Artificial protein cages have great potential in diverse fields including as vaccines and drug de... more Artificial protein cages have great potential in diverse fields including as vaccines and drug delivery vehicles. TRAP-cage is an artificial protein cage notable for the way in which the interface between its ring-shaped building blocks can be modified such that the conditions under which cages disassemble can be controlled. To date, TRAPcages have been constructed from homo-11mer rings, i.e., hendecamers. This is interesting as convex polyhedra with identical regular faces cannot be formed from hendecamers. TRAP-cage overcomes this limitation due to intrinsic flexibility, allowing slight deformation to absorb any error. The resulting TRAP-cage made from 24 TRAP 11mer rings is very close to regular with only very small errors necessary to allow the cage to form. The question arises as to the limits of the error that can be absorbed by a protein structure in this way before the formation of an apparently regular convex polyhedral becomes impossible. Here we use a naturally occurring TRAP variant consisting of twelve identical monomers (i.e., a dodecamer) to probe these limits. We show that it is able to form an apparently regular protein cage consisting of twelve TRAP rings. Comparison of the cryo-EM structure of the new cage with theoretical models and related cages gives insight into the rules of cage formation and allows us to predict other cages that may be formed given TRAP-rings consisting of different numbers of monomers.
We study the propagation of α-helix polarons in a model describing the non-adiabatic interaction ... more We study the propagation of α-helix polarons in a model describing the non-adiabatic interaction between an electron and a lattice of quantum mechanical oscillators at physiological temperature. We show that when excited by a sub-picosecond electric pulse, as induced by experimentally observed sub-picosecond charge separation, the polaron is displaced by up to hundreds of lattice sites before the electron becomes delocalised. We discuss biophysical implications of our results.
We study the long-range electron and energy transfer mediated by a polaron on an α-helix polypept... more We study the long-range electron and energy transfer mediated by a polaron on an α-helix polypeptide chain coupled to donor and acceptor molecules at opposite ends of the chain. We show that for specific parameters of the system, an electron initially located on the donor can tunnel onto the α-helix, forming a polaron, which then travels to the other extremity of the polypeptide chain where it is captured by the acceptor. We consider three families of couplings between the donor, acceptor and the chain, and show that one of them can lead to a 90% efficiency of the electron transport from donor to acceptor. We also show that this process remains stable at physiological temperatures in the presence of thermal fluctuations in the system.
We study the vibration modes of the Skyrme model within the rational map ansatz. We show that the... more We study the vibration modes of the Skyrme model within the rational map ansatz. We show that the vibrations of the radial profiles and the rational maps are decoupled and we consider explicitly the cases B ¼ 1, B ¼ 2, and B ¼ 4. We then compare our results with the vibration modes obtained numerically by Barnes et al. and show that qualitatively the rational map reproduces the vibration modes obtained numerically but that the vibration frequencies of these modes do not match very well.
We discuss an ansatz for Skyrme fields in three dimensions which uses rational maps between Riema... more We discuss an ansatz for Skyrme fields in three dimensions which uses rational maps between Riemann spheres, and produces shell-like structures of polyhedral form. Houghton, Manton and Sutcliffe showed that a single rational map gives good approximations to the minimal energy Skyrmions up to baryon number of order ten. We show how the method can be generalised by using two or more rational maps to give a double-shell or multi-shell structure. Particularly interesting examples occur at baryon numbers twelve and fourteen.
We consider the Einstein-Yang-Mills-Higgs equations for an SU(3) gauge group in a spherically sym... more We consider the Einstein-Yang-Mills-Higgs equations for an SU(3) gauge group in a spherically symmetric ansatz. Several properties of the gravitating monopole solutions are obtained an compared with their SU(2) counterpart.
We study Skyrmion dynamics in a (2+1)-dimensional Skyrme model. The system contains a dimensionle... more We study Skyrmion dynamics in a (2+1)-dimensional Skyrme model. The system contains a dimensionless parameter α, with α = 0 corresponding to the O(3) sigmamodel. If two Skyrmions collide head-on, then they can either coalesce or scatterthis depends on α and on the incident speed v, and is affected by transfer of energy to and from the internal vibrational modes of the Skyrmions. We classify these internal modes and compute their spectrum, for a range of values of α. In particular, we find that there is a fractal-like structure of scattering windows, analogous to those seen for kink-antikink scattering in 1+1 dimensions.
