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Mini-Reviews in Medicinal Chemistry, 2006, 6, 909-920
S-Layer Proteins as Key Components of a Versatile Molecular
Construction Kit for Biomedical Nanotechnology
B. Schuster*, D. Pum, M. Sára and U.B. Sleytr Center for NanoBiotechnology, University of Natural Resources and Applied Life Sciences, 1180 Vienna, Austria Abstract: Surface (S)-layer proteins and S-layer fusion proteins incorporating functional sequences, self-assemble into
monomolecular lattices on solid supports and on various lipid structures. Based on these S-layer proteins, supramolecular
assemblies can be constructed which are envisaged for label-free detection systems, as affinity matrix, as anti-allergic
immuno-therapeutics, as membrane protein-based screening devices, and as drug targeting and delivery systems.
Keywords: Crystalline surface layer proteins, artificial virus, biomimetics, bottom-up strategy, S-layer fusion protein,
microspheres-based detoxification system, nanobiotechnology.
2. GENERAL ASPECTS OF S-LAYER PROTEINS
The cross fertilization of biology, molecular biology, Crystalline bacterial cell surface layers, referred to as S- organic chemistry, material sciences, and physics has opened layers [1-3] have now been identified in hundreds of different up significant opportunities for innovation in previously species of bacteria and represent an almost universal feature unrelated fields. In this context, self-assembly is a new and of archaea (Fig. 1) [for reviews see 2, 4-10]. Since S-layers
rapidly growing scientific and engineering discipline that are composed of a single protein or glycoprotein species crosses the boundaries of numerous existing fields. Self- endowed with the ability to assemble into a monomolecular assembly can be defined as a "bottom-up" process by which lattice during all stages of cell growth and cell division, they individual molecules (ranging in size up to large polymers) can be considered as the simplest type of biological become spontaneously organized into supramolecular membranes developed in the course of evolution [for review structures. This alternative to "top-down" processing steps can lead to both, new materials and structures that are not S-layers can be associated with quite different supporting obtained by conventional techniques, and to the ultimate supramolecular structures. In most archaea, S-layers miniaturization of functional units.
represent the only wall component and can be so closely One of the great challenges for nano(bio)technology is associated with the plasma membrane that a hydrophobic the creation of supramolecular materials in which the domain of the constituent subunits is actually integrated into constituent units are highly regular molecular nanostructures.
the lipid layer [6, 8, 12]. In most Gram-positive bacteria the Thus, learning how to create complex and large supra- S-layer is attached to a rigid wall matrix involving lectin molecular structures and the elucidation of rules mediating binding between a glycan (referred to as secondary cell wall their organization into functional materials will offer a broad polymer, SCWP) covalently-attached to the peptidoglycan spectrum of new technologies.
meshwork [13]. In Gram-negative bacterial cell envelopes S-layers are linked to the lipopolysaccharide component of the It is now well-recognized that crystalline bacterial cell outer membrane. In most prokaryotic organisms S-layers surface layers (S-layers) composed of identical protein- have to be considered as non-conservative structures with the aceous subunits represent unique patterning elements and potential to fulfil a broad spectrum of functions [3, 4, 9].
scaffolding structures for nanobiotechnological applications.
Considering that S-layer carrying organisms are ubiquitous The possibility for incorporating single or multifunctional in the biosphere and even dwell under the most extreme domains to S-layer proteins by genetic engineering has led to environmental conditions, the supramolecular concept of a ultimate control and precision in the spatial distribution and dynamic closed crystalline surface layer could have the orientation of molecules and functional domains as required potential to fulfil a broad spectrum of functions. Because S- for life- and non-life science applications. Most relevant, S- layer lattices possess pores identical in size and morphology layers represent the base for very versatile self-assembly in the 2 to 8 nm range, they work as precise molecular sieves systems involving all major species of biological molecules providing sharp cut off levels for the bacterial cell [14]. As such as proteins, lipids, glycans, nucleic acids, and isoporous ultrafiltration membrane they can apparently combination of that.
provide the microorganisms with a selection advantage byfunctioning as protective coats, molecule and ion traps, andas a structure involved in cell adhesion, surface recognition *Address correspondence to this author at the Center for Nano- or antifouling [5, 11, 12, 15]. In those archaea which possess Biotechnology, University of Natural Resources and Applied Life Sciences S-layers as exclusive envelope component outside the Vienna, Gregor-Mendel-Strasse 33, 1180 Vienna, Austria; Tel: ++43-1-47654-2200; Fax: ++43-1-4789112; E-mail: [email protected] cytoplasmic membrane, the crystalline array acts as a frame 2006 Bentham Science Publishers Ltd.
Mini-Reviews in Medicinal Chemistry, 2006, Vol. 6, No. 8
Schuster et al.
Fig. (1). In (a), freeze-etching preparation of a whole cell of Bacillus sphaericus with a square S-layer lattice is shown. Bar corresponds to
200 nm. Schematic illustration of the supramolecular architecture of the three major classes of prokaryotic cell envelopes containing
crystalline bacterial cell surface layers (S-layers). (b) Cell envelope structure of gram-negative archaea with S-layers as the only component
external to the cytoplasmic membrane. (c) Cell envelope as observed in gram-positive archaea and bacteria. In bacteria the rigid wall
component is primarily composed of peptidoglycan. In archaea other wall polymers (e.g. pseudomurein) are found. (d) Cell envelope profile
of gram-negative bacteria composed of a thin peptidoglycan layer and an outer membrane. If present, the S-layer is closely associated with
the lipopolysaccharide of the outer membrane. Modified after Ref. [5], Copyright (1999) with permission from Wiley-VCH.
work that determines and maintains the cell shape and Contrary to the electron microscopical preparation tech- stabilizes the cytoplasmic membrane [16, 17].
niques, scanning force microscopy allows to investigate S-layer monolayers in their native environment [26-28].
From a general point of view S-layers as the most Contact mode microscopy in liquid is most frequently used abundant of bacterial cellular proteins are important model to investigate S-layer protein monolayers at sub-nanometer systems for studying the structure, synthesis, assembly, and resolution. S-layer proteins are highly susceptible towards function of these proteinaceous components. The investi- the applied tip loading forces which shall not exceed 0.5 to 1 gation of the general principles of S-layers also have nN. Ionic content and strength of the buffer solution in the revealed considerable application potential in biotechnology, liquid cell has to be carefully adjusted in order to minimize biomimetics, and nano(bio)technology [11, 15, 18-21].
electrostatic interactions between tip and sample. Silicon 2.1. Structural Analysis of S-Layer Lattices
wafers and mica are the most commonly used substrates forscanning force microscopical investigations since these High resolution transmission electron microscopy (TEM) provide hard and very flat surfaces. In particular, silicon and scanning force microscopy (SFM) are commonly used to surfaces are most relevant for nanobiotechnological characterize S-layer protein lattices. In particular, in TEM applications. S-layer proteins recrystallize into large scale the appropriate preparation method is most important for monomolecular protein lattices on silicon, whereas S-layer investigating the ultrastructure of S-layer protein lattices at fragments or self-assembly products are preferably deposited molecular resolution (Fig. 1). Freeze-etching and freeze-
on mica. If S-layer proteins are recrystallized on flat solid drying in combination with heavy metal shadowing are the supports such as silicon wafers, lattice formation can be most straight forward approaches for obtaining information followed in real time [26]. It could be demonstrated that about the lattice type and surface structure of S-layers on crystal growth starts at several distant nucleation points and bacterial cells and S-layer cell wall fragments [1, 22]. These proceeds in-plane until a closed layer of crystalline domains studies revealed that many S-layers show a smooth outer and is formed [26]. The scanning force microscope has been also a more corrugated inner face [23, 24]. This difference is of used as a nano-tool for inducing conformational changes in particular importance when the orientation (sidedness due to S-layer proteins [29, 30]. Furthermore, the capability of attachment of the S-layer subunits via the inner or outer scanning force microscopy to resolve molecular details on surface) of S-layers on artificial substrates has to be deter- biological samples together with its force detection mined. Nevertheless, TEM of frozen hydrated specimens sensitivity has led to the development of the so-called [23-25] yields the highest resolution among all microscopical "topography and recognition mode", a method suitable for techniques. In plane, a resolution of 0.35 nm and in the third visualizing the chemical composition of a sample while dimension 0.7 nm is attainable. In three-dimensional TEM, mapping its topography [31]. It is anticipated that the tilt series of the specimen is recorded under low electron simultaneous investigation of both, topography and dose conditions (usually not more than 1 to 2 electrons per recognition, will allow to elucidate the structure-function Å2). Although quantum noise governs image formation, such relationship of a broad spectrum of biological samples in an low electron doses are mandatory in order to maintain the unsurpassed way.
three-dimensional structure of the proteins [25]. Imageprocessing methods are used to enhance the signal-to-noise 2.2. Self-assembly Properties of S-Layer Proteins
ratio in low dose micrographs.
While many archaeal S-layer proteins are covalently Negative staining is an easy preparation technique in anchored, those of bacteria are non- covalently linked to TEM. Particularly in combination with two and three dimen- each other and to the supporting cell wall component. Thus, sional image reconstruction techniques, it allows high reso- a complete solubilization of S-layers into their constituent lution studies of the ultrastructure of S-layer lattices [23-25].
subunits and release from the bacterial cell envelope can be S-Layer Proteins as Key Components
Mini-Reviews in Medicinal Chemistry, 2006, Vol. 6, No. 8 911
achieved by treatment with high concentrations of hydrogen- 25 mol % are charged amino acids, and S-layer proteins bond breaking agents (e. g. urea, guanidinium hydro- possess little or no sulfur-containing amino acids. Secondary chloride), by dramatic changes in the pH-value or in the salt structure predictions of S-layer proteins indicate that about concentration. Upon removal of the disrupting agent, e. g. by 40 % occur as -sheets and approximately 20 % of the amino dialysis, S-layer proteins self-assemble into two dimensional acids are organized as -helices. Most -helical segments arrays [for review see ref. 32]. Such self-assembly products are arranged in the N-terminal part. Aperiodic foldings and may have the form of flat sheets or open-ended cylinders.
-turn content may vary between 5 % and 45 %.
Depending on the particular S-layer protein species used and In order to elucidate the structure-function relationship of on the environmental conditions, monolayers or double distinct segments of S-layer proteins, N- and / or C- layers are formed.
terminally truncated forms were produced and their self- Contrary to the reassembly in solution, prior to assembly and recrystallization properties investigated [44- recrystallization on artificial supports, S-layer proteins must 46]. Another approach was seen in performing a cysteine be kept in a water soluble state. This can either be achieved scanning mutagenesis and screening the accessibility of the in the absence of bivalent cations [33] or by maintaining a single introduced cysteine residue in the soluble, self- sub-critical protein concentration for self-assembly [34]. In assembled and recrystallized S-layer proteins [34]. This addition, in the presence of S-layer-specific SCWPs, the study elucidated which amino acid positions in the primary reassembly in suspension is inhibited, whereas the recrystalli- sequence are located on the outer or inner S-layer surface of zation of soluble S-layer proteins on artificial supports is the subunits, inside the pores, or at the subunit to subunit promoted [13, 34, 35]. The formation of coherent crystalline arrays strongly depends on the S-layer protein species, the The fact that no structural model at atomic resolution of environmental conditions of the bulk phase (e. g. temperature, an S-layer protein is available until now, may be explained pH-value, ion composition and ionic strength) and, in parti- by the molecular mass of the subunits being too large for cular, on the surface properties of the substrate. For example, nuclear magnetic resonance analysis, as well as by the the S-layer protein SbpA of Bacillus sphaericus CCM2177 intrinsic property of S-layer proteins to self-assemble into forms double layers with perfect long range order (up to two dimensional lattices, thereby hindering the formation of several micrometers in diameter) on hydrophilic silicon but isotropic three dimensional crystals as required for X-ray monolayers consisting of 200 to 500 nm sized patches on crystallography. In addition, the low solubility of S-layer proteins is a general hindrance for both methods.
