Adriana T. Dawes

PAR-3 Oligomerization May Provide an Actin-Independent Mechanism to Maintain Distinct Par Protein Domains in the Early Caenorhabditis elegans Embryo

Par proteins establish discrete intracellular spatial domains to polarize many different cell types. In the single-cell embryo of the nematode worm Caenorhabditis elegans, the segregation of Par proteins is crucial for proper division and cell fate specification. Actomyosin-based cortical flows drive the initial formation of anterior and posterior Par domains, but cortical actin is not required for the maintenance of these domains. Here we develop a model of interactions between the Par proteins that includes both mutual inhibition and PAR-3 oligomerization. We show that this model gives rise to a bistable switch mechanism, allowing the Par proteins to occupy distinct anterior and posterior domains seen in the early C. elegans embryo, independent of dynamics or asymmetries in the actin cortex. The model predicts a sharp loss of cortical Par protein asymmetries during gradual depletion of the Par protein PAR-6, and we confirm this prediction experimentally. Together, these results suggest both mutual inhibition and PAR-3 oligomerization are sufficient to maintain distinct Par protein domains in the early C. elegans embryo.

A mathematical model of α-catenin dimerization at adherens junctions in polarized epithelial cells

The protein alpha-catenin is found as a monomer or homodimer. As a monomer, alpha-catenin can bind to beta-catenin, which localizes to the plasma membrane at the site of adherens junctions (AJs) in polarized epithelial cells. As a dimer, alpha-catenin can bind to actin filaments, affecting the organization of the actin cytoskeleton. At usual cytoplasmic concentrations, alpha-catenin is found predominantly in monomeric form. It is currently thought that alpha-catenin cannot simultaneously bind beta-catenin and homodimerize, and that the dynamics of binding and unbinding from beta-catenin, possibly coupled with lower diffusion near an AJ, are sufficient to locally accumulate alpha-catenin monomers and homodimers. Using a mathematical model of alpha-catenin dynamics, I show that alpha-catenin must transiently homodimerize while bound to beta-catenin in order for homodimers to form, even in the presence of a spatial diffusion gradient.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos.

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.

Microtubule patterning in the presence of moving motor proteins.

Cytoskeletal polymers such as microtubules (MTs) interact with motor proteins to form higher-order structures. In vitro experiments have shown that MT patterns such as asters, bundles, and vortices can form under the influence of a single type of dynamic motor protein. MTs also can form anti-parallel bundles, similar to bundles that form the mitotic spindle during cell division, under the influence of two types of moving motors with opposite directionality. Despite the importance of MT structures, their mechanism of formation is not yet understood. We develop an integro-partial differential equation model to describe the dynamic interactions between MTs and moving motor proteins. Our model takes into account motor protein speed, processivity, density, and directionality, as well as MT treadmilling and reorganization due to interactions with motors. Simulation results show that plus-end directed motor proteins can form vortex patterns at low motor density, while minus-end directed motor proteins form aster patterns at similar densities. Also, motor proteins with opposite directionality are able to organize MTs into anti-parallel bundles. Our model is able to provide a quantitative and qualitative description of MT patterning, providing insights into possible mechanisms of spindle formation.

Microtubule Patterning in the Presence of Stationary Motor Distributions

In this paper, we construct a novel nonlocal transport model that describes the evolution of microtubules (MTs) as they interact with stationary distributions of motor proteins. An advection term accounts for directed MT transport (sliding due to motor protein action), and an integral term accounts for reorientation of MTs due to their interactions with cross-linking motor proteins. Simulations of our model show how MT patterns depend on boundary constraints, as well as model parameters that represent motor speed, cross-linking capability (motor activity), and directionality. In large domains, and using motor parameter values consistent with experimentally-derived values, we find that patterns such as asters, vortices, and bundles are able to persist. In vivo, MTs take on aster patterns during interphase and they form bundles in neurons and polarized epithelial cells. Vortex patterns have not been observed in vivo, however, are found in in vitro experiments. In constrained domains, we find that similar patterns form (asters, bundles, and vortices). However, we also find that when two opposing motors are present, anti-parallel bundles are able to form, resembling the mitotic spindle during cell division. This model demonstrates how MT sliding and MT reorientation are sufficient to produce experimentally observed patterns.