We present a one-parameter family of mathematical models describing the dynamics of polarons in p... more We present a one-parameter family of mathematical models describing the dynamics of polarons in periodic structures, such as linear polypeptides, which, by tuning the model parameter, can be reduced to the Davydov or the Scott model. We describe the physical significance of this parameter and, in the continuum limit, we derive analytical solutions which represent stationary polarons. On a discrete lattice, we compute stationary polaron solutions numerically. We investigate polaron propagation induced by several external forcing mechanisms. We show that an electric field consisting of a constant and a periodic component can induce polaron motion with minimal energy loss. We also show that thermal fluctuations can facilitate the onset of polaron motion. Finally, we discuss the bio-physical implications of our results.
We investigate some aspects of the soliton dynamics in an α-helical protein macromolecule within ... more We investigate some aspects of the soliton dynamics in an α-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability and dynamical properties of Davydov's solitons in an α-helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possess the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of an hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multi-hump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previosly been observed numerically in [A.C. Scott, Phys.Rev. A 26 578 (1982)]. Here we argue that the solitons studied by A. Scott, are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the α-helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.
We study the 3 + 1 dimensional Skyrme model with a mass term different from the usual one. We sho... more We study the 3 + 1 dimensional Skyrme model with a mass term different from the usual one. We show that this new model possesses domain walls solutions. We describe how, in the equivalent 2+1 dimensional model, the Skyrmion is absorbed by the wall.
Even the simplest organisms are too complex to have spontaneously arisen fully formed, yet precur... more Even the simplest organisms are too complex to have spontaneously arisen fully formed, yet precursors to first life must have emerged ab initio from their environment. A watershed event was the appearance of the first entity capable of evolution: the Initial Darwinian Ancestor. Here, we suggest that nucleopeptide reciprocal replicators could have carried out this important role and contend that this is the simplest way to explain extant replication systems in a mathematically consistent way. We propose short nucleic acid templates on which amino-acylated adapters assembled. Spatial localization drives peptide ligation from activated precursors to generate phosphodiester-bond-catalytic peptides. Comprising autocatalytic protein and nucleic acid sequences, this dynamical system links and unifies several previous hypotheses and provides a plausible model for the emergence of DNA and the operational code.
We study the mechanical response of an open cell dry foam subjected to periodic forcing using exp... more We study the mechanical response of an open cell dry foam subjected to periodic forcing using experiments and theory. Using the measurements of the static and dynamic stress-strain relationship, we derive an over-damped model of the foam, as a set of infinitesimal non-linear springs, where the damping term depends on the local foam strain. We then analyse the properties of the foam when subjected to large amplitudes periodic stresses and determine the conditions for which the foam becomes optimally absorbing.
We study numerically self-trapped (polaron) states of quasiparticles (electrons, holes or exciton... more We study numerically self-trapped (polaron) states of quasiparticles (electrons, holes or excitons) in a deformable nanotube formed by a hexagonal lattice, wrapped into a cylinder (carbon-and boron nitride-type nanotube structures). We present a Hamiltonian for such a system taking into account an electron-phonon interaction, and determine conditions under which the lowest energy states are polarons. We compute a large class of numerical solutions of this model for a wide range of the parameters. We show that at not too strong electron-phonon coupling, the system admits ring-like localized solutions wrapped around the nanotube (the charge carrier is localized along the nanotube axis and uniformly distributed with respect to the azimuthal coordinate). At stronger coupling, solutions are localized on very few lattice sites in both directions of the nanotube. The transition from one type solution to the other one depends on the diameter of the nanotube. We show that for the values of the carbon nanotube parameters, the polarons have a ring-like structure wrapped around the nanotube with a profile resembling that of the nonlinear Schrödinger soliton.
Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform... more Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform (CWT) based on the 2D Euclidean group with dilations. In this case, the wavelet transform of a 2D signal (an image) is a function of 4 parameters: two translation parameters b x , b ...
