Gabapentin: A Stereochemically Constrained gamma Amino Acid Residue in Hybrid Peptide Design

Nature has used the all-alpha-polypeptide backbone of proteins to create a remarkable diversity of folded structures. Sequential patterns of 20 distinct amino adds, which differ only in their side chains, determine the shape and form of proteins. Our understanding of these specific secondary structures is over half a century old and is based primarily on the fundamental elements: the Pauling alpha-helix and beta-sheet. Researchers can also generate structural diversity through the synthesis of polypeptide chains containing homologated (omega) amino acid residues, which contain a variable number of backbone atoms. However, incorporating amino adds with more atoms within the backbone introduces additional torsional freedom into the structure, which can complicate the structural analysis. Fortunately, gabapentin (Gpn), a readily available bulk drug, is an achiral beta,beta-disubstituted gamma amino add residue that contains a cyclohexyl ring at the C-beta carbon atom, which dramatically limits the range of torsion angles that can be obtained about the flanking C-C bonds. Limiting conformational flexibility also has the desirable effect of increasing peptide crystallinity, which permits unambiguous structural characterization by X-ray diffraction methods. This Account describes studies carried out in our laboratory that establish Gpn as a valuable residue in the design of specifically folded hybrid peptide structures. The insertion of additional atoms into polypeptide backbones facilitates the formation of intramolecular hydrogen bonds whose directionality is opposite to that observed in canonical alpha-peptide helices. If hybrid structures mimic proteins and biologically active peptides, the proteolytic stability conferred by unusual backbones can be a major advantage in the area of medicinal chemistry. We have demonstrated a variety of internally hydrogen-bonded structures in the solid state for Gpn-containing peptides, including the characterization of the C-7 and C-9 hydrogen bonds, which can lead to ribbons in homo-oligomeric sequences. In hybrid alpha gamma sequences, district C-12 hydrogen-bonded turn structures support formation of peptide helices and hairpins in longer sequences. Some peptides that include the Gpn residue have hydrogen-bond directionality that matches alpha-peptide helices, while others have the opposite directionality. We expect that expansion of the polypeptide backbone will lead to new classes of foldamer structures, which are thus far unknown to the world of alpha-polypeptides. The diversity of internally hydrogen-bonded structures observed in hybrid sequences containing Gpn shows promise for the rational design of novel peptide structures incorporating hybrid backbones.

Random network behaviour of protein structures

Geometric and structural constraints greatly restrict the selection of folds adapted by protein backbones, and yet, folded proteins show an astounding diversity in functionality. For structure to have any bearing on function, it is thus imperative that, apart from the protein backbone, other tunable degrees of freedom be accountable. Here, we focus on side-chain interactions, which non-covalently link amino acids in folded proteins to form a network structure. At a coarse-grained level, we show that the network conforms remarkably well to realizations of random graphs and displays associated percolation behavior. Thus, within the rigid framework of the protein backbone that restricts the structure space, the side-chain interactions exhibit an element of randomness, which account for the functional flexibility and diversity shown by proteins. However, at a finer level, the network exhibits deviations from these random graphs which, as we demonstrate for a few specific examples, reflect the intrinsic uniqueness in the structure and stability, and perhaps specificity in the functioning of biological proteins.

Lactobacillus bacteremia, with special focus on the safety of probiotic Lactobacillus rhamnosus GG

