The one-dimensional (1D) 𝐽1−𝐽2 quantum spin model can be viewed as a strong-coupling analog of the Schrieffer-Su-Heeger model with two inequivalent alternating Ising couplings along the wire, associated to the physics of resonating valence bonds. Similar to the quantum Ising model, which differently presents a long-range Néel ordered phase, this model also maps onto a 𝑝-wave superconducting wire which shows a topological phase transition with the emergence of low-energy Majorana fermions. We show how signatures of the topological phase transition for the 𝑝-wave superconducting wire, i.e., a half Skyrmion, are revealed through local (short-range) spin observables and their derivatives related to the capacitance of the pairing fermion model. Then, we present an “edge” correspondence through the edge spin susceptibility in the 𝐽1−𝐽2 model revealing that the topological phase transition is a metal of Majorana fermions. We justify that the spin magnetization at an edge at very small transverse magnetic field is a good marker of the topological invariant and of Majorana zero modes. We identify a correspondence between the quantum information of resonating valence bonds and the charge fluctuations in a 𝑝-wave superconductor through our method “the bipartite fluctuations”. Physical properties of this 1D model are in fact robust when including additional interactions, which is optimistic for practical applications, e.g., in quantum circuits.

The anomalous bulk-boundary correspondence in non-Hermitian systems featuring an intricate interplay between skin and boundary modes has attracted enormous theoretical and experimental attention. Still, in dimensions higher than one, this interplay remains much less understood. Here we provide insights from exact analytical solutions of a large class of models in any dimension 𝑑, with open boundaries in 𝑑_𝑐≤𝑑 directions, and by tracking their topological origin. Specifically, we show that amoeba theory accounting for the separation gaps of the bulk modes augmented with higher-dimensional generalizations of the biorthogonal polarization and the generalized Brillouin zone approaches accounting for the surface gaps of boundary modes provide a comprehensive understanding of these systems.

The interplay between dissipation, topology, and sensitivity to boundary conditions has recently attracted tremendous amounts of attention at the level of effective non-Hermitian descriptions. Here we exactly solve a quantum mechanical Lindblad master equation describing a dissipative topological Su-Schrieffer-Heeger (SSH) chain of fermions for both open boundary condition (OBC) and periodic boundary condition (PBC). We find that the extreme sensitivity on the boundary conditions associated with the non-Hermitian skin effect is directly reflected in the rapidities governing the time evolution of the density matrix giving rise to a Liouvillian skin effect. This leads to several intriguing phenomena including boundary sensitive damping behavior, steady state currents in finite periodic systems, and diverging relaxation times in the limit of large systems. We illuminate how the role of topology in these systems differs in the effective non-Hermitian Hamiltonian limit and the full master equation framework.

Biophysical chemistry deals with the structural behavior, properties and molecular function of biological macromolecules. A long-standing challenge is here to establish how these macromolecular features change upon transfer from simplified conditions in vitro to the crowded and molecularly complex environment of live cells.  

This thesis focuses on establishing a general overview of the structural behavior and interaction properties of the ALS-associated protein superoxide dismutase 1 (SOD1) in its natural cellular environment. Importantly, SOD1 constitutes also a multifaceted model system for the yet poorly understood mechanism of protein-aggregation disease, since it is readily amenable to protein-engineering analysis. The focus is on (i) SOD1 folding, (ii) the modulation of the SOD1 properties induced by intracellular interactions and (iii) the process of SOD1 fibrillation, all of which central to the understanding of the ALS disease mechanism. First, we investigate the biophysical role of the disordered catalytic loops in the apoSOD1 monomer, what is identified as the primary aggregation precursor. The results show that these loops play a pivotal role in modulation the apoSOD1 stability due to the generic Flory-entropy penalty, shedding new light to why this species is biased to be aggregation prone. Second, we target the diffusive interactions between SOD1 and the crowded intracellular environment by in-cell NMR. Our findings are that both the rotational tumbling and in-cell stability are controlled by basic physicochemical rules relating to the SOD1 surface properties. Finally, we analyze the kinetics of the SOD1-aggregation behavior in vitro. The observations confirm that the disordered SOD1 loops indeed accelerate the aggregation process because of their penalty to the apo state stability and show, additionally, that they influence the fibril stability.

