Label-free probe of HIV-1 TAT peptide binding to mimetic membranes

The transacting activator of transduction (TAT) protein plays a key role in the progression of AIDS. Studies have shown that a +8 charged sequence of amino acids in the protein, called the TAT peptide, enables the TAT protein to penetrate cell membranes. To probe mechanisms of binding and translocation of the TAT peptide into the cell, investigators have used phospholipid liposomes as cell membrane mimics. We have used the method of surface potential sensitive second harmonic generation (SHG), which is a label-free and interface-selective method, to study the binding of TAT to anionic 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-1′-rac-glycerol (POPG) and neutral 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) liposomes. It is the SHG sensitivity to the electrostatic field generated by a charged interface that enabled us to obtain the interfacial electrostatic potential. SHG together with the Poisson–Boltzmann equation yielded the dependence of the surface potential on the density of adsorbed TAT. We obtained the dissociation constants K_d for TAT binding to POPC and POPG liposomes and the maximum number of TATs that can bind to a given liposome surface. For POPC K_d was found to be 7.5 ± 2 μM, and for POPG K_d was 29.0 ± 4.0 μM. As TAT was added to the liposome solution the POPC surface potential changed from 0 mV to +37 mV, and for POPG it changed from −57 mV to −37 mV. A numerical calculation of K_d, which included all terms obtained from application of the Poisson–Boltzmann equation to the TAT liposome SHG data, was shown to be in good agreement with an approximated solution.The HIV type 1 (HIV-1) transacting activator of transduction (TAT) is an important regulatory protein for viral gene expression (1–3). It has been established that the TAT protein has a key role in the progression of AIDS and is a potential target for anti-HIV vaccines (4). For the TAT protein to carry out its biological functions, it needs to be readily imported into the cell. Studies on the cellular internalization of TAT have led to the discovery of the TAT peptide, a highly cationic 11-aa region (protein transduction domain) of the 86-aa full-length protein that is responsible for the TAT protein translocating across phospholipid membranes (5–8). The TAT peptide is a member of a class of peptides called cell-penetrating peptides (CPPs) that have generated great interest for drug delivery applications (ref. 9 and references therein). The exact mechanism by which the TAT peptide enters cells is not fully understood, but it is likely to involve a combination of energy-independent penetration and endocytosis pathways (8, 10). The first step in the process is high-affinity binding of the peptide to phospholipids and other components on the cell surface such as proteins and glycosaminoglycans (1, 9).The binding of the TAT peptide to liposomes has been investigated using a variety of techniques, each of which has its own advantages and limitations. Among the techniques are isothermal titration calorimetry (9, 11), fluorescence spectroscopy (12, 13), FRET (12, 14), single-molecule fluorescence microscopy (15, 16), and solid-state NMR (17). Second harmonic generation (SHG), as an interface-selective technique (18–24), does not require a label, and because SHG is sensitive to the interface potential, it is an attractive method to selectively probe the binding of the highly charged (+8) TAT peptide to liposome surfaces. Although coherent SHG is forbidden in centrosymmetric and isotropic bulk media for reasons of symmetry, it can be generated by a centrosymmetric structure, e.g., a sphere, provided that the object is centrosymmetric over roughly the length scale of the optical coherence, which is a function of the particle size, the wavelength of the incident light, and the refractive indexes at ω and 2ω (25–30). As a second-order nonlinear optical technique SHG has symmetry restrictions such that coherent SHG is not generated by the randomly oriented molecules in the bulk liquid, but can be generated coherently by the much smaller population of oriented interfacial species bound to a particle or planar surfaces. As a consequence the SHG signal from the interface is not overwhelmed by SHG from the much larger populations in the bulk media (25–28).The total second harmonic electric field, E_2ω, originating from a charged interface in contact with water can be expressed as (31–33)E2ω∝∑iχc,i(2)EωEω+∑jχinc,j(2)EωEω+χH2O(3)EωEωΦ,[1]where χc,i(2) represents the second-order susceptibility of the species i present at the interface; χinc,j(2) represents the incoherent contribution of the second-order susceptibility, arising from density and orientational fluctuations of the species j present in solution, often referred to as hyper-Rayleigh scattering; χH2O(3) is the third-order susceptibility originating chiefly from the polarization of the bulk water molecules polarized by the charged interface; Φ is the potential at the interface that is created by the surface charge; and E_ω is the electric field of the incident light at the fundamental frequency ω. The second-order susceptibility, χc,i(2), can be written as the product of the number of molecules, N, at the surface and the orientational ensemble average of the hyperpolarizability α_i⁽²⁾ of surface species i, yielding χc,i(2)=N〈αi(2)〉 (18). The bracket ?? indicates an orientational average over the interfacial molecules. The third term in Eq. 1 depicts a third-order process by which a second harmonic field is generated by a charged interface. This term is the focus of our work. The SHG signal is dependent on the surface potential created by the electrostatic field of the surface charges, often called the χ⁽³⁾ contribution to the SHG signal. The χ⁽³⁾ method has been used to extract the surface charge density of charged planar surfaces and microparticle surfaces, e.g., liposomes, polymer beads, and oil droplets in water (21, 25, 34–39).In this work, the χ⁽³⁾ SHG method is used to explore a biomedically relevant process. The binding of the highly cationic HIV-1 TAT peptide to liposome membranes changes the surface potential, thereby enabling the use of the χ⁽³⁾ method to study the binding process in a label-free manner. Two kinds of liposomes, neutral 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) and anionic 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-1′-rac-glycerol (POPG), were investigated. The chemical structures of TAT, POPC, and POPG lipids are shown in Scheme 1.Open in a separate windowScheme 1.Chemical structures of HIV-1 TAT (47–57) peptide and the POPC and POPG lipids.     

Fracture of model end-linked networks

Advances in polymer chemistry over the last decade have enabled the synthesis of molecularly precise polymer networks that exhibit homogeneous structure. These precise polymer gels create the opportunity to establish true multiscale, molecular to macroscopic, relationships that define their elastic and failure properties. In this work, a theory of network fracture that accounts for loop defects is developed by drawing on recent advances in network elasticity. This loop-modified Lake–Thomas theory is tested against both molecular dynamics (MD) simulations and experimental fracture measurements on model gels, and good agreement between theory, which does not use an enhancement factor, and measurement is observed. Insight into the local and global contributions to energy dissipated during network failure and their relation to the bond dissociation energy is also provided. These findings enable a priori estimates of fracture energy in swollen gels where chain scission becomes an important failure mechanism.

Models that link materials structure to macroscopic behavior can account for multiple levels of molecular structure. For example, the statistical, affine deformation model connects the elastic modulus E to the molecular structure of a polymer chain,Eaff=3νkbT(ϕo13Roϕ13R)2,[1]where ν is density of chains, ϕ is polymer volume fraction, R is end-to-end distance, ϕo and R_o represent the parameters taken in the reference state that is assumed to be the reaction concentration in this work, and k_bT is the available thermal energy where k_b is Boltzmann’s constant and T is temperature (1–6). Refinements to this model that account for network-level structure, such as the presence of trapped entanglements or number of connections per junction, have been developed (7–11). Further refinements to the theory of network elasticity have been developed to account for dynamic processes such as chain relaxation and solvent transport (12–17). Together these refinements link network elasticity to chain-level molecular structure, network-level structure, and the dynamic processes that occur at both size scales.While elasticity has been connected to multiple levels of molecular structure, models for network fracture have not developed to a similar extent. The fracture energy G_c typically relies upon the large strain deformation behavior of polymer networks, making it experimentally difficult to separate the elastic energy released upon fracture from that dissipated through dynamic processes (18–26). In fact, most fracture theories have been developed at the continuum scale and have focused on modeling dynamic dissipation processes (27). An exception to this is the theory of Lake and Thomas that connects the elastic energy released during chain scission to chain-level structure,Gc,LT=ChainsArea×Energy DissipatedChain=νRoNU,[2]where NU is the total energy released when a chain ruptures in which N represents the number of monomer segments in the chain and U the energy released per monomer (26).While this model was first introduced in 1967, experimental attempts to verify Lake–Thomas theory as an explicit model, as summarized in SI Appendix, have been unsuccessful. Ahagon and Gent (28) and Gent and Tobias (29) attempted to do this on highly swollen networks at elevated temperature but found that, while the scalings from Eq. 2 work well, an enhancement factor was necessary to observe agreement between theory and experiment. This led many researchers to conclude that Lake–Thomas theory worked only as a scaling argument. In 2008, Sakai et al. (30) introduced a series of end-linked tetrafunctional, star-like poly(ethylene glycol) (PEG) gels. Scattering measurements indicated a lack of nanoscale heterogeneities that are characteristic of most polymer networks (30–32). Fracture measurements on these well-defined networks were performed and it was again observed that an enhancement factor was necessary to realize explicit agreement between experiment and theory (33). Arora et al. (34) recently attempted to address this discrepancy by accounting for loop defects; however, different assumptions were used when inputting U to calculate Lake–Thomas theory values that again required the use of an enhancement factor to achieve quantitative agreement. In this work we demonstrate that refining the Lake–Thomas theory to account for loop defects while using the full bond dissociation energy to represent U yields excellent agreement between the theory and both simulation and experimental data without the use of any adjustable parameters.PEG gels synthesized via telechelic end-linking reactions create the opportunity to build upon previous theory to establish true multiscale, molecular to macroscopic relationships that define the fracture response of polymer networks. This paper combines pure shear notch tests, molecular dynamics (MD) simulations, and theory to quantitatively extend the concept of network fracture without the use of an enhancement factor. First, the control of molecular-level structure in end-linked gel systems is discussed. Then, the choice of molecular parameters used to estimate chain- and network-level properties is discussed. Experimental and MD simulation methods used when fracturing model end-linked networks are then presented. A theory of network fracture that accounts for loop defects is developed, in the context of other such models that have emerged recently, and tested against data from experiments and MD simulations. Finally, a discussion of the local and global energy dissipated during failure of the network is presented.     

