. jannaschii [16], Pseudomonas aeruginosa [22], Aquifex aeolicus [22], Thermococcus maritima [21], and B. subtilis [31]. On the tubulin dimer template, the molecular model for the complete structure of M. tuberculosis FtsZ, inclusive of the extreme C-terminal 66 residues [69], the co-ordinates of which were not visible in the crystal structure of M. tuberculosis FtsZ [62], also showed an unstructured C-terminus [69].
Like in the case of E. coli FtsZ and B. subtilis FtsZ, the C-terminal region of the FtsZ proteins of all the mycobacterial species contain charged residues towards the C-terminal end. FtsZ of diverse mycobacterial species also shows wide variations in the nature of the residues in the C-terminal region (Figure 2). These residues show high levels of divergence from the residues on the FtsZ of B. subtilis (Gram-positive) and E. coli (Gram-negative) and even from the FtsZ of Streptomyces coelicolor, which is also an Actinobacteria like mycobacteria. Many mycobacterial species have an insertion of GGIAD in the C-terminal variable region, the function of which is under investigation.
Homology comparison of the amino acid sequences in the C-terminal domain of FtsZ of various mycobacterial species.
The M. tuberculosis FtsZ dimer model showed a possible role for the C-terminal Arg residues (R378 and R379) on the stability of the dimer and hence on the polymerisation of the protein. In fact, biochemical studies showed that the deletion of the extreme C-terminal residues, R378 and R379 in the C-terminal extreme stretch of 373PPFMRR379 of M. tuberculosis FtsZ, completely abolished polymerisation in vitro [69]. Besides the deletion of R379, the deletion of the R378 or its replacement with Lys, His, Ala, or Asp completely abolished polymerisation activity, indicating the crucial nature of the residues in the unstructured region of the protein for polymerisation. However, the polymerisation potential of the protein was not affected by the deletion of the single R379 residue alone [69]. The C-terminus of most of the mycobacterial FtsZ ends with 377MRR379 and very few with 377MRH379, with the conservation of M377 (Figure 2).
M. tuberculosis FtsZ has been found to interact with M. tuberculosis FtsW in vitro [70] and in vivo [71]. The presence of three of the four aspartate residues, D367 to D370, was found to be critically required for the interaction of M. tuberculosis FtsZ with a cluster of positively charged residues in the C-terminal tail of M. tuberculosis FtsW in vitro [70] and in vivo [71] (Figure 2). Similarly, PknA-dependent phosphorylation of T343 in M. tuberculosis FtsZ is required for FtsZ function during oxidative stress [72]. Like in the case of FtsZ-SepF interaction in B. subtilis [35, 36], the C-terminal tail of M. tuberculosis FtsZ was found to be required for the interaction of SepF with FtsZ [73]. These and other studies showed that M. tuberculosis SepF is found to be an essential part of the mycobacterial cell division machinery [74], probably in assisting FtsZ localisation at the mid-cell site [73].
Like in the case of the mutations D212A\\N\\C, D212G (FtsZ2) in E. coli FtsZ, which impair GTPase activity [42, 43, 44], the equivalent mutant D210G of M. tuberculosis FtsZ also showed impaired GTPase activity [67] (Figure 3). Although M. tuberculosis FtsZ-D210G polymerised in vivo to the mid-cell ring in a merodiploid Mycobacterium smegmatis background, it failed to act as the sole source of FtsZ in vivo and showed 100-fold impaired GTPase activity in vitro [67]. The authors suggested that the mutation on the T7 loop might not prevent FtsZ self-association or association with the wild-type protein but rather specifically affect GTP hydrolysis, thereby uncoupling GTPase property from polymerisation. The specific residues, which have been found to be involved in the polymerisation or interaction of M. tuberculosis FtsZ with itself or with other cell division proteins, are listed in Table 1.
Homology comparison of the amino acid sequences at other parts of FtsZ of various mycobacterial species.
Residue in M. tuberculosis FtsZ | Interacting protein of M. tuberculosis | Functional role of the interaction | Reference |
---|---|---|---|
N22 | E136 of FtsZ | Polymerisation | [62] |
D94 | R181 of FtsZ | Polymerisation | [62] |
G103 | FtsZ | GTP binding | [67] |
C155 | FtsZ | Polymerisation | [68] |
A172 | FtsZ | Polymerisation | [66] |
D210 | FtsZ | GTPase | [67] |
T343 | PknA | FtsZ function* | [72] |
D367-D370 | FtsW | FtsZ function* | [70] |
R378, R379 | FtsZ | Polymerisation | [69] |
C-terminal tail | SepF | FtsZ function* | [73] |
Residues in M. tuberculosis FtsZ that interact with residues in another subunit or with other cell division proteins.
