Effects of Hydrophobicity on the Antifungal Activity of α‐Helical Antimicrobial Peptides

We utilized a series of analogs of D‐V13K (a 26‐residue amphipathic α‐helical antimicrobial peptide, denoted D1) to compare and contrast the role of hydrophobicity on antifungal and antibacterial activity to the results obtained previously with Pseudomonas aeruginosa strains. Antifungal activity for zygomycota fungi decreased with increasing hydrophobicity (D‐V13K/A12L/A20L/A23L, denoted D4, the most hydrophobic analog was sixfold less active than D1, the least hydrophobic analog). In contrast, antifungal activity for ascomycota fungi increased with increasing hydrophobicity (D4, the most hydrophobic analog was fivefold more active than D1). Hemolytic activity is dramatically affected by increasing hydrophobicity with peptide D4 being 286‐fold more hemolytic than peptide D1. The therapeutic index for peptide D1 is 1569‐fold and 62‐fold better for zygomycota fungi and ascomycota fungi, respectively, compared with peptide D4. To reduce the hemolytic activity of peptide D4 and improve/maintain the antifungal activity of D4, we substituted another lysine residue in the center of the non‐polar face (V16K) to generate D5 (D‐V13K/V16K/A12L/A20L/A23L). This analog D5 decreased hemolytic activity by 13‐fold, enhanced antifungal activity to zygomycota fungi by 16‐fold and improved the therapeutic index by 201‐fold compared with D4 and represents a unique approach to control specificity while maintaining high hydrophobicity in the two hydrophobic segments on the non‐polar face of D5.     

1H NMR studies of prototypical helical designer peptides A comparative study of the amide chemical shift dependency on temperature and polypeptide sequence

A series of designer α-helical peptides with hydrophobic residues located at different positions along the sequence (PH-1.0 =L YQEL QKL TQTL K, PH-1.19 =L YQEL QKL TQTL FK, PH-1.12 =L YQEL QKL L QTL K,PH-1.13 =L YQEL QKL TL TL K,PH-1.4 =L YQEL QKL TQTTK) were analyzed using one- and two-dimensional NMR methods (TOCSY and NOESY). The central feature of these designer peptides is the incorporation of a maximal hydrophobic strip which may play a role in antigen processing and the nucleation of α-helices in proteins (J. Immunol. 145 , 899, 1990). Using the 2D-NMR, sequence specific assignments and NOE connectivities were determined in all peptides when dissolved in H₂O/TFE mixtuRes NOE connectivities indicated that all these peptides are helical in this medium. An unusually large number of NOEs was found for all these designer peptides. This is in accord with ultracentrifugation studies that showed that PH-1.0 forms a trimer in 50% H₂O/TFE mixtures. Other peptides in the series behave in similar manner as PH-1.0. The structural differences among these peptides was addressed using the backbone amide chemical shift temperature coefficients, ?, and the differences between the observed and random coil values, Δδ_HN. The Δδ_HN patterns along the peptide sequence are consistent with those expected for amphiphilic α-helices, where most Δδ_HN values are below zero. However, no significant differences among the peptides in this series can be detected on the Δδ_HN patterns, with the exception of PH-1.12. The ? values reveal differences among the peptides of the series. The patterns of ? along the peptide sequences are similar to that found for Δδ_HN for PH-1.0, PH-1.19 and PH-1.4. The other peptides in the series, PH-1.12 and PH-1.13, showed different patterns for ?. The latter parameter was used to evaluate the helicity of this series of peptides. According to this parameter the relative helicity of this series is as follows: PH-1.12 < PH-I .O < PH-1.4 < PH-1.19 < PH-1.13 The NMR data shown here correlated well with the helical propensities predicted for polypeptide sequences using statistical arguments (Proc. Nutl. Acad Sci. USA, 90, 9100, 1993). 0 Munksgaard 1996.     

Synthesis of cyclic tryptathionine peptides

The helicity of the tryptathionine moiety of the phallotoxins has been recognized by comparison with cyclic tryptathionine tripeptides. In order to investigate the influence of the configuration of the component amino acids on the conformation of the cyclic peptides, six analogue thioether tripeptides containing L- and D-alanine and L- and D-cysteine, respectively, have been synthesized. The CD spectra of the peptides are very similar to each other, showing mirror images of the CD of phalloidin and, therefore, negative helicity. The spectra of the D-cysteine containing compounds differ from the L-cysteine containing compounds by their weakly positive ellipticity values around 270 nm. The cyclization reaction of Boc-Hpi-D-Ala-D-Cys(STrt)OCH₃, along with the cyclic tripeptide, afforded a cyclic hexapeptide by dimerization. The CD spectrum of the dimer is very similar to that of phalloidin, thus pointing to a positive helicity of its two tryptathionine moieties. The dimeric thioether peptide forms a rather strong complex with Cu²⁺ ions.     

