In a nonadhesive environment, cells will self-assemble into microtissues, a process relevant to tissue engineering. Although this has been recognized for some time, there is no basis for quantitative characterization of this complex process. Here we describe a recently developed assay designed to quantify aspects of the process and discuss its application in comparing behaviors between cell types. Cells were seeded in nonadhesive micromolded wells, each well with a circular trough at its base formed by the cylindrical sidewalls and by a central peg in the form of a right circular cone. Cells settled into the trough and coalesced into a toroid, which was then driven up the conical peg by the forces of self-assembly. The mass of the toroid and its rate of upward movement were used to calculate the cell power expended in the process against gravity. The power of the toroid was found to be 0.31 ± 0.01 pJ/h and 4.3 ± 1.7 pJ/h for hepatocyte cells and fibroblasts, respectively. Blocking Rho kinase by means of Y-27632 resulted in a 50% and greater reduction in power expended by each type of toroid, indicating that cytoskeletal-mediated contraction plays a significant role in the self-assembly of both cell types. Whereas the driving force for self-assembly has often been viewed as the binding of surface proteins, these data show that cellular contraction is important for cell-cell adhesion. The power measurement quantifies the contribution of cell contraction, and will be useful for understanding the concerted action of the mechanisms that drive self-assembly. T he self-assembly of cells into microtissues, often viewed as a passive process driven by chemical forces resulting from binding of cadherins expressed on cell surfaces (1, 2), is now thought to be a more complex process. Recent work has shown that the cytoskeleton also plays an important role in force generation, stability, and self-sorting of microtissues (3-6). Thus, self-assembly is a complex process involving multiple protein mechanisms working in concert.Self-assembly and the cell-cell adhesion that drives it are relevant to tissue formation, morphogenesis, and disease. However, little quantitative data are available on the forces driving self-assembly. At the molecular level, precise measurements of the binding strength and energy of adhesion of surface proteins (7,8) as well as the force of contraction of the actin cytoskeleton (9, 10) have been made. On the cellular level, traction-force microscopy has been used to measure the adhesion forces of single cells on flat substrates (11-13), and fibroblast-populated collagen gels have been used in the study of cell-matrix interactions (14). However, cell-cell interactions in a 3D environment have not been fully investigated. Doing so requires a quantitative systems biology approach that can be used not only to quantify the aggregation and self-assembly of groups of cells but also to quantify the contributions of specific proteins and protein systems to the process.In the past, we have reported observations on ...
We consider the equilibrium and stability of rotating axisymmetric fluid drops by appealing to a variational principle that characterizes the equilibria as stationary states of a functional containing surface energy and rotational energy contributions, augmented by a volume constraint. The linear stability of a drop is determined by solving the eigenvalue problem associated with the second variation of the energy functional. We compute equilibria corresponding to both oblate and prolate shapes, as well as toroidal shapes, and track their evolution with rotation rate. The stability results are obtained for two cases: (i) a prescribed rotational rate of the system (“driven drops”), or (ii) a prescribed angular momentum (“isolated drops”). For families of axisymmetric drops instabilities may occur for either axisymmetric or non-axisymmetric perturbations; the latter correspond to bifurcation points where non-axisymmetric shapes are possible. We employ an angle-arc length formulation of the problem which allows the computation of equilibrium shapes that are not single-valued in spherical coordinates. We are able to illustrate the transition from spheroidal drops with a strong indentation on the rotation axis to toroidal drops that do not extend to the rotation axis. Toroidal drops with a large aspect ratio (major radius to minor radius) are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. Prolate spheroidal shapes occur if a drop of lower density rotates within a denser medium; these drops appear to be linearly stable. This work is motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013].
Observation of the self-assembly of clusters of cells in three dimensions has raised questions about the forces that drive changes in the shape of the cell clusters. Cells that selfassemble into a toroidal cluster about the base of a conical pillar have been observed in the laboratory to spontaneously climb the conical pillar. Assuming that cell cluster reorganization is due solely to surface diffusion, a mathematical model based on the thermodynamics of an isothermal dissipative system is presented. The model shows that the cluster can reduce its surface area by climbing the conical pillar, however, this is at the expense of increasing its gravitational potential energy. As a result, the kinetics of the climb are affected by parameters that influence this energy competition, such as the slope of the conical pillar and a parameter of the model K that represents the influence of the surface energy of the cluster relative to its gravitational potential energy.
Equilibrium behavior of sessile drops under surface tension, applied external fields, and material variationsMotivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., "A model of force generation in a three-dimensional toroidal cluster of cells," J. Appl. Mech. 79, 051013 (2012)], we study a related problem of the determination of the equilibrium and stability of axisymmetric drops on a conical substrate in the presence of gravity. A variational principle is used to characterize equilibrium shapes that minimize surface energy and gravitational potential energy subject to a volume constraint, and the resulting Euler equation is solved numerically using an angle/arclength formulation. The resulting equilibria satisfy a Laplace-Young boundary condition that specifies the contact angle at the three-phase trijunction. The vertical position of the equilibrium drops on the cone is found to vary significantly with the dimensionless Bond number that represents the ratio of gravitational and capillary forces; a global force balance is used to examine the conditions that affect the drop positions. In particular, depending on the contact angle and the cone half-angle, we find that the vertical position of the drop can either increase ("the drop climbs the cone") or decrease due to a nominal increase in the gravitational force. Most of the equilibria correspond to upward-facing cones and are analogous to sessile drops resting on a planar surface; however, we also find equilibria that correspond to downward facing cones that are instead analogous to pendant drops suspended vertically from a planar surface. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. The drops with positive Bond number are generally found to be unstable to non-axisymmetric perturbations that promote a tilting of the drop. Additional points of marginal stability are found that correspond to limit points of the axisymmetric base state. Drops that are far from the tip are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. We have also found a range of completely stable solutions that correspond to small contact angles and cone half-angles. [http://dx.
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