The exact biochemical activity of the FtsZ, in which these residues participate, is not known.
Mycobacterium leprae is one of the slowest growing bacteria with a generation time of 13.5 days in vivo [11]. The generation time of M. tuberculosis is 18 h in vivo [9] and 24 h in vitro [10]. Similarly, while M. smegmatis divides once in 3 h [8], S. coelicolor A3(2), which is classified under Actinobacteria like mycobacteria, has a generation time of 2.31 h, except that some of the strains grow as slow as 28.9 h depending upon growth conditions [75]. Meanwhile, different strains of E. coli show generation time of only 18–55 min [6]. In agreement with the slower generation time of M. tuberculosis, compared to that of E. coli, M. tuberculosis FtsZ showed slower polymerisation in vitro [63, 65]. Interestingly, the FtsZ of M. leprae, which divides once in 13.5 days [11], did not polymerise at all in vitro even in the presence of DEAE-dextran [66]. Comparatively, the time taken by M. smegmatis and S. coelicolor to reach steady state of FtsZ polymerisation is about 4 min [76]. Similarly, the time taken by the FtsZ of Caulobacter crescentus, which has a generation time of 3 h [77], to reach steady state of polymerisation is 5 min [78]. On the contrary, while E. coli FtsZ takes only 1–6 s to reach steady state of polymerisation [18, 79], FtsZ of B. subtilis, which divides once in 120 min [7], takes approximately 200 s [47]. These observations on the comparative polymerisation kinetics of FtsZ proteins of E. coli, M. tuberculosis, S. coelicolor, M. smegmatis, and M. leprae and the generation time of the respective bacterium (Table 2) probably allude to the existence of a correlation between the generation time of the bacterium and the time taken by the respective FtsZ to reach steady state kinetics in vitro, which may hold true in vivo as well. Mechanistically, such a correlation needs to be necessarily in-built so that kinetics of FtsZ polymerisation during division of the cell keeps pace with the overall generation time of the bacterium. The presence of A172 in the FtsZ of all mycobacterial species, except in M. leprae where it is T172, is a typical example for such a correlation. The homology comparison of mycobacterial FtsZ sequence vividly shows the existence of several such minor differences in terms of specific amino acid residues at crucial positions that may play significant role in conferring differences in the polymerisation kinetics of the FtsZ protein of the respective bacterium. Structure-function studies on the polymerisation kinetics of a large number of FtsZ molecules from diverse bacterial genera differing widely in their generation time might establish such correlation.
Bacterium | Time taken to reach steady state of FtsZ polymerisation (from light scattering data)a [with reference] | Generation time of the bacteriuma [with reference] |
---|---|---|
M. smegmatis | ˜4 min [76] | 3 h [8] |
S. coelicolor | ˜4 min [76] | 2.31 h [75] |
M. tuberculosis | ˜6–10 min [65] | 24 h [10] |
M. leprae | NDb [66] | 13.5 days [11] |
E. coli | ˜1–6 s [18] | 18–55 min [6] |
B. subtilis | ˜200 s [47] | 120 min [7] |
C. crescentus | ˜3 min [78] | 3 h [77] |
Correlation between the time taken to reach FtsZ polymerisation steady state and generation time of the bacterium.
The studies from which the respective values were taken are given in the parenthesis.
Not determined as M. leprae FtsZ does not polymerise in vitro [66].
FtsZ being the principal essential cytokinetic protein in bacterial systems, it has been examined as a potential target for the design of inhibitory compounds that could be used as anti-bacterial drugs against diverse pathogenic bacteria (reviewed in [80]). Although the overall sequence conservation between FtsZ and β-tubulin is only 10–18% except at two stretches [15], it is important to ensure that anti-FtsZ compounds do not inhibit β-tubulin in humans. Owing to the overall conservation of the three-dimensional structure of FtsZ proteins from diverse bacterial genera, developing broad spectrum antibiotics, which are equally effective against the FtsZ of pathogenic bacteria of diverse genera, by designing inhibitor against the FtsZ molecule of a single bacterium, seems to be an attractive possibility [81].