Rational design of peptides with anti-HCV/HIV activities and enhanced specificity

Lack of vaccines for HCV and HIV makes the antiviral drug development urgently needed. The recently identified HCV NS5A-derived virucidal peptide (C5A) demonstrated a wide spectrum of activities against viruses. In this study, the C5A sequence SWLRDIWDWICEVLSDFK was utilized as the framework to study the effect of the modulation of peptide helicity and hydrophobicity on its anti-HCV and anti-HIV activities. Peptide helicity and hydrophobicity were altered by substitutions of varying amino acids on the non-polar face of C5A. Peptide hydrophobicity has been proved to play a crucial role in peptide anti-HCV or anti-HIV activities. Peptide helicity was relatively independent with antiviral activity. However, peptide analogs with dimerized structure in an aqueous medium while maintaining the ability to be induced into a more helical structure in a hydrophobic environment may tend to show comparable or improved antiviral activity and specificity to C5A. By modulating peptide helicity and hydrophobicity, we improved the specificity of C5A against HCV and HIV by 23- and 69-fold, respectively, in terms of the ratio of hemolytic activity to antiviral activity. We demonstrated that obtained by de novo design approach, peptide I6L/I10L/V13L may be a promising candidate as a new anti-HCV and anti-HIV therapeutic.     

A Chiral Supramolecular Polymer Membrane with no Chiral Substituents by Highly Selective Photocyclic Aromatization of a One‐Handed Helical Cis‐cisoidal Polyphenylacetylene

A one‐handed helical cis‐cisoidal polyphenylacetylene ( P ) is successfully synthesized by an asymmetric‐induced polymerization of a chiral‐substituted phenylacetylene having two hydroxymethyl groups. Quantitative photocyclic aromatization of P in a membrane‐state followed by in situ removal of the chiral substituents affords a one‐handed helical supramolecular polymer membrane of the corresponding cyclic trimer ( T) with no chiral substituents. The original one‐handed helicity of the precursor P is successfully transmitted to the resulting supra­molecular helicity of de‐ T . Both the hydrogen bonds and the π–π stacking are thought to contribute to the retention of the chirality during the two‐step reaction in the membrane state.

     

Perspective: peptides as mimics of transmembrane segments in proteins

Abstract: Peptide-based approaches to protein structure within membranes have proven enormously valuable. When one focusses on the detailed manner through which membrane proteins actually traverse the cell bilayer, a simple observation emerges: helical peptide segments of 20 amino acids each constitute the only tangible connection between the inside and outside of the cell. Thus, a major step towards understanding the key relationships between biological function and membrane protein structure can be taken through characterization, by composition, sequence, chain length, hydrophobicity and conformation, of hydrophobic peptides designed as mimics of transmembrane segments.     

From the Cover: Helicity conservation by flow across scales in reconnecting vortex links and knots