The physicochemical cues exposed by this thesis work provide not only fundamental clues to our understanding of protein properties, but shed also new light on disease-promoting properties ALS-associated protein SOD1.

In the cytosolic environment, protein crowding and Brownian motions result in numerous transient encounters. Each such encounter event increases the apparent size of the interacting molecules, leading to slower rotational tumbling. The extent of transient protein complexes formed in live cells can conveniently be quantified by an apparent viscosity, based on NMR-detected spin-relaxation measurements, that is, the longitudinal (T-1) and transverse (T-2) relaxation. From combined analysis of three different proteins and surface mutations thereof, we find that T-2 implies significantly higher apparent viscosity than T-1. At first sight, the effect on T-1 and T-2 seems thus nonunifiable, consistent with previous reports on other proteins. We show here that the T-1 and T-2 deviation is actually not a inconsistency but an expected feature of a system with fast exchange between free monomers and transient complexes. In this case, the deviation is basically reconciled by a model with fast exchange between the free-tumbling reporter protein and a transient complex with a uniform 143 kDa partner. The analysis is then taken one step further by accounting for the fact that the cytosolic content is by no means uniform but comprises a wide range of molecular sizes. Integrating over the complete size distribution of the cytosolic interaction ensemble enables us to predict both T-1 and T-2 from a single binding model. The result yields a bound population for each protein variant and provides a quantification of the transient interactions. We finally extend the approach to obtain a correction term for the shape of a database-derived mass distribution of the interactome in the mammalian cytosol, in good accord with the existing data of the cellular composition.

Random encounters between proteins in crowded cells are by no means passive, but found to be under selective control. This control enables proteome solubility, helps to optimise the diffusive search for interaction partners, and allows for adaptation to environmental extremes. Interestingly, the residues that modulate the encounters act mesoscopically through protein surface hydrophobicity and net charge, meaning that their detailed signatures vary across organisms with different intracellular constraints. To examine such variations, we use in-cell NMR relaxation to compare the diffusive behaviour of bacterial and human proteins in both human and Escherichia coli cytosols. We find that proteins that ‘stick’ in E. coli are generally less restricted in mammalian cells. Furthermore, the rotational diffusion in the mammalian cytosol is less sensitive to surface-charge mutations. This implies that, in terms of protein motions, the mammalian cytosol is more forgiving to surface alterations than E. coli cells. The cellular differences seem not linked to the proteome properties per se, but rather to a 6-fold difference in protein concentrations. Our results outline a scenario in which the tolerant cytosol of mammalian cells, found in long-lived multicellular organisms, provides an enlarged evolutionary playground, where random protein-surface mutations are less deleterious than in short-generational bacteria.

A conspicuous feature of the amyotrophic lateral sclerosis (ALS)-associated protein SOD1 is that its maturation into a functional enzyme relies on local folding of two disordered loops into a catalytic subdomain. To drive the disorder-to-order transition, the protein employs a single Zn2+ ion. The question is then if the entropic penalty of maintaining such disordered loops in the immature apoSOD1 monomer is large enough to explain its unusually low stability, slow folding, and pathological aggregation in ALS. To find out, we determined the effects of systematically altering the SOD1-loop lengths by protein redesign. The results show that the loops destabilize the apoSOD1 monomer by similar to 3 kcal/mol, rendering the protein marginally stable and accounting for its aggregation behavior. Yet the effect on the global folding kinetics remains much smaller with a transition-state destabilization of <1 kcal/mol. Notably, this 1/3 transition-state to folded-state stability ratio provides a clear-cut example of the enigmatic disagreement between the Leffler alpha value from loop-length alterations (typically 1/3) and the standard reaction coordinates based on solvent perturbations (typically >2/3). Reconciling the issue, we demonstrate that the disagreement disappears when accounting for the progressive loop shortening that occurs along the folding pathway. The approach assumes a consistent Flory loop entropy scaling factor of c = 1.48 for both equilibrium and kinetic data and has the added benefit of verifying the tertiary interactions of the folding nucleus as determined by phi-value analysis. Thus, SOD1 not only represents a case where evolution of key catalytic function has come with the drawback of a destabilized apo state but also stands out as a well-suited model system for exploring the physicochemical details of protein self-organization.