The vast regions of homology and overall conservation of the three-dimensional structure of FtsZ proteins of diverse bacterial systems may seem attractive to design common inhibitors directed against these regions expecting them to be effective against the FtsZ proteins. However, the structure-function studies on FtsZ revealing the subtle differences among the primary, secondary, and tertiary structures of FtsZ proteins from diverse bacteria give the hint that an anti-FtsZ inhibitor designed against the FtsZ of a select pathogenic bacterium may not act effectively with the same MIC/MBC against the FtsZ proteins of other pathogenic bacteria, as found [81]. Thus, the design of a common inhibitor that may be expected to act against FtsZ proteins from diverse pathogens remains a big challenge in the development of inhibitors against bacterial cytokinetic protein, FtsZ. Secondly, identification of the residues contributing to the structure that is required for the generation of force by FtsZ for cell wall/membrane constriction [82] during the physical division of the mother cell also remains a challenge for future studies.
The essential cytokinetic protein, FtsZ, of different Mycobacterial species and of other bacteria has evolved to possess specific amino acid residues at crucial positions on the protein to suit the polymerisation kinetics that befit cell division duration. Thus, each FtsZ protein is unique in terms of the specific types of amino acid residues at crucial positions on the protein in spite of the regions of homology and overall conservation of the three-dimensional structure. It is these unique differences in the residues at specific crucial positions on the FtsZ protein that confer subtle differences on the FtsZ structure and hence on the cytokinetic function of the protein in the respective bacterium.
The work described in this review was supported by funding from the Department of Biotechnology, Govt. of India (BT/R&D/15/35/94, BT/PR7790/BRB/10/500/2006 and BT/PR3787/MED/29/649/2012) to PA and infrastructure support from the DST-FIST, UGC Centre for Advanced Study in Molecular Microbiology, DBT-IISc Partnership Programme, ICMR Centre for Advanced Study in Molecular Medical Microbiology, and Indian Institute of Science. The authors apologise for missing citation of anyone’s work on FtsZ relevant to this review due to any inadvertent error and/or space constraints.
The eight-by-eight spacetime matrix operator
The partial derivative symbols are defined by the following:
The imaginary quantity i represents the square root of minus one, and the physical quantity c corresponds to the speed of light in free space.
Eight compact matrix equations are listed in Table 1, each containing the spacetime matrix operator
Compact matrix equation | Compact matrix equation description |
---|---|
Maxwell spacetime matrix equation for free space | |
Maxwell matrix equation with charges and currents | |
Charge continuity and electromagnetic wave equations | |
Lorentz conditions and electromagnetic potentials | |
Electromagnetic potential wave equations | |
Dirac spacetime matrix equation for free space | |
Klein-Gordon spacetime matrix equation for free space | |
Generalized spacetime matrix equation for free space |
Compact matrix equations where the spacetime matrix operator
The spacetime matrix operator
The four eight-by-eight matrices
Each matrix
These matrices satisfy the anti-commutation relation
Each matrix
In addition
The symbol
Some authors use the
The Maxwell field equations play a fundamental role in both classical electrodynamics and physical optics. The propagations of electromagnetic waves through free space (see [4], pp. 514–522), nonconducting media (see [3], pp. 295–309), thin-film optical filters [5], and solid-state crystalline materials [6] are just a few examples where the Maxwell field equations play an important role.
An earlier eight-by-eight matrix representation of the Maxwell field equations was first introduced by the authors back in 1993 [7]. An improved updated version using the spacetime matrix operator
The compact matrix form of Eq. (9) is given by
The wave function
The Maxwell spacetime matrix equation (9) when expanded is equivalent to two divergences and two curl equations, namely,
We recognize these four equations as the traditional Maxwell field equations (Gaussian units) for free space in the absence of charges, currents, and ordinary matter terms (see [8], pp. 362–368).