The conjecture that helicity (or knottedness) is a fundamental conserved quantity has a rich history in fluid mechanics, but the nature of this conservation in the presence of dissipation has proven difficult to resolve. Making use of recent advances, we create vortex knots and links in viscous fluids and simulated superfluids and track their geometry through topology-changing reconnections. We find that the reassociation of vortex lines through a reconnection enables the transfer of helicity from links and knots to helical coils. This process is remarkably efficient, owing to the antiparallel orientation spontaneously adopted by the reconnecting vortices. Using a new method for quantifying the spatial helicity spectrum, we find that the reconnection process can be viewed as transferring helicity between scales, rather than dissipating it. We also infer the presence of geometric deformations that convert helical coils into even smaller scale twist, where it may ultimately be dissipated. Our results suggest that helicity conservation plays an important role in fluids and related fields, even in the presence of dissipation.In addition to energy, momentum, and angular momentum, ideal (Euler) fluids have an additional conserved quantity—helicity (Eq. 1)—which measures the linking and knotting of the vortex lines composing a flow (1). For an ideal fluid, the conservation of helicity is a direct consequence of the Helmholtz laws of vortex motion, which both forbid vortex lines from ever crossing and preserve the flux of vorticity, making it impossible for linked or knotted vortices to ever untie (1, 2). Because conservation laws are of fundamental importance in understanding flows, the question of whether this topological conservation law extends to real, dissipative systems is of clear and considerable interest. The general importance of this question is further underscored by the recent and growing impact knots and links are having across a range of fields, including plasmas (3, 4), liquid crystals (5, 6), optical (7), electromagnetic (8), and biological structures (9–11), cosmic strings (12, 13), and beyond (14). Determining whether and how helicity is conserved in the presence of dissipation is therefore paramount in understanding the fundamental dynamics of real fluids and the connections between tangled fields across systems.The robustness of helicity conservation in real fluids is unclear because dissipation allows the topology of field lines to change. For example, in viscous flows vorticity will diffuse, allowing nearby vortex tubes to “reconnect” (Fig. 1 A–C), creating or destroying the topological linking of vortices. This behavior is not unique to classical fluids: analogous reconnection events have also been experimentally observed in superfluids (15) and coronal loops of plasma on the surface of the sun (16). In general, these observed reconnection events exhibit divergent, nonlinear dynamics that makes it difficult to resolve helicity dynamics theoretically (4, 17, 18). On the other hand, experimental tests of helicity conservation have been hindered by the lack of techniques to create vortices with topological structure. Thanks to a recent advance (19), this is finally possible.Open in a separate windowFig. 1.(A) A sketch of the evolution of vortex tube topology in ideal (Euler) and viscous (Navier–Stokes) flow. Dissipative flows allow for reconnections of vortex tubes, and so tube topology is not conserved. (B) Two frames of a 3D reconstruction of a vortex reconnection in experiment, which turns an initially linked pair of rings into a single twisted ring. (C) A close-up view of the reconnection in B. (D) If a tube is subdivided into multiple tubes, linking between the two may be created by introducing a twist into the pair. (E) Similarly, if a coiled tube is subdivided, linking can result even without adding twist. This can be seen either by calculating the linking number for the pair, or imagining trying to separate the two. (F) In a continuum fluid, the vortex tube may be regarded as a bundle of vortex filaments, which may be twisted. In this case, a twist of Δθ ～ 0.7 × 2π results in a total helicity of ? ～ 0.7Γ². (G) If the vortex tube is coiled, linking will also be introduced, as in E. Conceptually, this coiling can be regarded as producing a net rotation of the vortex bundle even when it is everywhere locally untwisted.By performing experiments on linked and knotted vortices in water, as well as numerical simulations of Bose–Einstein condensates [a compressible superfluid (20)] and Biot–Savart vortex evolution, we investigate the conservation of helicity, in so far as it can be inferred from the center-lines of reconnecting vortex tubes. We describe a new method for quantifying the storage of helicity on different spatial scales of a thin-core vortex: a “helistogram.” Using this analysis technique, we find a rich structure in the flow of helicity, in which geometric deformations and vortex reconnections transport helicity between scales. Remarkably, we find that helicity can be conserved even when vortex topology changes dramatically, and identify a system-independent geometric mechanism for efficiently converting helicity from links and knots into helical coils.     

