Network Hamiltonian Models for Unstructured Protein Aggregates, with Application to γD-Crystallin

The Functional Significance of High Cysteine Content in Eye Lens γ-Crystallins

Cataract disease is strongly associated with progressively accumulating oxidative damage to the extremely long-lived crystallin proteins of the lens. Cysteine oxidation affects crystallin folding, interactions, and light-scattering aggregation especially strongly due to the formation of disulfide bridges. Minimizing crystallin aggregation is crucial for lifelong lens transparency, so one might expect the ubiquitous lens crystallin superfamilies (α and βγ) to contain little cysteine. Yet, the Cys content of γ-crystallins is well above the average for human proteins. We review literature relevant to this longstanding puzzle and take advantage of expanding genomic databases and improved machine learning tools for protein structure prediction to investigate it further. We observe remarkably low Cys conservation in the βγ-crystallin superfamily; however, in γ-crystallin, the spatial positioning of Cys residues is clearly fine-tuned by evolution. We propose that the requirements of long-term lens transparency and high lens optical power impose competing evolutionary pressures on lens βγ-crystallins, leading to distinct adaptations: high Cys content in γ-crystallins but low in βB-crystallins. Aquatic species need more powerful lenses than terrestrial ones, which explains the high methionine content of many fish γ- (and even β-) crystallins. Finally, we discuss synergies between sulfur-containing and aromatic residues in crystallins and suggest future experimental directions.

Genetic Algorithm for Automated Parameterization of Network Hamiltonian Models of Amyloid Fibril Formation

The time scales of long-time atomistic molecular dynamics simulations are typically reported in microseconds, while the time scales for experiments studying the kinetics of amyloid fibril formation are typically reported in minutes or hours. This time scale deficit of roughly 9 orders of magnitude presents a major challenge in the design of computer simulation methods for studying protein aggregation events. Coarse-grained molecular simulations offer a computationally tractable path forward for exploring the molecular mechanism driving the formation of these structures, which are implicated in diseases such as Alzheimer's, Parkinson's, and type-II diabetes. Network Hamiltonian models of aggregation are centered around a Hamiltonian function that returns the total energy of a system of aggregating proteins, given the graph structure of the system as an input. In the graph, or network, representation of the system, each protein molecule is represented as a node, and noncovalent bonds between proteins are represented as edges. The parameter, i.e., a set of coefficients that determine the degree to which each topological degree of freedom is favored or disfavored, must be determined for each network Hamiltonian model, and is a well-known technical challenge. The methodology is first demonstrated by beginning with an initial set of randomly parametrized models of low fibril fraction (<5% fibrillar), and evolving to subsequent generations of models, ultimately leading to high fibril fraction models (>70% fibrillar). The methodology is also demonstrated by applying it to optimizing previously published network Hamiltonian models for the 5 key amyloid fibril topologies that have been reported in the Protein Data Bank (PDB). The models generated by the AI produced fibril fractions that surpass previously published fibril fractions in 3 of 5 cases, including the most naturally abundant amyloid fibril topology, the 1,2 2-ribbon, which features a steric zipper. The authors also aim to encourage more widespread use of the network Hamiltonian methodology for fitting a wide variety of self-assembling systems by releasing a free open-source implementation of the genetic algorithm introduced here.

Continuous Time Graph Processes with Known ERGM Equilibria: Contextual Review, Extensions, and Synthesis

Graph processes that unfold in continuous time are of obvious theoretical and practical interest. Particularly useful are those whose long-term behavior converges to a graph distribution of known form. Here, we review some of the conditions for such convergence, and provide examples of novel and/or known processes that do so. These include subfamilies of the well-known stochastic actor oriented models, as well as continuum extensions of temporal and separable temporal exponential family random graph models. We also comment on some related threads in the broader work on network dynamics, which provide additional context for the continuous time case. Graph processes that unfold in continuous time are natural models for social network dynamics: able to directly represent changes in structure as they unfold (rather than, e.g. as snapshots at discrete intervals), such models not only offer the promise of capturing dynamics at high temporal resolution, but are also easily mapped to empirical data without the need to preselect a level of granularity with respect to which the dynamics are defined. Although relatively few general frameworks of this type have been extensively studied, at least one (the stochastic actor-oriented models, or SAOMs) is arguably among the most successful and widely used families of models in the social sciences (see, e.g., Snijders (2001); Steglich et al. (2010); Burk et al. (2007); Sijtsema et al. (2010); de la Haye et al. (2011); Weerman (2011); Schaefer and Kreager (2020) among many others). Work using other continuous time graph processes has also found applications both within (Koskinen and Snijders, 2007; Koskinen et al., 2015; Stadtfeld et al., 2017; Hoffman et al., 2020) and beyond (Grazioli et al., 2019; Yu et al., 2020) the social sciences, suggesting the potential for further advances.