Modeling microtubule-based transport and anchoring of mRNA

Localization of messenger RNA (mRNA) at the vegetal cortex plays an important role in the early development of Xenopus laevis oocytes. While it is known that molecular motors are responsible for the transport of mRNA cargo along microtubules to the cortex, the mechanisms of localization remain unclear. We model cargo transport along microtubules using partial differential equations with spatially-dependent rates. A theoretical analysis of reduced versions of our model predicts effective velocity and diffusion rates for the cargo and shows that randomness of microtubule networks enhances effective transport. A more complex model using parameters estimated from fluorescence microscopy data reproduces the spatial and timescales of mRNA localization observed in Xenopus oocytes, corroborates experimental hypotheses that anchoring may be necessary to achieve complete localization, and shows that anchoring of mRNA complexes actively transported to the cortex is most effective in achieving robust accumulation at the cortex.

Stable spike clusters for the precursor Gierer-Meinhardt system in R 2

We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.

A GRAPH PARTITIONING APPROACH TO PREDICTING PATTERNS IN LATERAL INHIBITION SYSTEMS

We analyze spatial patterns on networks of cells where adjacent cells inhibit each other through contact signaling. We represent the network as a graph where each vertex represents the dynamics of identical individual cells and where graph edges represent cell-to-cell signaling. To predict steady-state patterns we find equitable partitions of the graph vertices and assign them into disjoint classes. We then use results from monotone systems theory to prove the existence of patterns that are structured in such a way that all the cells in the same class have the same final fate. To study the stability properties of these patterns, we rely on the graph partition to perform a block decomposition of the system. Then, to guarantee stability, we provide a small-gain type criterion that depends on the input-output properties of each cell in the reduced system. Finally, we discuss pattern formation in stochastic models. With the help of a modal decomposition we show that noise can enhance the parameter region where patterning occurs.