Lactobacilli are gram positive rods, which belong to normal oropharyngeal, gastrointestinal and urogenital flora. They are widely used in food industry and as food additives. Although their virulence is presumed to be very low, opportunistic bacteremic infections, especially in immunocompromised hosts, have been detected. In the present study, the possible effects of increasing probiotic use of Lactobacillus rhamnosus GG (LGG) on the occurrence of bacteremia due to lactobacilli was evaluated on population level. In Finland, 90 Lactobacillus bacteremia cases were reported to the National Infectious Disease Register maintained by National Public Health Institute, during 1995-2000. Their proportion of all bacteremia cases was on average 0.24%, corresponding to 0.3 cases/100 000 inhabitants annually. In the Helsinki University Central Hospital district the corresponding proportion of all bacteremia cases was observed during 1990-2000. Despite LGG intake increased six folded no increasing trend in the occurrence of lactobacilli bacteremia was seen. A total of 85 Lactobacillus sp. blood isolates collected from different human bacteremic cases were characterised and compared with the commercial probiotic LGG strain. In species characterisation 46 L. rhamnosus strains, 12 L. fermentum and L. casei strains each, three each of L. gasseri, L. salivarius and L. jensenii species, two L. curvatus, and one each of L. plantarum, L. sakei, L. zeae and L. reuteri species were detected. Nearly half of the L. rhamnosus findings turned out to be indistinguishable from the probiotic LGG strain. Common predisposing factors to Lactobacillus bacteremia were immunosuppression, prior prolonged hospitalisation and prior surgical interventions. Severe or fatal comorbidities were found in 82% of the patients. Mortality at one month was 26% and severe underlying diseases were a significant predictor of death (OR 15.8). Antimicrobial susceptibility of Lactobacillus strains was species dependent. The Lactobacillus isolates were generally susceptible to imipenem, piperacillin-tazobactam, clindamycin and erythromycin, whereas all other than L. gasseri and L. jensenii species were not at all susceptible to vancomycin. The susceptibility to cephalosporin varied greatly even within species why they might not be recommended for treatment of Lactobacillus infections. The effect and safety of probiotic LGG preparation in amelioration of gastric symptoms and diarrhea in HIV-infected patients was evaluated. No significant differences in gastrointestinal symptoms or diarrhoea in LGG treated patients compared to placebo could be found. LGG was well tolerated with no adverse effects including bacteremic outbreaks could be observed. The use of probiotic LGG can be regarded safe in this immunocompromised patient group.

Conformation of di-n-propylglycine residues (Dpg) in peptides: crystal structures of a type I-turn forming tetrapeptide and an -helical tetradecapeptide

The crystal structures of two oligopeptides containing di-n-propylglycine (Dpg) residues, Boc-Gly-Dpg-Gly-Leu-OMe (1) and Boc-Val-Ala-Leu-Dpg-Val-Ala-Leu-Val-Ala-Leu-Dpg-Val-Ala-Leu-OMe (2) are presented. Peptide 1 adopts a type I-turn conformation with Dpg(2)-Gly(3) at the corner positions. The 14-residue peptide 2 crystallizes with two molecules in the asymmetric unit, both of which adopt -helical conformations stabilized by 11 successive 5 1 hydrogen bonds. In addition, a single 4 1 hydrogen bond is also observed at the N-terminus. All five Dpg residues adopt backbone torsion angles (, ) in the helical region of conformational space. Evaluation of the available structural data on Dpg peptides confirm the correlation between backbone bond angle NCC() and the observed backbone , values. For > 106° , helices are observed, while fully extended structures are characterized by < 106° . The mean values for extended and folded conformations for the Dpg residue are 103.6° ± 1.7° and 109.9° ± 2.6° , respectively.

Two-Step Folding of Donor-Acceptor Foldamers

In a series of polymers containing alternately placed electron-rich dialkoxyilaphthalene (DAN) donors and electron-deficient pyromellitic diimide (PDI) acceptors linked by hexa(oxyethylene) (OE-6) segments, the ability to form a folded D-A stack was intentionally disrupted by random inclusion of varying amounts of a comonomer that is devoid of DAN donor units. NMR spectroscopic studies of folding in these copolymers, induced by NH4SCN that coordinates with the OE-6 segments and facilitates the charge-transfer (C-T) induced D-A stacking, clearly reveals the presence of PDI units that are isolated and those that are located at the ends of (D-A),, stacks. Similar conclusions regarding the presence of stacked and unstacked regions along the polymer chain were also inferred from UV-vis spectroscopic studies that probe the evolution of charge-transfer band. One fascinating aspect of these copolymers wits their ability to undergo it two-step folding: first, short (D-A),, stacks are formed by the interaction of the NH4+ ion with some specific regions of the polymer chain, and subsequently these Stacks are further stacked via a two-point interaction with it suitably designed external folding agent that carries a DAN unit and all ammonium group. In the second step, the interaction first occurs by the coordination of the ammonium group of the folding agent with the OE-6 segment, which in turn facilitates the C-T interaction of the DAN unit with the adjacent uncomplexed PDI units along the polymer chain, leading to an increase ill the slacking. Variations of several spectral features, during both UV-vis and NMR spectroscopic titrations, clearly reveal this novel two-step folding process.