We study the polaron (soliton) states of a quasiparticle (electron, hole, exciton) in a quasi-one... more We study the polaron (soliton) states of a quasiparticle (electron, hole, exciton) in a quasi-one-dimensional (quasi-1D) model which describes a carbon-type zigzag nanotube structure. In the Hamiltonian of the system we include the electron-phonon interaction that arises from the dependence of both the on-site and the hopping interaction energies on the lattice deformation. We derive, in the adiabatic approximation, the equations for the self-trapped states of a quasiparticle in a zigzag nanotube. We show that the ground state of such a system depends on the strength of the electron-phonon coupling and we find polaron-type solutions with different symmetries. Namely, at a relatively weak coupling a quasiparticle is self-trapped in a quasi-1D polaron state which has an azimuthal symmetry. When the coupling constant exceeds some critical value, the azimuthal symmetry breaks down and the quasiparticle state can be described as a two-dimensional small polaron on the nanotube surface. In the crossover region between the two solutions there is a range of intermediate couplings, in which the two structures, the quasi-1D polaron and the strongly localized 2D polaron, coexist as their energies are very close together. We note that the results of this analytical study are in quantitative agreement with what has recently been observed numerically.
We use a prescription to gauge the SU (2) Skyrme model with a U (1) field, characterised by a con... more We use a prescription to gauge the SU (2) Skyrme model with a U (1) field, characterised by a conserved Baryonic current. This model reverts to the usual Skyrme model in the limit of the gauge coupling constant vanishing. We show that there exist axially symmetric static solutions with zero magnetic charge, which can be electrically either charged or uncharged. The energies of the (uncharged) gauged Skyrmions are less than the energy of the (usual) ungauged Skyrmion. For physical values of the parameters the impact of the U (1) field is very small, so that it can be treated as a perturbation to the (ungauged) spherically symmetric Hedgehog. This allows the perturbative calculation of the magnetic moment.
Artificial protein cages have great potential in diverse fields including as vaccines and drug de... more Artificial protein cages have great potential in diverse fields including as vaccines and drug delivery vehicles. TRAP-cage is an artificial protein cage notable for the way in which the interface between its ring-shaped building blocks can be modified such that the conditions under which cages disassemble can be controlled. To date, TRAPcages have been constructed from homo-11mer rings, i.e., hendecamers. This is interesting as convex polyhedra with identical regular faces cannot be formed from hendecamers. TRAP-cage overcomes this limitation due to intrinsic flexibility, allowing slight deformation to absorb any error. The resulting TRAP-cage made from 24 TRAP 11mer rings is very close to regular with only very small errors necessary to allow the cage to form. The question arises as to the limits of the error that can be absorbed by a protein structure in this way before the formation of an apparently regular convex polyhedral becomes impossible. Here we use a naturally occurring TRAP variant consisting of twelve identical monomers (i.e., a dodecamer) to probe these limits. We show that it is able to form an apparently regular protein cage consisting of twelve TRAP rings. Comparison of the cryo-EM structure of the new cage with theoretical models and related cages gives insight into the rules of cage formation and allows us to predict other cages that may be formed given TRAP-rings consisting of different numbers of monomers.
We study the propagation of α-helix polarons in a model describing the non-adiabatic interaction ... more We study the propagation of α-helix polarons in a model describing the non-adiabatic interaction between an electron and a lattice of quantum mechanical oscillators at physiological temperature. We show that when excited by a sub-picosecond electric pulse, as induced by experimentally observed sub-picosecond charge separation, the polaron is displaced by up to hundreds of lattice sites before the electron becomes delocalised. We discuss biophysical implications of our results.
We study the long-range electron and energy transfer mediated by a polaron on an α-helix polypept... more We study the long-range electron and energy transfer mediated by a polaron on an α-helix polypeptide chain coupled to donor and acceptor molecules at opposite ends of the chain. We show that for specific parameters of the system, an electron initially located on the donor can tunnel onto the α-helix, forming a polaron, which then travels to the other extremity of the polypeptide chain where it is captured by the acceptor. We consider three families of couplings between the donor, acceptor and the chain, and show that one of them can lead to a 90% efficiency of the electron transport from donor to acceptor. We also show that this process remains stable at physiological temperatures in the presence of thermal fluctuations in the system.
