Abelian subgroups of pro-p Galois groups

It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois p-extension of a field F (where p is an odd prime) arise from p-henselian valuations with non-p-divisible value group, provided #(F/F-p) greater than or equal to p(2) and F contains a primitive p-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.35062473248

Quaternion algebras with the same subfields

G. Prasad and A. Rapinchuk asked if two quaternion division F -algebras that have the same subfields are necessarily isomorphic. The answer is known to be "no" for some very large fields. We prove that the answer is "yes" if F is an extension of a global field K so that F /K is unirational and has zero unramified Brauer group. We also prove a similar result for Pfister forms and give an application to tractable fields

A Difference Version of Nori's Theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).Comment: 29 page

A Multiwell Platform for Studying Stiffness-Dependent Cell Biology

Adherent cells are typically cultured on rigid substrates that are orders of magnitude stiffer than their tissue of origin. Here, we describe a method to rapidly fabricate 96 and 384 well platforms for routine screening of cells in tissue-relevant stiffness contexts. Briefly, polyacrylamide (PA) hydrogels are cast in glass-bottom plates, functionalized with collagen, and sterilized for cell culture. The Young's modulus of each substrate can be specified from 0.3 to 55 kPa, with collagen surface density held constant over the stiffness range. Using automated fluorescence microscopy, we captured the morphological variations of 7 cell types cultured across a physiological range of stiffness within a 384 well plate. We performed assays of cell number, proliferation, and apoptosis in 96 wells and resolved distinct profiles of cell growth as a function of stiffness among primary and immortalized cell lines. We found that the stiffness-dependent growth of normal human lung fibroblasts is largely invariant with collagen density, and that differences in their accumulation are amplified by increasing serum concentration. Further, we performed a screen of 18 bioactive small molecules and identified compounds with enhanced or reduced effects on soft versus rigid substrates, including blebbistatin, which abolished the suppression of lung fibroblast growth at 1 kPa. The ability to deploy PA gels in multiwell plates for high throughput analysis of cells in tissue-relevant environments opens new opportunities for the discovery of cellular responses that operate in specific stiffness regimes

Contractility Dominates Adhesive Ligand Density in Regulating Cellular De-adhesion and Retraction Kinetics

Cells that are enzymatically detached from a solid substrate rapidly round up as the tensile prestress in the cytoskeleton is suddenly unopposed by cell–ECM adhesions. We recently showed that this retraction follows sigmoidal kinetics with time constants that correlate closely with cortical stiffness values. This raises the promising prospect that these de-adhesion measurements may be used for high-throughput screening of cell mechanical properties; however, an important limitation to doing so is the possibility that the retraction kinetics may also be influenced and potentially rate-limited by the time needed to sever matrix adhesions. In this study, we address this open question by separating contributions of contractility and adhesion to cellular de-adhesion and retraction kinetics. We first develop serum-free conditions under which U373 MG glioma cells can be cultured on substrates of fixed fibronectin density without direct matrix contributions from the medium. We show that while spreading area increases with ECM protein density, cortical stiffness and the time constants of retraction do not. Conversely, addition of lysophosphatidic acid (LPA) to stimulate cell contractility strongly speeds retraction, independent of the initial matrix protein density and LPA’s contributions to spreading area. All of these trends hold in serum-rich medium commonly used in tissue culture, with the time constants of retraction much more closely tracking cortical stiffness than adhesive ligand density or cell spreading. These results support the use of cellular de-adhesion measurements to track cellular mechanical properties

Dynamic Mechanisms of Cell Rigidity Sensing: Insights from a Computational Model of Actomyosin Networks

Cells modulate themselves in response to the surrounding environment like substrate elasticity, exhibiting structural reorganization driven by the contractility of cytoskeleton. The cytoskeleton is the scaffolding structure of eukaryotic cells, playing a central role in many mechanical and biological functions. It is composed of a network of actins, actin cross-linking proteins (ACPs), and molecular motors. The motors generate contractile forces by sliding couples of actin filaments in a polar fashion, and the contractile response of the cytoskeleton network is known to be modulated also by external stimuli, such as substrate stiffness. This implies an important role of actomyosin contractility in the cell mechano-sensing. However, how cells sense matrix stiffness via the contractility remains an open question. Here, we present a 3-D Brownian dynamics computational model of a cross-linked actin network including the dynamics of molecular motors and ACPs. The mechano-sensing properties of this active network are investigated by evaluating contraction and stress in response to different substrate stiffness. Results demonstrate two mechanisms that act to limit internal stress: (i) In stiff substrates, motors walk until they exert their maximum force, leading to a plateau stress that is independent of substrate stiffness, whereas (ii) in soft substrates, motors walk until they become blocked by other motors or ACPs, leading to submaximal stress levels. Therefore, this study provides new insights into the role of molecular motors in the contraction and rigidity sensing of cells

