Fermion Mass Generation without Spontaneous Symmetry Breaking

The conventional mechanism of fermion mass generation in the Standard Model involves Spontaneous Symmetry Breaking (SSB). In this thesis, we study an alternate mechanism for the generation of fermion masses that does not require SSB, in the context of lattice field theories. Being inherently strongly coupled, this mechanism requires a non-perturbative approach like the lattice approach.

In order to explore this mechanism, we study a simple lattice model with a four-fermion interaction that has massless fermions at weak couplings and massive fermions at strong couplings, but without any spontaneous symmetry breaking. Prior work on this type of mass generation mechanism in 4D, was done long ago using either mean-field theory or Monte-Carlo calculations on small lattices. In this thesis, we have developed a new computational approach that enables us to perform large scale quantum Monte-Carlo calculations to study the phase structure of this theory. In 4D, our results confirm prior results, but differ in some quantitative details of the phase diagram. In contrast, in 3D, we discover a new second order critical point using calculations on lattices up to size $ 60^3$. Such large scale calculations are unprecedented. The presence of the critical point implies the existence of an alternate mechanism of fermion mass generation without any SSB, that could be of interest in continuum quantum field theory.

Mechanisms of Cdc42 Polarization in Yeast

Polarization is important for the function and morphology of many different cell types. The keys regulators of polarity in eukaryotes are the Rho-family GTPases. In the budding yeast Saccharomyces cerevisiae, which must polarize in order to bud and to mate, the master regulator is the highly conserved Rho GTPase, Cdc42. During polarity establishment, active Cdc42 accumulates at a site on the plasma membrane characterizing the “front” of the cell where the bud will emerge. The orientation of polarization is guided by upstream cues that dictate the site of Cdc42 clustering. However, in the absence of upstream cues, yeast can still polarize in a random direction during symmetry breaking. Symmetry breaking suggests cells possess an autocatalytic polarization mechanism that can amplify stochastic fluctuations of polarity proteins through a positive feedback mechanism.

Two different positive feedback mechanisms have been proposed to polarize Cdc42 in budding yeast. One model posits that Cdc42 activation must be localized to a site at the plasma membrane. Another model posits that Cdc42 delivery must be localized to a particular site at the plasma membrane. Although both mechanisms could work in parallel to polarize Cdc42, it is unclear which mechanism is critical to polarity establishment. We directly tested the predictions of the two positive feedback models using genetics and live microscopy. We found that localized Cdc42 activation is necessary for polarity establishment.

While this explains how active Cdc42 localizes to a particular site at the plasma membrane, it does not address how Cdc42 concentrates at that site. Several different mechanisms have been proposed to concentrate Cdc42. The GDI can extract Cdc42 from membranes and selective mobilize GDP-Cdc42 in the cytoplasm. It was proposed that selectively mobilizing GDP-Cdc42 in combination with local activation could locally concentrate total Cdc42 at the polarity site. Although the GDI is important for rapid Cdc42 accumulation at the polarity site, it is not essential to Cdc42 concentration. It was proposed that delivery of Cdc42 by actin-mediated vesicle can act as a backup pathway to concentrate Cdc42. However, we found no evidence for an actin-dependent concentrating pathway. Live microscopy experiments reveal that prenylated proteins are not restricted to membranes, and can enter the cytoplasm. We found that the GDI-independent concentrating pathway still requires Cdc42 to exchange between the plasma membrane and the cytoplasm, which is supported by computational modeling. In the absence of the GDI, we found that Cdc42 GAP became essential for polarization. We propose that the GAP limits GTP-Cdc42 leak into the cytoplasm, which would be prohibitive to Cdc42 polarization.

Latent Stick-Breaking Processes.

We develop a model for stochastic processes with random marginal distributions. Our model relies on a stick-breaking construction for the marginal distribution of the process, and introduces dependence across locations by using a latent Gaussian copula model as the mechanism for selecting the atoms. The resulting latent stick-breaking process (LaSBP) induces a random partition of the index space, with points closer in space having a higher probability of being in the same cluster. We develop an efficient and straightforward Markov chain Monte Carlo (MCMC) algorithm for computation and discuss applications in financial econometrics and ecology. This article has supplementary material online.