In the vertebrate retina, horizontal cells generate the inhibitory surround of bipolar cells, an essential step in contrast enhancement. For the last decades, the mechanism involved in this ...inhibitory synaptic pathway has been a major controversy in retinal research. One hypothesis suggests that connexin hemichannels mediate this negative feedback signal; another suggests that feedback is mediated by protons. Mutant zebrafish were generated that lack connexin 55.5 hemichannels in horizontal cells. Whole cell voltage clamp recordings were made from isolated horizontal cells and cones in flat mount retinas. Light-induced feedback from horizontal cells to cones was reduced in mutants. A reduction of feedback was also found when horizontal cells were pharmacologically hyperpolarized but was absent when they were pharmacologically depolarized. Hemichannel currents in isolated horizontal cells showed a similar behavior. The hyperpolarization-induced hemichannel current was strongly reduced in the mutants while the depolarization-induced hemichannel current was not. Intracellular recordings were made from horizontal cells. Consistent with impaired feedback in the mutant, spectral opponent responses in horizontal cells were diminished in these animals. A behavioral assay revealed a lower contrast-sensitivity, illustrating the role of the horizontal cell to cone feedback pathway in contrast enhancement. Model simulations showed that the observed modifications of feedback can be accounted for by an ephaptic mechanism. A model for feedback, in which the number of connexin hemichannels is reduced to about 40%, fully predicts the specific asymmetric modification of feedback. To our knowledge, this is the first successful genetic interference in the feedback pathway from horizontal cells to cones. It provides direct evidence for an unconventional role of connexin hemichannels in the inhibitory synapse between horizontal cells and cones. This is an important step in resolving a long-standing debate about the unusual form of (ephaptic) synaptic transmission between horizontal cells and cones in the vertebrate retina.
Supramolecular block copolymers are becoming attractive materials in nascent optoelectronic and catalytic technologies. However, their dynamic nature precludes the straightforward tuning and analysis ...of the polymer’s structure. Here we report the elucidation on the microstructure of triarylamine triamide-based supramolecular block copolymers through a comprehensive battery of spectroscopic, theoretical, and super-resolution microscopic techniques. Via spectroscopic analysis we demonstrate that the direct mixing of preassembled homopolymers and the copolymerization induced by slow cooling of monomers lead to the formation of the same copolymer’s architecture. The small but pronounced deviation of the experimental spectra from the linear combination of the homopolymers’ spectra hints at the formation of block copolymers. A mass balance model is introduced to further unravel the microstructure of the copolymers formed, and it confirms that stable multiblock supramolecular copolymers can be accessed from different routes. The multiblock structure of the supramolecular copolymers originates from the fine balance between favorable hydrogen-bonding interactions and a small mismatch penalty between two different monomers. Finally, we visualized the formation of the supramolecular block copolymers by adapting a recently developed super-resolution microscopy technique, interface point accumulation for imaging in nanoscale topography (iPAINT), for visualizing the architectures formed in organic media. Combining multiple techniques was crucial to unveil the microstructure of these complex dynamic supramolecular systems.