Hybrid peptide design. Hydrogen bonded conformations in peptides containing the stereochemically constrained gamma-amino acid residue, gabapentin

he crystal structure of 12 peptides containing the conformationally constrained 1-(aminomethyl)cyclohexaneacetic acid, gabapentin (Gpn), are reported. In all the 39 Gpn residues conformationally characterized so far, the torsion angles about the C-alpha-C-beta and C-beta-C-gamma bonds are restricted to the gauche conformation (+/- 60 degrees). The Gpn residue is constrained to adopt folded conformations resulting in the formation of intramolecularly hydrogen-bonded structures even in short peptides. The peptides Boc-Ac(6)c-Gpn-OMe 1 and Boc-Gpn-Aib-Gpn-Aib-OMe 2 provide examples of C-7 conformation; peptides Boc-Gpn-Aib-OH 3, Boc-Ac(6)c-Gpn-OH 4, Boc-Val-Pro-Gpn-OH 5, Piv-Pro-Gpn-Val-OMe 6, and Boc-Gpn-Gpn-Leu-OMe 7 provide examples of C-9 conformation; peptide Boc-Ala-Aib-Gpn-Aib-Ala-OMe 8 provides an example of C-12 conformation and peptides Boc-beta Leu-Gpn-Val-OMe 9 and Boc-beta Phe-Gpn-Phe-OMe 10 provide examples of C-13 conformation. Gpn peptides provide examples of backbone expanded mimetics for canonical alpha-peptide turns like the gamma (C-7) and the beta (C-10) turns. The hybrid beta gamma sequences provide an example of a mimetic of the C-13 alpha-turn formed by three contiguous alpha-amino acid residues. Two examples of folded tripeptide structures, Boc-Gpn-beta Phe-Leu-OMe 11 and Boc-Aib-Gpn-beta Phg-NHMe 12, lacking internal hydrogen bonds are also presented. An analysis of available Gpn residue conformations provides the basis for future design of folded hybrid peptides.

Cloning, overexpression, folding and purification of a biosynthetically derived three disulfide scorpion toxin (BTK-2) from Mesobuthus tamulus

BTK-2, a 32 residue scorpion toxin initially identified in the venom of red Indian scorpion Mesobuthus tamulus was cloned, overexpressed and purified using Cytochrome 155 fusion protein system developed in our laboratory. The synthetic gene coding for the peptide was designed taking into account optimal codon usage by Escherichia coli. High expression levels of the fusion protein enabled facile purification of this peptide. The presence of disulfide bonded isomers, occurring as distinctly populated states even in the fusion protein, were separated by gel filtration chromatography. The target peptide was liberated from the host protein by Tev protease cleavage and subsequent purification was achieved using RP-HPLC methods. Reverse phase HPLC clearly showed the presence of at least two isomeric forms of the peptide that were significantly populated. The oxidative folding of BTK-2 was achieved under ambient conditions during the course of purification. Structural characterization of the two forms, by solution homonuclear and heteronuclear NMR methods, has shown that these two forms exhibit significantly different structural properties, and represent the natively folded and a "misfolded" form of the peptide. The formation of properly folded BTK-2 as a major fraction without the use of in vitro oxidative refolding methods clearly indicate the versatility of the Cytochrome b(5) fusion protein system for the efficient production of peptides for high resolution NMR studies.

Dynamic nature of proteins - interpretation of residual dipolar couplings

Protein conformations and dynamics can be studied by nuclear magnetic resonance spectroscopy using dilute liquid crystalline samples. This work clarifies the interpretation of residual dipolar coupling data yielded by the experiments. It was discovered that unfolded proteins without any additional structure beyond that of a mere polypeptide chain exhibit residual dipolar couplings. Also, it was found that molecular dynamics induce fluctuations in the molecular alignment and doing so affect residual dipolar couplings. The finding clarified the origins of low order parameter values observed earlier. The work required the development of new analytical and computational methods for the prediction of intrinsic residual dipolar coupling profiles for unfolded proteins. The presented characteristic chain model is able to reproduce the general trend of experimental residual dipolar couplings for denatured proteins. The details of experimental residual dipolar coupling profiles are beyond the analytical model, but improvements are proposed to achieve greater accuracy. A computational method for rapid prediction of unfolded protein residual dipolar couplings was also developed. Protein dynamics were shown to modulate the effective molecular alignment in a dilute liquid crystalline medium. The effects were investigated from experimental and molecular dynamics generated conformational ensembles of folded proteins. It was noted that dynamics induced alignment is significant especially for the interpretation of molecular dynamics in small, globular proteins. A method of correction was presented. Residual dipolar couplings offer an attractive possibility for the direct observation of protein conformational preferences and dynamics. The presented models and methods of analysis provide significant advances in the interpretation of residual dipolar coupling data from proteins.