For electromagnetic waves, time-harmonic plane-wave solutions of the form
will next be substituted back into the previous four vector equations. This yields the following set of equations:
The quantities k and
These properties represent transverse electromagnetic waves. We also obtain the important results
and
The quantities k, f, and
The Maxwell spacetime matrix equation, with the addition of charge and current terms [1], is given by
The compact matrix form of the Maxwell spacetime matrix equation is given by
Eq. (19), when expanded, is equivalent to two divergences and two curl equations. The resulting four vector equations are referred to as the microscopic Maxwell field equations (see [8], pp. 283–290). They are given by
The various scalar and vector quantities appearing in the microscopic Maxwell vector equations are the electric field vector
Charge continuity equations for electric (see [8], p. 15) and magnetic charges as well as the electromagnetic wave equations involving electric and magnetic charges and currents may be easily obtained by simply multiplying both sides of the Maxwell spacetime matrix equation in compact form (20) by the spacetime matrix operator
Expanding this single matrix equation yields the charge continuity and electromagnetic wave equations:
By using the spacetime matrix operator
The compact matrix form of Eq. (26) is given by
The ket vector
The new scalar and vector quantities appearing in the above equations are the electric vector potential
It is well-known that the electromagnetic vector and scalar potentials satisfy wave equations (see [9], pp. 179–181). This can be easily shown by multiplying both sides of Eq. (27) by the spacetime matrix operator
Next replace the term
Expanding this single matrix equation yields eight partial differential equations which can be easily combined to form the following four potential wave equations:
The single compact matrix (Eq. (31)) is therefore equivalent to these four potential wave equations.
The nonrelativistic Schrödinger wave equation (see [10], pp. 143–146) plays a fundamental role in quantum mechanical phenomena where the spin property of nonrelativistic particles may be ignored. This equation is usually first met in modern physics textbooks. However, when a particle with half-integer spin and/or moving at relativistic speeds is involved, the relativistic Dirac equation [11] comes into play.
Using the spacetime matrix operator
The compact matrix form of Eq. (34) is given by
The wave function
Here
The Dirac spacetime matrix equation (34) when expanded is equivalent to eight partial differential equations. These eight equations can be rewritten as two divergence and two curl equations [1], namely,
We refer to these equations as the Dirac spacetime vector equations for free space. It is noted that these equations resemble the four Maxwell field equations for free space in the absence of charge, current, and ordinary matter terms.
The simplest solutions of these vector equations are time-harmonic plane-wave solutions of the form
The quantities p and E correspond to the linear momentum and the total energy of the associated matter-wave particle; r and t represent the position vector and the instantaneous time. For particles with nonzero rest mass
The quantities
From the previous equations we find the three vectors
These properties represent transverse waves. In addition, we also obtain the important result:
The magnitudes of the vectors
The
The Klein-Gordon equation (see [12], pp. 118–129) is yet another quantum mechanical relativistic equation which is the field equation of the quanta associated with spin-less (spin-0) particles. An example of a spin-less particle is the recently discovered Higgs boson.
A version of the Klein-Gordon equation can be easily derived by simply starting with the compact matrix form of the Dirac spacetime matrix equation for free space, namely, Eq. (35). Multiply both sides by the spacetime matrix operator
Next replace the term
We refer to this equation as the Klein-Gordon spacetime matrix equation for free space. Using the fourth property of the spacetime matrix operator
Therefore, the vectors
In this section, we will introduce for the first time a new matrix equation where again the spacetime operator
The number of unanswered questions and mysteries regarding the universe from the smallest to the largest, in the fields of physics and astronomy, is unimaginable. There are many references, too numerous to list here, which address this topic. However, an excellent comprehensive list of unsolved problems in physics appears in [13] for various broad areas of physics. These areas include general physics, quantum physics, cosmology, general relativity, quantum gravity, high-energy physics, particle physics, astronomy, astrophysics, nuclear physics, atomic physics, molecular physics, optical physics, classical mechanics, condensed matter physics, plasma physics, and biophysics. The following is a partial list of some of the most important questions and mysteries being addressed today by physicists and astronomers around the globe:
How did the universe begin and what is the ultimate fate of the universe?
Is the universe infinite or just very big?
Why is there more matter than antimatter in the universe?
What came before the big bang?
Why are the galaxies distributed in clumps and filaments?
Are there additional dimensions?
Is spacetime fundamentally continuous or discrete?
How can we create a quantum theory of gravity?
What is dark energy and dark matter?
Do dark gravity, dark charge, and dark antimatter exist?
What happens inside a black hole and do naked singularities exist?
Why does time seem to flow only in one direction?
Is time travel really possible?
Is string theory or M-theory a viable theory of everything?
What kind of physics underlies the standard model?
Are there really just three generations of leptons and quarks?
Do gravitons exist?
Are protons unstable?
Do magnetic monopoles exist?
What are the masses of neutrinos?
Do the quarks or leptons have any substructure?
Do tachyons exist and can information travel faster than light?
Why do the particles have the precise masses they do?
Do fundamental physical constants vary over time?
Why are the strengths of the fundamental forces what they are?
Do parallel universes exist and is there a multiverse?