Conspectus Recent years have witnessed increasing attention on supramolecular polymerization, i.e., the formation of one-dimensional aggregates in which the monomeric units bind together via ...reversible and usually highly directional non-covalent interactions. Because of the presence of these reversible interactions, such as hydrogen bonding, π–π interactions, or metal coordination, supramolecular polymers exhibit numerous desirable properties ranging from high thermoresponsiveness to self-healing and great capacity for processability and recycling. These properties relate to intriguing experimentally observed nonlinear effects such as the monomer-dependent presence of a critical temperature for aggregation and a solvent- and temperature-tunable aggregate morphology. For coassemblies this is complemented with monomer-ratio- and monomer-compatibility-dependent internal order as well as majority-rules-type chiral amplification. However, the dynamic nature of the (co)polymers and the intricate interplay of many interactions make these effects difficult to rationalize without theoretical models. This Account presents recent advances in the development and use of equilibrium models for supramolecular copolymerization based on mass balances, mainly developed by our group. The basic idea of these models is that we describe a supramolecular (co)polymerization by a set of independent equilibrium reactions, like monomer associations and dissociations, and that in thermodynamic equilibrium the concentrations of the reactants and products in each reaction are coupled via the equilibrium constant of that reaction. Recursion then allows the concentration of each possible aggregate to be written as a function of the free monomer concentrations. Because a monomer should be present either as a free monomer or in one of the aggregates, a set of n equations can be formed with the n free monomer concentrations as the only unknowns. This set of mass-balance equations can then be solved numerically, yielding the free monomer concentrations, from which the complete system can be reconstituted. By a step-by-step extension of the model for the aggregation of a single monomer type to include the formation of multiple aggregate types and the coassembly of multiple monomer types, we can capture increasingly complex supramolecular (co)polymerizations. In each step we illustrate how the extended model explains in detail another of the experimentally observed nonlinear effects, with the common denominator that small differences in association energies are intricately amplified at the supramolecular level. We finally arrive at our latest and most general approach to modeling (cooperative) supramolecular (co)polymerization, which encompasses all of our earlier models and shows great promise to help rationalize also future systems featuring ever-increasing complexity.
In the field of supramolecular (co)polymerizations, the ability to predict and control the composition and length of the supramolecular (co)polymers is a topic of great interest. In this work, we ...elucidate the mechanism that controls the polymer length in a two-component cooperative supramolecular polymerization and unveil the role of the second component in the system. We focus on the supramolecular copolymerization between two derivatives of benzene-1,3,5-tricarboxamide (BTA) monomers: a-BTA and Nle-BTA. As a single component, a-BTA cooperatively polymerizes into long supramolecular polymers, whereas Nle-BTA only forms dimers. By mixing a-BTA and Nle-BTA in different ratios, two-component systems are obtained, which are analyzed in-depth by combining spectroscopy and light-scattering techniques with theoretical modeling. The results show that the length of the supramolecular polymers formed by a-BTA is controlled by competitive sequestration of a-BTA monomers by Nle-BTA, while the obvious alternative Nle-BTA acts as a chain-capper is not operative. This sequestration of a-BTA leads to short, stable species coexisting with long cooperative aggregates. The analysis of the experimental data by theoretical modeling elucidates the thermodynamic parameters of the copolymerization, the distributions of the various species, and the composition and length of the supramolecular polymers at various mixing ratios of a-BTA and Nle-BTA. Moreover, the model was used to generalize our results and to predict the impact of adding a chain-capper or a competitor on the length of the cooperative supramolecular polymers under thermodynamic control. Overall, this work unveils comprehensive guidelines to master the nature of supramolecular (co)polymers and brings the field one step closer to applications.
Elucidating the microstructure of supramolecular copolymers remains challenging, despite the progress in the field of supramolecular polymers. In this work, we present a detailed approach to ...investigate supramolecular copolymerizations under thermodynamic control. Our approach provides insight into the interactions of different types of monomers and hereby allows elucidating the microstructure of copolymers. We select two monomers that undergo cooperative supramolecular polymerization by way of threefold intermolecular hydrogen bonding in a helical manner, namely, benzene-1,3,5-tricarboxamide (BTA) and benzene-1,3,5-tris(carbothioamide) (thioBTA). Two enantiomeric forms and an achiral analogue of BTA and thioBTA are synthesized and their homo- and copolymerizations are studied using light scattering techniques, infrared, ultraviolet, and circular dichroism spectroscopy. After quantifying the thermodynamic parameters describing the homopolymerizations, we outline a method to follow the self-assembly of thioBTA derivatives in the copolymerization with BTA, which involves monitoring a characteristic spectroscopic signature as a function of temperature and relative concentration. Using modified types of sergeants-and-soldiers and majority-rules experiments, we obtain insights into the degree of aggregation and the net helicity. In addition, we apply a theoretical model of supramolecular copolymerization to substantiate the experimental results. We find that the model describes the two-component system well and allows deriving the hetero-interaction energies. The interactions between the same kinds of monomers (BTA–BTA and thioBTA–thioBTA) are slightly more favorable than those between different monomers (BTA–thioBTA), corresponding to a nearly random copolymerization. Finally, to study the interactions of the monomers at the molecular level, we perform density functional theory-based computations. The results corroborate that the two-component system exhibits a random distribution of the two monomer units along the copolymer chain.