In accordance with S-layer proteins recrystallized on In the case of the S-layer protein SbsC of G . solid substrates, the orientation of the protein arrays stearothermophilus ATCC 12980, water soluble N- or C- (sidedness due to the attachment via the inner or outer terminally truncated forms were used for first three dimen- surface of the S-layer subunits) at liquid interfaces and at sional crystallization studies. Crystals of the C-terminally lipid films is determined by the anisotropy in the physico- chemical surface properties of the protein lattice [for review 31-844 diffracted to a resolution of 3 Å using synchrotron radiation [47]. Native and heavy atom derivative see ref. 36]. For example, the S-layer protein SbsB of data confirmed the results that the N-terminal region is Geobacillus stearothermophilus pv72/p2 reassembles with mainly organized as -helices, whereas the middle and C- its more hydrophobic outer face at the air-water interfaces terminal part of SbsC consist of loops and ß-sheets [47].
while at lipid films with zwitterionic head groups the S-layerlattice is attached with its inner face [37]. The unambiguous The N-terminal region was found to be responsible for determination of the orientation of the S-layer is possible anchoring the S-layer subunits to the underlying rigid cell since it shows oblique lattice symmetry with a characteristic envelope layer by binding to the SCWP. The polymer chains handedness of the proteins. In addition to the formation of are covalently linked to the peptidoglycan backbone which flat S-layer lattices it has also been demonstrated that S-layer occurs most probably via phosphodiester bonds [48].
proteins are able to cover liposomes and nanocapsules Basically, two types of binding mechanisms between the N- completely [38-49]. The S-layer shows facets and numerous terminal part of S-layer proteins and SCWPs have been lattice faults in order to follow the curvature of the spheres.
described [49]. The first one, which involves so-called S- According to the observations with planar lipid films, the layer-homologous (SLH) domains and pyruvylated SCWPs charge of the lipid head groups and the polyelectrolyte [33, 40, 44, 50, 51] has been found to be widespread among determines the orientation of the S-layer protein against the prokaryotes and is considered as having been conserved in liposome and the nanocapsule, respectively [41, 42].
the course of evolution [51]. The second type of bindingmechanism has been described for G. stearothermophilus 2.3. Chemical Properties and Molecular Biology of S-
PV72/p6 and ATCC 12980 [9, 52, 53], a temperature-derived strain variant from the latter [54], and G. stearothermophilus Chemical analyses and genetic studies revealed that the NRS 2004/3a [55]. This binding mechanism involves an S-layer lattices are composed of a single homogeneous SCWP that consists of N-acetyl glucosamine, glucose and protein or glycoprotein species with a molecular mass 2,3-dideoxy-diacetamido-D-mannosamine uronic acid in the ranging from 40 to 200 kDa [5, 9, 11, 43]. Most S-layer molar ratio of 1:1:2 (see compound (1) in Fig. 2) [55, 56]
proteins are weakly acidic with isoelectric points in the range and a highly conserved N-terminal region which does not of 4 to 6 [9]. In general, S-layer proteins consist of a large possess an SLH-domain [52-55]. Concerning the first portion of hydrophobic amino acids (40 - 60 mol %), about binding mechanism, the construction of knock-out mutants Mini-Reviews in Medicinal Chemistry, 2006, Vol. 6, No. 8
Fig. (2). Chemical structure of the repeating unit of the secondary cell wall polymer of G. stearothermophilus NRS 2004/3a (1).
in Bacillus anthracis and Thermus thermophilus in which the parameters of a = 10.4 nm, b = 7.9 nm and a base angle of gene encoding a putative pyruvyl transferase was deleted = 81°, whereas SbpA assembles into a square lattice with a demonstrated that the addition of pyruvic acid residues to the lattice constant of 13.1 nm.
peptidoglycan-associated cell wall polymer was a necessary For generating a universal affinity matrix for binding any modification to bind SLH-domain containing proteins [50, kind of biotinylated molecule, S-layer-streptavidin fusion proteins were constructed [60, 61]. Minimum-sized core 3. A MOLECULAR CONSTRUCTION KIT BASED ON
streptavidin (118 amino acids) was either fused to N- or C- terminal positions of SbsB or to the C-terminal end ofrSbpA31-1068 [45, 61].
The biomimetic approach learning from nature how to create supramolecular, layered structures by a bottom-up The genes encoding the fusion proteins and core process is one of the most challenging scientific tasks in streptavidin were expressed independently in E. coli and nanobiotechnology. Advantage can be taken of the self- isolated from the host cells. To obtain functional hetero- assembling nature of S-layer (fusion) proteins, SCWPs and tetramers (HTs), a refolding procedure was developed by natural and/or artificial lipids and their properties in the subjecting a mixture of fusion protein with excess core compartmentalization of components in nanoscale regions, streptavidin to denaturing and renaturing conditions and production of self-assembling biomaterials, construction of isolating functional HTs by size exclusion and affinity drug-targeting and delivery systems, and development of chromatography. HTs comprising the N-terminal rSbsB- smart biosensors [18, 19, 57-59].
streptavidin formed self-assembly products in suspensionand recrystallized on liposomes and silicon wafers [61],whereas HTs based on the C-terminal rSbpA 3.1. S-Layer Fusion Proteins and their Application
vidin fusion protein showed dirigible self-assembly formation, as lattice formation of SbpA is strongly dependent on the So far, the chimaeric genes encoding several S-layer presence of calcium ions. HTs based on the rSbpA31-1068- fusion proteins have been heterologously expressed in streptavidin fusion protein recrystallized on gold surfaces Escherichia coli. S-layer fusion proteins were based on the that were optionally pre-coated with SCWP [60]. Analysis of S-layer proteins SbsB, SbsC, and SbpA (Table 1). SbsB
negatively-stained preparations of self-assembly products forms an oblique S-layer lattice with p1 symmetry and lattice formed by HTs revealed that neither the oblique S-layer Summary of Various S-Layer Fusion Proteins (Selected from Various Constructs)
S-layer fusion protein
Length of functionality
rSbsB / core streptavidin / core streptavidin major birch pollen allergen rSbpA / Strep-tag I affinity tag for streptavidin IgG-binding domain green fluorescent protein heavy chain camel antibody Mature proteins: Bacillus sphaericus CCM2177 variant A (SbpA): 1238 amino acids (aa); Geobacillus stearothermophilus pv72/p2 (SbsB): 889 aa; Bacillus stearothermophilusATCC 12980 (SbsC): 1099 aa.
S-Layer Proteins as Key Components
lattice of SbsB, nor the square lattice of SbpA had changed monoclonal antibodies that recognize free, as well as PSA due to the presence of the fusion partner. Digital image complexed with alpha-1-anti-chymotrypsin. For application reconstructions of self-assembly products of HTs comprising in a PSA biosensor, VHHs recognizing free and complexed the N-terminal rSbsB-streptavidin fusion protein showed an PSA are desired. Moreover, kinetic requirements in the additional protein mass on the N-terminal SLH-domain biosensor impose a high probe density that can probably only which resulted from the fused streptavidin moiety [61]. As a be obtained with single domain VHHs.
first application approach, monolayers of HTs based on To generate a PSA-specific sensing layer for SPR rSbpA31-1068 were recrystallized on plain gold chips and on measurements, the S-layer fusion protein rSbpA those pre-coated with thiolated SbpA-specific SCWP, and PSA-N7 was recrystallized on gold chips pre-coated with the obtained affinity matrix was used to perform hybri- thiolated SCWP. The formation of the monomolecular dization experiments. In a first step, biotinylated oligo- protein lattice was confirmed by scanning force microscopy, nucleotides (30-mers) were bound to the streptavidin moiety as well as by the level of the measured SPR signal. As of the HTs, and complementary oligonucleotides were derived from response levels measured for binding of PSA to hybridized carrying no or one mismatch [60]. Evaluation of a monolayer consisting of rSbpA the hybridization experiments was performed by applying 31-1068 /cAb-PSA-N7, the molar ratio between bound PSA and the S-layer fusion protein was 0.78, which means that at least three PSA which combines the advantages of the high optical field molecules were bound per morphological unit of the square intensities of surface plasmon waves with the sensitive S-layer lattice with an area of 170 nm2. To summarize, by detection of fluorescence light emission. For hybridization using SbpA-specific SCWP as biomimetic linker to gold experiments on monolayers generated by recrystallization of chips, a sensing layer for SPR could be generated by HTs on gold chips pre-coated with thiolated SCWP, fluo- recrystallization of this S-layer fusion protein. Due to the rescently labelled oligonucleotides carrying one mismatch crystalline structure of the S-layer lattice, the fused ligands were used. The fluorescence intensity increased linearly at showed a well defined distance in the protein lattice, and the beginning of the hybridization reaction, so that the linear according to the selected fusion site in the S-layer protein, slope of the increase in the fluorescence intensity plotted they were located on the outermost surface, which should versus the concentration of the hybridizing oligonucleotides reduce diffusion limited reactions. A further advantage can led to a linear correlation [60]. In a different set of be seen in the constant and low distance of the ligands from hybridization experiments which were performed on the optically active gold layer, which is exclusively monolayers generated by direct recrystallization of HTs on determined by the thickness of the S-layer and lies in the plain gold chips, the concentration of oligonucleotides range of only 10 to 15 nm. Thus, S-layer fusion proteins carrying one mismatch was step-wise increased. The incorporating camel antibody sequences can be considered as Langmuir isotherm which indicated that oligonucleotides in key element for the development of label free detection solution were in equilibrium with those bound to the systems such as SPR, surface acoustic wave, or quartz monolayer carrying the biotinylated oligonucleotides could crystal microbalance, in which the binding event can be be established from the obtained fluorescence intensities measured directly by the mass increase without the need of [60]. The detection limit was found to be 1.57 pM on any labelled molecule.
monolayers generated by recrystallization of HTs on goldchips pre-coated with thiolated SCWP, whereas on plain The sequence encoding rSbpA31-1068 was also used as gold chips, the detection limit was determined to be at least base form for the construction of an IgG-binding fusion 8.2 pM. To conclude, the hybridization experiments protein [64]. As fusion partner, the sequence encoding the Z- indicated that a functional sensor surface could be generated domain, a synthetic analogue of the IgG-binding domain of by recrystallization of HTs on gold chips, which could find Protein A from Staphylococcus aureus, was used. To numerous applications in (nano)biotechnology.
generate the S-layer fusion protein, the 5´-end of thesequence encoding two copies of the Z-domain was fused via An S-layer fusion protein comprising the C-terminally a short linker to the gene encoding rSbpA truncated form rSbpA 31-1068 and the variable region of a heterologous expression in E. coli, the S-layer fusion protein heavy chain camel antibody directed against lysozyme was was isolated from the host cells, purified by size exclusion constructed. The Camelidae is the only taxonomic family chromatography under denaturing conditions, dialysed and known to possess functional heavy chain antibodies lacking recrystallized on gold chips which were pre-coated with light chains and the first constant region. These unique thiolated SbpA-specific SCWP. As shown by scanning force antibody isotypes interact with the antigen by virtue of a microscopy, a monomolecular protein lattice with square single variable domain, termed VHH. A single VHH domain symmetry was formed. Native monolayers or monolayers has a molecular mass of only 15,000 and is the smallest cross-linked with the bifunctional imidoester dimethyl- known complete antigen binding fragment from a functional pimelimidate (DMP, compound (2)) (Fig. 3) were finally
immunoglobulin. As proof-of-principle was provided with a exploited for binding of human IgG. The amount that could fusion protein comprising a VHH directed against lysozyme be bound by the native monolayer was 2.9 x 10-5 nM or 4.35 [62], an S-layer fusion protein incorporating the sequence of ng IgG / mm2, whereas in the case of the DMP-cross-linked a variable domain of a heavy chain camel antibody (cAb- monolayer (Fig. 3) 2.8 x 10-5 nM or 4.20 ng IgG / mm2 could