Was our spatially 3-D universe formed out of a vacuum by a 2-D hologram?
Was the hologram formed by a flow of information? If so, what form?
Does pair production formed, spontaneously, out of a vacuum?
Are they likewise formed out of a flow of information?
Do life processes, such as ion flows through cell membranes, form likewise as flows of information?
As we can see, even with all of the discoveries made over the past several hundred years, there is so much we do not understand and so much yet to be discovered about our universe and possibly beyond.
So far we have described the first seven compact matrix equations listed in Table 1 where the spacetime matrix operator
We define the generalized spacetime matrix equation for free space by the following equation:
The compact matrix form of Eq. (49) is given by
This is the eighth compact matrix equation in Table 1. Note the similarity between the generalized spacetime matrix equation for free space and the Dirac spacetime matrix equation for free space (34) when
In Eq. (49), we no longer restrict elements (4,1) and (8,1) to be equal to zero. The wave function
Our primary goal now is to determine the properties of time-harmonic plane-wave solutions satisfying the generalized spacetime matrix (Eq. (49)) for free space. The approach we will take is to cast Eq. (49) into an eigenvalue equation and use the methods of linear algebra to determine the set of orthonormal eigenvectors and corresponding eigenvalues satisfying this eigenvalue equation. (For an excellent book on linear algebra and the solution of eigenvalue equations; see [14], pp. 189–190.) For now let
We first multiply Eq. (49) by the factor
The compact matrix form of this equation is given by
This equation has the same identical form as the nonrelativistic Schrödinger equation (see [12], pp. 118–129). However, the Hamiltonian matrix operator
For time-harmonic plane-wave solutions, the ket vector
Again the quantities p and E correspond to the linear momentum vector and the total energy; r and t represent the position vector and the instantaneous time. After substituting the eight-by-one ket vector
We will refer to Eq. (54) as the eigenvalue spacetime matrix equation. The compact matrix form of Eq. (54) is represented by
The eight-by-eight matrix H is Hermitian which implies the eigenvalues E are real (see [14], p. 222). The following equations define various quantities appearing in Eq. (54):
The quantity p is the magnitude of the linear momentum vector p, and
Without loss of generality, let us consider matter-wave propagation along the +z direction, that is,
Eq. (54) reduces to the following simplified form:
where
The matrix in Eq. (58) is an eight-by-eight square matrix. A compact matrix version of Eq. (58) may be expressed as follows:
At this point we are now in a position to determine eight eigenvectors
From the special theory of relativity (see [10], pp. 21–25), the following relations may also be of use:
As before,
The symbol
+ | + | + | + | |||||
0 | + | 0 | + | 0 | 0 | 0 | 0 | |
+ | 0 | + | 0 | 0 | 0 | 0 | 0 | |
0 | 0 | 0 | 0 | + | 0 | + | 0 | |
0 | 0 | 0 | 0 | 0 | + | 0 | + | |
0 | + | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | + | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | + | ||
0 | 0 | 0 | 0 | 0 | + | 0 |
Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when
The constants
Inspection of the contents of Table 2 reveals the following important results:
For wave propagation in the +z direction, the transverse waves have eigenvector solutions
On the other hand, for wave propagation in the +z direction, the non-transverse waves have eigenvector solutions
The authors, in their most recent publication [1], indicated solutions of their Dirac spacetime matrix equation for free space could be mapped into solutions satisfying the traditional Dirac matrix equation. We wish to explore this in greater detail. The traditional Dirac equation, in the absence of electromagnetic potential terms, is given by
This equation corresponds to the special case employing the Dirac representation (see [12], pp. 694–706) for details. The compact matrix form of Eq. (65) is given by
The Dirac matrix operator
Substituting this time-harmonic plane-wave solution back into the traditional Dirac equation (65) ultimately leads to the corresponding eigenvalue equation:
For the special case of wave propagation in the +z direction, the preceding eigenvalue equation reduces to the following simplified form:
Again using the matrix software MATLAB, the four orthonormal eigenvectors and corresponding eigenvalues satisfying Eq. (69) are listed in the Table 3.