This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier–Stokes equations. The framework of the methodology and the correct-energy ...concept have been developed in the convective–diffusive context in the preceding paper M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective–diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259–280. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection–diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov–Galerkin (SUPG) and the Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.
•The energy behavior of the RBVMS for LES of incompressible flow is examined.•We present a novel GLS method with dynamic small-scales that has correct energy behavior.•The method conserves mass and linear and angular momentum and has divergence-free small-scales.•The correct-energy demand creates a link between VMS, GLS and SUPG.•The method shows excellent accuracy on a 3D Taylor–Green vortex flow case.
This paper presents the construction of novel stabilized finite element methods in the convective–diffusive context that exhibit correct-energy behavior. Classical stabilized formulations can create ...unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be H01-orthogonal. Using this orthogonality the VMS method can be converted into the streamline-upwind Petrov–Galerkin (SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large- and small-scales in the energy definition asks for dynamic behavior of the small-scales. Therefore, the large- and small-scales are treated as separate equations.
Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized-α time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, on both a local and a global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization.
•Two stabilized finite element methods that have correct energy behavior are proposed.•These methods use (i) dynamic small-scales and (ii) dynamic orthogonal small-scales.•The demand for correct energy behavior creates a link between VMS, GLS and SUPG.•Dissipation due to small-scales in classical stabilized methods can be negative.•The proposed GLSD and DO methods rectify this discrepancy without loss of accuracy.•The methodology can be generalized to the incompressible Navier–Stokes equations.
The coassembly of different building blocks into supramolecular copolymers provides a promising avenue to control their properties and to thereby expand the potential of supramolecular polymers in ...applications. However, contrary to covalent copolymerization which nowadays can be well controlled, the control over sequence, polymer length, and morphology in supramolecular copolymers is to date less developed, and their structures are more determined by the delicate balance in binding free energies between the distinct building blocks than by kinetics. Consequently, to rationalize the structures of supramolecular copolymers, a thorough understanding of their thermodynamic behavior is needed. Though this is well established for single-component assemblies and over the past years several models have been proposed for specific copolymerization cases, a generally applicable model for supramolecular cooperative copolymers is still lacking. Here, we provide a generalization of our earlier mass-balance models for supramolecular copolymerizations that encompasses all our earlier models. In this model, the binding free energies of each pair of monomer types in each aggregate type can be set independently. We provide scripts to solve the model numerically for any (co)polymerization of one or two types of monomer into an arbitrary number of distinct aggregate types. We illustrate the applicability of the model on data from literature as well as on new experimental data of triarylamine triamide-based copolymers in three distinct solvents. We show that apart from common properties such as the degree of polymerization and length distributions, our approach also allows us to investigate properties such as the copolymer microstructure, that is, the internal ordering of monomers within the copolymers. Moreover, we show that in some cases, also intriguing analytical approximations can be derived from the mass balances.
Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not ...energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.
•We derive a novel diffuse-interface energy-stable model for two-phase flow.•The model lies in between phase-field and level-set formulations.•We also provide a fully-discrete method that inherits the energy stable property.•The method is moreover point-wise incompressible and maximum-principle-preserving.•Numerical results in two and three dimensions confirm the energy-stable property.
PDF