PSA-N7) directed against the prostate-specific antigen (PSA) attach. These values corresponded to 65 and 67 % of the was constructed [63]. PSA is a useful marker to screen theoretical saturation capacity of a planar surface for IgG potential prostate cancer patients. The current diagnostic test (6.5 ng / mm2) with the Fab regions occurring in the systems determine the concentration of total PSA with
The major birch pollen allergen Bet v1 shares IgE epitopes with all tree pollen allergens from closely related species (e. g. alder hazel, hornbeam, beech). Because of highsequence identities among these allergens and well studied cross-reactions with B-cell epitopes, Bet v1 represents a model allergen. The gene encoding the chimaeric S-layerproteins rSbsC31-920/Bet v1 [64] and rSbpA31-1068/Bet v1 [45] carrying Bet v1 at the C-terminal end were cloned and expressed in E. coli. In a recent study, the applicability of rSbsC31-920/Bet v1 as a novel approach to design vaccines cross-linked S-layer protein rSbpA (3)
with reduced allergenicity in combination with strongimmune-modulating capacity for immunotherapy of type I allergy could be demonstrated [67]. This fusion proteinexhibited all relevant Bet v1-specific B and T cell epitopes, Fig. (3). Cross-linking of the recombinant S-layer protein rSbpA
but was significant less efficient in releasing histamine than with the homobifunctional imidoester dimethylpimelimidate (DMP; free Bet v1. In cells of birch pollen-allergic individuals, the compound (2)). The spacer arm length of DMP is 0.92 nm.
fusion protein was capable of modulating the allergen-specific Th2-dominated response into a more balanced Th1/ condensed state. As derived from these binding capacities, Th0-like phenotype accompanied by enhanced production of on average 2.7 and 2.6 IgG molecules were bound per IFN- and IL-10. To conclude, rSbsC31-920/Bet v1 could morphological unit of the square S-layer lattice consisting of find application as carrier/adjuvants to design vaccines for four identical subunits of the S-layer fusion protein. For specific immunotherapy of type 1 allergy with improved preparing biocompatible microparticles for the microspheres- efficacy and safety [67].
based detoxification system (MDS) [65] to remove auto-antibodies from patients´ sera suffering from auto-immune The nucleotide sequence encoding enhanced green disease, the S-layer fusion protein was recrystallized on fluorescent protein (EGFP), a red-shifted green fluorescent SCWP-coated, 3 m large cellulose-based microbeads (Fig.
protein (GFP)-derivative possessing a 30 times brighter 4). The MDS is an alternative approach to conventional
fluorescence intensity at 488 nm than wild-type GFP was immunoadsorption systems, in which the plasma does not fused to the 3´end of the sequence encoding the C-terminally perfuse on an adsorption column, but is recirculated into a truncated form rSbpA31-1068 [68]. The chimaeric gene filtrate compartment of a membrane module. The addition of encoding rSbpA31-1068/EGFP was expressed in E. coli, microbeads to the plasma circuit would allow the rapid whereby expression at 28°C instead of 37°C resulted in removal of the pathogenic substrates. In the case of clearly increased fluorescence intensity, indicating that the microbeads that were covered with a native monolayer, the folding process of the EGFP moiety was temperature binding capacity was 1,065 g human IgG / mg S-layer sensitive. Comparison of excitation and emission spectra of fusion protein. For DMP-treated microbeads, a binding rEGFP and rSbpA31-1068/EGFP indicated identical maxima at capacity of 870 g IgG / mg S-layer fusion protein was 488 and 507 nm, respectively. Furthermore, this fusion determined. These values corresponded to 78 or 65 % of the protein was used for recrystallization on silicon wafers theoretical saturation capacity of a planar surface for IgG covered with polyelectrolytes, as well as for coating of having the Fab regions in the condensed state. Bound IgG hollow polyelectrolyte capsules. Fluorescence spectroscopy could be eluted with glycine-HCl buffer at a pH value of 2.2 confirmed that the adsorption of rSbpA31-1068/EGFP on and the microbeads were used for further IgG-binding hollow capsules did not shift the fluorescence emission of experiments [64].
the chromophore [41]. Finally, the recrystallization of this Fig. (4). (a) Schematic drawing of the MDS, showing the primary circuit (labelled 1) containing the whole blood of the patient. The blood
cells are rejected by the plasma filter. In the second circuit (labelled 2), the plasma re-circulates together with the S-layer fusion protein-
coated microbeads, on which IgG is bound. After passing the plasma filter again, the purified plasma is combined with blood cells, and the
whole blood is re-infused into the patient. (b) Scanning electron micrograph of the cellulose-based microbeads used for recrystallization of
rSbpA31-1068/ZZ. (c) Schematic drawing showing the oriented recrystallization of the S-layer fusion protein rSbpA31-1068/ZZ on microbeads
pre-coated with SCWP and binding of IgG to the ZZ-domains. Reprinted with permission from Ref. [64], Copyright (2004) ASM.
fusion protein on liposomes and their application is exploited for covalent binding of functional macromolecules, described in the following section.
like biotinylated antibodies v i a the streptavidin – biotinbridge [40]. These immuno-S-liposomes comprise several 3.2. S-Layer Fusion Proteins on Liposomes as Model for
components with specific functions: the liposome as drug carrier, the antibody as homing device, the S-layer lattice as Biomolecular self-assembly can be used as powerful tool stabilizing structure for the liposome, as anchoring layer for for nanoscale engineering. One well known example is the the antibodies, and most probably as stealth coat for formation of liposomes, which are still very promising supra- prolonged blood circulation times (Fig. 5).
molecular structures for the application in nanobiotechno- To avoid chemical modification reactions and to prevent logy and nanobiomedicine.
diffusion of potentially toxic agents through the lipid bilayer Liposomes are colloidal, vesicular structures based on into the interior of the vesicles, S-layer fusion proteins (phospho)lipid bilayers or on tetraetherlipid monolayers [69] incorporating the sequence of core–streptavidin have been and they are widely used as delivery systems for enhancing constructed. Functional streptavidin HTs were prepared as the efficiency of various biologically active molecules and three of the four binding pockets remained accessible for for the transport of therapeutic agents to the site of disease in binding biotinylated molecules [61]. After recrystallization vivo [70, 71]. Liposomes can encapsulate water soluble of this streptavidin fusion protein on positively charged agents in their aqueous compartment and lipid soluble liposomes, the protein lattice was further functionalized by substances within the lipid bilayer itself [72]. These agents binding biotinylated peroxidase or biotinylated ferritin [61].
include small molecular drugs used in cancer chemotherapy Binding of biotinylated ligands to S-liposomes can be and genetic drugs as plasmids encoding therapeutic genes exploited for enabling receptor-mediated uptake into human [73]. Generally, liposomes release their contents by cells. A further promising application potential can be seen interaction with target cells, either by adsorption, in the development of drug targeting and delivery systems endocytosis, lipid exchange or fusion [71, 74].
based on lipid-plasmid complexes coated with functional In previous studies, wild-type SbsB has been recrystalli- HTs for transfection of human cells.
zed on positively charged liposomes composed of dipalmi- Another interesting approach can be seen in the toylphosphatidylcholine, cholesterol and hexadecylamine generation of a functional chimaeric rSbpA31-1068/EGFP [38-40, 75]. Such S-layer-coated liposomes (S-liposomes) fusion protein to follow the uptake of S-liposomes into with a diameter of 50–200 nm represent simple model mammalian cells [68]. Liposomes coated with a monolayer systems resembling the architecture of artificial virus of rSbpA31-1068/EGFP were incubated with HeLa cells.
envelopes (Fig. 5). For that reason, S-liposomes could reveal
Subsequently, confocal laser scanning microscopy was a broad application potential, particularly as drug delivery applied to investigate the ongoing interaction between the systems or in gene therapy [5].
fluorescently labelled cell membrane and the greenfluorescent S-liposomes. This study demonstrated that mostof the S-liposomes were internalized within 2 hours ofincubation and that the major part entered the HeLa cells byendocytosis [68]. To our knowledge, rSbpA31-1068/EGFP isthe first fusion protein that maintained the ability tofluorescence and to recrystallize into a monomolecularprotein lattice. Due to the intrinsic fluorescence, liposomescoated with rSbpA31-1068/EGFP represent a useful tool tovisualize the uptake of S-liposomes into mammalian cells.
The most interesting advantage can be seen in therecrystallization of fusion proteins incorporating EGFP incombination with HTs on the same liposome surface. In that Fig. (5). (a) Schematic drawing of (1) an S-liposome with
case, it would be possible to simultaneously investigate the entrapped functional molecules and (2) functionalized by uptake of these specially coated S-liposomes by target cells reconstituted integral proteins. S-liposomes can be used as and the functionality of transported drugs without the immobilization matrix for functional molecules (e.g. IgG) either by necessity of additional labelling procedures.
direct binding (3), by immobilization via the Fc-specific ligandprotein A (4), or biotinylated ligands can be bound to the S- 3.3. S-Layer Based Lipid Chips
liposome via the biotin–streptavidin system (5). (6) Alternatively, Biological membranes have attracted lively interest, as liposomes can be coated with genetically modified S layer proteins the advances in genome mapping revealed that incorporating functional domains. (b) Electron micrograph of a approximately one-third of all genes in an organism encode freeze-etched preparation of an S-liposome. Bar: 100 nm. Reprinted membrane proteins, such as ion channels, receptors, and with permission from Ref. [15], Copyright (2002) Wiley-VCH.
membrane-bound enzymes [76]. In addition, more than 60 % S-liposomes possess significantly enhanced stability of all consumed drugs act on membrane proteins [77].
towards thermal and mechanical stress factors [39]. For Therefore, the generation of stabilized lipid membranes with generating targeted S-liposomes, the S-layer lattice on functional membrane proteins represents a challenge to apply liposomes was cross-linked with bis(sulfosuccinimidyl) membrane proteins as key elements in drug discovery, protein-ligand screening and biosensors.
3; compound (4 )) (Fig. 6), biotinylated and
cross-linked S-layer protein SbsB (5)
Fig. (6). Cross-linking of the S-layer protein SbsB with the water-soluble, homobifunctional N-hydroxysuccinimide-ester bis(sulfosucci-
nimidyl) suberate (BS3; compound (4)). The spacer arm length of BS3 is 1.14 nm.
S-layer-supported lipid membranes (SLM) mimic the supramolecular assembly of archaeal cell envelope
structures, as they are composed of a cytoplasmic membrane
and a closely associated S-layer [36]. In this biomimetic
architecture, either a tetraetherlipid monolayer, or an
artificial phospholipid bilayer replaces the cytoplasmic
membrane and isolated bacterial S-layer proteins are
attached either on one or both sides of the lipid membrane
(Fig. 7). The most commonly used lipids to generate planar
SLMs are the phospholipid 1,2-diphytanoyl-sn-glycero-3-
phosphatidylcholine (6), and the membrane-spanning tetra-
etherlipids Main Phospholipid (7) isolated from Thermo-
plasma acidophilum and glycerol dialkyl nonitol tetra-
Fig. (7). Schematic drawing of an S-layer covered (modified) solid
etherlipid (8) extracted and purified from Sulfolobus and
support (e.g. a gold electrode) carrying a lipid bilayer generated by Metallosphaera archaea (Fig. 8).
vesicle fusion or by the Langmuir-Blodgett-technique. Integral Electrostatic interactions between exposed carboxylic membrane proteins can be reconstituted into this SLM. Further- acid groups on the inner surface of the S-layer lattice and the more, a second S-layer lattice can be recrystallized on the top of zwitterionic lipid head groups were found to be primarily this biomimetic structure to provide an enhanced long-term stability responsible for the defined binding of the S-layer subunits.
and to act as a protective coat with pores in the nanometer range.
As two to three contact points between the lipid film and theattached S-layer protein were determined, only few lipid In reconstitution experiments, the self-assembly of the molecules were anchored to protein domains on the S-layer staphylococcal pore-forming protein -hemolysin ( HL) lattice having a unit cell with a spacing of about 8 to 13 nm [86] was examined at plain and SLMs [87]. HL forms lytic [78]. The remaining scores of lipid molecules diffused freely pores when added to the lipid-exposed side of the S-layer- in the membrane between the pillars consisting of anchored supported membrane. No assembly was detected upon lipid molecules. Because of its widely retained fluid adding HL monomers to the S-layer-face of the composite characteristics, this nano-patterned lipid membrane was membrane. Therefore, it was concluded that the intrinsic termed "semifluid membrane" [79]. But most important, molecular sieving properties of the S-layer lattice did not although peptide side groups of the S-layer protein allow the formation of HL heptamers within the S-layer interpenetrated the phospholipid head group regions almost pores. Most interestingly, in SLMs the attached S-layer in its entire depth, no impact on the hydrophobic lipid alkyl lattice caused a decreased tendency to rupture in the presence chains was observed [80-83]. Thus, S-layer lattices constitute of HL, indicating an enhanced stability [87]. Even single unique supporting scaffoldings for lipid membranes [36, 56, HL pore recordings could be performed when reconstituted in S-layer supported lipid membranes [88].