Nature of dynamic gradients,glass formation,and collective effects in ultrathin freestanding films

Molecular, polymeric, colloidal, and other classes of liquids can exhibit very large, spatially heterogeneous alterations of their dynamics and glass transition temperature when confined to nanoscale domains. Considerable progress has been made in understanding the related problem of near-interface relaxation and diffusion in thick films. However, the origin of “nanoconfinement effects” on the glassy dynamics of thin films, where gradients from different interfaces interact and genuine collective finite size effects may emerge, remains a longstanding open question. Here, we combine molecular dynamics simulations, probing 5 decades of relaxation, and the Elastically Cooperative Nonlinear Langevin Equation (ECNLE) theory, addressing 14 decades in timescale, to establish a microscopic and mechanistic understanding of the key features of altered dynamics in freestanding films spanning the full range from ultrathin to thick films. Simulations and theory are in qualitative and near-quantitative agreement without use of any adjustable parameters. For films of intermediate thickness, the dynamical behavior is well predicted to leading order using a simple linear superposition of thick-film exponential barrier gradients, including a remarkable suppression and flattening of various dynamical gradients in thin films. However, in sufficiently thin films the superposition approximation breaks down due to the emergence of genuine finite size confinement effects. ECNLE theory extended to treat thin films captures the phenomenology found in simulation, without invocation of any critical-like phenomena, on the basis of interface-nucleated gradients of local caging constraints, combined with interfacial and finite size-induced alterations of the collective elastic component of the structural relaxation process.

Spatially heterogeneous dynamics in glass-forming liquids confined to nanoscale domains (1–7) play a major role in determining the properties of molecular, polymeric, colloidal, and other glass-forming materials (8), including thin films of polymers (9, 10) and small molecules (11–15), small-molecule liquids in porous media (2, 4, 16, 17), semicrystalline polymers (18, 19), polymer nanocomposites (20–22), ionomers (23–25), self-assembled block and layered (26–33) copolymers, and vapor-deposited ultrastable molecular glasses (34–36). Intense interest in this problem over the last 30 y has also been motivated by the expectation that its understanding could reveal key insights concerning the mechanism of the bulk glass transition.Considerable progress has been made for near-interface altered dynamics in thick films, as recently critically reviewed (1). Large amplitude gradients of the structural relaxation time, τ(z,T), converge to the bulk value, τ_bulk(T), in an intriguing double-exponential manner with distance, z, from a solid or vapor interface (13, 37–42). This implies that the corresponding effective activation barrier, F_total(z,T,H) (where H is film thickness), varies exponentially with z, as does the glass transition temperature, T_g (37). Thus the fractional reduction in activation barrier, ε(z,H), obeys the equation ε(z,H)≡1−Ftotal(z,T,H)/Ftotal,bulk(T)=ε0⁡exp(−z/ξF), where F_total,bulk(T) is the bulk temperature-dependent barrier and ξ_F a length scale of modest magnitude. Although the gradient of reduction in absolute activation barriers becomes stronger with cooling, the amplitude of the fractional reduction of the barrier gradient, quantified by ε₀, and the range ξ_F of this gradient, exhibit a weak or absent temperature dependence at the lowest temperatures accessed by simulations (typically with the strength of temperature dependence of ξ_F decreasing rather than increasing on cooling), which extend to relaxation timescales of order 10⁵ ps. This finding raises questions regarding the relevance of critical-phenomena–like ideas for nanoconfinement effects (1). Partially due to this temperature invariance, coarse-grained and all-atom simulations (1, 37, 42, 43) have found a striking empirical fractional power law decoupling relation between τ(z,T) and τ_bulk(T):τ(T,z)τbulk(T)∝(τbulk(T))−ε(z).[1]Recent theoretical analysis suggests (44) that this behavior is consistent with a number of experimental data sets as well (45, 46). Eq. 1 also corresponds to a remarkable factorization of the temperature and spatial location dependences of the barrier:Ftotal(z,T)=[1−ε(z)]Ftotal,bulk(T).[2]This finding indicates that the activation barrier for near-interface relaxation can be factored into two contributions: a z-dependent, but T-independent, “decoupling exponent,” ε(z), and a temperature-dependent, but position-insensitive, bulk activation barrier, F_total,bulk(T). Eq. 2 further emphasizes that ε(z) is equivalent to an effective fractional barrier reduction factor (for a vapor interface), 1−Ftotal(z,T,H)/Ftotal,bulk(T), that can be extracted from relaxation data.In contrast, the origin of “nanoconfinement effects” in thin films, and how much of the rich thick-film physics survives when dynamic gradients from two interfaces overlap, is not well understood. The distinct theoretical efforts for aspects of the thick-film phenomenology (44, 47–50) mostly assume an additive summation of one-interface effects in thin films, thereby ignoring possibly crucial cooperative and whole film finite size confinement effects. If the latter involve phase-transition–like physics as per recent speculations (14, 51), one can ask the following: do new length scales emerge that might be truncated by finite film size? Alternatively, does ultrathin film phenomenology arise from a combination of two-interface superposition of the thick-film gradient physics and noncritical cooperative effects, perhaps in a property-, temperature-, and/or thickness-dependent manner?Here, we answer these questions and establish a mechanistic understanding of thin-film dynamics for the simplest and most universal case: a symmetric freestanding film with two vapor interfaces. We focus on small molecules (modeled theoretically as spheres) and low to medium molecular weight unentangled polymers, which empirically exhibit quite similar alterations in dynamics under “nanoconfinement.” We do not address anomalous phenomena [e.g., much longer gradient ranges (29), sporadic observation of two distinct glass transition temperatures (52, 53)] that are sometimes reported in experiments with very high molecular weight polymers and which may be associated with poorly understood chain connectivity effects that are distinct from general glass formation physics (54–56).We employ a combination of molecular dynamics simulations with a zero-parameter extension to thin films of the Elastically Cooperative Nonlinear Langevin Equation (ECNLE) theory (57, 58). This theory has previously been shown to predict well both bulk activated relaxation over up to 14 decades (44–46) and the full single-gradient phenomenology in thick films (1). Here, we extend this theory to treat films of finite thickness, accounting for coupled interface and geometric confinement effects. We compare predictions of ECNLE theory to our previously reported (37, 43) and new simulations, which focus on translational dynamics of films comprised of a standard Kremer–Grest-like bead-spring polymer model (see SI Appendix). These simulations cover a wide range of film thicknesses (H, from 4 to over 90 segment diameters σ) and extend to low temperatures where the bulk alpha time is ∼0.1 μs (10⁵ Lennard Jones time units τ_LJ).The generalized ECNLE theory is found to be in agreement with simulation for all levels of nanoconfinement. We emphasize that this theory does not a priori assume any of the empirically established behaviors discovered using simulation (e.g., fractional power law decoupling, double-exponential barrier gradient, gradient flattening) but rather predicts these phenomena based upon interfacial modifications of the two coupled contributions to the underlying activation barrier– local caging constraints and a long-ranged collective elastic field. It is notable that this strong agreement is found despite the fact the dynamical ideas are approximate, and a simple hard sphere fluid model is employed in contrast to the bead-spring polymers employed in simulation. The basic unit of length in simulation (bead size σ) and theory (hard sphere diameter d) are expected to be proportional to within a prefactor of order unity, which we neglect in making comparisons.As an empirical matter, we find from simulation that many features of thin-film behavior can be described to leading order by a linear superposition of the thick-film gradients in activation barrier, that is:ε(z,H)=1−Ftotal(z,T,H)/Ftotal,bulk(T)≈ε0[exp(−z/ξF)+exp(−(H−z)/ξF)],[3]where the intrinsic decay length ξ_F is unaltered from its thick-film value and where ε₀ is a constant that, in the hypothesis of literal gradient additivity, is invariant to temperature and film thickness. We employ this functional form [originally suggested by Binder and coworkers (59)], which is based on a simple superposition of the two single-interface gradients, as a null hypothesis throughout this study: this form is what one expects if no new finite-size physics enters the thin-film problem relative to the thick film.However, we find that the superposition approximation progressively breaks down, and eventually entirely fails, in ultrathin films as a consequence of the emergence of a finite size confinement effect. The ECNLE theory predicts that this failure is not tied to a phase-transition–like mechanism but rather is a consequence of two key coupled physical effects: 1) transfer of surface-induced reduction of local caging constraints into the film, and 2) interfacial truncation and nonadditive modifications of the collective elastic contribution to the activation barrier.     

Residue level quantification of protein stability in living cells

The intracellular milieu differs from the dilute conditions in which most biophysical and biochemical studies are performed. This difference has led both experimentalists and theoreticians to tackle the challenging task of understanding how the intracellular environment affects the properties of biopolymers. Despite a growing number of in-cell studies, there is a lack of quantitative, residue-level information about equilibrium thermodynamic protein stability under nonperturbing conditions. We report the use of NMR-detected hydrogen–deuterium exchange of quenched cell lysates to measure individual opening free energies of the 56-aa B1 domain of protein G (GB1) in living Escherichia coli cells without adding destabilizing cosolutes or heat. Comparisons to dilute solution data (pH 7.6 and 37 °C) show that opening free energies increase by as much as 1.14 ± 0.05 kcal/mol in cells. Importantly, we also show that homogeneous protein crowders destabilize GB1, highlighting the challenge of recreating the cellular interior. We discuss our findings in terms of hard-core excluded volume effects, charge–charge GB1-crowder interactions, and other factors. The quenched lysate method identifies the residues most important for folding GB1 in cells, and should prove useful for quantifying the stability of other globular proteins in cells to gain a more complete understanding of the effects of the intracellular environment on protein chemistry.Proteins function in a heterogeneous and crowded intracellular environment. Macromolecules comprise 20–30% of the volume of an Escherichia coli cell and reach concentrations of 300–400 g/L (1, 2). Theory predicts that the properties of proteins and nucleic acids can be significantly altered in cells compared with buffer alone (3, 4). Nevertheless, most biochemical and biophysical studies are conducted under dilute (<10 g/L macromolecules) conditions. Here, we augment the small but growing list of reports probing the equilibrium thermodynamic stability of proteins in living cells (5–9), and provide, to our knowledge, the first measurement of residue-level stability under nonperturbing conditions.Until recently, the effects of macromolecular crowding on protein stability were thought to be caused solely by hard-core, steric repulsions arising from the impenetrability of matter (4, 10, 11). The expectation was that crowding enhances stability by favoring the compact native state over the ensemble of denatured states. Increased attention to transient, nonspecific protein-protein interactions (12–15) has led both experimentalists (16–19) and theoreticians (20–22) to recognize the effects of chemical interactions between crowder and test protein when assessing the net effect of macromolecular crowding. These weak, nonspecific interactions can reinforce or oppose the effect of hard-core repulsions, resulting in increased or decreased stability depending on the chemical nature of the test protein and crowder (23–26).We chose the B1 domain of streptococcal protein G (GB1) (27) as our test protein because its structure, stability and folding kinetics have been extensively studied in dilute solution (28–38). Its small size (56 aa; 6.2 kDa) and high thermal stability make GB1 well suited for studies by NMR spectroscopy.Quantifying the equilibrium thermodynamic stability of proteins relies on determining the relative populations of native and denatured states. Because the denatured state ensemble of a stable protein is sparsely populated under native conditions, stability is usually probed by adding heat or a cosolute to promote unfolding so that the concentration ratio of the two states can be determined (39). However, stability can be measured without these perturbations by exploiting the phenomenon of backbone amide H/D exchange (40) detected by NMR spectroscopy (41). The observed rate of amide proton (N–H) exchange, k_obs, is related to equilibrium stability by considering a protein in which each N–H exists in an open (exposed, exchange-competent) state, or a closed (protected, exchange-incompetent) state (40, 42):closed(N−H)⇌kclkopopen(N−H)→kintopen(N−D)⇌kopkclclosed(N−D).[1]Each position opens and closes with rate constants, k_op and k_cl (where K_op = k_op/k_cl), and exchange from the open state occurs with intrinsic rate constant, k_int. Values for k_int are based on exchange data from unstructured peptides (43, 44). If the test protein is stable (i.e., k_cl >> k_op), the observed rate becomes:kobs=kopkintkcl+kint.[2]Exchange occurs within two limits (42). At the EX1 limit, closing is rate determining, and k_obs = k_op. This limit is usually observed for less stable proteins and at basic pH (45). Most globular proteins undergo EX2 kinetics, where exchange from the open state is rate limiting (i.e., k_cl >> k_int), and k_obs values can be converted to equilibrium opening free energies, ΔGop°′ (46):kobs=kopkclkint=Kopkint[3]ΔGop°′=−RT⁡lnkobskint,[4]where RT is the molar gas constant multiplied by the absolute temperature.The backbone amides most strongly involved in H-bonded regions of secondary structure exchange only from the fully unfolded state, yielding a maximum value of ΔGop°′ (47–49). For these residues ΔGop°′ approximates the free energy of denaturation, ΔGden°′, providing information on global stability. Lower amplitude fluctuations of the native state can give rise to partially unfolded forms (50), resulting in residues with ΔGop°′ values less than those of the global unfolders.In summary, NMR-detected H/D exchange can measure equilibrium thermodynamic stability of a protein at the level of individual amino acid residues under nonperturbing conditions. Inomata et al. (51) used this technique to measure k_obs values in human cells for four residues in ubiquitin, but experiments confirming the exchange mechanism were not reported and opening free energies were not quantified. Our results fill this void and provide quantitative residue-level protein stability measurements in living cells under nonperturbing conditions.     

