Jin Dai

Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of time crystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete Hamiltonian variant of the Q-ball Lagrangian of time dependent non-topological solitons. We explain how a Q-ball becomes a time crystal, and we construct examples of time crystalline Q-balls.

A time crystal is a time dependent physical system that does not reach a standstill, even in state of minimum energy. Here we show that the stability of a time crystal can be enhanced by its topology. For this we simulate time crystals made of chainlike ensembles of mutually interacting point particles. When we tie the chain into a knot we find that its timecrystalline qualities improve. The theoretical models we consider are widely used in coarse grained descriptions of linear polymers. Thus we expect that physical realizations of time crystals can be found in terms of knotted molecules.

We identify circumstances where the effective descriptions of microscopic physical systems leads to a self-consistent reduced dynamics for a truncated subset of the original variables. The effective Hamiltonian involves unusual Poisson brackets that bring in noncommutative geometry. In idealized models of ring molecules, we find time crystal behavior is widespread.

A protein is traditionally visualised as a piecewise linear discrete curve, and its geometry is conventionally characterised by the extrinsically determined Ramachandran angles. However, a protein backbone has also two independent intrinsic geometric structures, due to the peptide planes and the side chains. Here we adapt and develop modern 3D virtual reality techniques to scrutinize the atomic geometry along a protein backbone, in the vicinity of a peptide plane. For this we compare backbone geometry-based (extrinsic) and structure-based (intrinsic) coordinate systems, and as an example we inspect the trans and cis peptide planes. We reveal systematics in the way how a cis peptide plane deforms the neighbouring atomic geometry, and we develop a virtual reality based visual methodology that can identify the presence of a cis peptide plane from the arrangement of atoms in its vicinity. Our approach can easily detect exceptionally placed atoms in crystallographic structures. Thus it can be employed as a powerful visual refinement tool which is applicable also in the case when resolution of the protein structure is limited and whenever refinement is needed. As concrete examples we identify a number of crystallographic protein structures in Protein Data Bank (PDB) that display exceptional atomic positions around their cis peptide planes.

We propose to combine a mean field approach with all atom molecular dynamics into a multistage algorithm that can model protein folding and dynamics over very long time periods yet with atomic level precision. As an example we investigate an isolated monomeric Myc oncoprotein that has been implicated in carcinomas including those in colon, breast and lungs. Under physiological conditions a monomeric Myc is presumed to be an example of intrinsically disordered proteins, that pose a serious challenge to existing modelling techniques. We argue that a room temperature monomeric Myc is in a dynamical state, it oscillates between different conformations that we identify. For this we adopt the C-alpha backbone of Myc in a crystallographic heteromer as an initial Ansatz for the monomeric structure. We construct a multisoliton of the pertinent Landau free energy, to describe the C-alpha profile with ultra high precision. We use Glauber dynamics to resolve how the multisoliton responds to repeated increases and decreases in ambient temperature. We confirm that the initial structure is unstable in isolation. We reveal a highly degenerate ground state landscape, an attractive set towards which Glauber dynamics converges in the limit of vanishing ambient temperature. We analyse the thermal stability of this Glauber attractor using room temperature molecular dynamics. We identify and scrutinise a particularly stable subset in which the two helical segments of the original multisoliton align in parallel, next to each other. During the MD time evolution of a representative structure from this subset, we observe intermittent quasiparticle oscillations along the C-terminal alpha-helix, some of which resemble a translating Davydov's Amide-I soliton. We propose that the presence of oscillatory motion is in line with the expected intrinsically disordered character of Myc.

We inquire how structure emerges during the process of protein folding. For this we scrutinise col- lective many-atom motions during all-atom molecular dynamics simulations. We introduce, develop and employ various topological techniques, in combination with analytic tools that we deduce from the concept of integrable models and structure of discrete nonlinear Schroedinger equation. The example we consider is an alpha-helical subunit of the HIV envelope glycoprotein gp41. The helical structure is stable when the subunit is part of the biological oligomer. But in isolation the helix becomes unstable, and the monomer starts deforming. We follow the process computationally. We interpret the evolving structure both in terms of a backbone based Heisenberg spin chain and in terms of a side chain based XY spin chain. We find that in both cases the formation of protein super-secondary structure is akin the formation of a topological Bloch domain wall along a spin chain. During the process we identify three individual Bloch walls and we show that each of them can be modelled with a very high precision in terms of a soliton solution to a discrete nonlinear Schroedinger equation.

Protein folding [1] is the process of formation of a functional 3D structure from a random coil — the shape in which amino-acid chains leave the ribosome. Anfinsen’s dogma states that the native 3D shape of a protein is completely determined by protein’s amino acid sequence. Despite the progress in understanding the process rate and the success in folding prediction for some small proteins, with presently available physics-based methods it is not yet possible to reliably deduce the shape of a biologically active protein from its amino acid sequence. The protein-folding problem endures as one of the most important unresolved problems in science; it addresses the origin of life itself. Furthermore, a wrong fold is a common cause for a protein to lose its function or even endanger the living organism. Soliton solutions of a generalized discrete non-linear Schrödinger equation (GDNLSE) obtained from the energy function in terms of bond and torsion angles κ and τ provide a constructive theoretical framework for describing protein folds and folding patterns [2]. Here we study the dynamics of this process by means of molecular-dynamics simulations. The soliton manifestation is the pattern helix–loop–helix in the secondary structure of the protein, which explains the importance of understanding loop formation in helical proteins. We performed in silico experiments for unfolding one subunit of the core structure of gp41 from the HIV envelope glycoprotein (PDB ID: 1AIK [3]) by molecular-dynamics simulations with the MD package GROMACS. We analyzed 80 ns trajectories, obtained with one united-atom and two different all-atom force fields, to justify the side-chain orientation quantification scheme adopted in the studies and to eliminate force-field based artifacts. Our results are compatible with the soliton model of protein folding and provide first insight into soliton-formation dynamics.

The human islet amyloid polypeptide (hIAPP) co-operates with insulin to maintain glycemic balance. It also constitutes the amyloid plaques that aggregate in the pancreas of type-II diabetic patients. We have performed extensive in silico investigations to analyse the structural landscape of monomeric hIAPP, which is presumed to be intrinsically disordered. For this we construct from first principles a highly predictive energy function that describes a monomeric hIAPP observed in a NMR experiment, as a local energy minimum. We subject our theoretical model of hIAPP to repeated heating and cooling simulations, back and forth between a high temperature regime where the conformation resembles a random walker and a low temperature limit where no thermal motions prevail. We find that the final low temperature conformations display a high level of degeneracy, in a manner which is fully in line with the presumed intrinsically disordered character of hIAPP. In particular, we identify an isolated family of alpha-helical conformations that might cause the transition to amyloidosis, by nucleation.