Main Phospholipid isolated from Thermoplasma acidophilum (7)
glycerol dialkyl nonitol tetraetherlipid extracted and purified
from Sulfolobus and Metallosphaera archaea (8)
OH OH OH OH OH OH Fig. (8). Chemical structures of the phospholipid 1,2-diphytanoyl-sn-glycero-3-phosphatidylcholine (6), the membrane-spanning tetraether-
lipids Main Phospholipid (7) isolated from Thermoplasma acidophilum, and glycerol dialkyl nonitol tetraetherlipid (8) extracted and purified
from Sulfolobus and Metallosphaera archaea.
The functionality of lipid membranes resting on S-layer surrounded by sodium buffer with incorporated valinomycin, covered filters and gold electrodes was demonstrated by the a potassium-selective ion carrier, revealed a resistance in the reconstitution of HL and membrane-active peptides [89, G -range. In contrast, for the same membrane bathed in 90]. In a first study, gramicidin A was incorporated into potassium buffer the resistance dropped almost three orders tetraetherlipid monolayers, but also in phospholipid bilayers of magnitude due to the valinomycin-mediated ion transport.
which were deposited on S-layer covered filters [89]. These These results demonstrated that the biomimetic approach of membranes revealed not only a remarkable stability, parti- copying the supramolecular architecture of archaeal cell cularly with an S-layer cover, but the most striking result envelopes opened new possibilities for exploiting functional was that high-resolution conductance measurements on lipid membranes at meso- and macroscopic scale [92].
single gramicidin pores were feasible. In addition, for thevery first time, with filter supported lipid membranes, even 4. CONCLUSION AND FUTURE PERSPECTIVES
single pore recordings were performed on reconstituted Basic and applied S-layer research has demonstrated that nature provides most elegant examples for nanometer sized, The functionality of lipid membranes resting on S-layer molecular self-assembly systems. There are only a few covered gold electrodes was demonstrated by the reconsti- examples in nature where proteins reveal the intrinsic capa- tution of alamethicin, gramicidin and valinomycin [90]. Due bility to self-assemble into crystalline arrays, in suspension to the formation of conductive alamethicin channels, the and on a great variety of surfaces and interfaces. Since S- membrane resistance dropped two orders of magnitude layer lattices are highly anisotropic structures with signi- whereas the capacitance was not altered. Partial inhibition of ficant differences in the topography and physicochemical the alamethicin channels with amiloride and analogues was properties of the inner and outer surface, it was most demonstrated, as increasing amounts of inhibitor gave rise to important to copy nature's solution for assembling a secreted an increased membrane resistance [90]. In addition, an SLM protein on the cell surface into lattices with defined
orientation. This biomimetic strategy is particularly essential molecular construction kit (Fig. 9). Although, a broad
to ensure that crystallization of genetically engineered S- spectrum of applications for S-layers has been developed, it layer proteins occurred in defined orientation on solid is expected that other areas will emerge particularly in areas supports (metals, polymers, silicon wafers), lipid membranes, where top-down and bottom-up strategies are commonly liposomes, and a great variety of nanoparticles [62-64].
Another line for exploiting the unique features of S- layers is directed to the use of lattices as support and
stabilizing structures for functionalized lipid films and
liposomes (Figs. 5 and 7). Again, composite, semifluid
SLMs are biomimetic structures copying the supramolecular
principle of archaeal cell envelopes or human or animal virus
envelopes optimized during biological evolution for a great
variety of functions [5, 19, 36, 92].
The numerous benefits generated by the attachment of coherent S-layer lattices on lipid vesicles and mono- orbilayer membranes already triggered innovative approachesfor membrane biosensors, high through-put screening,diagnostics, and different lab-on-a-chip designs. S-liposomesrevealed high potentials for the development of new drug- Fig. (9). Schematic drawing of an S-layer lattice (yellow
targeting, drug-delivery and transfection systems [11, 20, 38, chessboard) with regular and well orientated functional molecules 40, 41, 43, 93].
(grey knights). The S-layer lattice of SbpA provides an area of up to Moreover, S-layer self-assembly products have been 13 to 13 nm2 for each functional molecule.
demonstrated to be particularly well-suited for a geometri-cally defined covalent attachment of haptens and immuno- genic or immuno-stimulating substances [94]. Most recently, Financial support from the Austrian Science Fund (FWF, a remarkable immuno-modulating capacity of S-layers was projects 16295-B10 and 17170-B10), the Erwin-Schrödinger demonstrated for a fusion protein comprising an S-layer Society for Nanosciences, the FP6 EC STREP NASSAP protein from a Bacillaceae and the major birch pollen (project 13352), the Volkswagen Stiftung (project I/77710), allergen Bet v 1 [67]. It is expected that innovative and and the Air Force Office of Scientific Research, USA highly specific immunogenic components with intrinsic (AFOSR, project F49620-03-1-0222) is gratefully acknow- targeting and delivery functionalities can be developed combining recombinant S-layer proteins with the supra-
molecular construction principle of virus envelopes (Fig. 6).
Another attractiveness for S-layer self-assembly systems Major birch pollen allergen is seen for non-life science applications. Current state-of-the- art methods for self-assembly of nanoparticle arrays thatgenerally involve bifunctional linkers, molecular recognition, Variable domain of a heavy chain camel or Langmuir-Blodgett techniques do not offer the control and flexibility of the S-layer system. The S-layer approach for cAb directed against the prostate-specific the first time allows adjustable lattice constants and control over template surface properties by chemical or geneticmodifications [5, 11, 15, 18] as required in molecular electronics, biocatalysis, and non-linear optics.
Enhanced green fluorescent protein Currently, there is a strong need to improve and develop Green fluorescent protein procedures for high resolution structural analysis ofmembrane proteins which can not be recrystallized to a quality suitable for X-ray analysis studies. By using S-layer fusion proteins, such target proteins could be forced into Microspheres-based detoxification system order arrays (Fig. 9) [61] accessible for structural analysis
involving established methods for image reconstruction such
as high resolution (cryo) electron microscopy, X-ray and S-layer protein of Bacillus sphaericus neutron reflectivity, and grazing incidence X-ray diffraction [81]. Intrinsic in plane distortions of lattices formed by S-layer fusion proteins can be corrected following standard S-layer protein of Geobacillus stearo- procedures [23, 95].
S-layer protein o f Geobacillus stearo- It is now evident that S-layers represent unique thermophilus ATCC 12980 patterning elements or base plates for a complex supra- S-Layer Proteins as Key Components
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Received: November 10, 2005
Accepted: January 18, 2006
Source: http://homes.nano.aau.dk/fp/self-assembling/pdf-material/protein-drugdelivery.pdf
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text:
\section{Introduction}
AdS/CFT is a correspondance \cite{JM, GKP, EW, AGM} between a string theory in AdS space and a conformal field theory in physical space-time. It leads to an analytic semi-classical model for strongly coupled QCD. The moving open string with constant velocity solutions have been constructed by using Nambu-Goto action in $AdS_5$ black hole backgrounds \cite{HKY, SG}. In order to study the dynamics of quark moving in strongly coupled $N = 4$ SYM thermal plasma two cases are investigated. First case is The infinitely massive quark as the open string end at the boundary of the $AdS_5$-Schwarzschild space-time \cite{SG}. And second is the finitely massive quark as the open string end at the D7-brane \cite{KK} in the $AdS_5$-Schwarzschild space-time \cite{HKY}.\\
Mikhailov has introduced an analytic generic solution for the open string dual to a single infinitely massive quark moving on an arbitrary timelike trajectory in the $N = 4$ SYM theory. And he extracted a rate of the energy loss which agrees with the Lienard formula by using the Nambu-Goto action in the Poincare $AdS_5$ spacetime \cite{AM}. Based on the extension of this generic solution to the finite quark mass case it has been shown that an event horizon appears on the worldsheet whenever the single quark accelerates in any fashion \cite{CG}. By applying AdS/CFT correspondence, the dynamics of a composite quark or dressed quark \cite{CGG, MCGG} in strongly-coupled large-Nc $N = 4$ SYM has been studied. Bo-Wen Xiao has used the Nambu-Goto action in the Poincare $AdS_5$ spacetime and found an accelerating open string solution dual to a heavy quark-antiquark pair uniformly accelerated in opposite directions \cite{BX}. In that case the string configuration has expressed as
\begin{equation}
x = \pm \sqrt{t^2 + b^2 - z^2}.
\label{ex}\end{equation}
in this expression, the plus and minus sign of $x$ represents position of the quark and antiquark respectively and  $\frac{1}{b}$ is the acceleration of the quark and anti-quark pair \cite{BX, SRG}. Here the infinitely massive quark and antiquark are located on the hyperbolic trajectories $x = \sqrt{t^2 + b^2}$ at the AdS boundary z = 0 such that the plus and minus sign of (\ref{ex}) represents the right and left half of the accelerating string. The quark and antiquark first approach to each other in decelerating, and stop to return back in accelerating away from each other with proper acceleration $\frac{1}{b}$ \cite{SRG}. The Nambu-Goto action has been analysed in Rindler spacetime to construct the similar accelerating open string solution \cite{PPZ}. The energy loss is related with the appearance of the worldsheet horizon via the open string which is moving in $AdS_5$-Schwarzschild spacetime \cite{SSG,JCT, BMX} and the worldsheet Hawking radiation generates the stochastic motion of the quark \cite{BHR,ST}. \\
In \cite{BX} the accelerating string solution which is related to uniformly accelerating quark and anti-quark, the resulting stochastic fluctuations of the quark trajectory have been determined \cite{CCG}. The stochastic motion of the string is due to interactions with other Hawking quanta in de Sitter space has been studied in \cite{WPJW}. Also for the case where the flux tube is sourced by a uniformly accelerated quark it is determined that the dual string embedding known heretofore terminates unphysically at the worldsheet horizon \cite{JAP}. The generic string solution has been directly substituted into the string equation of motion and confirmed to solve it \cite{GP}. There are several researches about thermal effects of the worldsheet horizon on the accelerating string associated with Unruh temperature \cite{CCG, CP, HKK}. By applying the string solution \cite{AM} and suitable coordinate, the accelerating string solution dual to a single accelerating quark in the global $AdS_5$ spacetime has been constructed \cite{HS, FGP}.
The entanglement of the general quantum Einstein-Podolsky-Rosen (EPR) pair are connected through the interior via a wormhole, or Einstein-Rosen bridge \cite{MS}. There have been various investigations where the quark-antiquark pair is concretely regarded as a color singlet EPR pair in the $N = 4$ SYM theory and the related entanglement is encoded in a nontraversable wormhole on the worldsheet of the flux tube connecting the pair \cite{JK, JS, CGP, JKR}.In \cite{WPJ} the string solution dual to a quark and anti-quark pair with constant acceleration in de Sitter space has been constructed and shown that quark and anti-quark pair can be interpreted as a EPR pair or Hawking pair emanating from dS horizon. These results are related to the existence of horizon on the worldsheet
of the accelerating open string dual to a uniformly accelerating quark-antiquark pair. The entanglement entropy of the quark-antiquark pair has been studied in \cite{JK, LM, VHGS}. And the relevence of the entanglement entropy and the string surface describing gluon scattering in position space has been investigated \cite{SS}.\\