Unusual magnetotransport in twisted bilayer graphene

We present transport measurements of bilayer graphene with a 1.38^∘ interlayer twist. As with other devices with twist angles substantially larger than the magic angle of 1.1^∘, we do not observe correlated insulating states or band reorganization. However, we do observe several highly unusual behaviors in magnetotransport. For a large range of densities around half filling of the moiré bands, magnetoresistance is large and quadratic. Over these same densities, the magnetoresistance minima corresponding to gaps between Landau levels split and bend as a function of density and field. We reproduce the same splitting and bending behavior in a simple tight-binding model of Hofstadter’s butterfly on a triangular lattice with anisotropic hopping terms. These features appear to be a generic class of experimental manifestations of Hofstadter’s butterfly and may provide insight into the emergent states of twisted bilayer graphene.

The mesmerizing Hofstadter butterfly spectrum arises when electrons in a two-dimensional periodic potential are immersed in an out-of-plane magnetic field. When the magnetic flux Φ through a unit cell is a rational multiple p / q of the magnetic flux quantum Φ0=h/e, each Bloch band splits into q subbands (1). The carrier densities corresponding to gaps between these subbands follow straight lines when plotted as a function of normalized density n/ns and magnetic field (2). Here, n_s is the density of carriers required to fill the (possibly degenerate) Bloch band. These lines can be described by the Diophantine equation (n/ns)=t(Φ/Φ0)+s for integers s and t. In experiments, they appear as minima or zeros in longitudinal resistivity coinciding with Hall conductivity quantized at σxy=te2/h (3, 4). Hofstadter originally studied magnetosubbands emerging from a single Bloch band on a square lattice. In the following decades, other authors considered different lattices (5–7), the effect of anisotropy (6, 8–10), next-nearest-neighbor hopping (11–15), interactions (16, 17), density wave states (9), and graphene moirés (18, 19).It took considerable ingenuity to realize clean systems with unit cells large enough to allow conventional superconducting magnets to reach Φ/Φ0∼1. The first successful observation of the butterfly in electrical transport measurements was in GaAs/AlGaAs heterostructures with lithographically defined periodic potentials (20–22). These experiments demonstrated the expected quantized Hall conductance in a few of the largest magnetosubband gaps. In 2013, three groups mapped out the full butterfly spectrum in both density and field in heterostructures based on monolayer (23, 24) and bilayer (25) graphene. In all three cases, the authors made use of the 2% lattice mismatch between their graphene and its encapsulating hexagonal boron nitride (hBN) dielectric. With these layers rotationally aligned, the resulting moiré pattern was large enough in area that gated structures studied in available high-field magnets could simultaneously approach normalized carrier densities and magnetic flux ratios of 1. Later work on hBN-aligned bilayer graphene showed that, likely because of electron–electron interactions, the gaps could also follow lines described by fractional s and t (26).In twisted bilayer graphene (TBG), a slight interlayer rotation creates a similar-scale moiré pattern. Unlike with graphene–hBN moirés, in TBG there is a gap between lowest and neighboring moiré subbands (27). As the twist angle approaches the magic angle of 1.1^∘ the isolated moiré bands become flat (28, 29), and strong correlations lead to fascinating insulating (30–37), superconducting (31–33, 35–37), and magnetic (34, 35, 38) states. The strong correlations tend to cause moiré subbands within a fourfold degenerate manifold to move relative to each other as one tunes the density, leading to Landau levels that project only toward higher magnitude of density from charge neutrality and integer filling factors (37, 39). This correlated behavior obscures the single-particle Hofstadter physics that would otherwise be present.In this work, we present measurements from a TBG device twisted to 1.38^∘. When we apply a perpendicular magnetic field, a complicated and beautiful fan diagram emerges. In a broad range of densities on either side of charge neutrality, the device displays large, quadratic magnetoresistance. Within the magnetoresistance regions, each Landau level associated with ν=±8,±12,±16,… appears to split into a pair, and these pairs follow complicated paths in field and density, very different from those predicted by the usual Diophantine equation. Phenomenology similar in all qualitative respects appears in measurements on several regions of this same device with similar twist angles and in two separate devices, one at 1.59^∘ and the other at 1.70^∘ (see SI Appendix for details).We reproduce the unusual features of the Landau levels (LLs) in a simple tight-binding model on a triangular lattice with anisotropy and a small energetic splitting between two species of fermions. At first glance, this is surprising, because that model does not represent the symmetries of the experimental moiré structure. We speculate that the unusual LL features we experimentally observe can generically emerge from spectra of Hofstadter models that include the same ingredients we added to the triangular lattice model. With further theoretical work it may be possible to use our measurements to gain insight into the underlying Hamiltonian of TBG near the magic angle.     

n-type thermoelectric material Mg2Sn0.75Ge0.25 for high power generation

Thermoelectric power generation is one of the most promising techniques to use the huge amount of waste heat and solar energy. Traditionally, high thermoelectric figure-of-merit, ZT, has been the only parameter pursued for high conversion efficiency. Here, we emphasize that a high power factor (PF) is equivalently important for high power generation, in addition to high efficiency. A new n-type Mg₂Sn-based material, Mg₂Sn_0.75Ge_0.25, is a good example to meet the dual requirements in efficiency and output power. It was found that Mg₂Sn_0.75Ge_0.25 has an average ZT of 0.9 and PF of 52 μW⋅cm⁻¹⋅K⁻² over the temperature range of 25–450 °C, a peak ZT of 1.4 at 450 °C, and peak PF of 55 μW⋅cm⁻¹⋅K⁻² at 350 °C. By using the energy balance of one-dimensional heat flow equation, leg efficiency and output power were calculated with T_h = 400 °C and T_c = 50 °C to be of 10.5% and 6.6 W⋅cm⁻² under a temperature gradient of 150 °C⋅mm⁻¹, respectively.Thermoelectric power generation from waste heat is attracting more and more attention. Potential fuel efficiency enhancement by recovering the waste heat is beneficial for automobiles and many other applications (1, 2). In addition, solar thermoelectric generator provides an alternative route to convert solar energy into electrical power besides the photovoltaic conversion (3). Thermoelectric generator (TEG) can be regarded as a heat engine using electrons/holes as the energy carrier. The conversion efficiency of a TEG is related to the Carnot efficiency and the material’s average thermoelectric figure of merit ZT (4):η=Th−TcTh(1+ZT−11+ZT+Tc/Th),[1]where ZT = (S²σ/κ)T, and S, σ, κ, and T are Seebeck coefficient, electrical conductivity, thermal conductivity, and absolute temperature, respectively. Pursuing high ZT has been the focus of the entire thermoelectric community by applying various phonon engineering via nanostructuring approaches to reduce the thermal conductivity (5–7), or by exploring new compounds with intrinsically low thermal conductivity, such as compounds having complex crystalline structure, local rattlers, liquid-like sublattice, and highly distorted lattice (8–11). However, for practical applications, efficiency is not the only concern, and high output power density is as important as efficiency when the capacity of the heat source is huge (such as solar heat), or the cost of the heat source is not a big factor (such as waste heat from automobiles, steel industry, etc.). The output power density ω is defined as the output power W divided by the cross-sectional area A of the leg, i.e., ω = W/A, which is related to power factor PF = S²σ by the following:ω=14(Th−Tc)2LPF.[2]Eq. 2 contains two main parts: square of the temperature difference divided by leg length, and material power factor PF = S²σ. Clearly, to achieve higher power density for a given heat source, we have to either increase the power factor PF or decrease the leg length. However, decreasing the leg length could cause severe consequences such as increase of large heat flux that will increase the cost of the heat management at the cold end, increase of percentage of contact resistance in the device that will increase the parasitic loss and consequently decrease the energy conversion efficiency, increase of the thermal stress due to the larger thermal gradient leading to device failure, etc. Therefore, it is better to increase the power factor PF. Because PF is a pure material parameter, we can use it as a criterion in searching for new thermoelectric materials for high output power.A useful thermoelectric material should possess high ZT value for high efficiency, and also very importantly high PF for high output power. Ideally, temperature-independent ZT and PF over the whole temperature range from cold side to hot side are desired. However, both the ZT and PF of all materials show strong temperature dependency, usually increasing first with temperature and then decreasing when bipolar effect starts to play a role. The working temperature of thermoelectric materials is limited by the band energy gap E_g; e.g., Bi₂Te₃, a well-known thermoelectric material for applications below 200 °C, has an E_g of ∼0.13 eV (12). PbTe and associated materials have much higher peak ZT in the temperature range of 400–600 °C due to its larger E_g of 0.32 eV (13). However, the toxicity of lead, poor mechanical properties, and thermal instability above 400 °C seriously limit the application of Pb-based thermoelectric materials. Even though Mg₂Si, skutterudites, and half-Heuslers are promising for thermoelectric power generation at up to 500 °C [Mg₂Si (14) and skutterudites (15, 16)] or 600–700 °C [half-Heuslers (17, 18)], the ZT values of these materials below 400 °C is relatively low (ZT < 1). Other materials, such as n-type In₄Se_3-δ (19), n-type Ba₈Ga₁₆Sn₃₀ (20), and p-type Zn_4-δSb₃ (21) have higher average ZT values below 400 °C. However, the low power factors make them unsuitable for power generation applications below 400 °C. Because both efficiency and output power are equally important, new n- and p-type materials that can work up to 400 °C are more desirable for thermoelectric power generation.Here, we report a new Mg₂Sn-based n-type thermoelectric material that shows promise to work below 400 °C for power generation due to the narrow band gap of ∼0.26 eV. Historically, Mg₂Sn material has been investigated less than its analogous compound Mg₂Si for thermoelectric applications due to its lower ZT (22–25). Most of the research has been focused on the alloy of Mg₂Si-Mg₂Sn with a peak ZT value of ∼1 at 500 °C (26–28). Recently, different groups have improved the peak ZT value to 1.1–1.3 by adjusting the x value in the Mg₂Si_1−xSn_x solid solution (14, 29, 30). The challenges in preparing and handling these materials were the high vapor pressure and chemical activity of Mg. Methods of direct comelting with subsequent annealing, and solid-state reaction with subsequent annealing and Bridgman method were reported to synthesize Mg₂Si-Mg₂Sn alloys (22–30). Powder metallurgy route, e.g., ball milling plus hot pressing, was widely used to fabricate a variety of high-performance thermoelectric bulk materials such as Bi₂Te₃ (6, 31), PbTe (32), PbSe (33), and skutterudites CoSb₃ (16, 34). In fact, ball milling was reported to synthesize Mg₂Si and its alloys Mg₂Si-Mg₂Sn (35–38). However, the reported ZT was lower than 0.7 (27, 38), which may be due to the difficulty in avoiding oxidization of Mg. Here, we report a successful synthesis of an Sn-dominated composition Mg₂Sn_0.75Ge_0.25 through ball milling and hot pressing to achieve a ZT of 1.4 at 450 °C and power factor PF of 55 μW⋅cm⁻¹⋅K⁻² at 350 °C. Calculations show that these could yield a leg efficiency η of 10.5%, and output power density ω of 6.6 W·cm⁻² at T_h = 400 °C and T_c = 50 °C, which will be very useful for the vast amount of waste heat sources at up to 400 °C and concentrated solar energy conversion applications.     