Helicity is the only integral invariant of volume-preserving transformations

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional ? defined on exact divergence-free vector fields of class C¹ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that ? is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.Incompressible inviscid fluids are modeled by the 3D Euler equations, which assert that the velocity field u(x, t) of the fluid flow must satisfy the system of differential equations?_tu + (u ? ?)u = ??p, ?div?u = 0.Here the scalar function p(x, t) is another unknown of the problem, which physically corresponds to the pressure of the fluid.It is customary to introduce the vorticity ω: = curl?u to simplify the analysis of these equations, as it enables us to get rid of the pressure function. In terms of the vorticity, the Euler equations read as?_tω = [ω, u], [1]where [ω, u]: = (ω ? ?)u ? (u ? ?)w is the commutator of vector fields and u can be written in terms of ω, using the Biot–Savart lawu(x)=curl−1ω(x):=14π∫ℝ3ω(y)×(x−y)|x−y|3dy,[2]at least when the space variable is assumed to take values in the whole space ?³.The transport Eq. 1 was first derived by Helmholtz, who showed that the meaning of this equation is that the vorticity at time t is related to the vorticity at initial time t₀ via the flow of the velocity field, provided that the equation does not develop any singularities in the time interval [t₀, t]. More precisely, if ?_t,t0 denotes the (time-dependent) flow of the divergence-free field u, then the vorticity at time t is given by the action of the push forward of the volume-preserving diffeomorphism ?_t,t0 on the initial vorticity:ω( ? , t) = (?_t,t0)_??ω( ? , t₀).The phenomenon of the transport of vorticity gives rise to a new conservation law of the 3D Euler equations. Moffatt coined the term “helicity” for this conservation law in his influential paper (1) and exhibited its topological nature. Indeed, defining the helicity of a divergence-free vector field w in ?³ asℋ(w):=∫ℝ3w⋅curl−1 w dx,it turns out that the helicity of the vorticity ?(ω( ? , t)) is a conserved quantity for the Euler equations. In fact, helicity is also conserved for the compressible Euler equations provided the fluid is barotropic (i.e., the pressure is a function of the density).It is well known that the relevance of the helicity goes well beyond that of being a new (nonpositive) conserved quantity for the Euler equations. On the one hand, the helicity appears in other natural phenomena that are also described by a divergence-free field whose evolution is given by a time-dependent family of volume-preserving diffeomorphisms (2). For instance, the case of magnetohydrodynamics (MHD), where one is interested in the helicity of the magnetic field of a conducting plasma, has attracted considerable attention. On the other hand, it turns out that the helicity not only corresponds to a conserved quantity for evolution equations such as Euler or MHD, but also in fact defines an integral invariant for vector fields under any kind of volume-preserving diffeomorphisms (3).It is important to emphasize that conserved quantities of the Euler or MHD equations (e.g., the kinetic energy and the momentum) are not, in general, invariant under arbitrary volume-preserving diffeomorphisms, but they are invariant only under the very particular diffeomorphism defined by the flow of the velocity field of the fluid or conducting plasma. Perhaps the key feature of the helicity, which distinguishes it from other conserved quantities of Euler or MHD, is its invariance under any kind of volume-preserving transformations (in particular, it is invariant under the transport of the vorticity or the magnetic field by an arbitrary divergence-free vector field), so let us elaborate on this property.Helicity is often analyzed in the context of a compact three-dimensional manifold M without boundary, endowed with a Riemannian metric. The simplest case would be that of the flat 3-torus, which corresponds to fields on Euclidean space with periodic boundary conditions. To define the helicity in a general compact 3-manifold, let us introduce some notation. We denote by Xex1 the vector space of exact divergence-free vector fields on M of class C¹, endowed with its natural C¹ norm. We recall that a divergence-free vector field w is exact if its flux through any closed surface is zero (or, equivalently, if a vector field v exists such that w = curl?v). This is a topological condition, and in particular when the first homology group of the manifold is trivial (e.g., in the 3-sphere), every divergence-free field is automatically exact.As is well known, the reason to consider exact fields in this context is that, on exact fields, the curl operator has a well-defined inverse curl−1:Xex1→Xex1. The inverse of curl is a generalization to compact 3-manifolds of the Biot–Savart operator [2] and can also be written in terms of a (matrix-valued) integral kernel k(x, y) ascurl−1 w(x)=∫Mk(x,y) w(y) dy,[3]where dy now stands for the Riemannian volume measure. Using this integral operator, one can define the helicity of a vector field w on M asℋ(w):=∫Mw⋅curl−1 w dx.Here and in what follows the dot denotes the scalar product of two vector fields defined by the Riemannian metric on M. The helicity is then invariant under volume-preserving transformations; that is, ?(w) = ?(Φ_?w) for any diffeomorphism Φ of M that preserves volume.In view of expression [3] for the inverse of the curl operator, it is clear that the helicity is an integral invariant, meaning that it is given by the integral of a density of the form?(w) = ∫G(x, y, w(x), w(y))?dx?dy.Arnold and Khesin conjectured (ref. 3, section I.9) that, in fact, the helicity is the only integral invariant; that is, there are no other invariants of the form?(u): = ∫G(x₁, …, x_n, u(x₁), …, u(x_n))?dx₁ ? dx_n[4]with G a reasonably well-behaved function. Here all variables are assumed to be integrated over M.Our objective in this paper is to show, under some natural regularity assumptions, that the helicity is indeed the only integral invariant under volume-preserving diffeomorphisms. To this end, let us define a regular integral invariant as follows:

Definition:

Let ℐ:Xex1→ℝ be a C¹ functional. We say that ? is a regular integral invariant if (i) it is invariant under volume-preserving transformations, i.e., ?(w) = ?(Φ_?w) for any diffeomorphism Φ of M that preserves volume, and (ii) at any point w∈Xex1, the (Fréchet) derivative of ? is an integral operator with continuous kernel; that is,(Dℐ)w(u)=∫MK(w)⋅u,for any u∈Xex1, where K:Xex1→Xex1 is a continuous map.In the above definition and in what follows, we omit the Riemannian volume measure under the integral sign when no confusion can arise. Observe that any integral invariant of the form [4] is a regular integral invariant provided that the function G satisfies some mild technical assumptions. In particular, the helicity is a regular integral invariant.The following Theorem, which is the main result of this paper, shows that the helicity is essentially the only regular integral invariant in the above sense. The proof of this result is presented in Proof of the Main Theorem and is an extension to any closed 3-manifold of a theorem of Kudryavtseva (4), who proved an analogous result for divergence-free vector fields on 3-manifolds that are trivial bundles of a compact surface with boundary over the circle, which admit a cross-section and are tangent to the boundary. Kudryavtseva’s theorem is based on her work on the uniqueness of the Calabi invariant for area-preserving diffeomorphisms of the disk (5). We observe that our main result does not imply the aforementioned theorem because we consider manifolds without boundary.

