A Model of Force Generation in a Three-Dimensional Toroidal Cluster of Cells

Nurse, Asha K.; Freund, L. B.; Youssef, Jacquelyn

doi:10.1115/1.4006257

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…In recent years, the interest in non-trivial forms of fluid particles was stimulated by applications of non-spherical microparticles to have important potential as building blocks for self-assembled materials including clustering of cells, imaging probes for therapy, drug carriers (see, for example, Dean et al. (2007), Nurse, Freund & Youssef (2012) and a review by Champion, Katare & Mitragotri (2007)) and more. In particular, toroidal forms are advantageous compared with spherical and spheroidal shapes due to their relatively large surface-to-volume ratio.…”

Section: Introductionmentioning

confidence: 99%

Dynamic and stationary shapes of rotating toroidal drops in viscous linear flows

2021

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Section: Introductionmentioning

confidence: 99%

Dynamic and stationary shapes of rotating toroidal drops in viscous linear flows

2021

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“…6,7 The results for both equilibrium and stability in the fluid model presented here are more complicated than we originally anticipated; it would be useful to have experimental validations for some of the predicted behavior in this simpler case. In our solution procedure, we have generally prescribed the conical half-angle β, the contact angle α, and the drop height z 0 , and computed the corresponding Bond number and drop shape; equilibria for either upward-pointing cones (corresponding to positive Bond numbers Bo) or for downward-pointing cones (negative Bond numbers) are then obtained depending on the sign of the Bond number.…”

Section: Discussionmentioning

confidence: 91%

“…This study is motivated by recent experimental observations of the behavior of clusters of biological cells that self-assemble at the bottom of patterned containers. 6,7 Each compartment of the partitioned container includes a conical a) Author to whom correspondence should be addressed. Electronic mail: mcfadden@nist.gov.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium and stability of axisymmetric drops on a conical substrate under gravity

et al. 2015

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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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“…Experiments on neonatal fibroblast cells that have self assembled into a toroidal cluster about the base of a conical pillar, have shown that the toroidal cluster will actively do work to climb the pillar to become a sphere or will remain at the base of the pillar and break up to form smaller clusters [ 14 ]. Subsequent theoretical work on this self assembled system points to the surface energy as the configurational driving force for the climbing motion of the cluster [ 15 ]. This suggests that the fate of the cluster is determined by its size.…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Rotating Drops

Nurse¹,

Coriell²,

McFadden³

2015

J. RES. NATL. INST. STAN.

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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].

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A Model of Force Generation in a Three-Dimensional Toroidal Cluster of Cells

Cited by 4 publications

References 11 publications

Dynamic and stationary shapes of rotating toroidal drops in viscous linear flows

Dynamic and stationary shapes of rotating toroidal drops in viscous linear flows

Equilibrium and stability of axisymmetric drops on a conical substrate under gravity

On the Stability of Rotating Drops

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