Differing roles for sur-2/MED23 in C. elegans and C. briggsae vulval development

Normal vulval development in the nematode Caenorhabditis briggsae is identical to that in the related Caenorhabditis elegans. However, several experiments suggest that there are differences between the two species with respect to the contribution of EGF/Ras signaling. To investigate these differences genetically, we have characterized a C. briggsae mutant strain that phenocopies the effect observed when C. briggsae animals are treated with U0126, an inhibitor of the EGF pathway component MEK. We identify that the gene affected in the mutant strain is Cbr-sur-2, which encodes a MED23 mediator complex protein that acts downstream of EGF signaling in C. elegans and other organisms, such as mammals. When Cbr-sur-2 and Cel-sur-2 mutants are compared, we find that the production of additional vulval cells from P5.p and P7.p in C. elegans is dependent on proper development of P6.p, while C. briggsae does not have a similar requirement. Combined chemical and genetic interference with the EGF pathway completely eliminates vulval development in C. elegans but not in C. briggsae. Our results provide genetic evidence for the differing requirements for EGF signaling in the two species.

Existence and uniqueness for a coupled PDE model for motor-induced microtubule organization

ABSTRACT Microtubules (MTs) are protein filaments that provide structure to the cytoskeleton of cells and a platform for the movement of intracellular substances. The spatial organization of MTs is crucial for a cells form and function. MTs interact with a class of proteins called motor proteins that can transport and position individual filaments, thus contributing to overall organization. In this paper, we study the mathematical properties of a coupled partial differential equation (PDE) model, introduced by White et al. in 2015, that describes the motor-induced organization of MTs. The model consists of a nonlinear coupling of a hyperbolic PDE for bound motor proteins, a parabolic PDE for unbound motor proteins, and a transport equation for MT dynamics. We locally smooth the motor drift velocity in the equation for bound motor proteins. The mollification is not only critical for the analysis of the model, but also adds biological realism. We then use a Banach Fixed Point argument to show local existence and uniqueness of mild solutions. We highlight the applicability of the model by showing numerical simulations that are consistent with in vitro experiments.

Stronger net posterior cortical forces and asymmetric microtubule arrays produce simultaneous centration and rotation of the pronuclear complex in the early Caenorhabditis elegans embryo

Experimental and theoretical approaches are used to demonstrate the importance of asymmetries in microtubule arrays and cortical pulling forces mediated by dynein in positioning the pronuclear complex before nuclear envelope breakdown in the early Caenorhabditis elegans embryo.

Antagonistic Behaviors of NMY-1 and NMY-2 Maintain Ring Channels in the C. elegans Gonad

Contractile rings play critical roles in a number of biological processes, including oogenesis, wound healing, and cytokinesis. In many cases, the activity of motor proteins such as nonmuscle myosins is required for appropriate constriction of these contractile rings. In the gonad of the nematode worm Caenorhabditis elegans, ring channels are a specialized form of contractile ring that are maintained at a constant diameter before oogenesis. We propose a model of ring channel maintenance that explicitly incorporates force generation by motor proteins that can act normally or tangentially to the ring channel opening. We find that both modes of force generation are needed to maintain the ring channels. We demonstrate experimentally that the type II myosins NMY-1 and NMY-2 antagonize each other in the ring channels by producing force in perpendicular directions: the experimental depletion of NMY-1/theoretical decrease in orthogonal force allows premature ring constriction and cellularization, whereas the experimental depletion of NMY-2/theoretical decrease in tangential force opens the ring channels and prevents cellularization. Together, our experimental and theoretical results show that both forces, mediated by NMY-1 and NMY-2, are crucial for maintaining the appropriate ring channel diameter and dynamics throughout the gonad.

The importance of mechanical constraints for proper polarization and psuedo-cleavage furrow generation in the early Caenorhabditis elegans embryo

Intracellular polarization, where a cell specifies a spatial axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode worm Caenorhabditis elegans (C. elegans), polarization is often accompanied by deformations of the cortex, a highly contractile structure consisting of actin filaments cross-linked by the motor protein myosin (actomyosin). It has been suggested that the eggshell surrounding the early embryo plays a role in polarization although its function is not understood. Here we develop a mathematical model which couples a reaction-diffusion model of actomyosin dynamics with a phase field model of the cell cortex to implicitly track cell shape changes in the early C. elegans embryo. We investigate the potential rigidity effect of the geometric constraint imposed by the presence and size of the eggshell on polarization dynamics. Our model suggests that the geometric constraint of the eggshell is essential for proper polarization and the size of the eggshell also affects the dynamics of polarization. Therefore, we conclude that geometric constraint on a cell might affect the dynamics of a biochemical process.