Submesoscale dispersion in the vicinity of the Deepwater Horizon spill

Reliable forecasts for the dispersion of oceanic contamination are important for coastal ecosystems, society, and the economy as evidenced by the Deepwater Horizon oil spill in the Gulf of Mexico in 2010 and the Fukushima nuclear plant incident in the Pacific Ocean in 2011. Accurate prediction of pollutant pathways and concentrations at the ocean surface requires understanding ocean dynamics over a broad range of spatial scales. Fundamental questions concerning the structure of the velocity field at the submesoscales (100 m to tens of kilometers, hours to days) remain unresolved due to a lack of synoptic measurements at these scales. Using high-frequency position data provided by the near-simultaneous release of hundreds of accurately tracked surface drifters, we study the structure of submesoscale surface velocity fluctuations in the Northern Gulf of Mexico. Observed two-point statistics confirm the accuracy of classic turbulence scaling laws at 200-m to 50-km scales and clearly indicate that dispersion at the submesoscales is local, driven predominantly by energetic submesoscale fluctuations. The results demonstrate the feasibility and utility of deploying large clusters of drifting instruments to provide synoptic observations of spatial variability of the ocean surface velocity field. Our findings allow quantification of the submesoscale-driven dispersion missing in current operational circulation models and satellite altimeter-derived velocity fields.The Deepwater Horizon (DwH) incident was the largest accidental oil spill into marine waters in history with some 4.4 million barrels released into the DeSoto Canyon of the northern Gulf of Mexico (GoM) from a subsurface pipe over ∼84 d in the spring and summer of 2010 (1). Primary scientific questions, with immediate practical implications, arising from such catastrophic pollutant injection events are the path, speed, and spreading rate of the pollutant patch. Accurate prediction requires knowledge of the ocean flow field at all relevant temporal and spatial scales. Whereas ocean general circulation models were widely used during and after the DwH incident (2–6), such models only capture the main mesoscale processes (spatial scale larger than 10 km) in the GoM. The main factors controlling surface dispersion in the DeSoto Canyon region remain unclear. The region lies between the mesoscale eddy-driven deep water GoM (7) and the wind-driven shelf (8) while also being subject to the buoyancy input of the Mississippi River plume during the spring and summer months (9). Images provided by the large amounts of surface oil produced in the DwH incident revealed a rich array of flow patterns (10) showing organization of surface oil not only by mesoscale straining into the loop current “Eddy Franklin,” but also by submesoscale processes. Such processes operate at spatial scales and involve physics not currently captured in operational circulation models. Submesoscale motions, where they exist, can directly influence the local transport of biogeochemical tracers (11, 12) and provide pathways for energy transfer from the wind-forced mesoscales to the dissipative microscales (13–15). Dynamics at the submesoscales have been the subject of recent research (16–20). However, the investigation of their effect on ocean transport has been predominantly modeling based (13, 21–23) and synoptic observations, at adequate spatial and temporal resolutions, are rare (24, 25). The mechanisms responsible for the establishment, maintenance, and energetics of such features in the Gulf of Mexico remain unclear.Instantaneous measurement of all representative spatiotemporal scales of the ocean state is notoriously difficult (26). As previously reviewed (27), traditional observing systems are not ideal for synoptic sampling of near-surface flows at the submesoscale. Owing to the large spacing between ground tracks (28) and along-track signal contamination from high-frequency motions (29), gridded altimeter-derived sea level anomalies only resolve the largest submesoscale motions. Long time-series ship-track current measurements attain similar, larger than 2 km, spatial resolutions, and require averaging the observations over evolving ocean states (30). Simultaneous, two-point accoustic Doppler current profiler measurements from pairs of ships (25) provide sufficient resolution to show the existence of energetic submesoscale fluctuations in the mixed layer, but do not explicitly quantify the scale-dependent transport induced by such motions at the surface. Lagrangian experiments, centered on tracking large numbers of water-following instruments, provide the most feasible means of obtaining spatially distributed, simultaneous measurements of the structure of the ocean’s surface velocity field on 100-m to 10-km length scales.Denoting a trajectory by x(a, t), where x(a, t₀) = a, the relative separation of a particle pair is given by D(t,D0)=x(a1,t)−x(a2,t)=D0+∫t0tΔv(t′,D0)dt′, where the Lagrangian velocity difference is defined by Δv(t, D₀) = v(a₁, t) − v(a₂, t). The statistical quantities of interest, both practically and theoretically, are the scale-dependent relative dispersion D²(t) = 〈D ⋅ D〉 (averaged over particle pairs) and the average longitudinal or separation velocity, Δv(r), at a given separation, r. The velocity scale is defined by the second order structure function Δv(r)=〈δv2〉, where δv(r) = (v(x + r) − v(x)) ⋅ r/∥r∥ (31, 32) where the averaging is now conditioned on the pair separation r.The applicability of classical dispersion theories (32–34) developed in the context of homogeneous, isotropic turbulence with localized spectral forcing, to ocean flows subject to the effects of rotation, stratification, and complex forcing at disparate length and time scales remains unresolved. Turbulence theories broadly predict two distinct dispersion regimes depending upon the shape of the spatial kinetic energy spectrum, E(k) ∼ k^−β, of the velocity field (35). For sufficiently steep spectra (β ≥ 3) the dispersion is expected to grow exponentially, D ∼ e^λt with a scale-independent rate. At the submesoscales (∼ 100 m–10 km), this nonlocal growth rate will then be determined by the mesoscale motions currently resolved by predictive models. For shallower spectra (1 < β < 3), however, the dispersion is local, D ∼ t^2/(3−β), and the growth rate of a pollutant patch is dominated by advective processes at the scale of the patch. Accurate prediction of dispersion in this regime requires resolution of the advecting field at smaller scales than the mesoscale.Whereas compilations of data from dye measurements broadly support local dispersion in natural flows (36), the range of scales in any particular dye experiment is limited. A number of Lagrangian observational studies have attempted to fill this gap. LaCasce and Ohlmann (37) considered 140 pairs of surface drifters on the GoM shelf over a 5-y period and found evidence of a nonlocal regime for temporally smoothed data at 1-km scales. Koszalka et al. (38) using ??(100) drifter pairs with D₀ < 2 km launched over 18 mo in the Norwegian Sea, found an exponential fit for D²(t) for a limited time (t = 0.5 − 2 d), although the observed longitudinal velocity structure function is less clearly fit by a corresponding quadratic. They concluded that a nonlocal dispersion regime could not be identified. In contrast, Lumpkin and Elipot (39) found evidence of local dispersion at 1-km scales using 15-m drogued drifters launched in the winter-time North Atlantic. It is not clear how the accuracy of the Argos positioning system (150–1,000 m) used in these studies affects the submesoscale dispersion estimates. Schroeder et al. (40), specifically targeting a coastal front using a multiscale sampling pattern, obtained results consistent with local dispersion, but the statistical significance (maximum 64 pairs) remained too low to be definitive.     

Semiempirical method for examining asynchronicity in metal–oxido-mediated C–H bond activation

The oxidation of substrates via the cleavage of thermodynamically strong C–H bonds is an essential part of mammalian metabolism. These reactions are predominantly carried out by enzymes that produce high-valent metal–oxido species, which are directly responsible for cleaving the C–H bonds. While much is known about the identity of these transient intermediates, the mechanistic factors that enable metal–oxido species to accomplish such difficult reactions are still incomplete. For synthetic metal–oxido species, C–H bond cleavage is often mechanistically described as synchronous, proton-coupled electron transfer (PCET). However, data have emerged that suggest that the basicity of the M–oxido unit is the key determinant in achieving enzymatic function, thus requiring alternative mechanisms whereby proton transfer (PT) has a more dominant role than electron transfer (ET). To bridge this knowledge gap, the reactivity of a monomeric Mn^IV–oxido complex with a series of external substrates was studied, resulting in a spread of over 10⁴ in their second-order rate constants that tracked with the acidity of the C–H bonds. Mechanisms that included either synchronous PCET or rate-limiting PT, followed by ET, did not explain our results, which led to a proposed PCET mechanism with asynchronous transition states that are dominated by PT. To support this premise, we report a semiempirical free energy analysis that can predict the relative contributions of PT and ET for a given set of substrates. These findings underscore why the basicity of M–oxido units needs to be considered in C–H functionalization.