Modular design of synthetic protein mimics. Crystal structure of two seven-residue helical peptide segments linked by .epsilon.-aminocaproic acid

Two seven-residue helical segments, Val-Ala-Leu-Aib-Val-Ala-Leu, were linked synthetically with an epsilon-aminocaproic acid (Acp) linker with the intention of making a stable antiparallel helix-helix motif. The crystal structure of the linked peptide Boc-Val-Ala-Leu-Aib-Val-Ala-Leu-Acp-Val-Ala-Leu-Aib-Val-Ala-Leu-OMe (1) shows the two helices displaced laterally from each other by the linker, but the linker has not folded the molecule into a close-packed antiparallel conformation. Two strong intermolecular NH...O = C hydrogen bonds are formed between the top of the lower helix of one molecule and the bottom of the upper helix in a laterally adjacent molecule to give the appearance of an extended single helix. The composite peptide with Boc and OMe end groups, C76H137N15O18.H2O, crystallize in space group P2(1) with a = 8.802 (1) angstrom, b = 20.409 (4) angstrom, c = 26.315 (3) angstrom, and beta = 90.72 (1)degrees; overall agreement R = 7.86% for 5030 observed reflections (\F(o)\ > 3-sigma(F)); resolution = 0.93 angstrom. Limited evidence for a more compact conformation in solution consistent with an antiparallel helix arrangement is obtained by comparison of the HPLC retention times and CD spectra of peptide 1 with well-characterized continuous helices of similar length and sequence.

Conformation of di-n-propylglycine residues (Dpg) in peptides: crystal structures of a type I 'beta-turn forming tetrapeptide and an alpha-helical tetradecapeptide

The crystal structures of two oligopeptides containing di-n-propylglycine (Dpg) residues, Boc-Gly-Dpg-Gly-Leu-OMe (1) and Boc-Val-Ala-Leu-Dpg-Val-Ala-Leu-Val-Ala-Leu-Dpg-Val-Ala-Leu-OMe (2) are presented. Peptide 1 adopts a type I' beta-turn conformation with Dpg(2)-Gly(3) at the corner positions. The 14-residue peptide 2 crystallizes with two molecules in the asymmetric unit, both of which adopt alpha-helical conformations stabilized by 11 successive 5 -> 1 hydrogen bonds. In addition, a single 4 -> 1 hydrogen bond is also observed at the N-terminus. All live Dpg residues adopt backbone torsion angles (phi, psi) in the helical region of conformational space. Evaluation of the available structural data on Dpg peptides confirm the correlation between backbone bond angle N-C-alpha-C' (tau) and the observed backbone phi,psi values. For tau > 106 degrees, helices are observed, while fully extended structures are characterized by tau < 106 degrees. The mean r values for extended and folded conformations for the Dpg residue are 103.6 degrees +/- 1.7 degrees and 109.9 degrees +/- 2.6 degrees, respectively. Copyright (C) 2007 European Peptide Society and John Wiley & Sons, Ltd.

Design of a compact wideband bandpass filter

This article presents the analysis and design of a compact multi-layer layer, high selectivity wideband bandpass filter using stub loaded and `U' shaped resonators over a slotted bottom ground plane. While the resonators with folded open circuit stub loadings create the desired bandpass characteristics. the IT shaped resonators reduce the size of filter. The slotted bottom ground plane is used to enhance the coupling to achieve the desired bandwidth. The proposed filter has been analyzed using circuit model, and the results were verified through full wave simulations and measurements. The fabricated filter is compact and measures a size of 18 mm x 25 mm x 1.6 MM. (C) 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1387-1389, 2010: Published online in Wiley InterScience (www.interscience.wiley.com).

