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## Scholarly Works (53 results)

Compartmentalization of biochemical processes is a central principle in cell biology. Traditionally, understanding spatial organization of reactions has focused on their localization to membrane-bound organelles. However, recent work has highlighted the ability of proteins and RNAs to dynamically partition into large, membrane-less condensates. Formation of these assemblies have been observed in both eukaryotes and prokaryotes suggesting that this could be an ancient mechanism utilized for compartmentalizing and regulating specific processes. This thesis focuses on identifying novel mechanisms that regulate the assembly of two classes of membrane-less condensates: metabolic enzymes and stress granules (SGs). First, we further expanded the list of metabolic enzymes capable of forming a filament or foci structure to 60 proteins. This expansion allowed us to determine that metabolic enzymes, which acted at branch points or highly connected nodes in the metabolic network, had a higher propensity to assemble into intracellular structures than other enzymes. Our analysis of the de novo purine biosynthesis pathway not only revealed that assembly is based on the hierarchical position in a pathway, but also that a subset of enzymes in this pathway localizes to SGs. These results led us to reexamine the hits from our screen and conduct a secondary screen for SG localization. In total, we identify 17 metabolic enzymes that are associated with SGs providing a connection between metabolic activity and post-transcriptional gene regulation. We find that the product of the SG-localized enzyme Sam1, AdoMet, regulates composition and frequency of acute and chronic nutrient stress-induced SGs. Furthermore, AdoMet blocks fusion of SGs in proliferating cancer cell lines while suppressing SG formation in motor neurons derived from patients with amyotrophic lateral sclerosis. With the goal of understanding the role of RNA-RNA interactions in SG formation, we developed a reconstitution system using yeast cytoplasmic extracts and in vitro transcribed RNA. Our in vitro assembled SGs mimicked in vivo SGs and advanced the knowledge of how ATP levels regulate SG assembly, disassembly, morphology, and dynamics. Lastly, we find that building a canonical SG depends not only on the material state of a RNA, but also the composition of the RNA.

This dissertation will present two new rigidity theorems for manifolds with sectional curvature bounded below. The main new result is a new splitting theorem for Jacobi fields on manifolds with positive sectional curvature.

This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside.

Abstract of chapter II: If $\pi:M\rightarrow B$ is a Riemannian Submersion and $M$ has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.

Abstract of Chapter III: Though Riemannian submersions preserve non-negative sectional curvature this does not generalize to Riemannian submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian submersion $\pi: M\rightarrow B$ for which $\textrm{Ricci}_p(M)>0$ and at some point $p\in B$, $\text{Ricci}_p(B)<0$.

Abstract of Chapter IV: The smallest $r$ so that a metric $r$--ball covers a metric space $M$ is called the radius of $M.$ The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

In this work, we describe a method to construct new examples of collapse with a lower curvature bound inspired by Cheeger and Gromov. Unlike with collapse with an upper and lower curvature bound, which is now completely understood, the structure of collapse with a lower curvature bound is still a mystery.

In [4], Grove and Petersen showed that if a sequence of Riemannian manifolds (M i , g i ) has uniform lower curvature bound, k, and M i → X, then X is an Alexandrov space with lower curvature bound k. Petersen, Wilhelm, and Zhu then showed that the converse is false [7]. Perelman showed that given a sequence of n-dimensional Alexandrov spaces with a uniform lower curvature bound, with limit space X such that dim X = n, all but finitely many of the prelimits are homeomorphic to X.

In 1985, Cheeger and Gromov introduced the concept of an F-structure, which can be thought as a generalized torus action on a manifold [1]. They showed that a manifold collapses with bounded curvature if and only if it admits an F-structure [2]. An F-structure is an example of a more general construct known as a g-structure, which is a sheaf of Lie groups actions, and is one of the main tools used in our approach.

The second main tool we use was also defined by Cheeger and is known as a Cheeger deformation. Cheeger generalized the method used by Berger in his classic example now known as the Berger spheres, which helped create the study of collapse in Riemannian geometry. Berger showed that scaling the metric of S 3 along the Hopf circles collapses S 3 to the 2-sphere of radius 1/2.

In this work, using g-structures and Cheeger deformations, we construct new examples of collapse with a lower curvature bound.

The study was carried out to evaluate the effects of long-term application of different organic fertilizers (farmyard manure, compost and sewage sludge) as compared to mineral fertilizers on total sulphur (S) content as well as on inorganic (water soluble and adsorbed SO4-S) and organic S binding forms ( sulphate ester and carbon bonded S). Total S concentrations ranged between 99 and 263 mg kg-1 soil and were highest in the treatments with the high compost (COM2) and sewage sludge application rate (SS2), respectively. The share of inorganic S of total S varies between 4.6 and 11.7 %. It consists mainly of water-soluble sulphate with a concentration ranging between 11.7 mg and 17.9 mg kg-1 soil. The organic S pool is dominated by ester sulphates. The enrichment of this fraction is highest in COM2 (156 mg S kg-1) and SS2 (116 mg S kg-1). Carbon-bonded S ranges between 18.7 mg S kg-1 soil (SS1) and 98.9 mg S kg-1 soil (SS2) with the highest amounts in COM2 and SS2. Overall, the different forms and amounts of S applied for about 45 years lead to varying absolute amounts of organic and inorganic S forms in soil. Nevertheless, the relative proportions of organic and inorganic S only vary in a rather small range. Obviously, soil and site characteristics are more important for this partitioning than quality and quantity of S fertilization.

A fake real projective space is a manifold homotopy equivalent to real projective space, but not diffeomorphic to it. Equivalently, it is the orbit space of a free involu- tion on a (homotopy) sphere.

In this thesis, we show that some of the fake RP6s , constructed by Hirsch and Milnor in 1963, and the analogous fake RP14s admit metrics that simutaneously have almost nonnegative sectional curvature and positive Ricci curvature. These spaces are obtained by taking the Z2 quotients of the embedded images of the standard spheres of codimension one in some of Milnor’ s exotic 7−spheres and the analogous Shimada’s exotic 15−spheres. This part of my thesis is joint work with F. Wilhelm.

Hirsch and Milnor also constructed fake RP5s using invariant subspheres of codi- mension two. Octonionically, this construction yields closed 13−manifolds, that are homotopy equivalent to RP13s. The analog to their proof that fake RP5s are not diffeomorphic to standard RP5 breaks down; since in contrast to dimension 6, there is an exotic 14−sphere. We show that some of the Hirsch-Milnor RP13s are not diffeomorphic to standard RP13s. Here we obtain a complete diffeomorphism clas- sification of the Hirsch-Milnor RP13s. This part of my thesis is joint work with C. He.

Using techniques of group diagrams we provide an elementary proof that certain 3-dimensional Brieskorn varieties are SO(2) × SO(2)−equivariantly diffeomorphic to certain Lens Spaces.