EducationSpelman College, B.S. Physics Brown University, School of Engineering, Department of Solid Mechanics, Ph.D. Research Interests
My research interests lie in the formulation and analysis of models to enhance the understanding of complex emerging problems. Doctoral Research AbstractObservation of self assembly of cell aggregates in three dimensions has shed new light on the forces that drive the reorganization of cell clusters and the stability of the clusters formed. Specifically, experiments by Dean et al [1] on suspensions of cells confined in a chamber with a conical peg protruding from its base reveal that the cells first form a compact toroidal cluster around the base of the peg, after which that cluster tends to move spontaneously up the peg. Cell clusters that do not climb the peg are seen to undergo nonuniformdeformation or localized necking. 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 captures the effects of the relative influences of surface energy and gravitational potential energy, and the results imply that reduction of free energy is the underlying configurational force driving the upward motion of the cluster. To determine the stability of the initial self assembled cluster, the theory of shape change due to surface motion, driven by the Laplacian of mean curvature, is used to obtain numerical solutions for various surface fluctuations. This aims to quantify the overall tendency toward minimum surface area of a cluster versus the tendency for deformation localization observed for various configurations. Figure 1: Side view and top-down view microscopy images of the experimental chamber respectively showing the toroidal cluster on the conical pillar IntroductionThe precise nature of the forces driving the reorganization of the cells in a cluster remains obscure. In an attempt to gain clearer understanding of the origin and magnitude of the forces that drive morphology of a cell cluster, Professor Morgan and his students, at Brown University, have conducted a series of experiments in which cell suspensions are placed into self-fabricated experimental chambers. The cells contact each other and self assemble into micro-tissue structures that are observed over time in the presence of an applied force, usually gravity. Rather than evolving into the minimum surface area configuration, as predicted by the Steinberg DifferentialAdhesion Hypothesis [6], the behavior of the clusters was found to be more complex. [1] The toroidal self-assembled structures, as observed in these experiments, are particularly interesting as the aggregate may spontaneously climb the peg around which it had just self assembled. The simple geometry of the experimental chamber provides a starting point for a model which investigates the mechanical influences driving configurational changes in the cell cluster. From the rotational symmetry of the chamber, the boundary conditions on the cluster are known with minimal uncertainty. The upward motion of the cluster on the pillar is resisted only by gravity, a well characterized external force whose effects could be modulated by varying the slope of the pillar. As there are no other applied forces, morphological changes in the cluster can be driven only by interactions among the constituent cells. The goal of modeling this particular experiment is to understand the processes governing the observed shape changes in the cell cluster. FIGURE 2. Cartoon of the conical pillar and the toroidal cluster showing the physical parameters of the model Summary of ResearchThe mathematical model presumes that cells in the cluster are reorganizing through surface diffusion in such a way that the shape of the evolving cluster is continuously reducing its surface area while keeping its volume constant. For simplification, it is assumed that throughout the evolution of the cluster its cross section is always circular and that its shape is axially symmetric with respect to the vertical axis of the conical pillar. Principal geometric features of the model are represented by Fig. 2. The total surface area of the cluster s(t) is given in terms of the minor radius b(t) by, s(t) = 4pi 2a(t)b(t) = 2v0/b(t) where the major radius a(t) has been eliminated by means of the constant volume expression. Note that, for the surface area s(t) to decrease, b(t) has to increase in magnitude. As the volume is constant, this implies that a(t) decreases in magnitude which corresponds to movement up the conical pillar. The tendency to lower surface energy, and consequently system free energy, results in a configurational force on the toroidal cluster that pulls the cluster up the conical pillar. The system, being isothermal, has to dissipate the free energy that is lost in the process of evolution. Thermodynamics of dissipative systems at constant temperatures, as is demonstrated in [3] indicates that these systems evolve in such a way that This results in an ordinary differential equation governing the evolution of b(t), that is integrated numerically using the initial condition b(0) = b0 to determine the evolution of the cluster. However, experimental observation shows that not all self assembled toroidal clusters spontaneously climb the conical pillar around which it has self assembled. Some clusters have been observed to remain at the base of the chamber where localized necking at one or more points around the circumference has been observed. This may even result in the break up of the cluster into sub-bodies. It is believed that this is not a random statistical process but rather one that is based on the dimensions of the cluster. Presumably, the geometric dimensions of the aggregate delineates its stability and that stably formed clusters proceed to climb the conical peg while unstable clusters remain at the base of the chamber. To establish the values of these geometric benchmarks, a linear stability analysis of the toroidal shape is conducted. In the vein of work by Coleman, the Herring-Mullins equation is used to study the stability of an axially symmetric cylindrical cluster against a linear and nonlinear longitudinal perturbation. However, even the linear perturbation approach proved complicated for a toroidal structure as there is flux in two directions on the surface. A new model to investigate stability of the toroidal clusters is developed, that determines the change of the initial surface area of the toroidal cluster ˙A(0) under periodic surface perturbations b(q , t)= bk(t)−ck(t)(1−cos(kq) ). A negative value for ˙A(0) implies that the surface area decreases as the cluster changes from its original uniform torus shape to the perturbed shape. This inherently implies that the cluster is always stable under the applied perturbation. Conversely, a positive value for ˙A(0) implies that the cluster is always unstable under the applied perturbation. ConclusionsThe model captures the essential features of the climb of the cluster up the conical pillar and agrees with predicted outcomes when various input parameters are changed. This implies that reduction of the free energy of the cluster is indeed a major configurational driving force for the observed motion. Linear stability analysis reveals that the cell cluster is stable if the surface energy density is spatially uniform. However, if the surface energy density is allowed to vary from point to point around the circumference of the toroid, unstable configurations may develop. The stability of the cluster is observed to depend on its initial minor radius, the radius of the conical pillar and the wave number of the applied sinusoidal perturbation. AcknowledgementsThis work is supported primarily by the MRSEC Program of the National Science Foundation at brown University under award DMR-0520651. References[1] Dylan M. Dean, A.P. Napolitano, J. Youssef, and Jeffrey R. Morgan. Rods, tori, and honeycombs: the directed self assembly of microtissues with prescribed microscale geometries. FASEB J., 21(14):4005–4012, 2007. [2] B.D. Coleman, R.S. Falk, and M. Moakher. Space-time finite element methods for surface diffusion with applications to the theory of the stability of cylinders. SIAM Journal on Scientific Computing, 17(6):1434–1448, 1996. [3] L. B. Freund and S. Suresh. Thin film materials. Cambridge University Press, 1 edition, 2003. [4] Conyers Herring. Surface tension as a motivation for sintering. The Physics of Powder Metallurgy, 27(2):143–179, 1951. [5] W. W. Mullins. Theory of thermal grooving. Journal of Applied Physics, 28(3):333–339, 1957. [6] M.S. Steinberg. On the mechanism of tissue reconstruction by dissociated cells. iii: Free energy relations and the reorganisation of fused, heteronomic tissue fragments. Proc. Nat. Acad. Sc, 48:1769–1776, 1962. |