Long-distance intracellular axonal transport is predominantly microtubule-based, and its impairment is linked to neurodegeneration. Here we review recent theoretical and experimental evidence that suggest that near the axon boundaries (walls), the effective viscosity can become large enough to impede cargo transport in small (but not large) caliber axons. Theoretical work suggests that this opposition to motion increases rapidly as the cargo approaches the wall. However, having parallel microtubules close enough together to enable a cargo to simultaneously engage motors on more than one microtubule dramatically enhances motor activity, and thus decreases the effects due to such opposition. Experimental evidence supports this hypothesis: in small caliber axons, microtubule density is higher, increasing the probability of having parallel microtubules close enough that they can be used simultaneously by motors on a cargo. For transport toward the minus-end of microtubules, e.g., toward the cell body in an axon, a recently discovered force adaptation system can also contribute to overcoming such opposition to motion.

In developing tissues, cell polarization and proliferation are regulated by morphogens and signaling pathways. Cells throughout the Drosophila wing primordium typically show subcellular localization of the unconventional myosin Dachs on the distal side of cells (nearest the center of the disc). Dachs localization depends on the spatial distribution of bonds between the protocadherins Fat (Ft) and Dachsous (Ds), which form heterodimers between adjacent cells; and the Golgi kinase Four-jointed (Fj), which affects the binding affinities of Ft and Ds. The Fj concentration forms a linear gradient while the Ds concentration is roughly uniform throughout most of the wing pouch with a steep transition region that propagates from the center to the edge of the pouch during the third larval instar. Although the Fj gradient is an important cue for polarization, it is unclear how the polarization is affected by cell division and the expanding Ds transition region, both of which can alter the distribution of Ft-Ds heterodimers around the cell periphery. We have developed a computational model to address these questions. In our model, the binding affinity of Ft and Ds depends on phosphorylation by Fj. We assume that the asymmetry of the Ft-Ds bond distribution around the cell periphery defines the polarization, with greater asymmetry promoting cell proliferation. Our model predicts that this asymmetry is greatest in the radially-expanding transition region that leaves polarized cells in its wake. These cells naturally retain their bond distribution asymmetry after division by rapidly replenishing Ft-Ds bonds at new cell-cell interfaces. Thus we predict that the distal localization of Dachs in cells throughout the pouch requires the movement of the Ds transition region and the simple presence, rather than any specific spatial pattern, of Fj.

Long-distance intracellular axonal transport is predominantly microtubule-based, and its impairment is linked to neurodegeneration. In this study, we present theoretical arguments that suggest that near the axon boundaries (walls), the effective viscosity can become large enough to impede cargo transport in small (but not large) caliber axons. Our theoretical analysis suggests that this opposition to motion increases rapidly as the cargo approaches the wall. We find that having parallel microtubules close enough together to enable a cargo to simultaneously engage motors on more than one microtubule dramatically enhances motor activity, and thus minimizes the effects of any opposition to transport. Even if microtubules are randomly placed in axons, we find that the higher density of microtubules found in small-caliber axons increases the probability of having parallel microtubules close enough that they can be used simultaneously by motors on a cargo. The boundary effect is not a factor in transport in large-caliber axons where the microtubule density is lower.

Post-translational modification of tubulin provides differential functions to microtubule networks. Here, we address the role of tubulin acetylation on the penetrative capacity of cells undergoing radial intercalation, which is the process by which cells move apically, insert between outer cells, and join an epithelium. There are opposing forces that regulate intercalation, namely, the restrictive forces of the epithelial barrier versus the penetrative forces of the intercalating cell. Positively and negatively modulating tubulin acetylation in intercalating cells alters the developmental timing such that cells with more acetylation penetrate faster. We find that intercalating cells preferentially penetrate higher-order vertices rather than the more prevalent tricellular vertices. Differential timing in the ability of cells to penetrate different vertices reveals that lower-order vertices represent more restrictive sites of insertion. We shift the accessibility of intercalating cells toward more restrictive junctions by increasing tubulin acetylation, and we provide a geometric-based mathematical model that describes our results.

Favorable outcomes have been associated with high densities of tumor infiltrating lymphocytes (TILs) such as cytotoxic ([Formula: see text]) T cells. However, the clinical significance of the spatial distribution of TILs is less well understood. We have developed novel statistical techniques to characterize the spatial distribution of TILs at various length scales. These include a box counting method that we call “occupancy” and novel applications of fractal dimensions. We apply these techniques to the spatial distribution of [Formula: see text] T cells in the tumor microenvironment of tissue resected from 35 triple negative breast cancer patients. We find that there is a distinct difference in the spatial distribution of [Formula: see text] T cells between good clinical outcome (no recurrence within at least 5 years of diagnosis) and poor clinical outcome (recurrence within 3 years of diagnosis). The statistical significance of the difference between good and poor outcome in the occupancy, fractal dimension (FD), and FD difference of [Formula: see text] T cells is comparable to that of the [Formula: see text] T cell density. Even when we randomly exclude some of the cells so that the images have the same cell density, we still find that the fractal dimension at short length scales is correlated with cancer recurrence, implying that the actual spatial distribution of [Formula: see text] cells, and not just the [Formula: see text] cell density, is associated with clinical outcome. The occupancy and FD difference indicate that the [Formula: see text] T cells are more spatially dispersed in good outcome and more aggregated in poor outcome. We discuss possible interpretations.

The goal of immunotherapy is to mobilize the immune system to kill cancer cells. Immunotherapy is more effective and, in general, the prognosis is better, when more immune cells infiltrate the tumor. We explore the question of whether the spatial distribution rather than just the density of immune cells in the tumor is important in forecasting whether cancer recurs. After reviewing previous work on this issue, we introduce a novel application of maximum entropy to quantify the spatial distribution of discrete point-like objects. We apply our approach to B and T cells in images of tumor tissue taken from triple negative breast cancer patients. We find that the immune cells are more spatially dispersed in good clinical outcome (no recurrence of cancer within at least 5 years of diagnosis) compared to poor clinical outcome (recurrence within 3 years of diagnosis). Our results highlight the importance of spatial distribution of immune cells within tumors with regard to clinical outcome, and raise new questions on their role in cancer recurrence.

While tumor infiltration by CD8⁺ T cells is now widely accepted to predict outcomes, the clinical significance of intratumoral B cells is less clear. We hypothesized that spatial distribution rather than density of B cells within tumors may provide prognostic significance. We developed statistical techniques (fractal dimension differences and a box-counting method 'occupancy') to analyze the spatial distribution of tumor-infiltrating lymphocytes (TILs) in human triple-negative breast cancer (TNBC). Our results indicate that B cells in good outcome tumors (no recurrence within 5 years) are spatially dispersed, while B cells in poor outcome tumors (recurrence within 3 years) are more confined. While most TILs are located within the stroma, increased numbers of spatially dispersed lymphocytes within cancer cell islands are associated with a good prognosis. B cells and T cells often form lymphocyte clusters (LCs) identified via density-based clustering. LCs consist either of T cells only or heterotypic mixtures of B and T cells. Pure B cell LCs were negligible in number. Compared to tertiary lymphoid structures (TLS), LCs have fewer lymphocytes at lower densities. Both types of LCs are more abundant and more spatially dispersed in good outcomes compared to poor outcome tumors. Heterotypic LCs in good outcome tumors are smaller and more numerous compared to poor outcome. Heterotypic LCs are also closer to cancer islands in a good outcome, with LC size decreasing as they get closer to cancer cell islands. These results illuminate the significance of the spatial distribution of B cells and LCs within tumors.