A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $\lambda_s$ (the algebraic convergence) and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $\lambda_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $\lambda_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement "The convergence $\lambda_{ls}$ on the collapsing algebra B=\ro ((\omega_2)^{<\omega}) is weakly topological" is independent of ZFC

A mathematical model of mechanotransduction reveals how mechanical memory regulates mesenchymal stem cell fate decisions

Abstract Background Mechanical and biophysical properties of the cellular microenvironment regulate cell fate decisions. Mesenchymal stem cell (MSC) fate is influenced by past mechanical dosing (memory), but the mechanisms underlying this process have not yet been well defined. We have yet to understand how memory affects specific cell fate decisions, such as the differentiation of MSCs into neurons, adipocytes, myocytes, and osteoblasts. Results We study a minimal gene regulatory network permissive of multi-lineage MSC differentiation into four cell fates. We present a continuous model that is able to describe the cell fate transitions that occur during differentiation, and analyze its dynamics with tools from multistability, bifurcation, and cell fate landscape analysis, and via stochastic simulation. Whereas experimentally, memory has only been observed during osteogenic differentiation, this model predicts that memory regions can exist for each of the four MSC-derived cell lineages. We can predict the substrate stiffness ranges over which memory drives differentiation; these are directly testable in an experimental setting. Furthermore, we quantitatively predict how substrate stiffness and culture duration co-regulate the fate of a stem cell, and we find that the feedbacks from the differentiating MSC onto its substrate are critical to preserve mechanical memory. Strikingly, we show that re-seeding MSCs onto a sufficiently soft substrate increases the number of cell fates accessible. Conclusions Control of MSC differentiation is crucial for the success of much-lauded regenerative therapies based on MSCs. We have predicted new memory regions that will directly impact this control, and have quantified the size of the memory region for osteoblasts, as well as the co-regulatory effects on cell fates of substrate stiffness and culture duration. Taken together, these results can be used to develop novel strategies to better control the fates of MSCs in vitro and following transplantation