Mechanical Stress Inference for Two Dimensional Cell Arrays

Many morphogenetic processes involve mechanical rearrangement of epithelial tissues that is driven by precisely regulated cytoskeletal forces and cell adhesion. The mechanical state of the cell and intercellular adhesion are not only the targets of regulation, but are themselves likely signals that coordinate developmental process. Yet, because it is difficult to directly measure mechanical stress {\it in vivo} on sub-cellular scale, little is understood about the role of mechanics of development. Here we present an alternative approach which takes advantage of the recent progress in live imaging of morphogenetic processes and uses computational analysis of high resolution images of epithelial tissues to infer relative magnitude of forces acting within and between cells. We model intracellular stress in terms of bulk pressure and interfacial tension, allowing these parameters to vary from cell to cell and from interface to interface. Assuming that epithelial cell layers are close to mechanical equilibrium, we use the observed geometry of the two dimensional cell array to infer interfacial tensions and intracellular pressures. Here we present the mathematical formulation of the proposed Mechanical Inverse method and apply it to the analysis of epithelial cell layers observed at the onset of ventral furrow formation in the {\it Drosophila} embryo and in the process of hair-cell determination in the avian cochlea. The analysis reveals mechanical anisotropy in the former process and mechanical heterogeneity, correlated with cell differentiation, in the latter process. The method opens a way for quantitative and detailed experimental tests of models of cell and tissue mechanics

Stiffness Gradients Mimicking In Vivo Tissue Variation Regulate Mesenchymal Stem Cell Fate

Mesenchymal stem cell (MSC) differentiation is regulated in part by tissue stiffness, yet MSCs can often encounter stiffness gradients within tissues caused by pathological, e.g., myocardial infarction ∼8.7±1.5 kPa/mm, or normal tissue variation, e.g., myocardium ∼0.6±0.9 kPa/mm; since migration predominantly occurs through physiological rather than pathological gradients, it is not clear whether MSC differentiate or migrate first. MSCs cultured up to 21 days on a hydrogel containing a physiological gradient of 1.0±0.1 kPa/mm undergo directed migration, or durotaxis, up stiffness gradients rather than remain stationary. Temporal assessment of morphology and differentiation markers indicates that MSCs migrate to stiffer matrix and then differentiate into a more contractile myogenic phenotype. In those cells migrating from soft to stiff regions however, phenotype is not completely determined by the stiff hydrogel as some cells retain expression of a neural marker. These data may indicate that stiffness variation, not just stiffness alone, can be an important regulator of MSC behavior

Differential Matrix Rigidity Response in Breast Cancer Cell Lines Correlates with the Tissue Tropism

Metastasis to a variety of distant organs, such as lung, brain, bone, and liver, is a leading cause of mortality in the breast cancer patients. The tissue tropism of breast cancer metastasis has been recognized and studied extensively, but the cellular processes underlying this phenomenon, remain elusive. Modern technologies have enabled the discovery of a number of the genetic factors determining tissue tropism of malignant cells. However, the effect of these genetic differences on the cell motility and invasiveness is poorly understood. Here, we report that cellular responses to the mechanical rigidity of the extracellular matrix correlate with the rigidity of the target tissue. We tested a series of single cell populations isolated from MDA-MB-231 breast cancer cell line in a variety of assays where the extracellular matrix rigidity was varied to mimic the environment that these cells might encounter in vivo. There was increased proliferation and migration through the matrices of rigidities corresponding to the native rigidities of the organs where metastasis was observed. We were able to abolish the differential matrix rigidity response by knocking down Fyn kinase, which was previously identified as a critical component of the FN rigidity response pathway in healthy cells. This result suggests possible molecular mechanisms of the rigidity response in the malignant cells, indicating potential candidates for therapeutic interventions

Soft Substrates Promote Homogeneous Self-Renewal of Embryonic Stem Cells via Downregulating Cell-Matrix Tractions

Maintaining undifferentiated mouse embryonic stem cell (mESC) culture has been a major challenge as mESCs cultured in Leukemia Inhibitory Factor (LIF) conditions exhibit spontaneous differentiation, fluctuating expression of pluripotency genes, and genes of specialized cells. Here we show that, in sharp contrast to the mESCs seeded on the conventional rigid substrates, the mESCs cultured on the soft substrates that match the intrinsic stiffness of the mESCs and in the absence of exogenous LIF for 5 days, surprisingly still generated homogeneous undifferentiated colonies, maintained high levels of Oct3/4, Nanog, and Alkaline Phosphatase (AP) activities, and formed embryoid bodies and teratomas efficiently. A different line of mESCs, cultured on the soft substrates without exogenous LIF, maintained the capacity of generating homogeneous undifferentiated colonies with relatively high levels of Oct3/4 and AP activities, up to at least 15 passages, suggesting that this soft substrate approach applies to long term culture of different mESC lines. mESC colonies on these soft substrates without LIF generated low cell-matrix tractions and low stiffness. Both tractions and stiffness of the colonies increased with substrate stiffness, accompanied by downregulation of Oct3/4 expression. Our findings demonstrate that mESC self-renewal and pluripotency can be maintained homogeneously on soft substrates via the biophysical mechanism of facilitating generation of low cell-matrix tractions