The energy landscape, which organizes microstates by energies, has shed light on many cellular processes governed by dynamic biological macromolecules leveraging their structural dynamics to regulate interactions with molecular partners. In particular, the protein energy landscape has been central to understanding the relationship between protein structure, dynamics, and function. The landscape view, however, remains underutilized in an important problem in protein modeling, decoy selection in template-free protein structure prediction. Given the amino-acid sequence of a protein, template-free methods compute thousands of structures, known as decoys, as part of an optimization process that seeks minima of an energy function. Selecting biologically-active/native structures from the computed decoys remains challenging. Research has shown that energy is an unreliable indicator of nativeness. In this paper, we advocate that, while comparison of energies is not informative for structures that already populate minima of an energy function, the landscape view exposes the overall organization of generated decoys. As we demonstrate, such organization highlights macrostates that contain native decoys. We present two different computational approaches to extracting such organization and demonstrate through the presented findings that a landscape-driven treatment is promising in furthering research on decoy selection.

The protein energy landscape underscores the inherent nature of proteins as dynamic molecules interconverting between structures with varying energies. Reconstructing a protein’s energy landscape holds the key to characterizing a protein’s equilibrium conformational dynamics and its relationship to function. Many pathogenic mutations in protein sequences alter the equilibrium dynamics that regulates molecular interactions and thus protein function. In principle, reconstructing energy landscapes of a protein’s healthy and diseased variants is a central step to understanding how mutations impact dynamics, biological mechanisms, and function. Recent computational advances are yielding detailed, sample-based representations of protein energy landscapes. In this paper, we propose and describe two novel methods that leverage computed, sample-based representations of landscapes to reconstruct them and extract from them informative local structures that reveal the underlying organization of an energy landscape. Such structures constitute landscape features that, as we demonstrate here, can be utilized to detect alterations of landscapes upon mutation. The proposed methods detect altered protein energy landscape features in response to sequence mutations. By doing so, the methods allow formulating hypotheses on the impact of mutations on specific biological activities of a protein. This work demonstrates that the availability of energy landscapes of healthy and diseased variants of a protein opens up new avenues to harness the quantitative information embedded in landscapes to summarize mechanisms via which mutations alter protein dynamics to percolate to dysfunction.

This paper establishes a strong correspondence between two important clustering approaches that emerged in the 1970s: clustering by level sets or cluster tree as proposed by Hartigan, and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. This correspondence is drawn by showing that the gradient ascent flow provides a natural way to move up the cluster tree.

We develop large sample theory including nonparametric confidence regions for $r$-dimensional ridges of probability density functions on $\mathbb{R}^d$, where $1\leq r<d$. We view ridges as the intersections of level sets of some special functions. The vertical variation of the plug-in kernel estimators for these functions constrained on the ridges is used as the measure of maximal deviation for ridge estimation. Our confidence regions for the ridges are determined by the asymptotic distribution of this maximal deviation, which is established by utilizing the extreme value distribution of nonstationary $\chi$-fields indexed by manifolds.

We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical methods that have played important roles in these area -- and showcase their use in the context of functional data analysis. In the process, we highlight the crucial role that the ambient metric plays.

This paper studies and critically discusses the construction of nonparametric confidence regions for density level sets. Methodologies based on both vertical variation and horizontal variation are considered. The investigations provide theoretical insight into the behavior of these confidence regions via large sample theory. We also discuss the geometric relationships underlying the construction of horizontal and vertical methods, and how finite sample performance of these confidence regions is influenced by geometric or topological aspects. These discussions are supported by numerical studies.