We study the vibration modes of the Skyrme model within the rational map ansatz. We show that the... more We study the vibration modes of the Skyrme model within the rational map ansatz. We show that the vibrations of the radial profiles and the rational maps are decoupled and we consider explicitly the cases B ¼ 1, B ¼ 2, and B ¼ 4. We then compare our results with the vibration modes obtained numerically by Barnes et al. and show that qualitatively the rational map reproduces the vibration modes obtained numerically but that the vibration frequencies of these modes do not match very well.
We discuss an ansatz for Skyrme fields in three dimensions which uses rational maps between Riema... more We discuss an ansatz for Skyrme fields in three dimensions which uses rational maps between Riemann spheres, and produces shell-like structures of polyhedral form. Houghton, Manton and Sutcliffe showed that a single rational map gives good approximations to the minimal energy Skyrmions up to baryon number of order ten. We show how the method can be generalised by using two or more rational maps to give a double-shell or multi-shell structure. Particularly interesting examples occur at baryon numbers twelve and fourteen.
We consider the Einstein-Yang-Mills-Higgs equations for an SU(3) gauge group in a spherically sym... more We consider the Einstein-Yang-Mills-Higgs equations for an SU(3) gauge group in a spherically symmetric ansatz. Several properties of the gravitating monopole solutions are obtained an compared with their SU(2) counterpart.
We study Skyrmion dynamics in a (2+1)-dimensional Skyrme model. The system contains a dimensionle... more We study Skyrmion dynamics in a (2+1)-dimensional Skyrme model. The system contains a dimensionless parameter α, with α = 0 corresponding to the O(3) sigmamodel. If two Skyrmions collide head-on, then they can either coalesce or scatterthis depends on α and on the incident speed v, and is affected by transfer of energy to and from the internal vibrational modes of the Skyrmions. We classify these internal modes and compute their spectrum, for a range of values of α. In particular, we find that there is a fractal-like structure of scattering windows, analogous to those seen for kink-antikink scattering in 1+1 dimensions.
We present a one-parameter family of mathematical models describing the dynamics of polarons in p... more We present a one-parameter family of mathematical models describing the dynamics of polarons in periodic structures, such as linear polypeptides, which, by tuning the model parameter, can be reduced to the Davydov or the Scott model. We describe the physical significance of this parameter and, in the continuum limit, we derive analytical solutions which represent stationary polarons. On a discrete lattice, we compute stationary polaron solutions numerically. We investigate polaron propagation induced by several external forcing mechanisms. We show that an electric field consisting of a constant and a periodic component can induce polaron motion with minimal energy loss. We also show that thermal fluctuations can facilitate the onset of polaron motion. Finally, we discuss the bio-physical implications of our results.
We investigate some aspects of the soliton dynamics in an α-helical protein macromolecule within ... more We investigate some aspects of the soliton dynamics in an α-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability and dynamical properties of Davydov's solitons in an α-helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possess the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of an hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multi-hump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previosly been observed numerically in [A.C. Scott, Phys.Rev. A 26 578 (1982)]. Here we argue that the solitons studied by A. Scott, are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the α-helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity.
We study the 3 + 1 dimensional Skyrme model with a mass term different from the usual one. We sho... more We study the 3 + 1 dimensional Skyrme model with a mass term different from the usual one. We show that this new model possesses domain walls solutions. We describe how, in the equivalent 2+1 dimensional model, the Skyrmion is absorbed by the wall.
Even the simplest organisms are too complex to have spontaneously arisen fully formed, yet precur... more Even the simplest organisms are too complex to have spontaneously arisen fully formed, yet precursors to first life must have emerged ab initio from their environment. A watershed event was the appearance of the first entity capable of evolution: the Initial Darwinian Ancestor. Here, we suggest that nucleopeptide reciprocal replicators could have carried out this important role and contend that this is the simplest way to explain extant replication systems in a mathematically consistent way. We propose short nucleic acid templates on which amino-acylated adapters assembled. Spatial localization drives peptide ligation from activated precursors to generate phosphodiester-bond-catalytic peptides. Comprising autocatalytic protein and nucleic acid sequences, this dynamical system links and unifies several previous hypotheses and provides a plausible model for the emergence of DNA and the operational code.