Eigenvalues and orthonormal eigenvectors associated with the traditional Dirac equation for wave propagation in the +z direction when
The quantities a and b appearing in Table 3 are defined by
Note, the quantities a and b appearing in the traditional Dirac equation eigenvectors listed in Table 3 are the same a and b quantities appearing in the generalized spacetime matrix equation eigenvectors listed in Table 2 for
For the special case of a matter wave traveling through free space in the + z direction, we found the orthonormal set of eigenvectors and corresponding eigenvalues, for both the generalized spacetime matrix (Eq. (49)) and the traditional Dirac equation (65), when
The compact matrix form of Eq. (71) is given by
When we substitute each the eight eigenvectors
The four transverse eigenvectors in Table 2 map into the four eigenvectors in Table 3:
The four non-transverse eigenvectors in Table 2 map into the same four eigenvectors in Table 3:
Therefore, whether we use the four transverse eigenvector solutions or the four non-transverse eigenvector solutions satisfying the generalized spacetime matrix (Eq. (49)), the same four eigenvector solutions satisfying the traditional Dirac equation (65) are obtained using Eq. (71). It is noted the four transverse eigenvector solutions could have been obtained from the four Dirac vector equations (37) and (38).
For the special case of wave propagation in the +z direction, when
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
Eigenvalues and orthonormal eigenvectors associated with the generalized spacetime matrix equation for wave propagation in the +z direction when
The constants a and b appearing in Table 4 are now defined by
Inspection of the contents of Table 4 reveals the following important results:
For wave propagation in the +z direction, the transverse waves have eigenvector solutions
On the other hand, for wave propagation in the +z direction, the non-transverse waves have eigenvector solutions
The eigenvectors and eigenvalues associated with the generalized spacetime matrix equation, for the special case of a time-harmonic plane-wave propagating in free space in the +z direction, have been determined for both
For the case when
For the case when
Therefore, for the case when
For the case when
For the case when
The generalized spacetime matrix equation for
In the de Broglie-Bohm picture of quantum mechanics, Hardy [16] and Bell [17] suggest empty waves represented by wave functions propagating in spacetime, but not carrying energy or momentum, can exist. This same concept was called ghost waves or ghost fields by Albert Einstein (see [18]). The controversy as to whether matter waves correspond to real waves or ghost waves has been and is still a subject of debate and controversy.
In Section 5.1, we mentioned that the number of unanswered questions and mysteries regarding the universe from the smallest to the largest, in the fields of physics and astronomy, is unimaginable. Allowing the elements
For relativistic quantum mechanics—matter waves:
What class of particles do the transverse eigenvectors represent?
Do the transverse eigenvectors represent real or ghost waves?
What class of particles do the non-transverse eigenvectors represent?
Do non-transverse eigenvectors represent real or ghost waves?
Are the transverse and non-transverse eigenvectors equivalent in some way?
For classical electrodynamics—electromagnetic waves:
What can be said about those waves propagating with speed -c?
Do these represent a new type of electromagnetic wave?
What can be said about those waves having a longitudinal component?
What can be said about those waves having a fourth component?
Could these be associated with undiscovered electromagnetic waves?
And two last questions:
Why do the
Does the spacetime matrix operator
The four classical electromagnetic microscopic Maxwell field equations have been rewritten as a single matrix equation, referred to as the Maxwell spacetime matrix equation, using the spacetime matrix operator
The square eight-by-eight matrix operator
The traditional relativistic Dirac equation for free space has been expressed as a new matrix equation, referred to as the Dirac spacetime matrix equation for free space, using the same spacetime matrix operator
Solutions of the new Dirac spacetime matrix equation can be easily transformed into solutions satisfying the traditional relativistic Dirac equation using the linear transformation matrix Z.
The Dirac spacetime matrix equation is equivalent to four new relativistic quantum mechanical vector equations. We referred to these equations as the Dirac spacetime vector equations. In the absence of electromagnetic potentials, these vector equations resemble the four classical electromagnetic microscopic Maxwell field vector equations in the absence of charge and current densities.
Multiplication of the Dirac spacetime matrix equation by the spacetime matrix operator
Four transverse orthonormal eigenvectors as well as the four non-transverse orthonormal eigenvectors satisfying the Dirac spacetime matrix equation map, via the linear transformation matrix Z, into the same set of four orthonormal eigenvectors satisfying the traditional Dirac equation.
A new generalized spacetime matrix equation employing the operator
We are most appreciative of the help by Ms. Trin Riojas of the Optical Sciences Center in coordinating computer station inputs/outputs between authors and publishers. The past informal discussions with Dr. Arvind S. Marathay of the Optical Sciences Center are also greatly appreciated. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Supporting women in scientific research and encouraging more women to pursue careers in STEM fields has been an issue on the global agenda for many years. But there is still much to be done. And IntechOpen wants to help.
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