The functionalization of C–H bonds is one of the most challenging synthetic transformations in chemistry, in part because of the inherent large bond dissociation-free energies (BDFEs) associated with C–H bonds (BDFE_C–H) (1, 2). Monomeric metal–oxido species can overcome these barriers and cleave C–H bonds in a diverse set of substrates. The utility of metal–oxido species is exemplified within the active sites of metalloenzymes, such as heme and nonheme Fe monooxygenases (3–6) and in related, synthetic Mn– (7–15), Co– (16), and Fe–oxido complexes (17–25). Even with the advances made with these natural and synthetic metal–oxido species, key mechanistic questions about which properties of the metal complexes contribute to productive C–H cleavage persist (26–30). Much of our current understanding of metal–oxido-mediated C–H bond activation is predicated on a relationship between ground-state thermodynamics and the height of the activation barrier. Reactions of metal–oxido species with organic substrates often show a correlation between variations in substrate C–H bond strength, (BDFE_C–H, Eq. 1, where C_G is a constant that is dependent on the reference electrode and solvent) and the log of the rate constant for C–H bond cleavage. Systems for which these correlations are highly linear are said to follow a linear free energy relationship or display Bell–Evans–Polanyi (BEP)-like correlations (31, 32).BDFEC–H= 23.06(E˚) + 1.37(pKa) + CG.[1]The ground-state thermodynamics for C–H bond cleavage are determined by comparing the BDFE_C–H of the C–H bond to be cleaved to that of the O–H bond formed in the resulting M^n-1–OH species; this BDFE_O–H value is obtained from the reduction potential of the M = O species and its basicity, in a manner analogous to Eq. 1 (1, 33). The relationship between BDFE_O–H, reduction potential and basicity can be conveniently represented using a thermodynamic square scheme that contains three limiting, mechanistic paths (Fig. 1) (26): 1) proton transfer/electron transfer (PT-ET); 2) electron transfer/proton transfer (ET-PT); and 3) synchronous, proton-coupled electron transfer (PCET). An important outcome of the approach of comparing BDFE values is the recognition that the basicity of the M–oxido unit can be a key contributor to C–H bond cleavage. This concept is exemplified by the high-valent Fe–oxido intermediate in cytochrome P450s (compound I). The reduced intermediate is highly basic, promoting the abstraction of H atoms from strong C–H bonds at relatively low, one-electron reduction potentials (34, 35).Open in a separate windowFig. 1.Thermodynamic square scheme for M–oxido complexes involved in C–H bond cleavage. For 1, E_1/2'' = −1.0 V; E_1/2 = −0.18 V; pK_a = 15; pK_a'' = 28.1(2); and BDFE_O–H = 87(2) kcal/mol (SI Appendix, Eq. S1). Values measured in DMSO at room temperature. Potentials are referenced to [Fe^III/IICp₂]^+/0.We have also shown that the basicity of the oxido ligand drives the reactivity of a low-valent Mn^III–oxido complex that is competent at cleaving C–H bonds, even though its reduction potential is less than −2.0 V versus [Fe^III/IICp₂]^+/0 (7). Our initial mechanistic suggestion for this Mn^III–oxido complex was a stepwise PT-ET pathway with proton transfer (PT) being rate limiting. However, subsequent kinetic studies on a related series of Mn^III–oxido complexes showed that electron transfer (ET) must also be involved in the rate-determining step (8). To reconcile these results, we proposed a mechanism with an imbalanced transition state, in which PT precedes ET (8). This type of asynchronous PCET mechanism (36) was examined computationally (37) and experimentally to explain the cleavage of C–H bonds with Co^III–oxido (16), Ru^IV–oxido (38), and Cu^III−O₂CAr complexes (39). The involvement of asynchronous PCET processes in the cleavage of C–H bonds by high-valent M–oxido complexes, such as Fe^IV/Mn^IV–oxido species, is still uncertain. In fact, most reactions have synchronous PCET mechanisms that follow the BEP principle (31, 32) and exhibit the correlations between the BDFE_C–H values of the substrates and the log of the second-order rate constants [log (k)]. However, C–H bond cleavage via an asynchronous PCET mechanism that is driven by the basicity of the M–oxido unit would be beneficial because reactivity could occur at lower redox potentials, while still achieving the efficiency that is often associated with PCET processes. We reasoned that this type of mechanism could occur if the high-valent M–oxido complex was relatively basic and could therefore favor mechanisms that involve PT-driven, asynchronous transition states. The high-valent Mn^IV–oxido complex [Mn^IVH₃buea(O)]⁻ (1) is a suitable candidate for testing this premise because it has a relatively high basicity, as gauged by its conjugate acid, [Mn^IVH₃buea(OH)], which has an estimated pK_a of ∼15 in DMSO (7).We report here the kinetic studies of 1 with a variety of substrates, and the results of these studies support the involvement of PT-dominated, asynchronous transition states and show the applicability of this type of mechanism in the cleavage of C–H bonds by high-valent M–oxido complexes. To further evaluate our findings, we developed a semiempirical free energy analysis to estimate the relative contributions from the free energies of PT and ET within an asymmetric transition state. This analysis supports the assertion that the observed reactivity of 1 is driven by the pK_a, leading to PT-controlled, asynchronous transition states for the substrates examined. Moreover, we demonstrate that this analysis is useful in predicting whether a PCET reaction will be synchronous or asynchronous based on reactivity and thermodynamic parameters. Our method deviates from traditional, BEP-like analyses, in that it examines the behavior of log (k) versus a linear combination of PT and ET terms, rather than bond strengths. The appropriate combination of these free energy terms is determined by the linearity of plots versus log (k).     

Transient conformational fluctuation of TePixD during a reaction

Knowledge of the dynamical behavior of proteins, and in particular their conformational fluctuations, is essential to understanding the mechanisms underlying their reactions. Here, transient enhancement of the isothermal partial molar compressibility, which is directly related to the conformational fluctuation, during a chemical reaction of a blue light sensor protein from the thermophilic cyanobacterium Thermosynechococcus elongatus BP-1 (TePixD, Tll0078) was investigated in a time-resolved manner. The UV-Vis absorption spectrum of TePixD did not change with the application of high pressure. Conversely, the transient grating signal intensities representing the volume change depended significantly on the pressure. This result implies that the compressibility changes during the reaction. From the pressure dependence of the amplitude, the compressibility change of two short-lived intermediate (I₁ and I₂) states were determined to be +(5.6 ± 0.6) × 10⁻² cm³⋅mol⁻¹⋅MPa⁻¹ for I₁ and +(6.6 ± 0.7)×10⁻² cm³⋅mol⁻¹⋅MPa⁻¹ for I₂. This result showed that the structural fluctuation of intermediates was enhanced during the reaction. To clarify the relationship between the fluctuation and the reaction, the compressibility of multiply excited TePixD was investigated. The isothermal compressibility of I₁ and I₂ intermediates of TePixD showed a monotonic decrease with increasing excitation laser power, and this tendency correlated with the reactivity of the protein. This result indicates that the TePixD decamer cannot react when its structural fluctuation is small. We concluded that the enhanced compressibility is an important factor for triggering the reaction of TePixD. To our knowledge, this is the first report showing enhanced fluctuations of intermediate species during a protein reaction, supporting the importance of fluctuations.Proteins often transfer information through changes in domain–domain (or intermolecular) interactions. Photosensor proteins are an important example. They have light-sensing domains and function by using the light-driven changes in domain–domain interactions (1). The sensor of blue light using FAD (BLUF) domain is a light-sensing module found widely among the bacterial kingdom (2). The BLUF domain initiates its photoreaction by the light excitation of the flavin moiety inside the protein, which changes the domain–domain interaction, causing a quaternary structural change and finally transmitting biological signals (3, 4). It has been an important research topic to elucidate how the initial photochemistry occurring in the vicinity of the chromophore leads to the subsequent large conformation change in other domains, which are generally apart from the chromophore.It may be reasonable to consider that the conformation change in the BLUF domain is the driving force in its subsequent reaction; that is, the change in domain–domain interaction. However, sometimes, clear conformational changes have not been observed for the BLUF domain; its conformation is very similar before and after photo-excitation (5–13). The circular dichroism (CD) spectra of BLUF proteins AppA and PixD from thermophilic cyanobacterium Thermosynechococcus elongatus BP-1 (TePixD) did not change on illumination (5, 13). Similarly, solution NMR studies of AppA and BlrB showed only small chemical shifts on excitation (9, 10). The solution NMR structure of BlrP1 showed a clear change, but this was limited in its C-terminal extension region and not core BLUF (11). Furthermore, the diffusion coefficient (D) of the BLUF domain of YcgF was not changed by photo-excitation (12), although D is sensitive to global conformational changes. These results imply that a minor structural change occurs in the BLUF domain. In such cases, how does the BLUF domain control its interdomain interaction? Recently, a molecular dynamics (MD) simulation on another light-sensing domain, the light-oxygen-voltage (LOV) sensing domain, suggested that fluctuation of the LOV core structure could be a key to understanding the mechanism of information transfer (14–16).Because proteins work at room temperature, they are exposed to thermal fluctuations. The importance of such structural fluctuations for biomolecular reactions has been also pointed out: for example, enzymatic activity (17–20). Experimental detections of such conformation fluctuations using single molecular detection (21) or NMR techniques such as the hydrogen-deuterium (H-D) exchange, relaxation dispersion method, and high-pressure NMR (22–24) have succeeded. However, these techniques could not detect the fluctuation of short-lived transient species. Indeed, single molecule spectroscopy can trace the fluctuation in real time, but it is still rather difficult to detect rapid fluctuations for a short-lived intermediate during a reaction. Therefore, information about the fluctuation of intermediates is thus far limited.A thermodynamic measurement is another way to characterize the fluctuation of proteins. In particular, the partial molar isothermal compressibility [K¯T=−(∂V¯/∂P)T] is essential, because this property is directly linked to the mean-square fluctuations of the protein partial molar volume by 〈(V¯−〈V¯〉)2〉≡〈δV¯2〉=kBTK¯T (25). (Here, <X> means the averaged value of a quantity of X.) Therefore, isothermal compressibility is thought to reflect the structural fluctuation of molecules (26). However, experimental measurement of this parameter of proteins in a dilute solution is quite difficult. Indeed, this quantity has been determined indirectly from the theoretical equation using the adiabatic compressibility of a protein solution, which was determined by the sound velocity in the solution (26–31). Although the relation between volume fluctuations and isothermal compressibility is rigorously correct only with respect to the intrinsic part of the volume compressibility, and not the partial molar volume compressibility (32), we considered that this partial molar volume compressibility is still useful for characterizing the fluctuation of the protein structure including its interacting water molecules. In fact, the relationship between β¯T and the volume fluctuation has been often used to discuss the fluctuation of proteins (17, 26–28), and the strong correlation of β¯T of reactants with the functioning for some enzymes (17, 33, 34) has been reported. These studies show the functional importance of the structural fluctuation represented by β¯T. However, thermodynamic techniques lack time resolution, and it has been impossible to measure the fluctuations of short-lived intermediate species.Recently, we developed a time-resolving method for assessing thermodynamic properties using the pulsed laser induced transient grating (TG) method. Using this method, we thus far succeeded in measuring the enthalpy change (ΔH) (35–38), partial molar volume change (ΔV¯) (12, 35, 37), thermal expansion change (Δα¯th) (12, 37), and heat capacity change (ΔC_p) (36–38) for short-lived species. Therefore, in principle, the partial molar isothermal compressibility change (ΔK¯T) of a short-lived intermediate become observable if we conduct the TG experiment under the high-pressure condition and detect ΔV¯ with varying external pressure.There are several difficulties in applying the traditional high-pressure cell to the TG method to measure thermodynamic parameters quantitatively. The most serious problem is ensuring the quantitative performance of the intensity of TG signals measured under the high-pressure condition. On this point, our group has developed a new high-pressure cell specially designed for TG spectroscopy (39) and overcome this problem. In this paper, by applying this high-pressure TG system to the BLUF protein TePixD, we report the first measurement, to our knowledge, of ΔK¯T of short-lived intermediates to investigate the mechanism underlying signal transmission by BLUF proteins, from the view point of the transient fluctuation.TePixD is a homolog of the BLUF protein PixD, which regulates the phototaxis of cyanobacterium (40) and exists in a thermophilic cyanobacterium Thermocynechococcus elongates BP-1 (Tll0078). TePixD is a relatively small (17 kDa) protein that consists only of the BLUF domain with two extended helices in the C-terminal region. In crystals and solutions, it forms a decamer that consists of two pentameric rings (41). The photochemistry of TePixD is typical among BLUF proteins (42–45); on blue light illumination, the absorption spectrum shifts toward red by about 10 nm within a nanosecond. The absorption spectrum does not change further, and the dark state is recovered with a time constant of ∼5 s at room temperature (40, 43). The spectral red shift was explained by the rearrangement of the hydrogen bond network around the chromophore (6, 46–48). The TG method has revealed the dynamic photoreaction mechanism, which cannot be detected by conventional spectroscopic methods. The TG signal of TePixD (Fig. S1) showed that there are two spectrally silent reaction phases: a partial molar volume expansion with the time constant of ∼40 μs and the diffusion coefficient (D) change with a time constant of ∼4 ms. Furthermore, it was reported that the pentamer and decamer states of TePixD are in equilibrium and that the final photoproduct of the decamer is pentamers generated by its dissociation (13, 49). On the basis of these studies, the reaction scheme has been identified as shown in Fig. 1. Here, I₁ is the intermediate of the spectrally red-shifted species (generated within a nanosecond) and I₂ is the one created on the subsequent volume expansion process of +4 cm³⋅mol⁻¹ (∼40 μs). Furthermore, an experiment of the excitation laser power dependence of its TG signal revealed that the TePixD decamer undergoes the original dissociation reaction when only one monomer in the decamer is excited (50). In this study, we investigated the transient compressibility of the intermediates I₁ and I₂ of the photoreaction of TePixD and found a direct link between their fluctuation and reactivity.Open in a separate windowFig. 1.Schematic illustration of the photoreaction of TePixD. Yellow circles represent the TePixD monomer in the ground state, which constructs the decamer and pentamer states. In the dark state, these two forms are in equilibrium. The excited, spectral red-shifted state of the TePixD monomer is indicated by a red circle. The square represents the I₂ state of the monomer, which is created by the volume expansion process.     