The Structure and Functions of Hantavirus Nucleocapsid Protein

Hantaviruses, members of the genus Hantavirus in the Bunyaviridae family, are enveloped single-stranded RNA viruses with tri-segmented genome of negative polarity. In humans, hantaviruses cause two diseases, hemorrhagic fever with renal syndrome (HFRS) and hantavirus pulmonary syndrome (HPS), which vary in severity depending on the causative agent. Each hantavirus is carried by a specific rodent host and is transmitted to humans through excreta of infected rodents. The genome of hantaviruses encodes four structural proteins: the nucleocapsid protein (N), the glycoproteins (Gn and Gc), and the polymerase (L) and also the nonstructural protein (NSs). This thesis deals with the functional characterization of hantavirus N protein with regard to its structure. Structural studies of the N protein have progressed slowly and the crystal structure of the whole protein is still not available, therefore biochemical assays coupled with bioinformatical modeling proved essential for studying N protein structure and functions. Presumably, during RNA encapsidation, the N protein first forms intermediate trimers and then oligomers. First, we investigated the role of N-terminal domain in the N protein oligomerization. The results suggested that the N-terminal region of the N protein forms a coiled-coil, in which two antiparallel alpha helices interact via their hydrophobic seams. Hydrophobic residues L4, I11, L18, L25 and V32 in the first helix and L44, V51, L58 and L65 in the second helix were crucial for stabilizing the structure. The results were consistent with the head-to-head, tail-to-tail model for hantavirus N protein trimerization. We demonstrated that an intact coiled-coil structure of the N terminus is crucial for the oligomerization capacity of the N protein. We also added new details to the head-to-head, tail-to-tail model of trimerization by suggesting that the initial step is based on interaction(s) between intact intra-molecular coiled-coils of the monomers. We further analyzed the importance of charged aa residues located within the coiled-coil for the N protein oligomerization. To predict the interacting surfaces of the monomers we used an upgraded in silico model of the coiled-coil domain that was docked into a trimer. Next the predicted target residues were mutated. The results obtained using the mammalian two-hybrid assay suggested that conserved charged aa residues within the coiled-coil make a substantial contribution to the N protein oligomerization. This contribution probably involves the formation of interacting surfaces of the N monomers and also stabilization of the coiled-coil via intramolecular ionic bridging. We proposed that the tips of the coiled-coils are the first to come into direct contact and thus initiate tight packing of the three monomers into a compact structure. This was in agreement with the previous results showing that an increase in ionic strength abolished the interaction between N protein molecules. We also showed that residues having the strongest effect on the N protein oligomerization are not scattered randomly throughout the coiled-coil 3D model structure, but form clusters. Next we found evidence for the hantaviral N protein interaction with the cytoplasmic tail of the glycoprotein Gn. In order to study this interaction we used the GST pull-down assay in combination with mutagenesis technique. The results demonstrated that intact, properly folded zinc fingers of the Gn protein cytoplasmic tail as well as the middle domain of the N protein (that includes aa residues 80 248 and supposedly carries the RNA-binding domain) are essential for the interaction. Since hantaviruses do not have a matrix protein that mediates the packaging of the viral RNA in other negatve stranded viruses (NSRV), hantaviral RNPs should be involved in a direct interaction with the intraviral domains of the envelope-embedded glycoproteins. By showing the N-Gn interaction we provided the evidence for one of the crucial steps in the virus replication at which RNPs are directed to the site of the virus assembly. Finally we started analysis of the N protein RNA-binding region, which is supposedly located in the middle domain of the N protein molecule. We developed a model for the initial step of RNA-binding by the hantaviral N protein. We hypothesized that the hantaviral N protein possesses two secondary structure elements that initiate the RNA encapsidation. The results suggest that amino acid residues (172-176) presumably act as a hook to catch vRNA and that the positively charged interaction surface (aa residues 144-160) enhances the initial N-RNA interacation. In conclusion, we elucidated new functions of hantavirus N protein. Using in silico modeling we predicted the domain structure of the protein and using experimental techniques showed that each domain is responsible for executing certain function(s). We showed that intact N terminal coiled-coil domain is crucial for oligomerization and charged residues located on its surface form a interaction surface for the N monomers. The middle domain is essential for interaction with the cytoplasmic tail of the Gn protein and RNA binding.