We study the mechanical response of an open cell dry foam subjected to periodic forcing using exp... more We study the mechanical response of an open cell dry foam subjected to periodic forcing using experiments and theory. Using the measurements of the static and dynamic stress-strain relationship, we derive an over-damped model of the foam, as a set of infinitesimal non-linear springs, where the damping term depends on the local foam strain. We then analyse the properties of the foam when subjected to large amplitudes periodic stresses and determine the conditions for which the foam becomes optimally absorbing.
We study numerically self-trapped (polaron) states of quasiparticles (electrons, holes or exciton... more We study numerically self-trapped (polaron) states of quasiparticles (electrons, holes or excitons) in a deformable nanotube formed by a hexagonal lattice, wrapped into a cylinder (carbon-and boron nitride-type nanotube structures). We present a Hamiltonian for such a system taking into account an electron-phonon interaction, and determine conditions under which the lowest energy states are polarons. We compute a large class of numerical solutions of this model for a wide range of the parameters. We show that at not too strong electron-phonon coupling, the system admits ring-like localized solutions wrapped around the nanotube (the charge carrier is localized along the nanotube axis and uniformly distributed with respect to the azimuthal coordinate). At stronger coupling, solutions are localized on very few lattice sites in both directions of the nanotube. The transition from one type solution to the other one depends on the diameter of the nanotube. We show that for the values of the carbon nanotube parameters, the polarons have a ring-like structure wrapped around the nanotube with a profile resembling that of the nonlinear Schrödinger soliton.
Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform... more Images may be analyzed and reconstructed with a two-dimensional (2D) continuous wavelet transform (CWT) based on the 2D Euclidean group with dilations. In this case, the wavelet transform of a 2D signal (an image) is a function of 4 parameters: two translation parameters b x , b ...
We study the polaron (soliton) states of a quasiparticle (electron, hole, exciton) in a quasi-one... more We study the polaron (soliton) states of a quasiparticle (electron, hole, exciton) in a quasi-one-dimensional (quasi-1D) model which describes a carbon-type zigzag nanotube structure. In the Hamiltonian of the system we include the electron-phonon interaction that arises from the dependence of both the on-site and the hopping interaction energies on the lattice deformation. We derive, in the adiabatic approximation, the equations for the self-trapped states of a quasiparticle in a zigzag nanotube. We show that the ground state of such a system depends on the strength of the electron-phonon coupling and we find polaron-type solutions with different symmetries. Namely, at a relatively weak coupling a quasiparticle is self-trapped in a quasi-1D polaron state which has an azimuthal symmetry. When the coupling constant exceeds some critical value, the azimuthal symmetry breaks down and the quasiparticle state can be described as a two-dimensional small polaron on the nanotube surface. In the crossover region between the two solutions there is a range of intermediate couplings, in which the two structures, the quasi-1D polaron and the strongly localized 2D polaron, coexist as their energies are very close together. We note that the results of this analytical study are in quantitative agreement with what has recently been observed numerically.
We use a prescription to gauge the SU (2) Skyrme model with a U (1) field, characterised by a con... more We use a prescription to gauge the SU (2) Skyrme model with a U (1) field, characterised by a conserved Baryonic current. This model reverts to the usual Skyrme model in the limit of the gauge coupling constant vanishing. We show that there exist axially symmetric static solutions with zero magnetic charge, which can be electrically either charged or uncharged. The energies of the (uncharged) gauged Skyrmions are less than the energy of the (usual) ungauged Skyrmion. For physical values of the parameters the impact of the U (1) field is very small, so that it can be treated as a perturbation to the (ungauged) spherically symmetric Hedgehog. This allows the perturbative calculation of the magnetic moment.
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Papers by Bernard Piette