Life history effects on the molecular clock of autosomes and sex chromosomes

One of the foundational results in molecular evolution is that the rate at which neutral substitutions accumulate on a lineage equals the rate at which mutations arise. Traits that affect rates of mutation therefore also affect the phylogenetic “molecular clock.” We consider the effects of sex-specific generation times and mutation rates in species with two sexes. In particular, we focus on the effects that the age of onset of male puberty and rates of spermatogenesis have likely had in hominids (great apes), considering a model that approximates features of the mutational process in mammals, birds, and some other vertebrates. As we show, this model can account for a number of seemingly disparate observations: notably, the puzzlingly low X-to-autosome ratios of substitution rates in humans and chimpanzees and differences in rates of autosomal substitutions among hominine lineages (i.e., humans, chimpanzees, and gorillas). The model further suggests how to translate pedigree-based estimates of human mutation rates into split times among extant hominoids (apes), given sex-specific life histories. In so doing, it largely bridges the gap reported between estimates of split times based on fossil and molecular evidence, in particular suggesting that the human–chimpanzee split may have occurred as recently as 6.6 Mya. The model also implies that the “generation time effect” should be stronger in short-lived species, explaining why the generation time has a major influence on yearly substitution rates in mammals but only a subtle one in human pedigrees.Most of our inferences about species split times on short phylogenetic timescales rely on the neutral molecular clock. According to the neutral theory, the number of substitutions K that accumulate in a lineage over T years (e.g., since the split from another species) is K=(u¯/G¯)T, where u¯ and G¯ are the average mutation rate per generation and average generation time, respectively (1). Inferring split times therefore requires estimates of the yearly mutation rate u¯/G¯ on the lineage in question. Such estimates generally derive from securely dated fossils on other lineages or from measurements of mutation rates in extant species (2–5). Using these estimates for dating thus necessitates an understanding of the way that yearly mutation rates may change over time.Neutral substitution patterns in mammals offer some insights. Variation in yearly mutation rates on phylogenetic timescales can be assessed by comparing the number of neutral substitutions along two branches leading from a common ancestor to extant species. These comparisons show marked variation in yearly rates on autosomes. For example, there are 50% fewer substitutions on the human branch compared with rodents (6) and 25% fewer compared with baboons, with more moderate differences among hominine lineages (6–9). The average yearly rates are also negatively correlated with generation times (and their correlates) in extant mammals, leading to the notion of a “generation time effect” on the molecular clock (6, 10, 11).Neutral substitutions rates vary not only among taxa but also between sex chromosomes and autosomes. For brevity, we consider the relative rates on X and autosomes, but these considerations extend naturally to Y (or ZW). Because autosomes spend the same number of generations in both sexes, whereas the X spends twice as many generations in females, rates of neutral substitutions on autosomes reflect a greater relative contribution of male mutations than on the X. In a wide range of taxa, neutral substitutions rates on autosomes are greater than on the X (or lower than on the Z), suggesting a male biased contribution to yearly mutation rates (12). Moreover, observed X-to-autosome ratios are extremely variable, ranging between 0.76 and 0.9 in hominids and up to 1.0 in surveyed mammals, indicating that the degree of male bias itself varies greatly on phylogenetic timescales (12, 13).Our current understanding of mutation can help tie these observations together (5). Pedigree studies in humans and chimpanzees establish that most mutations are paternal in origin and that the paternal but not the maternal contribution increases strongly with age (4, 14). This has long been thought to be true because germ-cell division is arrested before birth in females but proceeds continuously postpuberty in males (5, 15–17). The same reasoning may extend to mammals, birds, and other vertebrate taxa in which oogenesis ceases before birth or hatching (18–20). These considerations suggest that maternal and paternal generation times should affect the molecular clock differently. They also imply that the age of puberty in males and the rate of spermatogenic germ cell divisions should affect yearly mutation rates (5). The variation observed among closely related extant species indicates that these parameters change over phylogenetic timescales. Here we ask how such changes would affect the molecular clock on X and autosomes.     