Folding a Polymer via Two-Point Interaction with an External Folding Agent: Use of H-Bonding and Charge-Transfer Interactions

A polymer containing electron-rich aromatic donors (1,5-dialkoxynaphthalene (DAN)) was coerced into a folded state by an external folding agent that contained an electron-deficient aromatic acceptor (pyromellitic diimide (PM)) unit. The donor-containing polymer was designed to carry a tertiary amine moiety in the linking segment, which served as an H-bonding site for reinforcing the interaction with the acceptor containing folding agent that also bore a carboxylic acid group. The H-bonding interaction of the carboxylic acid and the tertiary amine brings the PDI unit between two adjacent DAN units along the polymer backbone to induce charge-transfer (C-T) interactions, and this in turn causes the polymer chain to form a pleated structure. Evidence for the formation of such a pleated structure was obtained from NMR titration studies and also by monitoring the C-T band in their UV-visible spectra. By varying the length of the segment that links the PDI acceptor to the carboxylic acid group, we showed that the most effective folding agent was the one that had a single carbon spacer, as evident from the highest value of the association constant. Control experiments with propionic acid clearly demonstrated the importance of the additional C-T interactions for venerating the folded structures. Further, solution viscosity measurements in the presence of varying amounts of the folding agent revealed a gradual stiffening of the chain in the case of the PDI carrying carboxylic acid, whereas no such affect was seen in the case of simple propionic acid. These observations were supported by D FT calculations of the interactions of a dimeric model of the polymer with the various folding agents; here too the stability of the complex was seen to be highest in the case of the single carbon spacer.

Helical Conformations of Hexapeptides Containing N-Terminus Diproline Segments

The role of N-terminus diproline segments in facilitating helical folding in short peptides has been investigated in a set of model hexapeptides of the type Piv-Xxx-Yyy-Aib-Leu-Aib-Phe-OMe (Piv, pivaloyl). Nine sequences have been investigated with the following N-terminus dipeptide segments: (D)Pro-Ala (4) and Pro-Psi Pro (5, Psi, pseudoproline), Ala-Ala (6), Ala-Pro (7), Pro-Ala (8), Aib-Ala (9), Ala-Aib (10). The analog sequences Piv-Pro-Pro-Ala-Leu-Aib-Phe-OMe (2) and Piv-Pro-Pro-Ala-Aib-Ala-Aib-OMe (3) have also been studied. Solid state conformations have been determined by X-ray crystallography for peptides 4, 6, and 8 and compared with the previously determined crystal structure of peptide 1 (Boc-Pro-Pro-Aib-Leu-Aib-Val-OMe); (Rai et al., JACS 2006, 128, 7916-7928). Peptides 1 and 6 adopt almost identical helical conformations with unfolding of the helix at the N-terminus Pro (1) residue. Peptide 4 reveals the anticipated (D)Pro-Ala type II' beta-turn, followed by a stretch of 3(10)-helix. Peptide 8 adopts a folded conformation stabilized by four successive 4 -> 1 intramolecular hydrogen bonds. Ala (2) adopts an alpha(L) conformation, resulting in a type II beta-turn conformation followed by a stretch of 3(10)-helix. Conformational properties in solution were probed using solvent perturbation of NH chemical shifts which permit delineation of hydrogen bonded NH groups and nuclear Overhauser effects (NOEs) between backbone protons, which are diagnostic of local residue conformations. The results suggest that continuous helical conformations are indeed significantly populated for peptides 2 and 3. Comparison of the results for peptides 1 and 2, suggest that there is a significant influence of the residue that follows diproline segments in influencing backbone folding. (C) 2010 Wiley Periodicals, Inc. Biopolymers (Pept Sci) 94: 360-370, 2010.

Negative differential conductance and effective electron mass in highly asymmetric ballistic bilayer graphene nanoribbon

We present a simplified theory of the effective momentum mass (EMM) and ballistic current–voltage relationship in a degenerate two-folded highly asymmetric bilayer graphene nanoribbon. With an increase in the gap, the density-of-states in the lower set of subbands increases more than that of the upper set. This results in a phenomenological population inversion of carriers, which is reflected through a net negative differential conductance (NDC). It is found that with the increase of the ribbon width, the NDC also increases. The population inversion also signatures negative values of EMM above a certain ribbon-width for the lower set of subbands, which increases in a step-like manner with the applied longitudinal static bias. The well-known result for symmetric conditions has been obtained as a special case.