Genome-wide Profiling of Transcription Factor-DNA Binding Interactions in <em>Candida albicans</em>: A Comprehensive CUT&amp;RUN Method and Data Analysis Workflow

Regulatory transcription factors control many important biological processes, including cellular differentiation, responses to environmental perturbations ...

UNIVERSITY OF CALIFORNIA, MERCED

Prions are proteins most commonly associated with fatal neurodegenerative diseases in mammals but are also responsible for a number of harmless heritable phenotypes in yeast. These states arise when a misfolded form of a protein appears and, rather than be removed by cellular quality control mechanisms, persists. The misfolded prion protein forms aggregates and is capable of converting normally folded protein to the misfolded state through direct interaction between the two forms. The dominant mathematical model for prion aggregate dynamics has been the nucleated polymerization model (NPM) which considers the dynamics of only the normal protein and the aggregates. However, for yeast prions the molecular chaperone Hsp104 is essential for prion propagation. Further, although mammals do not express Hsp104, experimental assays have shown Hsp104 also interacts with mammalian prion aggregates. In this study, we generalize the NPM to account for molecular chaperones and develop what we call the enzyme-limited nucleated polymerization model (ELNPM). We discuss existence, uniqueness and stability of solutions to our model and demonstrate that the NPM represents a quasi-steady-state reduction of our model. We validate the ELNPM by demonstrating agreement with experimental results on the yeast prion PSI(+) that could not be supported by the NPM. Finally, we demonstrate that, in contrast to the NPM, the ELNPM permits the coexistence of multiple prion strains.

A mathematical model of the dynamics of prion aggregates with chaperone-mediated fragmentation.

In biological evolution traits may rise and fall in frequency due to genetic drift, where variant frequencies change by chance, or by selection where advantageous variants will rise in frequency. The neutral model of evolution, first developed by Kimura in the 1960s, has become the standard against which selection is detected. While the balance between these two important forces - drift and selection - has been well established in biology there are other domains where the contribution of these processes is still coming together. Although the idea of natural selection has been applied to the cultural domain since the time of Darwin, it has proven more challenging to positively identify cultural traits under selection both because of a lack of established tests for selection and a lack of large cultural data sets. However, in recent years with the accumulation of large cultural data sets many cultural features from pre-history pottery to modern baby names have been shown to evolve according to the neutral theory. But there is accumulating empirical evidence from cultural processes suggesting that the neutral theory alone cannot account for all features of the data. As such, there has been a renewed interest in determining whether there is selection amidst drift. Here we analyze a subset English word frequencies, and determine whether frequency change reveals processes of selection. Inspired by the Moran and Wright-Fisher models in population genetics, we developed a neutral model of word frequency variation to assess when linguistic data appears to depart from neutral evolution. As such, our model represents a possible "test for selection" in the linguistic domain. We explore how the distribution of word use has changed for sets of words in English for more than 100 years (1901-2008) as expressed in vocabulary usage in published books, made available by Google Ngram. When comparing empirical word frequency changes to our neutral model we find pervasive and systematic departures from neutrality.

Culturomics as a data playground for tests of selection: Mathematical approaches to detecting selection in word use.

Transposable elements (TEs), segments of DNA capable of self-replication, are abundant in the genomes of most organisms and thus serve as a record of past mutational events. While some work suggests TEs may serve a regulatory function for the host, most empirical and theoretical studies have shown that TEs often have deleterious effects on a host. Because they are not essential, the host genome consists of both full-length (actively replicating) and partial length (inactive remnant) copies of TEs. We developed a novel mathematical formulation of TE dynamics by modeling the density of full and partial length copies resulting from mutations (insertions and deletions) and TE replication within the host genome. More specifically, we model the time-evolution of the complete TE length distribution (full and partial elements) in a genome using fragmentation equations in both a discrete and continuous framework under two models of TE replication. In the first case, we assume that full-length TEs replicate at a constant rate regardless of the number of full-length TEs present in the genome. While this assumption simplifies the underlying biological processes, it allows us to derive an explicit analytical form of the time-varying TE density, as well as the steady-state behavior, under both discrete and continuous formulations. Next, we take into account the potential deleterious effects of TEs by modeling TE replication with a logistic growth equation. Under this assumption, the number of actively replicating TEs in a genome is limited by a carrying capacity. While we are still able to derive to derive analytical forms for the time-varying TE density, for both discrete and continuous length formulations, these solutions are implicit. For all four proposed models, we prove existence and uniqueness of these solutions describing TE length distributions. We compare both models and note that the logistic and exponential models initially agree. Since most TEs have not reached carrying capacity, we use the explicit exponential solution to quantify rates of replication to mutations. We apply our model to present day annotated collections of TEs from the genomes of species of fruit-flies, birds, and primates to uncover quantitative relationships of TE dynamics. With the increasing availability of complete genomes, our method is likely to uncover relationships of biological drivers of genomic variation in many species.