How amide hydrogens exchange in native proteins

Amide hydrogen exchange (HX) is widely used in protein biophysics even though our ignorance about the HX mechanism makes data interpretation imprecise. Notably, the open exchange-competent conformational state has not been identified. Based on analysis of an ultralong molecular dynamics trajectory of the protein BPTI, we propose that the open (O) states for amides that exchange by subglobal fluctuations are locally distorted conformations with two water molecules directly coordinated to the N–H group. The HX protection factors computed from the relative O-state populations agree well with experiment. The O states of different amides show little or no temporal correlation, even if adjacent residues unfold cooperatively. The mean residence time of the O state is ∼100 ps for all examined amides, so the large variation in measured HX rate must be attributed to the opening frequency. A few amides gain solvent access via tunnels or pores penetrated by water chains including native internal water molecules, but most amides access solvent by more local structural distortions. In either case, we argue that an overcoordinated N–H group is necessary for efficient proton transfer by Grotthuss-type structural diffusion.Before the tightly packed and densely H-bonded structure of globular proteins had been established, Hvidt and Linderstrøm-Lang (1) showed that all backbone amide hydrogens of insulin exchange with water hydrogens, implying that all parts of the polypeptide backbone are, at least transiently, exposed to solvent. In the following 60 y, hydrogen exchange (HX), usually monitored by NMR spectroscopy (2) or mass spectrometry (3), has been widely used to study protein folding and stability (4–10), structure (11, 12), flexibility and dynamics (13–15), and solvent accessibility and binding (16, 17), often with single-residue resolution. However, because the exchange mechanism is unclear, HX data from proteins can, at best, be interpreted qualitatively (18–25).Under most conditions, amide HX is catalyzed by hydroxide ions (26, 27) at a rate that is influenced by inductive and steric effects from adjacent side chains (28). For unstructured peptides, HX is a slow process simply because the hydroxide concentration is low. For example, at 25° C and pH 4, HX occurs on a time scale of minutes. Under similar conditions, amides buried in globular proteins exchange on a wide range of time scales, extending up to centuries. HX can only occur if the amide is exposed to solvent, so conformational fluctuations must be an integral part of the HX mechanism (18).Under sufficiently destabilizing conditions HX occurs from the denatured-state ensemble, but under native conditions few amides exchange by such global unfolding (9, 29–31). For example, in bovine pancreatic trypsin inhibitor (BPTI), 8 amides in the core β-sheet exchange by global unfolding under native conditions (7, 32), whereas the remaining 45 amides require less extensive conformational fluctuations. Much of the debate in the protein HX field over the past half-century has concerned the nature of these subglobal fluctuations and their frequency, duration, amplitude, and cooperativity (18–25).According to the standard HX model (18), each amide can exist in a closed (C) state, where exchange cannot occur, or in an open (O) state, where exchange proceeds at a rate k_int. The kinetic scheme for H exchange into D₂O then reads as(N−H)C⇌kclkop(N−H)O→kint(N−D)Oand the measured steady-state HX rate is k_HX = k_op?k_int/(k_op + k_cl + k_int). To make this phenomenological model practically useful, two auxiliary assumptions are needed to disentangle the conformational and intrinsic parts of the process: (i) The conformational fluctuations (k_op and k_cl) are independent of pH, and (ii) HX from the O state proceeds at the same rate as in model peptides with the same neighboring side chains, so that kint=kHX0.Two HX regimes are distinguished with reference to the pH dependence of k_HX (18). If k_HX is constant in some pH range, it follows that k_int ? k_op + k_cl so that k_HX = k_op. In this so-called EX1 limit, the HX experiment measures the opening rate, or the mean residence time (MRT), of the C state, τ_C = 1/k_op. For BPTI, such pH invariance has only been observed for the eight core amides, and then only in a narrow pH interval (32).More commonly, HX experiments are performed in the EX2 limit, where k_int ? k_op + k_cl. Then k_HX = k_int/(κ + 1), where κ ≡ k_cl/k_op = τ_C/τ_O is the protection factor (PF). At equilibrium, the fractional populations, f_C and f_O, and the rates are linked by detailed balance, k_op?f_C = k_cl?f_O, so the PF may also be expressed as κ = f_C/f_O. Clearly, 1/(κ + 1) is the probability of finding the amide in the O state, 1/κ is the C  ?  O equilibrium constant, and β?ΔG = ln?κ is the free energy difference between the O and C states in units of k_B?T ≡ 1/β. The PF can thus be deduced from the HX rates measured (under EX2 conditions) for the amide in the protein and in a model peptide as κ=kHX0/kHX−1.The vast majority of the available protein HX data pertains to the EX2 regime and thus provides no information about the time scales, τ_C and τ_O, of the conformational fluctuations, except for the EX2 bound: 1/τC+1/τO≫kint≈kHX0. In the typical case where kHX≪kHX0, so that τ_C ? τ_O, we therefore only know that τO≪1/kHX0, which is in the millisecond range at pH 9 (EX2 HX data are usually measured at lower pH, where 1/kHX0 is even longer). Our analysis indicates that τ_O is seven orders of magnitude shorter than this upper bound estimate.The HX experiment is unique in probing sparsely populated conformational states with single-residue resolution. However, the physical significance of the PF is obscured by our ignorance about the structure and dynamics of the O state. Several attempts have been made to correlate experimental PFs with physical attributes of the amides, such as solvent contact (33–37), burial depth (38), intramolecular H-bonds (35, 38–40), packing density (38, 41), or electric field (42). Where significant correlations have been found, they suggest that the chosen attribute can serve as a proxy for the propensity for C → O fluctuations. However, whether based on crystal structures or molecular dynamics (MD) trajectories, these studies examined the time-averaged protein structure, which is dominated by the C state and therefore provides little or no information about the nature of the C → O fluctuations.In principle, the O state can be identified from molecular simulations, but this requires extensive conformational sampling because most C → O transitions are exceedingly rare. To date, this approach has been tried only with coarse-grained and/or empirical protein models without explicit solvent (43–45), or for HX from the denatured-state ensemble (46). The recent availability of ultralong MD simulations with realistic force fields opens up new opportunities in the search for the elusive O state. We have thus analyzed the millisecond MD trajectory of fully solvated native BPTI performed by Shaw et al. (47). Fortunately, BPTI is also among the proteins that have been most thoroughly studied by HX experiments.     

Photon gating in four-dimensional ultrafast electron microscopy

Ultrafast electron microscopy (UEM) is a pivotal tool for imaging of nanoscale structural dynamics with subparticle resolution on the time scale of atomic motion. Photon-induced near-field electron microscopy (PINEM), a key UEM technique, involves the detection of electrons that have gained energy from a femtosecond optical pulse via photon–electron coupling on nanostructures. PINEM has been applied in various fields of study, from materials science to biological imaging, exploiting the unique spatial, energy, and temporal characteristics of the PINEM electrons gained by interaction with a “single” light pulse. The further potential of photon-gated PINEM electrons in probing ultrafast dynamics of matter and the optical gating of electrons by invoking a “second” optical pulse has previously been proposed and examined theoretically in our group. Here, we experimentally demonstrate this photon-gating technique, and, through diffraction, visualize the phase transition dynamics in vanadium dioxide nanoparticles. With optical gating of PINEM electrons, imaging temporal resolution was improved by a factor of 3 or better, being limited only by the optical pulse widths. This work enables the combination of the high spatial resolution of electron microscopy and the ultrafast temporal response of the optical pulses, which provides a promising approach to attain the resolution of few femtoseconds and attoseconds in UEM.In ultrafast electron microscopy (UEM) (1–3), electrons generated by photoemission at the cathode of a transmission electron microscope are accelerated down the microscope column to probe the dynamic evolution of a specimen initiated by an ultrafast light pulse. The use of femtosecond lasers to generate the electron probe and excite the specimen has made it possible to achieve temporal resolution on the femtosecond time scale, as determined by the cross-correlation of the optical and electron pulses. One important method in the UEM repertoire is photon-induced near-field electron microscopy (PINEM) (4, 5), in which the dynamic response detected by the electron probe is the pump-induced charge density redistribution in nanoscale specimens (6).Photon–electron coupling is the basic building block of PINEM, which takes place in the presence of nanostructures when the energy-momentum conservation condition is satisfied (4, 5). This coupling leads to inelastic gain/loss of photon quanta by electrons in the electron packet, which can be resolved in the electron energy spectrum (5, 7, 8). This spectrum consists of discrete peaks, spectrally separated by multiples of the photon energy (n?ω), on the higher and lower energy sides of the zero loss peak (ZLP) (4) (Fig. 1). The development of PINEM enables the visualization of the spatiotemporal dielectric response of nanostructures (9), visualization of plasmonic fields (4, 5) and their spatial interferences (10), imaging of low atomic number nanoscale materials (11), characterization of ultrashort electron packets (12, 13), and imaging of different biological structures (14).Open in a separate windowFig. 1.Concept of photon gating in 4D electron microscopy. (A) The microscope column with one electron (dark blue) and two optical (red) pulses focused onto the specimen. The wavefunctions of the three pulses are schematically shown at the top. One optical pulse is coincident with the electron pulse at the specimen to generate a PINEM signal. The resulting light blue PINEM pulse is sliced out from other electrons for detection as an energy spectrum, an image, or a diffraction signal (see the text). The second optical pulse initiates the dynamics to be probed. (B) Electron energy spectrum generated at the specimen plane when optical and electron pulses arrive simultaneously. The gain energy range is shaded light blue. (C) Illustration for the temporal pulse sequence, two optical and one electron pulse for ultrafast time-resolved PINEM measurements.As shown by Park et al. (5), the PINEM intensity (I_PINEM) is given by the square modulus of the field integral F˜0 (i.e., IPINEM∝|F˜0|2), in the weak interaction limit. The near field of a nanoparticle leads to the scattering of the electron packet, which can be treated rigorously using the Schrödinger equation/Mie scattering theory. It follows that PINEM images the object and displays its field characteristics depending on its shape, the polarization and wavelength of optical excitation, and the width of pulses used. For a spherical nanoparticle, the field integral at point (x, y) in the specimen plane is simplified to give (6)F˜0≈−iE˜0⁡cos⁡ϕχs 23 a3(Δk)2K[Δkb],[1]where E˜0 is the electric field amplitude of the incident light, ? the light polarization angle, a the particle radius, b=x2+y2 the impact parameter, K the modified Bessel function of the second type, Δk the momentum change of the electron, and χ_s = 3(ε ? 1)/(ε + 2), where χ_s is the material susceptibility and ε the dielectric function.In previous studies of the parameters in Eq. 1, only E˜0 was time dependent. The PINEM intensity, at a given point in space, was a function only of the time delay between the optical and electron pulses, providing, for the pulse lengths currently used, a cross-correlation profile when this delay was scanned across the time of temporal coincidence, or t = 0 (4, 5, 9, 13). Hitherto, PINEM has not been used to study the ultrafast dynamics of matter. Here, we follow the strategy of using the PINEM gain electrons generated by a first optical pulse, whose delay relative to the electron pulse is maintained at t = 0, to probe dynamics initiated by introduction of a second optical pulse on the specimen, as proposed theoretically in ref. 15. By this approach, we were able to optically gate the electron pulse (i.e., create an electron pulse that only lasts for the duration of the optical pulse) and achieve significant enhancement of the temporal resolution (see the second paragraph below).The concept of the experiment is illustrated by Fig. 1A, in which the electron pulse in blue and one optical pulse (P₁) in red are shown arriving at the specimen plane simultaneously. Interaction between photon and electron in the presence of the specimen “slices out” the light blue pulse of gain electrons, which are separated from all other electrons by energy dispersion or filtering to be detected according to microscope settings in spectroscopy, imaging, or diffraction mode, as illustrated schematically at the bottom of the column. Note, it is possible to obtain PINEM diffraction, but this is not the subject of this paper. A second, or pump, optical pulse (P₂) is shown below the specimen, having already triggered the dynamics of interest. A series of time axes is plotted in Fig. 1C showing examples of characteristic sequences of pulse arrival times at the specimen plane during the experiment, with the pump arrival defining the zero of time.A striking feature of this technique that was alluded to above is the potential for high temporal resolution, unlimited by the electron pulse duration, because the optical pulse acts as a temporal gate for a longer electron pulse. In the weak interaction limit, the duration of the pulse of PINEM electrons emulates that of the optical pulse that created it (15), as clearly shown in Fig. 1A. When these photon-gated electrons are used to probe dynamics triggered by a second ultrafast optical pulse, the time resolution is determined by the cross-correlation of the two optical pulses. This paves the way for the realization of attosecond electron microscopy, as done in all-optical spectroscopy (16) but with the spatial resolution being that of atomic motions. As suggested in Fig. 1A, we envisage the use of the photon-gated electron pulses, in imaging or in diffraction mode, for the study of a variety of optically initiated material processes, either of the nanostructure or of its surrounding media.The PINEM signal can be directly monitored to detect changes in any of the specimen optical or physical properties expressed in Eq. 1. Here, we demonstrate the use of the time-resolved PINEM technique where it is shown that the photoinduced dielectric response of VO₂—which is strongly related to the lattice symmetry (17)—manifests itself in a change in PINEM intensity. We relate the changes in optical properties of the polycrystalline VO₂ nanoparticles to the phase transition dynamics from initial (monoclinic) insulator phase to (tetragonal) metal phase, the subject of numerous previous studies.Vanadium dioxide has been discussed as an active metamaterial (18) and one of the best candidates for solid-state ultrafast optical switches in photonics applications (19, 20) due to its unique structural photoinduced phase transition behavior (21). This phase transition has been examined by investigating the change in the heat capacity through thermal excitation (22, 23), whereas its ultrafast dynamics has been studied by optical spectroscopy (24, 25), THz spectroscopy (26, 27), X-ray diffraction (28, 29), ultrafast electron crystallography (30), and electron microscopy (31).     

Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

The investigation of the potential energy surface (PES) V(r1(t)…rN(t)) of a liquid (made up of N particles with positions r1(t)…rN(t) at a time instant t) and the corresponding instantaneous normal modes (INMs) of the (Hessian) matrix of curvatures has been a focus of liquid and glass science since the appearance of Goldstein’s seminal article (1) on the relation between the PES and the liquid dynamics in the viscous regime above the glass transition (2–27).The PES has been shown to form a rather ragged landscape in configuration space (8, 28, 29) characterized by its stationary points. In a glass these points are minima and are called “inherent structures.” The PES is believed to contain important information on the liquid–glass transformation mechanism. For the latter a complete understanding is still missing (28, 30, 31). The existing molecular theory of the liquid–glass transformation is mode-coupling theory (MCT) (32, 33) and its mean-field Potts spin version (28, 34). MCT predicts a sharp transition at a temperature TMCT>Tg, where T_g is the temperature of structural arrest (glass transition temperature). MCT completely misses the heterogeneous activated relaxation processes (dynamical heterogeneities), which are evidently present around and below T_MCT and which are related to the unstable (negative-λ) part of the INM spectrum (28, 30).Near and above T_MCT, apparently, there occurs a fundamental change in the PES. Numerical studies of model liquids have shown that minima present below T_MCT change into saddles, which then explains the absence of activated processes above T_MCT (2–24). Very recently, it was shown that T_MCT is related to a localization–delocalization transition of the unstable INM modes (25, 26).The INM spectrum is obtained in molecular dynamic simulations by diagonalizing the Hessian matrix of the interaction potential, taken at a certain time instant t:Hijαβ(t)=∂2∂xi(α)∂xj(β)V{r1(t)…rN(t)},[1]with ri=(xi(1),xi(2),xi(3)). For large positive values of the eigenvalues λ_j (j=1…N, N being the number of particles in the system) they are related to the square of vibrational frequencies λj=ωj2, and one can consider the Hessian as the counterpart of the dynamical matrix of a solid. In this high-frequency regime one can identify the spectrum with the density of vibrational states (DOS) of the liquid viag(ω)=2 ωρ(λ(ω))=13N∑jδ(ω−ωj) .[2]For small and negative values of λ this identification is not possible. For the unstable part of the spectrum (λ<0) it has become common practice to call the imaginary number λ=iω˜ and define the corresponding DOS asg(ω˜)≡2 ω˜ρ(λ(ω˜)).[3]This function is plotted on the negative ω axis and the stable g(ω), according to [2], on the positive axis. However, the (as we shall see, very interesting) details of the spectrum ρ(λ) near λ = 0 become almost completely hidden by multiplying the spectrum with |ω|. In fact, it has been demonstrated by Sastry et al. (6) and Taraskin and Elliott (7) already 2 decades ago that the INM spectrum of liquids, if plotted as ρ(λ) and not as g(ω) according to [2] and [3], exhibits a characteristic cusp-like maximum at λ = 0. The shape of the spectrum changes strongly with temperature. This is what we find as well in our simulation and what we want to explore further in our present contribution.In the present contribution we demonstrate that the strong change of the spectrum with temperature can be rather well explained in terms of a model, in which the instantaneous harmonic spectrum of the liquid is interpreted to be that of an elastic medium, in which the local shear moduli exhibit strong spatial fluctuations, which includes a large number of negative values. Because these fluctuations are just a snapshot of thermal fluctuations, we assume that they are obeying Gaussian statistics, the variance of which is proportional to the temperature.Evidence for a characteristic change in the liquid configurations in the temperature range above T_g has been obtained in recent simulation studies of the low-frequency vibrational spectrum of glasses, which have been rapidly quenched from a certain parental temperature T*. If T* is decreased from high temperatures toward T_MCT, the low-frequency exponent of the vibrational DOS of the daughter glass (quenched from T* to T = 0) changed from Debye-like g(ω)∝ω2 to g(ω)∝ωs with s > 2. In our numerical investigation of the INM spectra we show a correlation of some details of the low-eigenvalue features of these spectra with the low-frequency properties of the daughter glasses obtained by quenching from the parental temperatures.The stochastic Helmholtz equations (Eq. 7) of an elastic model with spatially fluctuating shear moduli can be readily solved for the averaged Green’s functions by field theoretical techniques (35–37). Via a saddle point approximation with respect to the resulting effective field theory one arrives at a mean-field theory (self-consistent Born approximation [SCBA]) for the self-energy of the averaged Green’s functions. The SCBA predicts a stable spectrum below a threshold value of the variance. Restricted to this stable regime, this theory, called heterogeneous elasticity theory (HET), was rather successful in explaining several low-frequency anomalies in the vibrational spectrum of glasses, including the so-called boson peak, which is an enhancement at finite frequencies over the Debye behavior of the DOS g(ω)∝ω2 (37–41). We now explore the unstable regime of this theory and compare it to the INM spectrum of our simulated soft-sphere liquid.^*We start Results by presenting a comparison of the simulated spectra of the soft-sphere liquid with those obtained by the unstable version of HET-SCBA theory. We then concentrate on some specific features of the INM spectra, namely, the low-eigenvalue slopes and the shift of the spectral maximum from λ = 0. Both features are accounted for by HET-SCBA. In particular, we find an interesting law for the difference between the slopes of the unstable and the stable parts of the spectrum, which behaves as T−2/3, which, again, is accounted for by HET-SCBA.In the end we compare the shift of the spectral maximum with the low-frequency exponent of the DOS of the corresponding daughter glasses and find an empirical correlation. We discuss these results in connection with the saddle to minimum transformation near T_MCT.     

A new type of half-quantum circulation in a macroscopic polariton spinor ring condensate

We report the observation of coherent circulation in a macroscopic Bose–Einstein condensate of polaritons in a ring geometry. Because they are spinor condensates, half-quanta are allowed in where there is a phase rotation of π in connection with a polarization vector rotation of π around a closed path. This half-quantum behavior is clearly seen in the experimental observations of the polarization rotation around the ring. In our ring geometry, the half-quantum state that we see is one in which the handedness of the spin flips from one side of the ring to the other side in addition to the rotation of the linear polarization component; such a state is allowed in a ring geometry but will not occur in a simply connected geometry. This state is lower in energy than a half-quantum state with no change of the spin direction and corresponds to a superposition of two different elementary half-quantum states. The direction of circulation of the flow around the ring fluctuates randomly between clockwise and counterclockwise from one shot to the next; this fluctuation corresponds to spontaneous breaking of time-reversal symmetry in the system. This type of macroscopic polariton ring condensate allows for the possibility of direct control of the circulation to excite higher quantized states and the creation of Josephson junction tunneling barriers.Ring condensates, analogous to superconducting rings, have received much attention lately (1–9); among other predictions, a ring condensate allows the possibility of macroscopic superposition of states with different circulation. A ring condensate is topologically distinct from a condensate in a simply connected region.With the advance of the field of polariton condensates in the past few years, it is a natural step to create a condensate ring in a microcavity polariton system. The polariton system allows direct, nondestructive observation of the momentum distribution, energy distribution, and spatial distribution of the particles as well as direct measurement of the coherence properties through interferometry. To make a macroscopic ring requires macroscopic transport distances as well as macroscopic coherence length. Macroscopic coherence has been achieved with polaritons with coherent motion over tens of micrometers with lifetimes of 10–20 ps (10, 11) and coherent motion over hundreds of micrometers with lifetimes of 150–200 ps (12–14). One advantage of the long-lifetime polariton systems is that the polaritons can move well away from the laser spot where they are generated, so that the laser can be viewed as a simple source term and does not interact with the condensate. General reviews of previous polariton work with shorter transport distances are in refs. 15–19.The polaritons can be viewed as photons that have been given a small effective mass of the order of 10^?4 times the mass of a vacuum electron and repulsive interactions, which are about 10⁴ times stronger than the typical χ⁽³⁾ nonlinearities of photons in solids. The effective mass comes from the dispersion of the photons in a planar cavity, ℏω=ℏc(k∥2+k⊥)1/2, where k_⊥ is fixed by the width of the cavity, which implies that ℏω≃E0+ℏ2k∥2/2meff with m_eff = ?k_⊥/c for low k_∥. There are two circular polarization modes of the cavity photons corresponding to m = ±1 for the projection of the angular momentum on the z axis perpendicular to the plane. The strong interactions between photons are generated by mixing the photon states with a sharp excitonic resonance in a semiconductor inside the cavity, so that the photons pick up a fraction of the exciton–exciton interaction. Although their interactions are much stronger than the interactions of typical photons in a solid medium, the polaritons are still in the weakly interacting Bose gas regime.The structure for these experiments is a planar cavity, in which the mirrors are distributed Bragg reflectors of AlAs/AlGaAs and the exciton medium consists of GaAs/AlGaAs quantum wells embedded in this cavity. This structure has the same design as that used in previous experiments, which allows coherent transport of polaritons over hundreds of micrometers in the 2D plane of the cavity (12–14). Recent measurements (14) give the cavity lifetime as 135 ps, which corresponds to a polariton lifetime of 200 ps or more. Although this lifetime may seem to be short compared with atoms evaporating from an optical trap on timescales of seconds, the polariton lifetime is sufficient for them to interact many times with each other. In these long-lifetime polariton systems, the ratio of lifetime in the trap to the particle–particle collision time can be of the order of 500:1, comparable with the ratio for cold atom condensates.The lifetime of the polaritons and the strength of the interaction between the polaritons can be tuned by varying the energy difference between the photon states and the exciton states (known as the “detuning”), which leads to a varying degree of mixing of the photons and excitons. Because the planar cavity has a wedge that gives a gradient of cavity width, we can tune the strength of the polariton–polariton interactions simply by choosing different locations on the sample with different cavity width. There is a tradeoff in how much excitonic interaction character to give to the polaritons. Fewer interactions (more photon-like) allow long transport length, whereas more interactions allow better thermalization of the polariton gas through collisions and longer population lifetime.