Theorem.

Let ? be a regular integral invariant. Then ? is a function of the helicity; i.e., a C¹ function f:? → ? exists such that ? = f(?).We remark that this Theorem does not exclude the existence of other invariants of divergence-free vector fields under volume-preserving diffeomorphisms that are not C¹ or whose derivative is not an integral operator of the type described in the Definition above. For example, the Kolmogorov–Arnold–Moser (KAM)-type invariants recently introduced in ref. 6 are in no way related to the helicity, but they are not even continuous functionals on Xex1.Other types of invariants that have attracted considerable attention are the asymptotic invariants of divergence-free vector fields (7–13). These invariants are of nonlocal nature because they are defined in terms of a knot invariant (e.g., the linking number) and the flow of the vector field. In some cases, it turns out that the asymptotic invariant can be expressed as a regular integral invariant, as happens with the asymptotic linking number for divergence-free vector fields (8), the asymptotic signature (11), and the asymptotic Vassiliev invariants (10, 13) for ergodic divergence-free vector fields. In these cases, the authors prove that the corresponding asymptotic invariant is a function of the helicity, which is in perfect agreement with our main Theorem.The so-called higher-order helicities (14–16) are also invariants under volume-preserving diffeomorphisms. However, they are not defined for any divergence-free vector field, but just for vector fields supported on a disjoint union of solid tori. This property is, of course, not even continuous in Xex1, so these functionals do not fall in the category of the regular integral invariants considered in this paper.Our main Theorem is reminiscent of Serre’s theorem (17) showing that any conserved quantity of the 3D Euler equations that is the integral of a density depending on the velocity field and its first derivatives,ℐ(u):=∫ℝ3G(u(x,t),Du(x,t)) dx,is a function of the energy, the momentum, and the helicity. From a technical point of view, the proof of our main Theorem is totally different from the proof of Serre’s theorem, which is purely analytic, holds only in the Euclidean space, and is based on integral identities that the density G must satisfy to define a conservation law of the Euler equations.Even more importantly, from a conceptual standpoint it should be emphasized that Serre’s theorem applies to conserved quantities of the Euler equations, whereas our Theorem concerns the existence of functionals that are invariant under any kind of volume-preserving diffeomorphisms, which is a much stronger requirement, as explained in a previous paragraph. In particular, the fact that the energy and the momentum are not functions of the helicity does not contradict our main Theorem, because they are conserved by the evolution determined by the Euler equations but they are not invariant under the flow of an arbitrary divergence-free vector field. Accordingly, our Theorem does not mean that there are no other integrals of motion of the Euler (or MHD) equations.It is worth noting that one can construct well-behaved integral invariants of Lagrangian type that are invariant under general volume-preserving diffeomorphisms but are not functions of the helicity. These functionals arise in a natural manner in the analysis of the Euler or MHD equations especially when one considers integrable fields, that is, fields whose integral curves are tangent to a family of invariant surfaces. For example, one can define a partial helicity as the helicity integral taken over the region Ω bounded by an invariant surface of the field. In this context, if f is any well-behaved function (e.g., a smooth function supported on the region Ω covered by invariant surfaces) that is assumed to be transported under the action of the diffeomorphism group, the functionalℱ(f,w):=∫Mf w⋅curl−1 w dxis invariant under volume-preserving diffeomorphisms (and it is not a function of the helicity). The key point here is that the assumption that f is transformed in a Lagrangian way means that the action of the volume-preserving diffeomorphism group is not the one considered in this paper, which would beΦ ? ?(f, w): = ?(f, Φ_?w), but the one given byΦ ? ?(f, w): = ?(f?°?Φ^?1, Φ_?w).In this sense, this new action is defined on functionals mapping a function and a vector field (rather than just a vector field) to a number, so it does not fall within the scope of our Theorem. In the context of the partial helicity defined above, this action means that not only the vector field w, but also the function f and the region Ω where it is supported are transported by the fluid flow.