In \cite{SRY} the two string solutions associated with One-cusp Wilson loop \cite{MK} has been reconstructed which are dual to an accelerating quark-antiquark pair \cite{BX}. The $SL(2,R)L \times SL(2,R)R$ isometry group of the $AdS_3$ space-time has been studied in \cite{JMS,BHT}.By performing the conformal SO(2,4) transformation, one-cusp Wilson loop solution has been studied, which provides chances to study the planar four-gluon scattering amplitude by using four-cusp Wilson loop solution in the T-dual AdS space-time \cite{LAM}. An accelerating open string solution dual to a heavy accelerating quark and anti-quark pair in opposite direction has been constructed by performing three kinds of conformal transformations in \cite{SRG}, where different accelerations have been found for accelerating quark and anti-quark pair.
In this paper, based on the Nambu-Goto action we will consider an open string in $AdS_3$ with the following Poincare metric with $R=1$,
\begin{equation}
ds^2 = \frac{dz^2 - dt^2 + dx^2}{z^2}.
\label{me}\end{equation}
We make following ansatz \cite{JSH} for a rotating string with angular velocity $\omega$ which is moving with constant velocity $v$ in x direction from the AdS boundry at $z=0$ to the Poincare Horizon at $z=\infty$.
\begin{equation}
x=vt+r \sin{\omega t}
\label{xt}\end{equation}
In this work, the procedure of \cite{SRG} is followed to evaluate the effects of angular velocity on the accelerating quark and anti-quark pair. By performing the two mappings, the special conformal transformation and the conformal SO(2,2) transformation, we will obtain the acceleration of quark and anti-quark pair in six cases and we will see that quark and anti quark in some cases tend to deccelerate in order to approach each other then stop in accelerating in order to back away from each other.\\
So, we organize the paper as follows. In section 2 we discuss the case where we perform the special conformal transformations in four cases and we find the acceleration of quark and anti-quark and also we perform numerical calculation in order to investigate the effects of the angular velocity on the acceleration of the pair. In section 3 we perform the conformal SO(2,2) transformations as well. By the end, section 4 would be our conclusion.\\
\section{Special conformal transformations and rotating string}
As we know from \cite{SRG}, we can construct an open string configuration dual to accelerating quark and anti-quark by performing the special conformal transformations. In (\ref{ex}), $\frac{1}{b}$ is the acceleration of the quark and anti-quark pair \cite{BX} (see also \cite{SRG}).
Now, let us consider (\ref{xt}) where the string extends straight from the AdS boundary at $z=0$ to Poincare Horizon at $z=\infty$ in the $z$ direction and in addition to moving with constant velocity $v$ in the $x$ direction also it has rotation with angular velocity $\omega$.
If we consider $x=vt+r\sin{\omega t}$, we will face infinite loops in our numerical calculations, so we must use first order expression of extensions. On the other hand, our string profile becomes unstable then we take $\omega<<1$, which means that we are allowed to study very small fluctuations in this work. So our focuse is on the infenitesimal values of $\omega$, therefore we have,
\begin{equation}
\sin{\omega t}\approx\omega t.
\label{st}\end{equation}
Now, we intend to perform special conformal transformations of the Poincare $AdS_3$ spacetime coordinates $z, x^{\mu}=(t,x)$,
\begin{equation}
{x^{\mu}}' = \frac{x^{\mu} + a^{\mu}( z^2 + x^2 )}{1 + 2a\cdot x
+ a^2( z^2 + x^2 )}, \hspace{1cm} z' = \frac{z}{1 + 2a\cdot x
+ a^2( z^2 + x^2 )},
\label{pt}\end{equation}
the interested reader can see more details in \cite{BCF} that it has been studied for circular Wilson Loop.
As we know, we can construct particular $SL(2,R)_L$ and $SL(2,R)_R$ transformations by using special conditions. In order to obtain these particular transformations let us consider $SL(2,R)_L \times SL(2,R)_R$ isometry group which represents relations for $SL(2,R)_L$ and $SL(2,R)_R$ \cite{JMS, SR}. We can see both transformations map $AdS_3$ at $z=0$ to itself and act on the boundary as a usual conformal transformations of $1+1$ dimentional Minkowski space.\\
The $SL(2,R)_L$ transformation is,
\begin{eqnarray}
w^+ \rightarrow {w^+}' &=& \frac{\alpha w^+ + \beta}{\gamma w^+ + \delta}, \nonumber \\
w^- \rightarrow {w^-}' &=& w^- + \frac{\gamma z^2}{\gamma w^+ + \delta}, \nonumber \\
z \rightarrow z' &=& \frac{z}{\gamma w^+ + \delta}
\end{eqnarray}
and $SL(2,R)_R$ transformation is
\begin{eqnarray}
w^+ \rightarrow {w^+}' &=& w^+ + \frac{\gamma z^2}{\gamma w^- + \delta}, \nonumber \\
w^- \rightarrow {w^-}' &=& \frac{\alpha w^- + \beta}{\gamma w^- + \delta}, \nonumber \\
z \rightarrow z' &=& \frac{z}{\gamma w^- + \delta}.
\end{eqnarray}
With real $\alpha, \beta, \gamma, \delta$ obeying $\alpha \delta - \beta \gamma = 1$.
Here we consider two special cases $a^{\mu} = (-\frac{1}{l},\frac{1}{l})$ and $a^{\mu} = (\frac{1}{l},\frac{1}{l})$ in (\ref{pt}) which cioncide with the particular $SL(2,R)_L$ and $SL(2,R)_R$ transformations with $\alpha = 1, \beta = 0, \gamma =2a$ and $\delta = 1$ respectively.
Let us make a special conformal transformation with $a^{\mu} = (-\frac{1}{l},\frac{1}{l})$ for the rotating string with angular velocity $\omega$ which is also moving with constant velocity $v$ in the $x$ direction
\begin{equation}
x=vt+r\omega t,
\label{om}\end{equation}
to have
\begin{eqnarray}
t' &=& \frac{1}{P}\left[t-\frac{1}{l}\left(z^2-t^2+v^2 t^2+r^2 \omega^2t^2+2vt^2r\omega\right)\right],\hspace{2cm}\nonumber \\
x' &=& \frac{1}{P}\left[vt+r\omega+\frac{1}{l}\left(z^2-t^2+v^2 t^2+r^2 \omega^2t^2+2vt^2r\omega\right)\right],\hspace{1cm}\nonumber \\
z' &=& \frac{z}{P},\hspace{6cm}
\label{op}\end{eqnarray}
where P is expressed in the following form
\begin{equation}
P = 1 + \frac{2}{l}\left(t+vt+r\omega t\right).
\label{rp}\end{equation}
By using $t'$, $x'$ in (\ref{op}) we obtain following expression for t
\begin{equation}
t=\frac{P\left(x'+t'\right)}{1+v+r\omega},
\label{ft}\end{equation}
and with substitution of former expression in (\ref{rp}) we have P in terms of $t'$ and $x'$
\begin{equation}
P=\frac{l}{l-2t'-2x'}.
\end{equation}
Combining (2.6) and (2.9) leads to the following expanding string solution
\begin{equation}
\left(x'-\frac{l}{2\left(v+r\omega+1\right)}\right)^2=\left(t'-\frac{l\left(v+r\omega\right)}{2\left(v+r\omega+1\right)}\right)^2+\frac{l^2}{4}\frac{1-\left(v+r\omega\right)^2}{\left(v+r\omega+1\right)^2}-z'^2
\label{lr}\end{equation}
So we obtain an accelerating string configuration associated with proper acceleration $\frac{2\left(v+r\omega+1\right)}{l\sqrt{1-\left(v+r\omega\right)^2}}$ of a quark-antiquark pair.
Then we have found the term of acceleration of quark and antiquark pair by performing particular $SL(2,R)_L$ transformation. If we compare this expression with what it has been constructed in \cite{SRG} we will find that the angular velocity in this case acts like $v$. We perform some useful numerical calculations in order to show behaviors of the acceleration of the pair. We plot our results as Figure 1.\\
\begin{figure}
\centerline{\includegraphics[width=18cm]{SLL}} \caption{This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we have applied special conformal transformation with $a^{\mu}=(-\frac{1}{l},\frac{1}{l})$ which coincides with particular $SL(2,R)_L$ transformation. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dashed red curve corresponds to $r\omega=0.45$, the dotted green curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
In figure 1, we show the behavior of $\frac{1}{b}$ for this case as a function of $x$ for several choices of $r\omega$. In this figure and other figures in this paper, uper curves are ploted for plus sign and downer curves are ploted for minus sign, as one can see in (\ref{lr}) for this case. This figure shows that the maximum value of the acceleration is constant for different $r\omega$, it can be interpreted as permanent behavior of the accelerating quark and anti-quark pair with respect to any angular velocity in this transformation with $ a^{\mu}=(-\frac{1}{l},\frac{1}{l})$. So the pair has no feel to rotation but one can see that with increasing $r\omega$ curves are fined to be close to each other because quark and anti-quark get away from each other and it is not interesting for them. Also we see curves are shifted to left as a result of particular $SL(2,R)_L$ transformation. Shifting to the left means that existence of the maximum value of acceleration takes place in a shorter distance at more value of $\omega$.\\
Now we apply particular $SL(2,R)_R$ transformation by considering $a^{\mu}=(\frac{1}{l},\frac{1}{l})$, so we have,
\begin{eqnarray}
t' &=& \frac{1}{P}\left[t+\frac{1}{l}\left(z^2-t^2+v^2 t^2+r^2 \omega^2t^2+2vt^2r\omega\right)\right],\hspace{2cm}\nonumber \\
x' &=& \frac{1}{P}\left[vt+r\omega+\frac{1}{l}\left(z^2-t^2+v^2 t^2+r^2 \omega^2t^2+2vt^2r\omega\right)\right],\hspace{1cm}\nonumber \\
z' &=& \frac{z}{P},\hspace{6cm}
\label{sl}\end{eqnarray}
where P expressed in terms of $x'$ and $t'$ through
\begin{equation}
P=\frac{l}{l+2t'-2x'}.
\end{equation}
This expressions lead to following expanding string
\begin{equation}
\left(x'-\frac{l}{2\left(v+r\omega-1\right)}\right)^2=\left(t'-\frac{l\left(v+r\omega\right)}{2\left(v+r\omega-1\right)}\right)^2+\frac{l^2}{4}\frac{1-\left(v+r\omega\right)^2}{\left(v+r\omega-1\right)^2}-z'^2
\end{equation}
then the special conformal transformation with $a^{\mu}=(\frac{1}{l},\frac{1}{l})$ which coincides the particular $SL(2,R)_R$ transformation, generates the expanding string with acceleration $\frac{2(v+r\omega-1)}{l\sqrt{1-(v+r\omega)^2}}$.\\
\begin{figure}
\centerline{\includegraphics[width=18cm]{SLR}}
\caption{This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we have applied special conformal transformation with $a^{\mu}=(\frac{1}{l},\frac{1}{l})$ which coincides with particular $SL(2,R)_R$ transformation. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dotted light green curve corresponds to $r\omega=0.35$, the dashed red curve corresponds to $r\omega=0.45$, the dashed orange curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
According to what we see in figure 2, we consider that when there is no angular velocity, which is studied in \cite{SRG}, maximum value of the acceleration happens in $x=0$, on the other hand there is no habitual relation between increasing $r\omega$ and maximum value of acceleration. The figure shows that in $r\omega=0.55$, unlike former curves, has a shift to left side and start decreasing. Then in this transformation quark and anti-quark aproach to each other in deccelerating and stop to return back in accelerating away from each other with proper acceleration. There is an interesting point in this figure, there is an exact position where the pair has constant value of acceleration, with any value for $r\omega$, it seems that in this position pair has no feel to $r\omega$ and just has its constant acceleration anyway.\\
We make another special conformal transformation with $a^{\mu} = (0,\frac{1}{l})$ for the rotating string moving with constant velocity v to have,
\begin{eqnarray}
t' &=& \frac{t}{P}, \nonumber \\
x' &=& \frac{1}{P}\left[ vt +r\omega t+\frac{1}{l}\left(z^2-t^2+v^2t^2+r^2\omega^2 t^2+2vt^2r\omega\right)\right], \nonumber \\
z' &=& \frac{z}{P},
\end{eqnarray}
We optain $x'$ in terms of $t'$ and $z'$ through
\begin{equation}
x' = \frac{l}{2} \pm \sqrt{\left( t' - \frac{l\left(v+r\omega\right)}{2}\right)^2 +
\frac{(1-\left(v+r\omega\right)^2)l^2}{4} - {z'}^2 }
\end{equation}
So this special conformal transformation generates the expanding string with acceleration $\frac{2}{l\sqrt{1-(v+r\omega)^2}}$.
\begin{figure}
\centerline{\includegraphics[width=18cm]{SCT}}
\caption{ This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we have applied special conformal transformation with $a^{\mu}=(0,\frac{1}{l})$. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dashed red curve corresponds to $r\omega=0.45$, the dotted green curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
In this case we set time component in $a^{\mu}$ to zero, the Figure 3 shows that with increasing $r\omega$, acceleration reduces, so if $r\omega$ be large enough, pair has no acceleration any more. It means that quark and anti-quark aproach to each other in deccelerating and stop.
On the other hand, pair has acceleration in a finite range of $x$. thus, out of this region we consider curves are coincident, it means that the pair doesn't feel rotation.
On the other hand a special conformal transformation with $a^{\mu}=(\frac{1}{l},0)$ gives
\begin{equation}
x' = -\frac{l}{2\left(v+r\omega\right)} \pm \sqrt{\left( t' + \frac{l}{2}\right)^2 +
\frac{(1-\left(v+r\omega\right)^2)l^2}{4\left(v+r\omega\right)^2} - {z'}^2 }.
\end{equation}
Which represents the expanding string with acceleration $\frac{2(v+r\omega/)}{l\sqrt{1-\left(v+r\omega\right)^2}}$.\\
\begin{figure}
\centerline{\includegraphics[width=18cm]{SCTT}}
\caption{This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we have applied special conformal transformation with $a^{\mu}=(\frac{1}{l},0)$. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dashed red curve corresponds to $r\omega=0.45$, the dotted green curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
In figure 4, unlike Figure 1, we see that increasing in angular velocity leads to shifts in positive $x$ direction (right side), also the maximum value of acceleration has no change. Shifting to the right side in larger values of $x$ shows that pair reaches to it's maximum value of acceleration in a larger distance $x$ with increasing $\omega$. It means that quark and antiquark move with constant acceleration.\\
As we show the accelerating quark and anti-quark pair which is constructed by former types of the special conformal transformations, can be affected by increasing in angular velocity, and they can aproach to each other or return back from each other in deccelerating or accelerating respectively.\\