Modeling transposable element dynamics with fragmentation equations.

Prions are proteins capable of adopting misfolded conformations and transmitting these conformations to other normally folded proteins. Prions are most commonly known for causing fatal neurodegenerative diseases in mammals but are also associated with several harmless phenotypes in yeast. A distinct feature of prion propagation is the existence of different phenotypical variants, called strains. It is widely accepted that these strains correspond to different conformational states of the protein, but the mechanisms driving their interactions remain poorly understood. This study uses mathematical modeling to provide insight into this problem. We show that the classical model of prion dynamics allows at most one conformational strain to stably propagate. In order to conform to biological observations of strain coexistence and co-stability, we develop an extension of the classical model by introducing a novel prion species consistent with biological studies. Qualitative analysis of this model reveals a new variety of behavior. Indeed, it allows for stable coexistence of different strains in a wide parameter range, and it also introduces intricate initial condition dependency. These new behaviors are consistent with experimental observations of prions in both mammals and yeast. As such, our model provides a valuable tool for investigating the underlying mechanisms of prion propagation and the link between prion strains and strain specific phenotypes. The consideration of a novel prion species brings a change in perspective on prion biology and we use our model to generate hypotheses about prion infectivity.

Generalizing a mathematical model of prion aggregation allows strain coexistence and co-stability by including a novel misfolded species.

Biological systems are inherently complex, and the increasing level of detail with which we are able to experimentally probe such systems continually reveals new complexity. Fortunately, mathematical models are uniquely positioned to provide a tool suitable for rigorous analysis, hypothesis generation, and connecting results from isolated  experiments with results from  and whole-organism studies. However, developing useful mathematical models is challenging because of the often different domains of knowledge required in both math and biology. In this work, we endeavor to provide a useful guide for researchers interested in incorporating mathematical modeling into their scientific process. We advocate for the use of conceptual diagrams as a starting place to anchor researchers from both domains. These diagrams are useful for simplifying the biological process in question and distinguishing the essential components. Not only do they serve as the basis for developing a variety of mathematical models, but they ensure that any mathematical formulation of the biological system is led primarily by scientific questions. We provide a specific example of this process from our own work in studying prion aggregation to show the power of mathematical models to synergistically interact with experiments and push forward biological understanding. Choosing the most suitable model also depends on many different factors, and we consider how to make these choices based on different scales of biological organization and available data. We close by discussing the many opportunities that abound for both experimentalists and modelers to take advantage of collaborative work in this field.

How and why to build a mathematical model: A case study using prion aggregation.

Amyloid appearance is a rare event that is promoted in the presence of other aggregated proteins. These aggregates were thought to act by templating the formation of an assembly-competent nucleation seed, but we find an unanticipated role for them in enhancing the persistence of amyloid after it arises. Specifically, Saccharomyces cerevisiae Rnq1 amyloid reduces chaperone-mediated disassembly of Sup35 amyloid, promoting its persistence in yeast. Mathematical modeling and corresponding in vivo experiments link amyloid persistence to the conformationally defined size of the Sup35 nucleation seed and suggest that amyloid is actively cleared by disassembly below this threshold to suppress appearance of the [PSI] prion in vivo. Remarkably, this framework resolves multiple known inconsistencies in the appearance and curing of yeast prions. Thus, our observations establish the size of the nucleation seed as a previously unappreciated characteristic of prion variants that is key to understanding transitions between prion states.

Nucleation seed size determines amyloid clearance and establishes a barrier to prion appearance in yeast.

Agent-based models (ABM), also called individual-based models, first appeared several decades ago with the promise of nearly real-time simulations of active, autonomous individuals such as animals or objects. The goal of ABMs is to represent a population of individuals (agents) interacting with one another and their environment. Because of their flexible framework, ABMs have been widely applied to study systems in engineering, economics, ecology, and biology. This chapter is intended to guide the users in the development of an agent-based model by discussing conceptual issues, implementation, and pitfalls of ABMs from first principles. As a case study, we consider an ABM of the multi-scale dynamics of cellular interactions in a microbial community. We develop a lattice-free agent-based model of individual cells whose actions of growth, movement, and division are influenced by both their individual processes (cell cycle) and their contact with other cells (adhesion and contact inhibition).

Suzanne S. Sindi

Department of Applied Mathematics

UNIVERSITY OF CALIFORNIA, MERCED

Genome-wide Profiling of Transcription Factor-DNA Binding Interactions in Candida albicans: A Comprehensive CUT&RUN Method and Data Analysis Workflow

Suzanne S. Sindi

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UNIVERSITY OF CALIFORNIA, MERCED

Genome-wide Profiling of Transcription Factor-DNA Binding Interactions in Candida albicans: A Comprehensive CUT&RUN Method and Data Analysis Workflow

Department of Applied Mathematics