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Experiments on tumor spheroids have shown that compressive stress from their environment can reversibly decrease tumor expansion rates and final sizes. Stress release experiments show that nonuniform anisotropic elastic stresses can be distributed throughout. The elastic stresses are maintained by structural proteins and adhesive molecules, and can be actively relaxed by a variety of biophysical processes. In this paper, we present a new continuum model to investigate how the growth-induced elastic stresses and active stress relaxation, in conjunction with cell size control feedback machinery, regulate the cell density and stress distributions within growing tumors as well as the tumor sizes in the presence of external physical confinement and gradients of growth-promoting chemical fields. We introduce an adaptive reference map that relates the current position with the reference position but adapts to the current position in the Eulerian frame (lab coordinates) via relaxation. This type of stress relaxation is similar to but simpler than the classical Maxwell model of viscoelasticity in its formulation. By fitting the model to experimental data from two independent studies of tumor spheroid growth and their cell density distributions, treating the tumors as incompressible, neo-Hookean elastic materials, we find that the rates of stress relaxation of tumor tissues can be comparable to volumetric growth rates. Our study provides insight on how the biophysical properties of the tumor and host microenvironment, mechanical feedback control and diffusion-limited differential growth act in concert to regulate spatial patterns of stress and growth. When the tumor is stiffer than the host, our model predicts tumors are more able to change their size and mechanical state autonomously, which may help to explain why increased tumor stiffness is an established hallmark of malignant tumors., Author summary The mechanical state of cells can modulate their growth and division dynamics via mechanotransduction, which affects both the cell size distribution and the tissue size as a whole. Experiments on tumor spheroids have shown that compressive stress from their environment can reversibly decrease tumor expansion rates and final sizes. Besides external confinement and compression on the tumor border, a heterogeneous stress field can be generated inside the tumor by nutrient-driven differential growth. Such growth-induced mechanical stresses can be relaxed by tissue rearrangement, which happens during cell neighbor exchanges, cell divisions, and extracellular matrix renewal. In this study, we have developed a continuum model that describes the above mechanical interactions and the dynamics of tissue rearrangement explicitly. Motivated by published experimental data, we consider mechanotransduction where the local compressive stress slows down cell growth and cell size reduction limits cell division. We have analyzed how external mechanical stimuli and internal processes influence the outcome of cell-and-tissue sizes and spatial patterns of cell density and mechanical stress in growing tumors.

The conserved adapter protein Scribble (Scrib) plays essential roles in a variety of cellular processes, including polarity establishment, proliferation, and directed cell migration. While the mechanisms through which Scrib promotes epithelial polarity are beginning to be unraveled, its roles in other cellular processes including cell migration remain enigmatic. In C. elegans, the Scrib ortholog LET-413 is essential for apical–basal polarization and junction formation in embryonic epithelia. However, whether LET-413 is required for postembryonic development or plays a role in migratory events is not known. Here, we use inducible protein degradation to investigate the functioning of LET-413 in larval epithelia. We find that LET-413 is essential in the epidermal epithelium for growth, viability, and junction maintenance. In addition, we identify a novel role for LET-413 in the polarized outgrowth of the epidermal seam cells. These stem cell-like epithelial cells extend anterior and posterior directed apical protrusions in each larval stage to reconnect to their neighbors. We show that the role of LET-413 in seam cell outgrowth is likely mediated largely by the junctional component DLG-1 discs large, which we demonstrate is also essential for directed outgrowth of the seam cells. Our data uncover multiple essential functions for LET-413 in larval development and show that the polarized outgrowth of the epithelial seam cells is controlled by LET-413 Scribble and DLG-1 Discs large., Author summary Most cells in multicellular organisms are organized along a directional axis of cell polarity. One protein that is important for this polarized organization is the conserved polarity regulator Scribble. This protein has several functions, including forming the basolateral domains of cells, promoting the formation of cell junctions, and promoting cell migration. How Scribble performs these functions is not fully understood. In this paper we study the role of Scribble during larval development of the small nematode Caenorhabditis elegans using an inducible protein degradation system. We show that Scribble, called LET-413 in C. elegans, is essential in the epidermal epithelium for animal development, as depletion of LET-413 in only this tissue blocks growth. We also demonstrate that LET-413 is required for the polarized outgrowth of an epithelial cell type called the seam cells, a process resembling cell migration. Finally, we show that one major function of LET-413 in seam cell outgrowth is the localization of the junctional component Discs large (DLG-1), which we demonstrate is also essential for this process. Our data thus uncover multiple essential functions for LET-413 in larval development and provide new insights into how the directional outgrowth of epithelial seam cells is controlled.

The mean square displacement (MSD) is an important statistical measure on a stochastic process or a trajectory. In this paper we find an approximation to the mean square displacement for a model of cell motion. The model is a discrete-time jump process which approximates a force-based model for cell motion. In cell motion, the mean square displacement not only gives a measure of overall drift, but it is also an indicator of mode of transport. The key to finding the approximation is to find the mean square displacement for a subset of the state space and use it as an approximation for the entire state space. We give some intuition as to why this is an unexpectedly good approximation. A lower bound and upper bound for the mean square displacement are also given. We show that, although the upper bound is far from the computed mean square displacement, in rare cases the large displacements are approached.