\section{Conformal $SO(2,2)$ transformations and rotating string}
In this section we perform two conformal $SO(2,2)$ transformations on the rotating string which is also moving with constant velocity in the $x$ direction in order to study effects of angular velocity on the acceleration of quark and anti-quark.
Here we restore the AdS radius $R$ to express the following relations between the Poincare coordinates in $AdS_3$ and the embedding coordinates $X^M\; (M = -1, 0, 1, 2 )$ on which the conformal SO(2,2) transformation is acting linearly \cite{SRG},
\begin{eqnarray}
X^{\mu} &=& \frac{x^{\mu}}{z}R, \; (\mu =0, 1), \nonumber \\
X^{-1} &=& \frac{R^2 + z^2 + x_{\mu}x^{\mu} }{2z}, \hspace{1cm}
X^2 = \frac{R^2 - z^2 - x_{\mu}x^{\mu} }{2z}, \nonumber \\
- R^2 &=& - (X^{-1})^2 - (X^0)^2 + (X^1)^2 + (X^2)^2 .
\end{eqnarray}
For the string configuration (\ref{xt}), which is described
by $X^1 = (v+r\omega) X^0$, we perform following  conformal SO(2,2) transformation
\begin{equation}
{X^{-1}}' = - X^0, \;\; {X^0}' = X^{-1}, \;\; {X^1}' = X^1, \;\;
{X^2}' = X^2,
\label{os}\end{equation}
which interchanges $X^{-1}$ and $X^0$. The transformed configuration
is specified by ${X^1}' = - (v+r\omega) {X^{-1}}'$ which is expressed in terms of
the Poincare coordinates as
\begin{equation}
\left(x' + \frac{R}{v+r\omega}\right)^2 = {t'}^2 +
\frac{1 - \left(v+r\omega\right)^2}{\left(v+r\omega\right)^2}R^2 - {z'}^2
\end{equation}
This expression represents the expanding string with acceleration $\frac{v}{R\sqrt{1-(v+r\omega)^2}}$.\\
\begin{figure}
\centerline{\includegraphics[width=18cm]{SOT}}
\caption{This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we consider ${X^1}' = - (v+r\omega) {X^{-1}}'$. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dashed red curve corresponds to $r\omega=0.45$, the dotted green curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
In figure 5 we see the similar behavior with Figure 4, it shows that these transformations act similarly. As one can see, like the special conformal transformation with $a^{\mu}=(\frac{1}{l},0)$, this conformal $SO(2,2)$ transformation result in constant maximum value of acceleration in quark and anti-quark pair.\\
Now, let us apply the other conformal SO(2,2) transformation which has defined as the interchange
between $X^1$ and $X^2$
\begin{equation}
{X^{-1}}' = X^{-1}, \;\; {X^0}' = X^{0}, \;\; {X^1}' = - X^2, \;\;
{X^2}' = X^1
\label{so}\end{equation}
So the transformed string configuration which is specified by  ${X^2}' = (v+r\omega) {X^0}'$, becomes\\
\begin{equation}
{x'}^2 = ( t' - \left(v+r\omega\right)R )^2 + (1 - (v+r\omega)^2)R^2 - {z'}^2
\end{equation}
thus we constructed the expanding string with acceleration $\frac{1}{R\sqrt{1-(v+r\omega)^2}}$.\\
\begin{figure}
\centerline{\includegraphics[width=18cm]{SOTT}}
\caption{This figure shows the behavior of acceleration of the pair, with respect to the $x$, where we consider${X^2}' = (v+r\omega) {X^0}'$. The solid black curve corresponds to $r\omega=0$, the solid blue curve corresponds to $r\omega=0.25$, the dashed red curve corresponds to $r\omega=0.45$, the dotted green curve coresponds to $r\omega=0.65$ and the purple dashed-dotted curve corresponds to $r\omega=0.85$.}
\end{figure}
Figure 6 shows that this conformal transformation with ${X^2}' = (v+r\omega) {X^0}'$, unlike the special conformal transformation with $a^{\mu}=(0,\frac{1}{l})$, is the uniqe one in which maximum value of acceleration increases with increasing in $r\omega$, so maybe it is a chance for the quark and anti-quark pair to rotate with more values of $r\omega$ to achieve the large values of acceleration when it is sitting at $x=0$ position.\\

\section{Concolusion}
In this paper we have used the method of \cite{SRG} to investigate the effects of angular velocity on the accelerating quark and anti-quark in $AdS_3$ spacetime, and in general we have shown that increases in angular velocity can have effects on quark and anti-quark accelerating or deccelerating. We have considered and performed special conformal transformations and conformal SO(2,2) transformations in order to construct accelerating string configuration dual to accelerating quark and anti-quar pair. According to our numerical calculations and plots it is clarified that special conformal transformations whit $a^{\mu}=(-\frac{1}{l},\frac{1}{l})$ and $a^{\mu}=(\frac{1}{l},\frac{1}{l})$ which are coincide with $SL(2,R)_L$ and $SL(2,R)_R$ respectively, show behaviors which are not related or like other transformations. But we have shown that the special conformal transformation with $a^{\mu}=(0,\frac{1}{l})$ act contrariwise when angular velocity is increasing and it seems that if angular velocity becomes large enough deccelerating can happen, and conformal SO(2,2) transformation, where we consider ${X^2}' = (v+r\omega) {X^0}'$, vice versa. The special conformal transformation with $a^{\mu}=(\frac{1}{l},0)$ and the conformal SO(2,2) transformation, where we consider ${X^1}' = - (v+r\omega) {X^{-1}}'$, act similarly, as increases in angular velocity leads to constant maximum acceleration of quark and anti-quark pair. \\
\newpage
\textbf{Acknowledgement}\\
We are grateful to Sara Tahery for useful discussions and also for comments on the manuscript.

redpajama_set_name: RedPajamaArXiv


We've thoroughly enjoyed tasting this Kenyan Peaberry coffee from our friends at Legado Coffee Roasters. A peaberry is a naturally occurring mutation where the coffee bean (which is actually a seed) grows alone in its cherry rather than in the usual pair. The round result of this natural mutation is prized for its quality, flavour and ease of even roasting due to its shape.
This particular peaberry lot has everything you'd want from this type of coffee. It's sweet, fruity and features a subtle, integrated acidity than its often more tart AA compatriots. We particularly enjoyed it in brew methods with longer extractions like a French press or a Chemex. The dominant flavours to our palates were of the stone fruit variety, with the occasional hint of tangy green apple acidity.


This was an issue that a client raised with me when they needed Unreal Engine and Studio installed along with the Datasmith Exporter in order to take work from Sketchup Pro into Unreal. The ability to load from Sketchup was actually a new feature to me as I had recently done a little bit of work with Unreal so I was very interested to find out more. However this process was stopped at the point we tried to install the engine with the error message DP-06.
Using the normal approach as "run as administrator" to run the application we still weren't successful. After some research I logged into the machine as an administrator and was able to complete this install and the installation of the Datasmith exporter. The release in question was Unreal Engine 4.21 and I assume this is potentially a bug. Hopefully that will help you to get to the stage of having fun with your sketchup work in Unreal Engine.


text:
Q: Can complex numbers be ordered in the way of Pythagorean triples? This post did not quite seem to answer my question. If we have
$\quad a+bi\in\mathbb{C}\quad$ can ordering be done the way of Pythagorean triples $\quad A^2+B^2=C^2?\quad$
There are infinite triples for each of
$\quad
(C-B),\space
 (C-A),\space\text{and}\space
(B\pm A)\quad$
but these may be further ordered  as sets of sets
$
(C-B)=(2n-1)^2\in\big\{1,3,5,\cdots\big\}
$
$
(C=A)=2k^2\in\big\{2,8,18,\cdots\big\}
$
\begin{align*}
&(B\pm A)=P:\\
&P=p_1,\cdots ,p_n),\space p_k\equiv \pm 1 \text{ (mod }8) \in\big\{1,7,17, \cdots\big\}
\end{align*}
For each of $(A\text{ or } B\space\text{ or } \space C),\space$
there are $2^{n-1}\space$
primitive triples where $\space n\space $ is the number of unique prime factors of $\space(A\text{ or } B\space\text{ or } \space C)\space $ and these may be of vastly  different sizes. Perhaps these may be ordered by the pairs of natural numbers $\space(n,k)\space$ that generate them in this formula
\begin{align*}
  A=(2n-1)^2+                     & 2(2n-1)k            \tag{a}  \\
  B=  \qquad\qquad\quad   & 2(2n-1)k+ ]]    2k^2        \tag{b}  \\
  C=(2n-1)^2+                    & 2(2n-1)k+       2k^2 \tag{c}
\end{align*}
where $\space n\space $ is a set number and $\space k\space$ is the ordinal triple  within each set.
For example, $C=65\space$ represents triples
$\space f(4,1)=(63,16,65)\quad  f(5,4)=(33,56,65).\space$ where $\space(63,16,65)\space$ is the first member of $Set_4$ and $\space(33,56,65)\space$  is the fourth member of $Set_5.$
Can $\space (4+1i),\space (5+4i)\space$ be treated as natural ordering of these complex numbers or are their other considerations such as the size of the resulting squares or products to consider?

A: There are an infinite number of ways to order the complex numbers as a set.
There is absolutely no way to order the complex numbers as a field.
I didn't bother to read your explanation of the ordering of pythogorean triples but it seems similar to "lexigraphic ordering".
For two complex number as numbers $w,z$ we can define the lexigraphic order as:

*

*If $Re(w) < Re(z)$ then $w \prec z$

*If $Re(w) = Re(z)$ but $Im(w) < Im(z)$ then $w\prec z$.

*Otherwise $w \not \prec z$.

(In other words:  If $w = a+bi; z = c+di; a,b,c,d \in \mathbb R$ to compare $w$ to $z$ compare $a$ to $c$.  If $a = c$ then compare $b$ to $d$.)
That satisfies everything we need to have an ordered set.  To be an order on a set we need:

*

*Given complex $w$ and $z$ then exactly one of the following is true:  Either $w \prec z$ or $z\prec w$ or $w = z$.

*if $w\prec z$ and $z\prec \alpha$ then it always follows that $w \prec \alpha$.
It's easy to see that those are both always true.  So $\prec$ is an order on $\mathbb C$ as a set.
......
But it does not satisfy an order on a field.
To be an order on a field we need to be able to do math on these numbers and have the resulting inequalities be preserved.  We need:

*If $w \prec z$ then for every $a$ we have $w + a \prec z+a$.

This is actually true.
and

*If $w \prec z$ and $0 \prec a$ then $aw \prec az$.

This fails.  This fails big time!
Let $w = 4 - 6i$ and $z = 5-3i$ and $a = i$.
Then $4 - 6i \prec 5-3i$ and $0\prec i$ but $(4-6i)i = 6 + 4i \not\prec 3 + 5i = (5-3i)i$.
.....
Now having an order as a field is impossible.
Why.  Consider any $w$ in a field.  We either have $w > 0$ or $w = 0$ or $w< 0$.
If $w > 0$ then $w^2 = w\times w > 0$.  If $w=0$ then $w^2 = 0$. And if $w < 0$ then $0 =w +(-w) < 0 + (-w) =-w$ and so $w^2 = (-w)\cdot (-w) > 0$.
So we have $w^2 \ge 0$ and we never have $w^2 < 0$.
But in the complex we have $1^2 = 1$ so we would have to have $1^2 = 1 > 0$. Then we'd have to have $0 < 1$ so $-1 =0+(-1) < 1 +(-1) = 0$ so $-1 < 0$.
But we also have  And we have $i^2 = -1$ so we would HAVE to have $i^2 = -1 > 0$.  So we have BOTH $-1 < 0$ AND $-1 > 0$.  That's utterly impossible.
So the complex numbers can not be ordered as a field.

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